Daily Papers

Forward Gradients - the idea of using directional derivatives in forward differentiation mode - have recently been shown to be utilizable for neural network training while avoiding problems generally associated with backpropagation gradient computation, such as locking and memorization requirements. The cost is the requirement to guess the step direction, which is hard in high dimensions. While current solutions rely on weighted averages over isotropic guess vector distributions, we propose to strongly bias our gradient guesses in directions that are much more promising, such as feedback obtained from small, local auxiliary networks. For a standard computer vision neural network, we conduct a rigorous study systematically covering a variety of combinations of gradient targets and gradient guesses, including those previously presented in the literature. We find that using gradients obtained from a local loss as a candidate direction drastically improves on random noise in Forward Gradient methods.

The growing Synthetic Aperture Radar (SAR) data has the potential to build a foundation model through Self-Supervised Learning (SSL) methods, which can achieve various SAR Automatic Target Recognition (ATR) tasks with pre-training in large-scale unlabeled data and fine-tuning in small labeled samples. SSL aims to construct supervision signals directly from the data, which minimizes the need for expensive expert annotation and maximizes the use of the expanding data pool for a foundational model. This study investigates an effective SSL method for SAR ATR, which can pave the way for a foundation model in SAR ATR. The primary obstacles faced in SSL for SAR ATR are the small targets in remote sensing and speckle noise in SAR images, corresponding to the SSL approach and signals. To overcome these challenges, we present a novel Joint-Embedding Predictive Architecture for SAR ATR (SAR-JEPA), which leverages local masked patches to predict the multi-scale SAR gradient representations of unseen context. The key aspect of SAR-JEPA is integrating SAR domain features to ensure high-quality self-supervised signals as target features. Besides, we employ local masks and multi-scale features to accommodate the various small targets in remote sensing. By fine-tuning and evaluating our framework on three target recognition datasets (vehicle, ship, and aircraft) with four other datasets as pre-training, we demonstrate its outperformance over other SSL methods and its effectiveness with increasing SAR data. This study showcases the potential of SSL for SAR target recognition across diverse targets, scenes, and sensors.Our codes and weights are available in \url{https://github.com/waterdisappear/SAR-JEPA.

To adopt convolutional neural networks (CNN) for a range of resource-constrained targets, it is necessary to compress the CNN models by performing quantization, whereby precision representation is converted to a lower bit representation. To overcome problems such as sensitivity of the training dataset, high computational requirements, and large time consumption, post-training quantization methods that do not require retraining have been proposed. In addition, to compensate for the accuracy drop without retraining, previous studies on post-training quantization have proposed several complementary methods: calibration, schemes, clipping, granularity, and mixed-precision. To generate a quantized model with minimal error, it is necessary to study all possible combinations of the methods because each of them is complementary and the CNN models have different characteristics. However, an exhaustive or a heuristic search is either too time-consuming or suboptimal. To overcome this challenge, we propose an auto-tuner known as Quantune, which builds a gradient tree boosting model to accelerate the search for the configurations of quantization and reduce the quantization error. We evaluate and compare Quantune with the random, grid, and genetic algorithms. The experimental results show that Quantune reduces the search time for quantization by approximately 36.5x with an accuracy loss of 0.07 ~ 0.65% across six CNN models, including the fragile ones (MobileNet, SqueezeNet, and ShuffleNet). To support multiple targets and adopt continuously evolving quantization works, Quantune is implemented on a full-fledged compiler for deep learning as an open-sourced project.

Machine learning systems typically assume that the distributions of training and test sets match closely. However, a critical requirement of such systems in the real world is their ability to generalize to unseen domains. Here, we propose an inter-domain gradient matching objective that targets domain generalization by maximizing the inner product between gradients from different domains. Since direct optimization of the gradient inner product can be computationally prohibitive -- requires computation of second-order derivatives -- we derive a simpler first-order algorithm named Fish that approximates its optimization. We demonstrate the efficacy of Fish on 6 datasets from the Wilds benchmark, which captures distribution shift across a diverse range of modalities. Our method produces competitive results on these datasets and surpasses all baselines on 4 of them. We perform experiments on both the Wilds benchmark, which captures distribution shift in the real world, as well as datasets in DomainBed benchmark that focuses more on synthetic-to-real transfer. Our method produces competitive results on both benchmarks, demonstrating its effectiveness across a wide range of domain generalization tasks.

Lip-reading aims to infer the speech content from the lip movement sequence and can be seen as a typical sequence-to-sequence (seq2seq) problem which translates the input image sequence of lip movements to the text sequence of the speech content. However, the traditional learning process of seq2seq models always suffers from two problems: the exposure bias resulted from the strategy of "teacher-forcing", and the inconsistency between the discriminative optimization target (usually the cross-entropy loss) and the final evaluation metric (usually the character/word error rate). In this paper, we propose a novel pseudo-convolutional policy gradient (PCPG) based method to address these two problems. On the one hand, we introduce the evaluation metric (refers to the character error rate in this paper) as a form of reward to optimize the model together with the original discriminative target. On the other hand, inspired by the local perception property of convolutional operation, we perform a pseudo-convolutional operation on the reward and loss dimension, so as to take more context around each time step into account to generate a robust reward and loss for the whole optimization. Finally, we perform a thorough comparison and evaluation on both the word-level and sentence-level benchmarks. The results show a significant improvement over other related methods, and report either a new state-of-the-art performance or a competitive accuracy on all these challenging benchmarks, which clearly proves the advantages of our approach.

We present Natural Gradient Boosting (NGBoost), an algorithm for generic probabilistic prediction via gradient boosting. Typical regression models return a point estimate, conditional on covariates, but probabilistic regression models output a full probability distribution over the outcome space, conditional on the covariates. This allows for predictive uncertainty estimation -- crucial in applications like healthcare and weather forecasting. NGBoost generalizes gradient boosting to probabilistic regression by treating the parameters of the conditional distribution as targets for a multiparameter boosting algorithm. Furthermore, we show how the Natural Gradient is required to correct the training dynamics of our multiparameter boosting approach. NGBoost can be used with any base learner, any family of distributions with continuous parameters, and any scoring rule. NGBoost matches or exceeds the performance of existing methods for probabilistic prediction while offering additional benefits in flexibility, scalability, and usability. An open-source implementation is available at github.com/stanfordmlgroup/ngboost.

Large language models (LLMs) encode a diverse range of linguistic features within their latent representations, which can be harnessed to steer their output toward specific target characteristics. In this paper, we modify the internal structure of LLMs by training sparse autoencoders to learn a sparse representation of the query embedding, allowing precise control over the model's attention distribution. We demonstrate that manipulating this sparse representation effectively transforms the output toward different stylistic and cognitive targets. Specifically, in an educational setting, we show that the cognitive complexity of LLM-generated feedback can be systematically adjusted by modifying the encoded query representation at a specific layer. To achieve this, we guide the learned sparse embedding toward the representation of samples from the desired cognitive complexity level, using gradient-based optimization in the latent space.

Structure-Based molecule optimization (SBMO) aims to optimize molecules with both continuous coordinates and discrete types against protein targets. A promising direction is to exert gradient guidance on generative models given its remarkable success in images, but it is challenging to guide discrete data and risks inconsistencies between modalities. To this end, we leverage a continuous and differentiable space derived through Bayesian inference, presenting Molecule Joint Optimization (MolJO), the gradient-based SBMO framework that facilitates joint guidance signals across different modalities while preserving SE(3)-equivariance. We introduce a novel backward correction strategy that optimizes within a sliding window of the past histories, allowing for a seamless trade-off between explore-and-exploit during optimization. MolJO achieves state-of-the-art performance on CrossDocked2020 benchmark (Success Rate 51.3%, Vina Dock -9.05 and SA 0.78), more than 4x improvement in Success Rate compared to the gradient-based counterpart, and 2x "Me-Better" Ratio as much as 3D baselines. Furthermore, we extend MolJO to a wide range of optimization settings, including multi-objective optimization and challenging tasks in drug design such as R-group optimization and scaffold hopping, further underscoring its versatility. Code is available at https://github.com/AlgoMole/MolCRAFT.

To enable learning on edge devices with fast convergence and low memory, we present a novel backpropagation-free optimization algorithm dubbed Target Projection Stochastic Gradient Descent (tpSGD). tpSGD generalizes direct random target projection to work with arbitrary loss functions and extends target projection for training recurrent neural networks (RNNs) in addition to feedforward networks. tpSGD uses layer-wise stochastic gradient descent (SGD) and local targets generated via random projections of the labels to train the network layer-by-layer with only forward passes. tpSGD doesn't require retaining gradients during optimization, greatly reducing memory allocation compared to SGD backpropagation (BP) methods that require multiple instances of the entire neural network weights, input/output, and intermediate results. Our method performs comparably to BP gradient-descent within 5% accuracy on relatively shallow networks of fully connected layers, convolutional layers, and recurrent layers. tpSGD also outperforms other state-of-the-art gradient-free algorithms in shallow models consisting of multi-layer perceptrons, convolutional neural networks (CNNs), and RNNs with competitive accuracy and less memory and time. We evaluate the performance of tpSGD in training deep neural networks (e.g. VGG) and extend the approach to multi-layer RNNs. These experiments highlight new research directions related to optimized layer-based adaptor training for domain-shift using tpSGD at the edge.

We introduce a new family of particle evolution samplers suitable for constrained domains and non-Euclidean geometries. Stein Variational Mirror Descent and Mirrored Stein Variational Gradient Descent minimize the Kullback-Leibler (KL) divergence to constrained target distributions by evolving particles in a dual space defined by a mirror map. Stein Variational Natural Gradient exploits non-Euclidean geometry to more efficiently minimize the KL divergence to unconstrained targets. We derive these samplers from a new class of mirrored Stein operators and adaptive kernels developed in this work. We demonstrate that these new samplers yield accurate approximations to distributions on the simplex, deliver valid confidence intervals in post-selection inference, and converge more rapidly than prior methods in large-scale unconstrained posterior inference. Finally, we establish the convergence of our new procedures under verifiable conditions on the target distribution.

Efficiently approximating local curvature information of the loss function is a key tool for optimization and compression of deep neural networks. Yet, most existing methods to approximate second-order information have high computational or storage costs, which can limit their practicality. In this work, we investigate matrix-free, linear-time approaches for estimating Inverse-Hessian Vector Products (IHVPs) for the case when the Hessian can be approximated as a sum of rank-one matrices, as in the classic approximation of the Hessian by the empirical Fisher matrix. We propose two new algorithms as part of a framework called M-FAC: the first algorithm is tailored towards network compression and can compute the IHVP for dimension d, if the Hessian is given as a sum of m rank-one matrices, using O(dm^2) precomputation, O(dm) cost for computing the IHVP, and query cost O(m) for any single element of the inverse Hessian. The second algorithm targets an optimization setting, where we wish to compute the product between the inverse Hessian, estimated over a sliding window of optimization steps, and a given gradient direction, as required for preconditioned SGD. We give an algorithm with cost O(dm + m^2) for computing the IHVP and O(dm + m^3) for adding or removing any gradient from the sliding window. These two algorithms yield state-of-the-art results for network pruning and optimization with lower computational overhead relative to existing second-order methods. Implementations are available at [9] and [17].

Gradient descent optimization algorithms, while increasingly popular, are often used as black-box optimizers, as practical explanations of their strengths and weaknesses are hard to come by. This article aims to provide the reader with intuitions with regard to the behaviour of different algorithms that will allow her to put them to use. In the course of this overview, we look at different variants of gradient descent, summarize challenges, introduce the most common optimization algorithms, review architectures in a parallel and distributed setting, and investigate additional strategies for optimizing gradient descent.

Inferring causal structures from experimentation is a central task in many domains. For example, in biology, recent advances allow us to obtain single-cell expression data under multiple interventions such as drugs or gene knockouts. However, the targets of the interventions are often uncertain or unknown and the number of observations limited. As a result, standard causal discovery methods can no longer be reliably used. To fill this gap, we propose a Bayesian framework (BaCaDI) for discovering and reasoning about the causal structure that underlies data generated under various unknown experimental or interventional conditions. BaCaDI is fully differentiable, which allows us to infer the complex joint posterior over the intervention targets and the causal structure via efficient gradient-based variational inference. In experiments on synthetic causal discovery tasks and simulated gene-expression data, BaCaDI outperforms related methods in identifying causal structures and intervention targets.

Explaining the output of a deep network remains a challenge. In the case of an image classifier, one type of explanation is to identify pixels that strongly influence the final decision. A starting point for this strategy is the gradient of the class score function with respect to the input image. This gradient can be interpreted as a sensitivity map, and there are several techniques that elaborate on this basic idea. This paper makes two contributions: it introduces SmoothGrad, a simple method that can help visually sharpen gradient-based sensitivity maps, and it discusses lessons in the visualization of these maps. We publish the code for our experiments and a website with our results.

It is often advantageous to train models on a subset of the available train examples, because the examples are of variable quality or because one would like to train with fewer examples, without sacrificing performance. We present Gradient Information Optimization (GIO), a scalable, task-agnostic approach to this data selection problem that requires only a small set of (unlabeled) examples representing a target distribution. GIO begins from a natural, information-theoretic objective that is intractable in practice. Our contribution is in showing that it can be made highly scalable through a simple relaxation of the objective and a highly efficient implementation. In experiments with machine translation, spelling correction, and image recognition, we show that GIO delivers outstanding results with very small train sets. These findings are robust to different representation models and hyperparameters for GIO itself. GIO is task- and domain-agnostic and can be applied out-of-the-box to new datasets and domains.

In this paper, we propose a general deep learning training framework XGrad which introduces weight prediction into the popular gradient-based optimizers to boost their convergence and generalization when training the deep neural network (DNN) models. In particular, ahead of each mini-batch training, the future weights are predicted according to the update rule of the used optimizer and are then applied to both the forward pass and backward propagation. In this way, during the whole training period, the optimizer always utilizes the gradients w.r.t. the future weights to update the DNN parameters, making the gradient-based optimizer achieve better convergence and generalization compared to the original optimizer without weight prediction. XGrad is rather straightforward to implement yet pretty effective in boosting the convergence of gradient-based optimizers and the accuracy of DNN models. Empirical results concerning the most three popular gradient-based optimizers including SGD with momentum, Adam, and AdamW demonstrate the effectiveness of our proposal. The experimental results validate that XGrad can attain higher model accuracy than the original optimizers when training the DNN models. The code of XGrad will be available at: https://github.com/guanleics/XGrad.

We present a class of novel optimisers for training neural networks that makes use of the Riemannian metric naturally induced when the loss landscape is embedded in higher-dimensional space. This is the same metric that underlies common visualisations of loss landscapes. By taking this geometric perspective literally and using the induced metric, we develop a new optimiser and compare it to existing methods, namely: SGD, Adam, AdamW, and Muon, across a range of tasks and architectures. Empirically, we conclude that this new class of optimisers is highly effective in low dimensional examples, and provides slight improvement over state-of-the-art methods for training neural networks. These new optimisers have theoretically desirable properties. In particular, the effective learning rate is automatically decreased in regions of high curvature acting as a smoothed out form of gradient clipping. Similarly, one variant of these optimisers can also be viewed as inducing an effective scheduled learning rate and decoupled weight decay is the natural choice from our geometric perspective. The basic method can be used to modify any existing preconditioning method. The new optimiser has a computational complexity comparable to that of Adam.

It is believed that Gradient Descent (GD) induces an implicit bias towards good generalization in training machine learning models. This paper provides a fine-grained analysis of the dynamics of GD for the matrix sensing problem, whose goal is to recover a low-rank ground-truth matrix from near-isotropic linear measurements. It is shown that GD with small initialization behaves similarly to the greedy low-rank learning heuristics (Li et al., 2020) and follows an incremental learning procedure (Gissin et al., 2019): GD sequentially learns solutions with increasing ranks until it recovers the ground truth matrix. Compared to existing works which only analyze the first learning phase for rank-1 solutions, our result provides characterizations for the whole learning process. Moreover, besides the over-parameterized regime that many prior works focused on, our analysis of the incremental learning procedure also applies to the under-parameterized regime. Finally, we conduct numerical experiments to confirm our theoretical findings.

The generalization capability of deep neural networks has been substantially improved by applying a wide spectrum of regularization methods, e.g., restricting function space, injecting randomness during training, augmenting data, etc. In this work, we propose a simple yet effective regularization method named progressive self-knowledge distillation (PS-KD), which progressively distills a model's own knowledge to soften hard targets (i.e., one-hot vectors) during training. Hence, it can be interpreted within a framework of knowledge distillation as a student becomes a teacher itself. Specifically, targets are adjusted adaptively by combining the ground-truth and past predictions from the model itself. We show that PS-KD provides an effect of hard example mining by rescaling gradients according to difficulty in classifying examples. The proposed method is applicable to any supervised learning tasks with hard targets and can be easily combined with existing regularization methods to further enhance the generalization performance. Furthermore, it is confirmed that PS-KD achieves not only better accuracy, but also provides high quality of confidence estimates in terms of calibration as well as ordinal ranking. Extensive experimental results on three different tasks, image classification, object detection, and machine translation, demonstrate that our method consistently improves the performance of the state-of-the-art baselines. The code is available at https://github.com/lgcnsai/PS-KD-Pytorch.

While previous distribution shift detection approaches can identify if a shift has occurred, these approaches cannot localize which specific features have caused a distribution shift -- a critical step in diagnosing or fixing any underlying issue. For example, in military sensor networks, users will want to detect when one or more of the sensors has been compromised, and critically, they will want to know which specific sensors might be compromised. Thus, we first define a formalization of this problem as multiple conditional distribution hypothesis tests and propose both non-parametric and parametric statistical tests. For both efficiency and flexibility, we then propose to use a test statistic based on the density model score function (i.e. gradient with respect to the input) -- which can easily compute test statistics for all dimensions in a single forward and backward pass. Any density model could be used for computing the necessary statistics including deep density models such as normalizing flows or autoregressive models. We additionally develop methods for identifying when and where a shift occurs in multivariate time-series data and show results for multiple scenarios using realistic attack models on both simulated and real world data.

Gradient-based optimization has been critical to the success of machine learning, updating a single set of parameters to minimize a single loss. A growing number of applications rely on a generalization of this, where we have a bilevel or nested optimization of which subsets of parameters update on different objectives nested inside each other. We focus on motivating examples of hyperparameter optimization and generative adversarial networks. However, naively applying classical methods often fails when we look at solving these nested problems on a large scale. In this thesis, we build tools for nested optimization that scale to deep learning setups.

Advancing towards generalist agents necessitates the concurrent processing of multiple tasks using a unified model, thereby underscoring the growing significance of simultaneous model training on multiple downstream tasks. A common issue in multi-task learning is the occurrence of gradient conflict, which leads to potential competition among different tasks during joint training. This competition often results in improvements in one task at the expense of deterioration in another. Although several optimization methods have been developed to address this issue by manipulating task gradients for better task balancing, they cannot decrease the incidence of gradient conflict. In this paper, we systematically investigate the occurrence of gradient conflict across different methods and propose a strategy to reduce such conflicts through sparse training (ST), wherein only a portion of the model's parameters are updated during training while keeping the rest unchanged. Our extensive experiments demonstrate that ST effectively mitigates conflicting gradients and leads to superior performance. Furthermore, ST can be easily integrated with gradient manipulation techniques, thus enhancing their effectiveness.

A major challenge in training large-scale machine learning models is configuring the training process to maximize model performance, i.e., finding the best training setup from a vast design space. In this work, we unlock a gradient-based approach to this problem. We first introduce an algorithm for efficiently calculating metagradients -- gradients through model training -- at scale. We then introduce a "smooth model training" framework that enables effective optimization using metagradients. With metagradient descent (MGD), we greatly improve on existing dataset selection methods, outperform accuracy-degrading data poisoning attacks by an order of magnitude, and automatically find competitive learning rate schedules.

Robust validation metrics remain essential in contemporary deep learning, not only to detect overfitting and poor generalization, but also to monitor training dynamics. In the supervised classification setting, we investigate whether interactions between training data and model weights can yield such a metric that both tracks generalization during training and attributes performance to individual training samples. We introduce Gradient-Weight Alignment (GWA), quantifying the coherence between per-sample gradients and model weights. We show that effective learning corresponds to coherent alignment, while misalignment indicates deteriorating generalization. GWA is efficiently computable during training and reflects both sample-specific contributions and dataset-wide learning dynamics. Extensive experiments show that GWA accurately predicts optimal early stopping, enables principled model comparisons, and identifies influential training samples, providing a validation-set-free approach for model analysis directly from the training data.

Classifier-free guidance (CFG) is a cornerstone of text-to-image diffusion models, yet its effectiveness is limited by the use of static guidance scales. This "one-size-fits-all" approach fails to adapt to the diverse requirements of different prompts; moreover, prior solutions like gradient-based correction or fixed heuristic schedules introduce additional complexities and fail to generalize. In this work, we challeng this static paradigm by introducing a framework for dynamic CFG scheduling. Our method leverages online feedback from a suite of general-purpose and specialized small-scale latent-space evaluations, such as CLIP for alignment, a discriminator for fidelity and a human preference reward model, to assess generation quality at each step of the reverse diffusion process. Based on this feedback, we perform a greedy search to select the optimal CFG scale for each timestep, creating a unique guidance schedule tailored to every prompt and sample. We demonstrate the effectiveness of our approach on both small-scale models and the state-of-the-art Imagen 3, showing significant improvements in text alignment, visual quality, text rendering and numerical reasoning. Notably, when compared against the default Imagen 3 baseline, our method achieves up to 53.8% human preference win-rate for overall preference, a figure that increases up to to 55.5% on prompts targeting specific capabilities like text rendering. Our work establishes that the optimal guidance schedule is inherently dynamic and prompt-dependent, and provides an efficient and generalizable framework to achieve it.

Gradient regularization (GR) is a method that penalizes the gradient norm of the training loss during training. While some studies have reported that GR can improve generalization performance, little attention has been paid to it from the algorithmic perspective, that is, the algorithms of GR that efficiently improve the performance. In this study, we first reveal that a specific finite-difference computation, composed of both gradient ascent and descent steps, reduces the computational cost of GR. Next, we show that the finite-difference computation also works better in the sense of generalization performance. We theoretically analyze a solvable model, a diagonal linear network, and clarify that GR has a desirable implicit bias to so-called rich regime and finite-difference computation strengthens this bias. Furthermore, finite-difference GR is closely related to some other algorithms based on iterative ascent and descent steps for exploring flat minima. In particular, we reveal that the flooding method can perform finite-difference GR in an implicit way. Thus, this work broadens our understanding of GR for both practice and theory.

We introduce Gravity, another algorithm for gradient-based optimization. In this paper, we explain how our novel idea change parameters to reduce the deep learning model's loss. It has three intuitive hyper-parameters that the best values for them are proposed. Also, we propose an alternative to moving average. To compare the performance of the Gravity optimizer with two common optimizers, Adam and RMSProp, five standard datasets were trained on two VGGNet models with a batch size of 128 for 100 epochs. Gravity hyper-parameters did not need to be tuned for different models. As will be explained more in the paper, to investigate the direct impact of the optimizer itself on loss reduction no overfitting prevention technique was used. The obtained results show that the Gravity optimizer has more stable performance than Adam and RMSProp and gives greater values of validation accuracy for datasets with more output classes like CIFAR-100 (Fine).

Gradient descent is the method of choice for training large artificial intelligence systems. As these systems become larger, a better understanding of the mechanisms behind gradient training would allow us to alleviate compute costs and help steer these systems away from harmful behaviors. To that end, we suggest utilizing the circuit perspective brought forward by mechanistic interpretability. After laying out our intuition, we illustrate how it enables us to design a curriculum for efficient learning in a controlled setting. The code is available at https://github.com/facebookresearch/pal.

We derive a simple and model-independent formula for the change in the generalization gap due to a gradient descent update. We then compare the change in the test error for stochastic gradient descent to the change in test error from an equivalent number of gradient descent updates and show explicitly that stochastic gradient descent acts to regularize generalization error by decorrelating nearby updates. These calculations depends on the details of the model only through the mean and covariance of the gradient distribution, which may be readily measured for particular models of interest. We discuss further improvements to these calculations and comment on possible implications for stochastic optimization.

Motivated by a wide variety of applications, ranging from stochastic optimization to dimension reduction through variable selection, the problem of estimating gradients accurately is of crucial importance in statistics and learning theory. We consider here the classic regression setup, where a real valued square integrable r.v. Y is to be predicted upon observing a (possibly high dimensional) random vector X by means of a predictive function f(X) as accurately as possible in the mean-squared sense and study a nearest-neighbour-based pointwise estimate of the gradient of the optimal predictive function, the regression function m(x)=E[Ymid X=x]. Under classic smoothness conditions combined with the assumption that the tails of Y-m(X) are sub-Gaussian, we prove nonasymptotic bounds improving upon those obtained for alternative estimation methods. Beyond the novel theoretical results established, several illustrative numerical experiments have been carried out. The latter provide strong empirical evidence that the estimation method proposed works very well for various statistical problems involving gradient estimation, namely dimensionality reduction, stochastic gradient descent optimization and quantifying disentanglement.

We study the problem of attributing the prediction of a deep network to its input features, a problem previously studied by several other works. We identify two fundamental axioms---Sensitivity and Implementation Invariance that attribution methods ought to satisfy. We show that they are not satisfied by most known attribution methods, which we consider to be a fundamental weakness of those methods. We use the axioms to guide the design of a new attribution method called Integrated Gradients. Our method requires no modification to the original network and is extremely simple to implement; it just needs a few calls to the standard gradient operator. We apply this method to a couple of image models, a couple of text models and a chemistry model, demonstrating its ability to debug networks, to extract rules from a network, and to enable users to engage with models better.

We propose a new approach for generative modeling based on training a neural network to be idempotent. An idempotent operator is one that can be applied sequentially without changing the result beyond the initial application, namely f(f(z))=f(z). The proposed model f is trained to map a source distribution (e.g, Gaussian noise) to a target distribution (e.g. realistic images) using the following objectives: (1) Instances from the target distribution should map to themselves, namely f(x)=x. We define the target manifold as the set of all instances that f maps to themselves. (2) Instances that form the source distribution should map onto the defined target manifold. This is achieved by optimizing the idempotence term, f(f(z))=f(z) which encourages the range of f(z) to be on the target manifold. Under ideal assumptions such a process provably converges to the target distribution. This strategy results in a model capable of generating an output in one step, maintaining a consistent latent space, while also allowing sequential applications for refinement. Additionally, we find that by processing inputs from both target and source distributions, the model adeptly projects corrupted or modified data back to the target manifold. This work is a first step towards a ``global projector'' that enables projecting any input into a target data distribution.

We show that in a variety of large-scale deep learning scenarios the gradient dynamically converges to a very small subspace after a short period of training. The subspace is spanned by a few top eigenvectors of the Hessian (equal to the number of classes in the dataset), and is mostly preserved over long periods of training. A simple argument then suggests that gradient descent may happen mostly in this subspace. We give an example of this effect in a solvable model of classification, and we comment on possible implications for optimization and learning.

In this paper we introduce Feature Gradients, a gradient-based search algorithm for feature selection. Our approach extends a recent result on the estimation of learnability in the sublinear data regime by showing that the calculation can be performed iteratively (i.e., in mini-batches) and in linear time and space with respect to both the number of features D and the sample size N . This, along with a discrete-to-continuous relaxation of the search domain, allows for an efficient, gradient-based search algorithm among feature subsets for very large datasets. Crucially, our algorithm is capable of finding higher-order correlations between features and targets for both the N > D and N < D regimes, as opposed to approaches that do not consider such interactions and/or only consider one regime. We provide experimental demonstration of the algorithm in small and large sample-and feature-size settings.

Momentum based optimizers are central to a wide range of machine learning applications. These typically rely on an Exponential Moving Average (EMA) of gradients, which decays exponentially the present contribution of older gradients. This accounts for gradients being local linear approximations which lose their relevance as the iterate moves along the loss landscape. This work questions the use of a single EMA to accumulate past gradients and empirically demonstrates how this choice can be sub-optimal: a single EMA cannot simultaneously give a high weight to the immediate past, and a non-negligible weight to older gradients. Building on this observation, we propose AdEMAMix, a simple modification of the Adam optimizer with a mixture of two EMAs to better take advantage of past gradients. Our experiments on language modeling and image classification show -- quite surprisingly -- that gradients can stay relevant for tens of thousands of steps. They help to converge faster, and often to lower minima: e.g., a 1.3B parameter AdEMAMix LLM trained on 101B tokens performs comparably to an AdamW model trained on 197B tokens (+95%). Moreover, our method significantly slows-down model forgetting during training. Our work motivates further exploration of different types of functions to leverage past gradients, beyond EMAs.

We study the properties of common loss surfaces through their Hessian matrix. In particular, in the context of deep learning, we empirically show that the spectrum of the Hessian is composed of two parts: (1) the bulk centered near zero, (2) and outliers away from the bulk. We present numerical evidence and mathematical justifications to the following conjectures laid out by Sagun et al. (2016): Fixing data, increasing the number of parameters merely scales the bulk of the spectrum; fixing the dimension and changing the data (for instance adding more clusters or making the data less separable) only affects the outliers. We believe that our observations have striking implications for non-convex optimization in high dimensions. First, the flatness of such landscapes (which can be measured by the singularity of the Hessian) implies that classical notions of basins of attraction may be quite misleading. And that the discussion of wide/narrow basins may be in need of a new perspective around over-parametrization and redundancy that are able to create large connected components at the bottom of the landscape. Second, the dependence of small number of large eigenvalues to the data distribution can be linked to the spectrum of the covariance matrix of gradients of model outputs. With this in mind, we may reevaluate the connections within the data-architecture-algorithm framework of a model, hoping that it would shed light into the geometry of high-dimensional and non-convex spaces in modern applications. In particular, we present a case that links the two observations: small and large batch gradient descent appear to converge to different basins of attraction but we show that they are in fact connected through their flat region and so belong to the same basin.

Classifier guidance -- using the gradients of an image classifier to steer the generations of a diffusion model -- has the potential to dramatically expand the creative control over image generation and editing. However, currently classifier guidance requires either training new noise-aware models to obtain accurate gradients or using a one-step denoising approximation of the final generation, which leads to misaligned gradients and sub-optimal control. We highlight this approximation's shortcomings and propose a novel guidance method: Direct Optimization of Diffusion Latents (DOODL), which enables plug-and-play guidance by optimizing diffusion latents w.r.t. the gradients of a pre-trained classifier on the true generated pixels, using an invertible diffusion process to achieve memory-efficient backpropagation. Showcasing the potential of more precise guidance, DOODL outperforms one-step classifier guidance on computational and human evaluation metrics across different forms of guidance: using CLIP guidance to improve generations of complex prompts from DrawBench, using fine-grained visual classifiers to expand the vocabulary of Stable Diffusion, enabling image-conditioned generation with a CLIP visual encoder, and improving image aesthetics using an aesthetic scoring network. Code at https://github.com/salesforce/DOODL.

We propose a categorical semantics of gradient-based machine learning algorithms in terms of lenses, parametrised maps, and reverse derivative categories. This foundation provides a powerful explanatory and unifying framework: it encompasses a variety of gradient descent algorithms such as ADAM, AdaGrad, and Nesterov momentum, as well as a variety of loss functions such as as MSE and Softmax cross-entropy, shedding new light on their similarities and differences. Our approach to gradient-based learning has examples generalising beyond the familiar continuous domains (modelled in categories of smooth maps) and can be realized in the discrete setting of boolean circuits. Finally, we demonstrate the practical significance of our framework with an implementation in Python.

In this work, we develop new optimization algorithms that use approximate second-order information combined with the gradient regularization technique to achieve fast global convergence rates for both convex and non-convex objectives. The key innovation of our analysis is a novel notion called Gradient-Normalized Smoothness, which characterizes the maximum radius of a ball around the current point that yields a good relative approximation of the gradient field. Our theory establishes a natural intrinsic connection between Hessian approximation and the linearization of the gradient. Importantly, Gradient-Normalized Smoothness does not depend on the specific problem class of the objective functions, while effectively translating local information about the gradient field and Hessian approximation into the global behavior of the method. This new concept equips approximate second-order algorithms with universal global convergence guarantees, recovering state-of-the-art rates for functions with H\"older-continuous Hessians and third derivatives, quasi-self-concordant functions, as well as smooth classes in first-order optimization. These rates are achieved automatically and extend to broader classes, such as generalized self-concordant functions. We demonstrate direct applications of our results for global linear rates in logistic regression and softmax problems with approximate Hessians, as well as in non-convex optimization using Fisher and Gauss-Newton approximations.

We consider minimizing functions for which it is expensive to compute the (possibly stochastic) gradient. Such functions are prevalent in reinforcement learning, imitation learning and adversarial training. Our target optimization framework uses the (expensive) gradient computation to construct surrogate functions in a target space (e.g. the logits output by a linear model for classification) that can be minimized efficiently. This allows for multiple parameter updates to the model, amortizing the cost of gradient computation. In the full-batch setting, we prove that our surrogate is a global upper-bound on the loss, and can be (locally) minimized using a black-box optimization algorithm. We prove that the resulting majorization-minimization algorithm ensures convergence to a stationary point of the loss. Next, we instantiate our framework in the stochastic setting and propose the SSO algorithm, which can be viewed as projected stochastic gradient descent in the target space. This connection enables us to prove theoretical guarantees for SSO when minimizing convex functions. Our framework allows the use of standard stochastic optimization algorithms to construct surrogates which can be minimized by any deterministic optimization method. To evaluate our framework, we consider a suite of supervised learning and imitation learning problems. Our experiments indicate the benefits of target optimization and the effectiveness of SSO.

In multi-goal Reinforcement Learning, an agent can share experience between related training tasks, resulting in better generalization for new tasks at test time. However, when the goal space has discontinuities and the reward is sparse, a majority of goals are difficult to reach. In this context, a curriculum over goals helps agents learn by adapting training tasks to their current capabilities. In this work we propose Stein Variational Goal Generation (SVGG), which samples goals of intermediate difficulty for the agent, by leveraging a learned predictive model of its goal reaching capabilities. The distribution of goals is modeled with particles that are attracted in areas of appropriate difficulty using Stein Variational Gradient Descent. We show that SVGG outperforms state-of-the-art multi-goal Reinforcement Learning methods in terms of success coverage in hard exploration problems, and demonstrate that it is endowed with a useful recovery property when the environment changes.

How to ensure fairness is an important topic in federated learning (FL). Recent studies have investigated how to reward clients based on their contribution (collaboration fairness), and how to achieve uniformity of performance across clients (performance fairness). Despite achieving progress on either one, we argue that it is critical to consider them together, in order to engage and motivate more diverse clients joining FL to derive a high-quality global model. In this work, we propose a novel method to optimize both types of fairness simultaneously. Specifically, we propose to estimate client contribution in gradient and data space. In gradient space, we monitor the gradient direction differences of each client with respect to others. And in data space, we measure the prediction error on client data using an auxiliary model. Based on this contribution estimation, we propose a FL method, federated training via contribution estimation (FedCE), i.e., using estimation as global model aggregation weights. We have theoretically analyzed our method and empirically evaluated it on two real-world medical datasets. The effectiveness of our approach has been validated with significant performance improvements, better collaboration fairness, better performance fairness, and comprehensive analytical studies.

This paper studies generalization capabilities of neural networks (NNs) using new and improved PyTorch library Loss Landscape Analysis (LLA). LLA facilitates visualization and analysis of loss landscapes along with the properties of NN Hessian. Different approaches to NN loss landscape plotting are discussed with particular focus on normalization techniques showing that conventional methods cannot always ensure correct visualization when batch normalization layers are present in NN architecture. The use of Hessian axes is shown to be able to mitigate this effect, and methods for choosing Hessian axes are proposed. In addition, spectra of Hessian eigendecomposition are studied and it is shown that typical spectra exist for a wide range of NNs. This allows to propose quantitative criteria for Hessian analysis that can be applied to evaluate NN performance and assess its generalization capabilities. Generalization experiments are conducted using ImageNet-1K pre-trained models along with several models trained as part of this study. The experiment include training models on one dataset and testing on another one to maximize experiment similarity to model performance in the Wild. It is shown that when datasets change, the changes in criteria correlate with the changes in accuracy, making the proposed criteria a computationally efficient estimate of generalization ability, which is especially useful for extremely large datasets.

Given an unconditional diffusion model and a predictor for a target property of interest (e.g., a classifier), the goal of training-free guidance is to generate samples with desirable target properties without additional training. Existing methods, though effective in various individual applications, often lack theoretical grounding and rigorous testing on extensive benchmarks. As a result, they could even fail on simple tasks, and applying them to a new problem becomes unavoidably difficult. This paper introduces a novel algorithmic framework encompassing existing methods as special cases, unifying the study of training-free guidance into the analysis of an algorithm-agnostic design space. Via theoretical and empirical investigation, we propose an efficient and effective hyper-parameter searching strategy that can be readily applied to any downstream task. We systematically benchmark across 7 diffusion models on 16 tasks with 40 targets, and improve performance by 8.5% on average. Our framework and benchmark offer a solid foundation for conditional generation in a training-free manner.

We consider the training of the first layer of vision models and notice the clear relationship between pixel values and gradient update magnitudes: the gradients arriving at the weights of a first layer are by definition directly proportional to (normalized) input pixel values. Thus, an image with low contrast has a smaller impact on learning than an image with higher contrast, and a very bright or very dark image has a stronger impact on the weights than an image with moderate brightness. In this work, we propose performing gradient descent on the embeddings produced by the first layer of the model. However, switching to discrete inputs with an embedding layer is not a reasonable option for vision models. Thus, we propose the conceptual procedure of (i) a gradient descent step on first layer activations to construct an activation proposal, and (ii) finding the optimal weights of the first layer, i.e., those weights which minimize the squared distance to the activation proposal. We provide a closed form solution of the procedure and adjust it for robust stochastic training while computing everything efficiently. Empirically, we find that TrAct (Training Activations) speeds up training by factors between 1.25x and 4x while requiring only a small computational overhead. We demonstrate the utility of TrAct with different optimizers for a range of different vision models including convolutional and transformer architectures.

Normalization techniques are a boon for modern deep learning. They let weights converge more quickly with often better generalization performances. It has been argued that the normalization-induced scale invariance among the weights provides an advantageous ground for gradient descent (GD) optimizers: the effective step sizes are automatically reduced over time, stabilizing the overall training procedure. It is often overlooked, however, that the additional introduction of momentum in GD optimizers results in a far more rapid reduction in effective step sizes for scale-invariant weights, a phenomenon that has not yet been studied and may have caused unwanted side effects in the current practice. This is a crucial issue because arguably the vast majority of modern deep neural networks consist of (1) momentum-based GD (e.g. SGD or Adam) and (2) scale-invariant parameters. In this paper, we verify that the widely-adopted combination of the two ingredients lead to the premature decay of effective step sizes and sub-optimal model performances. We propose a simple and effective remedy, SGDP and AdamP: get rid of the radial component, or the norm-increasing direction, at each optimizer step. Because of the scale invariance, this modification only alters the effective step sizes without changing the effective update directions, thus enjoying the original convergence properties of GD optimizers. Given the ubiquity of momentum GD and scale invariance in machine learning, we have evaluated our methods against the baselines on 13 benchmarks. They range from vision tasks like classification (e.g. ImageNet), retrieval (e.g. CUB and SOP), and detection (e.g. COCO) to language modelling (e.g. WikiText) and audio classification (e.g. DCASE) tasks. We verify that our solution brings about uniform gains in those benchmarks. Source code is available at https://github.com/clovaai/AdamP.

Gradient methods have become mainstream techniques for Bi-Level Optimization (BLO) in learning fields. The validity of existing works heavily rely on either a restrictive Lower- Level Strong Convexity (LLSC) condition or on solving a series of approximation subproblems with high accuracy or both. In this work, by averaging the upper and lower level objectives, we propose a single loop Bi-level Averaged Method of Multipliers (sl-BAMM) for BLO that is simple yet efficient for large-scale BLO and gets rid of the limited LLSC restriction. We further provide non-asymptotic convergence analysis of sl-BAMM towards KKT stationary points, and the comparative advantage of our analysis lies in the absence of strong gradient boundedness assumption, which is always required by others. Thus our theory safely captures a wider variety of applications in deep learning, especially where the upper-level objective is quadratic w.r.t. the lower-level variable. Experimental results demonstrate the superiority of our method.

Early stopping monitors global validation loss and halts all parameter updates simultaneously, which is computationally costly for large transformers due to the extended time required for validation inference. We propose GradES, a novel gradient-based early stopping approach that operates within transformer components (attention projections and Feed-Forward layer matrices). We found that different components converge at varying rates during fine-tuning. GradES tracks the magnitude of gradients in backpropagation for these matrices during training. When a projection matrix's gradients fall below a convergence threshold tau, we exclude that projection matrix from further updates individually, eliminating costly validation passes while allowing slow converging matrices to continue learning. By strategically freezing parameters when their gradients converge, GradES speeds up training time by 1.57--7.22times while simultaneously enhancing generalization through early prevention of overfitting, resulting in 1.2% higher average accuracy.

We introduce ORIGEN, the first zero-shot method for 3D orientation grounding in text-to-image generation across multiple objects and diverse categories. While previous work on spatial grounding in image generation has mainly focused on 2D positioning, it lacks control over 3D orientation. To address this, we propose a reward-guided sampling approach using a pretrained discriminative model for 3D orientation estimation and a one-step text-to-image generative flow model. While gradient-ascent-based optimization is a natural choice for reward-based guidance, it struggles to maintain image realism. Instead, we adopt a sampling-based approach using Langevin dynamics, which extends gradient ascent by simply injecting random noise--requiring just a single additional line of code. Additionally, we introduce adaptive time rescaling based on the reward function to accelerate convergence. Our experiments show that ORIGEN outperforms both training-based and test-time guidance methods across quantitative metrics and user studies.

Recent work shows that path gradient estimators for normalizing flows have lower variance compared to standard estimators for variational inference, resulting in improved training. However, they are often prohibitively more expensive from a computational point of view and cannot be applied to maximum likelihood training in a scalable manner, which severely hinders their widespread adoption. In this work, we overcome these crucial limitations. Specifically, we propose a fast path gradient estimator which improves computational efficiency significantly and works for all normalizing flow architectures of practical relevance. We then show that this estimator can also be applied to maximum likelihood training for which it has a regularizing effect as it can take the form of a given target energy function into account. We empirically establish its superior performance and reduced variance for several natural sciences applications.

This work examines whether activating latent subspaces in language models (LLMs) can steer scientific code generation toward a specific programming language. Five causal LLMs were first evaluated on scientific coding prompts to quantify their baseline bias among four programming languages. A static neuron-attribution method, perturbing the highest activated MLP weight for a C++ or CPP token, proved brittle and exhibited limited generalization across prompt styles and model scales. To address these limitations, a gradient-refined adaptive activation steering framework (G-ACT) was developed: per-prompt activation differences are clustered into a small set of steering directions, and lightweight per-layer probes are trained and refined online to select the appropriate steering vector. In LLaMA-3.2 3B, this approach reliably biases generation towards the CPP language by increasing the average probe classification accuracy by 15% and the early layers (0-6) improving the probe classification accuracy by 61.5% compared to the standard ACT framework. For LLaMA-3.3 70B, where attention-head signals become more diffuse, targeted injections at key layers still improve language selection. Although per-layer probing introduces a modest inference overhead, it remains practical by steering only a subset of layers and enables reproducible model behavior. These results demonstrate a scalable, interpretable and efficient mechanism for concept-level control for practical agentic systems.

Optimization and generalization are two essential aspects of statistical machine learning. In this paper, we propose a framework to connect optimization with generalization by analyzing the generalization error based on the optimization trajectory under the gradient flow algorithm. The key ingredient of this framework is the Uniform-LGI, a property that is generally satisfied when training machine learning models. Leveraging the Uniform-LGI, we first derive convergence rates for gradient flow algorithm, then we give generalization bounds for a large class of machine learning models. We further apply our framework to three distinct machine learning models: linear regression, kernel regression, and two-layer neural networks. Through our approach, we obtain generalization estimates that match or extend previous results.

This paper provides a review and commentary on the past, present, and future of numerical optimization algorithms in the context of machine learning applications. Through case studies on text classification and the training of deep neural networks, we discuss how optimization problems arise in machine learning and what makes them challenging. A major theme of our study is that large-scale machine learning represents a distinctive setting in which the stochastic gradient (SG) method has traditionally played a central role while conventional gradient-based nonlinear optimization techniques typically falter. Based on this viewpoint, we present a comprehensive theory of a straightforward, yet versatile SG algorithm, discuss its practical behavior, and highlight opportunities for designing algorithms with improved performance. This leads to a discussion about the next generation of optimization methods for large-scale machine learning, including an investigation of two main streams of research on techniques that diminish noise in the stochastic directions and methods that make use of second-order derivative approximations.

Every data selection method inherently has a target. In practice, these targets often emerge implicitly through benchmark-driven iteration: researchers develop selection strategies, train models, measure benchmark performance, then refine accordingly. This raises a natural question: what happens when we make this optimization explicit? To explore this, we propose benchmark-targeted ranking (BETR), a simple method that selects pretraining documents based on similarity to benchmark training examples. BETR embeds benchmark examples and a sample of pretraining documents in a shared space, scores this sample by similarity to benchmarks, then trains a lightweight classifier to predict these scores for the full corpus. We compare data selection methods by training over 500 models spanning 10^{19} to 10^{22} FLOPs and fitting scaling laws to them. From this, we find that simply aligning pretraining data to evaluation benchmarks using BETR achieves a 2.1x compute multiplier over DCLM-Baseline (4.7x over unfiltered data) and improves performance on 9 out of 10 tasks across all scales. BETR also generalizes well: when targeting a diverse set of benchmarks disjoint from our evaluation suite, it still matches or outperforms baselines. Our scaling analysis further reveals a clear trend: larger models require less aggressive filtering. Overall, our findings show that directly matching pretraining data to target tasks precisely shapes model capabilities and highlight that optimal selection strategies must adapt to model scale.

Reinforcement learning with verifiable rewards (RLVR) has become a standard recipe for improving large language models (LLMs) on reasoning tasks, with Group Relative Policy Optimization (GRPO) widely used in practice. Yet GRPO wastes substantial compute on negative groups: groups in which no sampled response is correct yield zero advantage and thus no gradient. We ask whether negative groups can be leveraged without extra supervision. Starting from a maximum-likelihood (MLE) objective in reward modeling, we show that the MLE gradient is equivalent to a policy gradient for a modified value function. This value function adds a confidence-weighted penalty on incorrect responses, imposing larger penalties on more confident mistakes. We refer to this as Likelihood Estimation with Negative Samples (LENS). LENS modifies GRPO to assign non-zero, confidence-dependent rewards to incorrect generations, making negative groups informative and converting previously wasted samples into useful gradient updates. On the MATH benchmark with Llama-3.1-8B and Qwen-2.5-3B, the proposed variant consistently outperforms GRPO baseline, with significant gains on harder items. These results demonstrate a principled and practical way to "rescue" negative groups, improving efficiency and performance in RLVR.

Sparse training is emerging as a promising avenue for reducing the computational cost of training neural networks. Several recent studies have proposed pruning methods using learnable thresholds to efficiently explore the non-uniform distribution of sparsity inherent within the models. In this paper, we propose Gradient Annealing (GA), where gradients of masked weights are scaled down in a non-linear manner. GA provides an elegant trade-off between sparsity and accuracy without the need for additional sparsity-inducing regularization. We integrated GA with the latest learnable pruning methods to create an automated sparse training algorithm called AutoSparse, which achieves better accuracy and/or training/inference FLOPS reduction than existing learnable pruning methods for sparse ResNet50 and MobileNetV1 on ImageNet-1K: AutoSparse achieves (2x, 7x) reduction in (training,inference) FLOPS for ResNet50 on ImageNet at 80% sparsity. Finally, AutoSparse outperforms sparse-to-sparse SotA method MEST (uniform sparsity) for 80% sparse ResNet50 with similar accuracy, where MEST uses 12% more training FLOPS and 50% more inference FLOPS.

Today's deep learning methods focus on how to design the most appropriate objective functions so that the prediction results of the model can be closest to the ground truth. Meanwhile, an appropriate architecture that can facilitate acquisition of enough information for prediction has to be designed. Existing methods ignore a fact that when input data undergoes layer-by-layer feature extraction and spatial transformation, large amount of information will be lost. This paper will delve into the important issues of data loss when data is transmitted through deep networks, namely information bottleneck and reversible functions. We proposed the concept of programmable gradient information (PGI) to cope with the various changes required by deep networks to achieve multiple objectives. PGI can provide complete input information for the target task to calculate objective function, so that reliable gradient information can be obtained to update network weights. In addition, a new lightweight network architecture -- Generalized Efficient Layer Aggregation Network (GELAN), based on gradient path planning is designed. GELAN's architecture confirms that PGI has gained superior results on lightweight models. We verified the proposed GELAN and PGI on MS COCO dataset based object detection. The results show that GELAN only uses conventional convolution operators to achieve better parameter utilization than the state-of-the-art methods developed based on depth-wise convolution. PGI can be used for variety of models from lightweight to large. It can be used to obtain complete information, so that train-from-scratch models can achieve better results than state-of-the-art models pre-trained using large datasets, the comparison results are shown in Figure 1. The source codes are at: https://github.com/WongKinYiu/yolov9.

Gradients can be employed for sensitivity analysis. Here, we leverage the advantages of the Loss Landscape to comprehend which independent variables impact the dependent variable. We seek to grasp the loss landscape by utilizing first, second, and third derivatives through automatic differentiation. we know that Spearman's rank correlation coefficient can detect the monotonic relationship between two variables. However, I have found that second-order gradients, with certain configurations and parameters, provide information that can be visualized similarly to Spearman results, In this approach, we incorporate a loss function with an activation function, resulting in a non-linear pattern. Each exploration of the loss landscape through retraining yields new valuable information. Furthermore, the first and third derivatives are also beneficial, as they indicate the extent to which independent variables influence the dependent variable.

Diffusion models have demonstrated strong generative performance when using guidance methods such as classifier-free guidance (CFG), which enhance output quality by modifying the sampling trajectory. These methods typically improve a target output by intentionally degrading another, often the unconditional output, using heuristic perturbation functions such as identity mixing or blurred conditions. However, these approaches lack a principled foundation and rely on manually designed distortions. In this work, we propose Adversarial Sinkhorn Attention Guidance (ASAG), a novel method that reinterprets attention scores in diffusion models through the lens of optimal transport and intentionally disrupt the transport cost via Sinkhorn algorithm. Instead of naively corrupting the attention mechanism, ASAG injects an adversarial cost within self-attention layers to reduce pixel-wise similarity between queries and keys. This deliberate degradation weakens misleading attention alignments and leads to improved conditional and unconditional sample quality. ASAG shows consistent improvements in text-to-image diffusion, and enhances controllability and fidelity in downstream applications such as IP-Adapter and ControlNet. The method is lightweight, plug-and-play, and improves reliability without requiring any model retraining.

Multi-Task Learning (MTL) is a widely-used and powerful learning paradigm for training deep neural networks that allows learning more than one objective by a single backbone. Compared to training tasks separately, MTL significantly reduces computational costs, improves data efficiency, and potentially enhances model performance by leveraging knowledge across tasks. Hence, it has been adopted in a variety of applications, ranging from computer vision to natural language processing and speech recognition. Among them, there is an emerging line of work in MTL that focuses on manipulating the task gradient to derive an ultimate gradient descent direction to benefit all tasks. Despite achieving impressive results on many benchmarks, directly applying these approaches without using appropriate regularization techniques might lead to suboptimal solutions on real-world problems. In particular, standard training that minimizes the empirical loss on the training data can easily suffer from overfitting to low-resource tasks or be spoiled by noisy-labeled ones, which can cause negative transfer between tasks and overall performance drop. To alleviate such problems, we propose to leverage a recently introduced training method, named Sharpness-aware Minimization, which can enhance model generalization ability on single-task learning. Accordingly, we present a novel MTL training methodology, encouraging the model to find task-based flat minima for coherently improving its generalization capability on all tasks. Finally, we conduct comprehensive experiments on a variety of applications to demonstrate the merit of our proposed approach to existing gradient-based MTL methods, as suggested by our developed theory.

We introduce Gradient Descent with Adaptive Momentum Scaling (Grams), a novel optimization algorithm that decouples the direction and magnitude of parameter updates in deep learning. Unlike traditional optimizers that directly integrate momentum into updates, Grams separates the update direction, derived from current gradients, from momentum, which is used solely for adaptive magnitude scaling. This approach enables Grams to achieve improved loss descent compared to state-of-the-art cautious and momentum-based optimizers. We establish a global convergence guarantee for Grams and validate its effectiveness through extensive empirical evaluations. The results demonstrate Grams' superior performance, including faster convergence and better generalization, compared to widely-used optimizers such as Adam, Lion, and their cautious variants. Our results highlight Grams' potential as a transformative approach for efficient optimization in large-scale machine learning.

Two major sources of training data exist for post-training modern language models: online (model-generated rollouts) data, and offline (human or other-model demonstrations) data. These two types of data are typically used by approaches like Reinforcement Learning (RL) and Supervised Fine-Tuning (SFT), respectively. In this paper, we show that these approaches are not in contradiction, but are instances of a single optimization process. We derive a Unified Policy Gradient Estimator, and present the calculations of a wide spectrum of post-training approaches as the gradient of a common objective under different data distribution assumptions and various bias-variance tradeoffs. The gradient estimator is constructed with four interchangeable parts: stabilization mask, reference policy denominator, advantage estimate, and likelihood gradient. Motivated by our theoretical findings, we propose Hybrid Post-Training (HPT), an algorithm that dynamically selects different training signals. HPT is designed to yield both effective exploitation of demonstration and stable exploration without sacrificing learned reasoning patterns. We provide extensive experiments and ablation studies to verify the effectiveness of our unified theoretical framework and HPT. Across six mathematical reasoning benchmarks and two out-of-distribution suites, HPT consistently surpasses strong baselines across models of varying scales and families.

A supervised learning algorithm searches over a set of functions A to B parametrised by a space P to find the best approximation to some ideal function fcolon A to B. It does this by taking examples (a,f(a)) in Atimes B, and updating the parameter according to some rule. We define a category where these update rules may be composed, and show that gradient descent---with respect to a fixed step size and an error function satisfying a certain property---defines a monoidal functor from a category of parametrised functions to this category of update rules. This provides a structural perspective on backpropagation, as well as a broad generalisation of neural networks.

Model-Agnostic Meta-Learning (MAML) has become increasingly popular for training models that can quickly adapt to new tasks via one or few stochastic gradient descent steps. However, the MAML objective is significantly more difficult to optimize compared to standard non-adaptive learning (NAL), and little is understood about how much MAML improves over NAL in terms of the fast adaptability of their solutions in various scenarios. We analytically address this issue in a linear regression setting consisting of a mixture of easy and hard tasks, where hardness is related to the rate that gradient descent converges on the task. Specifically, we prove that in order for MAML to achieve substantial gain over NAL, (i) there must be some discrepancy in hardness among the tasks, and (ii) the optimal solutions of the hard tasks must be closely packed with the center far from the center of the easy tasks optimal solutions. We also give numerical and analytical results suggesting that these insights apply to two-layer neural networks. Finally, we provide few-shot image classification experiments that support our insights for when MAML should be used and emphasize the importance of training MAML on hard tasks in practice.

We present an algorithm for minimizing an objective with hard-to-compute gradients by using a related, easier-to-access function as a proxy. Our algorithm is based on approximate proximal point iterations on the proxy combined with relatively few stochastic gradients from the objective. When the difference between the objective and the proxy is delta-smooth, our algorithm guarantees convergence at a rate matching stochastic gradient descent on a delta-smooth objective, which can lead to substantially better sample efficiency. Our algorithm has many potential applications in machine learning, and provides a principled means of leveraging synthetic data, physics simulators, mixed public and private data, and more.

Several recently proposed stochastic optimization methods that have been successfully used in training deep networks such as RMSProp, Adam, Adadelta, Nadam are based on using gradient updates scaled by square roots of exponential moving averages of squared past gradients. In many applications, e.g. learning with large output spaces, it has been empirically observed that these algorithms fail to converge to an optimal solution (or a critical point in nonconvex settings). We show that one cause for such failures is the exponential moving average used in the algorithms. We provide an explicit example of a simple convex optimization setting where Adam does not converge to the optimal solution, and describe the precise problems with the previous analysis of Adam algorithm. Our analysis suggests that the convergence issues can be fixed by endowing such algorithms with `long-term memory' of past gradients, and propose new variants of the Adam algorithm which not only fix the convergence issues but often also lead to improved empirical performance.

Denoising diffusion models excel at generating high-quality images conditioned on text prompts, yet their effectiveness heavily relies on careful guidance during the sampling process. Classifier-Free Guidance (CFG) provides a widely used mechanism for steering generation by setting the guidance scale, which balances image quality and prompt alignment. However, the choice of the guidance scale has a critical impact on the convergence toward a visually appealing and prompt-adherent image. In this work, we propose an annealing guidance scheduler which dynamically adjusts the guidance scale over time based on the conditional noisy signal. By learning a scheduling policy, our method addresses the temperamental behavior of CFG. Empirical results demonstrate that our guidance scheduler significantly enhances image quality and alignment with the text prompt, advancing the performance of text-to-image generation. Notably, our novel scheduler requires no additional activations or memory consumption, and can seamlessly replace the common classifier-free guidance, offering an improved trade-off between prompt alignment and quality.

While deep learning and deep reinforcement learning (RL) systems have demonstrated impressive results in domains such as image classification, game playing, and robotic control, data efficiency remains a major challenge. Multi-task learning has emerged as a promising approach for sharing structure across multiple tasks to enable more efficient learning. However, the multi-task setting presents a number of optimization challenges, making it difficult to realize large efficiency gains compared to learning tasks independently. The reasons why multi-task learning is so challenging compared to single-task learning are not fully understood. In this work, we identify a set of three conditions of the multi-task optimization landscape that cause detrimental gradient interference, and develop a simple yet general approach for avoiding such interference between task gradients. We propose a form of gradient surgery that projects a task's gradient onto the normal plane of the gradient of any other task that has a conflicting gradient. On a series of challenging multi-task supervised and multi-task RL problems, this approach leads to substantial gains in efficiency and performance. Further, it is model-agnostic and can be combined with previously-proposed multi-task architectures for enhanced performance.

We introduce a new family of physics-inspired generative models termed PFGM++ that unifies diffusion models and Poisson Flow Generative Models (PFGM). These models realize generative trajectories for N dimensional data by embedding paths in N{+}D dimensional space while still controlling the progression with a simple scalar norm of the D additional variables. The new models reduce to PFGM when D{=}1 and to diffusion models when D{to}infty. The flexibility of choosing D allows us to trade off robustness against rigidity as increasing D results in more concentrated coupling between the data and the additional variable norms. We dispense with the biased large batch field targets used in PFGM and instead provide an unbiased perturbation-based objective similar to diffusion models. To explore different choices of D, we provide a direct alignment method for transferring well-tuned hyperparameters from diffusion models (D{to} infty) to any finite D values. Our experiments show that models with finite D can be superior to previous state-of-the-art diffusion models on CIFAR-10/FFHQ 64{times}64 datasets, with FID scores of 1.91/2.43 when D{=}2048/128. In class-conditional setting, D{=}2048 yields current state-of-the-art FID of 1.74 on CIFAR-10. In addition, we demonstrate that models with smaller D exhibit improved robustness against modeling errors. Code is available at https://github.com/Newbeeer/pfgmpp

Recent work by Baratin et al. (2021) sheds light on an intriguing pattern that occurs during the training of deep neural networks: some layers align much more with data compared to other layers (where the alignment is defined as the euclidean product of the tangent features matrix and the data labels matrix). The curve of the alignment as a function of layer index (generally) exhibits an ascent-descent pattern where the maximum is reached for some hidden layer. In this work, we provide the first explanation for this phenomenon. We introduce the Equilibrium Hypothesis which connects this alignment pattern to signal propagation in deep neural networks. Our experiments demonstrate an excellent match with the theoretical predictions.

The presence of distribution shifts poses a significant challenge for deploying modern machine learning models in real-world applications. This work focuses on the target shift problem in a regression setting (Zhang et al., 2013; Nguyen et al., 2016). More specifically, the target variable y (also known as the response variable), which is continuous, has different marginal distributions in the training source and testing domain, while the conditional distribution of features x given y remains the same. While most literature focuses on classification tasks with finite target space, the regression problem has an infinite dimensional target space, which makes many of the existing methods inapplicable. In this work, we show that the continuous target shift problem can be addressed by estimating the importance weight function from an ill-posed integral equation. We propose a nonparametric regularized approach named ReTaSA to solve the ill-posed integral equation and provide theoretical justification for the estimated importance weight function. The effectiveness of the proposed method has been demonstrated with extensive numerical studies on synthetic and real-world datasets.

Saliency methods seek to explain the predictions of a model by producing an importance map across each input sample. A popular class of such methods is based on backpropagating a signal and analyzing the resulting gradient. Despite much research on such methods, relatively little work has been done to clarify the differences between such methods as well as the desiderata of these techniques. Thus, there is a need for rigorously understanding the relationships between different methods as well as their failure modes. In this work, we conduct a thorough analysis of backpropagation-based saliency methods and propose a single framework under which several such methods can be unified. As a result of our study, we make three additional contributions. First, we use our framework to propose NormGrad, a novel saliency method based on the spatial contribution of gradients of convolutional weights. Second, we combine saliency maps at different layers to test the ability of saliency methods to extract complementary information at different network levels (e.g.~trading off spatial resolution and distinctiveness) and we explain why some methods fail at specific layers (e.g., Grad-CAM anywhere besides the last convolutional layer). Third, we introduce a class-sensitivity metric and a meta-learning inspired paradigm applicable to any saliency method for improving sensitivity to the output class being explained.

Using backpropagation to compute gradients of objective functions for optimization has remained a mainstay of machine learning. Backpropagation, or reverse-mode differentiation, is a special case within the general family of automatic differentiation algorithms that also includes the forward mode. We present a method to compute gradients based solely on the directional derivative that one can compute exactly and efficiently via the forward mode. We call this formulation the forward gradient, an unbiased estimate of the gradient that can be evaluated in a single forward run of the function, entirely eliminating the need for backpropagation in gradient descent. We demonstrate forward gradient descent in a range of problems, showing substantial savings in computation and enabling training up to twice as fast in some cases.

Current Reinforcement Fine-tuning (RFT) paradigms for Large Language Models (LLMs) suffer from sample inefficiency due to the redundant exposure of identical queries under uniform data sampling. While previous work has explored curriculum learning via heuristic difficulty metrics, these strategies exhibit limitations by neglecting the intrinsic learning signals generated by the model itself, thus leading to suboptimal training regimes. In this paper, we identify a model-inherent signal termed angle concentration that effectively reflects an LLM's capacity to learn from specific data. We theoretically and empirically demonstrate a correlation between the angular distribution of token hidden state vectors and the resulting gradient, revealing a learning preference for data exhibiting higher angle concentration. Inspired by this finding, we propose GAIN-RL, a Gradient-driven Angle-Informed Navigated RL framework. By leveraging the model's intrinsic angle concentration signal, GAIN-RL dynamically selects training data in each epoch, ensuring consistently impactful gradient updates and thus significantly enhancing overall training efficiency. Empirical evaluations show that GAIN-RL (GRPO) achieves over a 2.5x acceleration in training efficiency across diverse mathematical and coding tasks and varying model scales. Furthermore, GAIN-RL (GRPO)'s efficient sampling yields data-efficient training, achieving better performance with half the original data compared to vanilla GRPO with full training data. Code is realsed at https://github.com/wangqinsi1/GAINRL/tree/main.

The non-convexity of the artificial neural network (ANN) training landscape brings inherent optimization difficulties. While the traditional back-propagation stochastic gradient descent (SGD) algorithm and its variants are effective in certain cases, they can become stuck at spurious local minima and are sensitive to initializations and hyperparameters. Recent work has shown that the training of an ANN with ReLU activations can be reformulated as a convex program, bringing hope to globally optimizing interpretable ANNs. However, naively solving the convex training formulation has an exponential complexity, and even an approximation heuristic requires cubic time. In this work, we characterize the quality of this approximation and develop two efficient algorithms that train ANNs with global convergence guarantees. The first algorithm is based on the alternating direction method of multiplier (ADMM). It solves both the exact convex formulation and the approximate counterpart. Linear global convergence is achieved, and the initial several iterations often yield a solution with high prediction accuracy. When solving the approximate formulation, the per-iteration time complexity is quadratic. The second algorithm, based on the "sampled convex programs" theory, is simpler to implement. It solves unconstrained convex formulations and converges to an approximately globally optimal classifier. The non-convexity of the ANN training landscape exacerbates when adversarial training is considered. We apply the robust convex optimization theory to convex training and develop convex formulations that train ANNs robust to adversarial inputs. Our analysis explicitly focuses on one-hidden-layer fully connected ANNs, but can extend to more sophisticated architectures.

Deep multitask networks, in which one neural network produces multiple predictive outputs, can offer better speed and performance than their single-task counterparts but are challenging to train properly. We present a gradient normalization (GradNorm) algorithm that automatically balances training in deep multitask models by dynamically tuning gradient magnitudes. We show that for various network architectures, for both regression and classification tasks, and on both synthetic and real datasets, GradNorm improves accuracy and reduces overfitting across multiple tasks when compared to single-task networks, static baselines, and other adaptive multitask loss balancing techniques. GradNorm also matches or surpasses the performance of exhaustive grid search methods, despite only involving a single asymmetry hyperparameter alpha. Thus, what was once a tedious search process that incurred exponentially more compute for each task added can now be accomplished within a few training runs, irrespective of the number of tasks. Ultimately, we will demonstrate that gradient manipulation affords us great control over the training dynamics of multitask networks and may be one of the keys to unlocking the potential of multitask learning.

Neural network training relies on our ability to find "good" minimizers of highly non-convex loss functions. It is well-known that certain network architecture designs (e.g., skip connections) produce loss functions that train easier, and well-chosen training parameters (batch size, learning rate, optimizer) produce minimizers that generalize better. However, the reasons for these differences, and their effects on the underlying loss landscape, are not well understood. In this paper, we explore the structure of neural loss functions, and the effect of loss landscapes on generalization, using a range of visualization methods. First, we introduce a simple "filter normalization" method that helps us visualize loss function curvature and make meaningful side-by-side comparisons between loss functions. Then, using a variety of visualizations, we explore how network architecture affects the loss landscape, and how training parameters affect the shape of minimizers.

Recently, flat minima are proven to be effective for improving generalization and sharpness-aware minimization (SAM) achieves state-of-the-art performance. Yet the current definition of flatness discussed in SAM and its follow-ups are limited to the zeroth-order flatness (i.e., the worst-case loss within a perturbation radius). We show that the zeroth-order flatness can be insufficient to discriminate minima with low generalization error from those with high generalization error both when there is a single minimum or multiple minima within the given perturbation radius. Thus we present first-order flatness, a stronger measure of flatness focusing on the maximal gradient norm within a perturbation radius which bounds both the maximal eigenvalue of Hessian at local minima and the regularization function of SAM. We also present a novel training procedure named Gradient norm Aware Minimization (GAM) to seek minima with uniformly small curvature across all directions. Experimental results show that GAM improves the generalization of models trained with current optimizers such as SGD and AdamW on various datasets and networks. Furthermore, we show that GAM can help SAM find flatter minima and achieve better generalization.

Wasserstein gradient flow has emerged as a promising approach to solve optimization problems over the space of probability distributions. A recent trend is to use the well-known JKO scheme in combination with input convex neural networks to numerically implement the proximal step. The most challenging step, in this setup, is to evaluate functions involving density explicitly, such as entropy, in terms of samples. This paper builds on the recent works with a slight but crucial difference: we propose to utilize a variational formulation of the objective function formulated as maximization over a parametric class of functions. Theoretically, the proposed variational formulation allows the construction of gradient flows directly for empirical distributions with a well-defined and meaningful objective function. Computationally, this approach replaces the computationally expensive step in existing methods, to handle objective functions involving density, with inner loop updates that only require a small batch of samples and scale well with the dimension. The performance and scalability of the proposed method are illustrated with the aid of several numerical experiments involving high-dimensional synthetic and real datasets.

There are two widely known issues with properly training Recurrent Neural Networks, the vanishing and the exploding gradient problems detailed in Bengio et al. (1994). In this paper we attempt to improve the understanding of the underlying issues by exploring these problems from an analytical, a geometric and a dynamical systems perspective. Our analysis is used to justify a simple yet effective solution. We propose a gradient norm clipping strategy to deal with exploding gradients and a soft constraint for the vanishing gradients problem. We validate empirically our hypothesis and proposed solutions in the experimental section.

In current deep learning tasks, Adam style optimizers such as Adam, Adagrad, RMSProp, Adafactor, and Lion have been widely used as alternatives to SGD style optimizers. These optimizers typically update model parameters using the sign of gradients, resulting in more stable convergence curves. The learning rate and the batch size are the most critical hyperparameters for optimizers, which require careful tuning to enable effective convergence. Previous research has shown that the optimal learning rate increases linearly or follows similar rules with batch size for SGD style optimizers. However, this conclusion is not applicable to Adam style optimizers. In this paper, we elucidate the connection between optimal learning rates and batch sizes for Adam style optimizers through both theoretical analysis and extensive experiments. First, we raise the scaling law between batch sizes and optimal learning rates in the sign of gradient case, in which we prove that the optimal learning rate first rises and then falls as the batch size increases. Moreover, the peak value of the surge will gradually move toward the larger batch size as training progresses. Second, we conducted experiments on various CV and NLP tasks and verified the correctness of the scaling law.

Stochastic gradient descent method and its variants constitute the core optimization algorithms that achieve good convergence rates for solving machine learning problems. These rates are obtained especially when these algorithms are fine-tuned for the application at hand. Although this tuning process can require large computational costs, recent work has shown that these costs can be reduced by line search methods that iteratively adjust the stepsize. We propose an alternative approach to stochastic line search by using a new algorithm based on forward step model building. This model building step incorporates second-order information that allows adjusting not only the stepsize but also the search direction. Noting that deep learning model parameters come in groups (layers of tensors), our method builds its model and calculates a new step for each parameter group. This novel diagonalization approach makes the selected step lengths adaptive. We provide convergence rate analysis, and experimentally show that the proposed algorithm achieves faster convergence and better generalization in well-known test problems. More precisely, SMB requires less tuning, and shows comparable performance to other adaptive methods.

Large language models (LLMs) have revolutionized lots of fields of research. Although it is well-known that fine-tuning is essential for enhancing the capabilities of LLMs, existing research suggests that there is potential redundancy in the fine-tuning process and therefore proposes to update only a subset of parameters. However, these methods fail to leverage the task-specific information to identify important parameters during training. Based on the insight that gradients inherently contain information on task-specific data, we propose Gradient-Mask Tuning (GMT), a method that selectively updates parameters during training based on their gradient information. Specifically, we compute the absolute values of the gradients and apply masking to those with relatively smaller magnitudes. Our empirical results across various tasks demonstrate that GMT not only outperforms traditional fine-tuning methods but also elevates the upper limits of LLM performance. Further analysis indicates that GMT exhibits insensitivity to mask ratio and possesses computational efficiency comparable to vanilla SFT.

Classifier-free guidance (CFG) is crucial for improving both generation quality and alignment between the input condition and final output in diffusion models. While a high guidance scale is generally required to enhance these aspects, it also causes oversaturation and unrealistic artifacts. In this paper, we revisit the CFG update rule and introduce modifications to address this issue. We first decompose the update term in CFG into parallel and orthogonal components with respect to the conditional model prediction and observe that the parallel component primarily causes oversaturation, while the orthogonal component enhances image quality. Accordingly, we propose down-weighting the parallel component to achieve high-quality generations without oversaturation. Additionally, we draw a connection between CFG and gradient ascent and introduce a new rescaling and momentum method for the CFG update rule based on this insight. Our approach, termed adaptive projected guidance (APG), retains the quality-boosting advantages of CFG while enabling the use of higher guidance scales without oversaturation. APG is easy to implement and introduces practically no additional computational overhead to the sampling process. Through extensive experiments, we demonstrate that APG is compatible with various conditional diffusion models and samplers, leading to improved FID, recall, and saturation scores while maintaining precision comparable to CFG, making our method a superior plug-and-play alternative to standard classifier-free guidance.

In this article, we introduce a novel normalization technique for neural network weight matrices, which we term weight conditioning. This approach aims to narrow the gap between the smallest and largest singular values of the weight matrices, resulting in better-conditioned matrices. The inspiration for this technique partially derives from numerical linear algebra, where well-conditioned matrices are known to facilitate stronger convergence results for iterative solvers. We provide a theoretical foundation demonstrating that our normalization technique smoothens the loss landscape, thereby enhancing convergence of stochastic gradient descent algorithms. Empirically, we validate our normalization across various neural network architectures, including Convolutional Neural Networks (CNNs), Vision Transformers (ViT), Neural Radiance Fields (NeRF), and 3D shape modeling. Our findings indicate that our normalization method is not only competitive but also outperforms existing weight normalization techniques from the literature.

Aligning language models with preferences can be posed as approximating a target distribution representing some desired behavior. Existing approaches differ both in the functional form of the target distribution and the algorithm used to approximate it. For instance, Reinforcement Learning from Human Feedback (RLHF) corresponds to minimizing a reverse KL from an implicit target distribution arising from a KL penalty in the objective. On the other hand, Generative Distributional Control (GDC) has an explicit target distribution and minimizes a forward KL from it using the Distributional Policy Gradient (DPG) algorithm. In this paper, we propose a new approach, f-DPG, which allows the use of any f-divergence to approximate any target distribution that can be evaluated. f-DPG unifies both frameworks (RLHF, GDC) and the approximation methods (DPG, RL with KL penalties). We show the practical benefits of various choices of divergence objectives and demonstrate that there is no universally optimal objective but that different divergences present different alignment and diversity trade-offs. We show that Jensen-Shannon divergence strikes a good balance between these objectives, and frequently outperforms forward KL divergence by a wide margin, leading to significant improvements over prior work. These distinguishing characteristics between divergences persist as the model size increases, highlighting the importance of selecting appropriate divergence objectives.

In training neural networks, it is common practice to use partial gradients computed over batches, mostly very small subsets of the training set. This approach is motivated by the argument that such a partial gradient is close to the true one, with precision growing only with the square root of the batch size. A theoretical justification is with the help of stochastic approximation theory. However, the conditions for the validity of this theory are not satisfied in the usual learning rate schedules. Batch processing is also difficult to combine with efficient second-order optimization methods. This proposal is based on another hypothesis: the loss minimum of the training set can be expected to be well-approximated by the minima of its subsets. Such subset minima can be computed in a fraction of the time necessary for optimizing over the whole training set. This hypothesis has been tested with the help of the MNIST, CIFAR-10, and CIFAR-100 image classification benchmarks, optionally extended by training data augmentation. The experiments have confirmed that results equivalent to conventional training can be reached. In summary, even small subsets are representative if the overdetermination ratio for the given model parameter set sufficiently exceeds unity. The computing expense can be reduced to a tenth or less.

Training-free guided generation is a widely used and powerful technique that allows the end user to exert further control over the generative process of flow/diffusion models. Generally speaking, two families of techniques have emerged for solving this problem for gradient-based guidance: namely, posterior guidance (i.e., guidance via projecting the current sample to the target distribution via the target prediction model) and end-to-end guidance (i.e., guidance by performing backpropagation throughout the entire ODE solve). In this work, we show that these two seemingly separate families can actually be unified by looking at posterior guidance as a greedy strategy of end-to-end guidance. We explore the theoretical connections between these two families and provide an in-depth theoretical of these two techniques relative to the continuous ideal gradients. Motivated by this analysis we then show a method for interpolating between these two families enabling a trade-off between compute and accuracy of the guidance gradients. We then validate this work on several inverse image problems and property-guided molecular generation.

The gradients of convex functions are expressive models of non-trivial vector fields. For example, Brenier's theorem yields that the optimal transport map between any two measures on Euclidean space under the squared distance is realized as a convex gradient, which is a key insight used in recent generative flow models. In this paper, we study how to model convex gradients by integrating a Jacobian-vector product parameterized by a neural network, which we call the Input Convex Gradient Network (ICGN). We theoretically study ICGNs and compare them to taking the gradient of an Input-Convex Neural Network (ICNN), empirically demonstrating that a single layer ICGN can fit a toy example better than a single layer ICNN. Lastly, we explore extensions to deeper networks and connections to constructions from Riemannian geometry.

This paper extends the classic theory of convex optimization to the minimization of functions that are equal to the negated logarithm of what we term as a sum-log-concave function, i.e., a sum of log-concave functions. In particular, we show that such functions are in general not convex but still satisfy generalized convexity inequalities. These inequalities unveil the key importance of a certain vector that we call the cross-gradient and that is, in general, distinct from the usual gradient. Thus, we propose the Cross Gradient Descent (XGD) algorithm moving in the opposite direction of the cross-gradient and derive a convergence analysis. As an application of our sum-log-concave framework, we introduce the so-called checkered regression method relying on a sum-log-concave function. This classifier extends (multiclass) logistic regression to non-linearly separable problems since it is capable of tessellating the feature space by using any given number of hyperplanes, creating a checkerboard-like pattern of decision regions.

A central challenge to many fields of science and engineering involves minimizing non-convex error functions over continuous, high dimensional spaces. Gradient descent or quasi-Newton methods are almost ubiquitously used to perform such minimizations, and it is often thought that a main source of difficulty for the ability of these local methods to find the global minimum is the proliferation of local minima with much higher error than the global minimum. Here we argue, based on results from statistical physics, random matrix theory, and neural network theory, that a deeper and more profound difficulty originates from the proliferation of saddle points, not local minima, especially in high dimensional problems of practical interest. Such saddle points are surrounded by high error plateaus that can dramatically slow down learning, and give the illusory impression of the existence of a local minimum. Motivated by these arguments, we propose a new algorithm, the saddle-free Newton method, that can rapidly escape high dimensional saddle points, unlike gradient descent and quasi-Newton methods. We apply this algorithm to deep neural network training, and provide preliminary numerical evidence for its superior performance.

Various data augmentation techniques have been recently proposed in image-based deep reinforcement learning (DRL). Although they empirically demonstrate the effectiveness of data augmentation for improving sample efficiency or generalization, which technique should be preferred is not always clear. To tackle this question, we analyze existing methods to better understand them and to uncover how they are connected. Notably, by expressing the variance of the Q-targets and that of the empirical actor/critic losses of these methods, we can analyze the effects of their different components and compare them. We furthermore formulate an explanation about how these methods may be affected by choosing different data augmentation transformations in calculating the target Q-values. This analysis suggests recommendations on how to exploit data augmentation in a more principled way. In addition, we include a regularization term called tangent prop, previously proposed in computer vision, but whose adaptation to DRL is novel to the best of our knowledge. We evaluate our proposition and validate our analysis in several domains. Compared to different relevant baselines, we demonstrate that it achieves state-of-the-art performance in most environments and shows higher sample efficiency and better generalization ability in some complex environments.

Fairness in visual recognition is becoming a prominent and critical topic of discussion as recognition systems are deployed at scale in the real world. Models trained from data in which target labels are correlated with protected attributes (e.g., gender, race) are known to learn and exploit those correlations. In this work, we introduce a method for training accurate target classifiers while mitigating biases that stem from these correlations. We use GANs to generate realistic-looking images, and perturb these images in the underlying latent space to generate training data that is balanced for each protected attribute. We augment the original dataset with this perturbed generated data, and empirically demonstrate that target classifiers trained on the augmented dataset exhibit a number of both quantitative and qualitative benefits. We conduct a thorough evaluation across multiple target labels and protected attributes in the CelebA dataset, and provide an in-depth analysis and comparison to existing literature in the space.

The backpropagation algorithm has long been the canonical training method for neural networks. Modern paradigms are implicitly optimized for it, and numerous guidelines exist to ensure its proper use. Recently, synthetic gradients methods -where the error gradient is only roughly approximated - have garnered interest. These methods not only better portray how biological brains are learning, but also open new computational possibilities, such as updating layers asynchronously. Even so, they have failed to scale past simple tasks like MNIST or CIFAR-10. This is in part due to a lack of standards, leading to ill-suited models and practices forbidding such methods from performing to the best of their abilities. In this work, we focus on direct feedback alignment and present a set of best practices justified by observations of the alignment angles. We characterize a bottleneck effect that prevents alignment in narrow layers, and hypothesize it may explain why feedback alignment methods have yet to scale to large convolutional networks.

In recent years, various powerful policy gradient algorithms have been proposed in deep reinforcement learning. While all these algorithms build on the Policy Gradient Theorem, the specific design choices differ significantly across algorithms. We provide a holistic overview of on-policy policy gradient algorithms to facilitate the understanding of both their theoretical foundations and their practical implementations. In this overview, we include a detailed proof of the continuous version of the Policy Gradient Theorem, convergence results and a comprehensive discussion of practical algorithms. We compare the most prominent algorithms on continuous control environments and provide insights on the benefits of regularization. All code is available at https://github.com/Matt00n/PolicyGradientsJax.

Classifier-Free Guidance (CFG) enhances the quality and condition adherence of text-to-image diffusion models. It operates by combining the conditional and unconditional predictions using a fixed weight. However, recent works vary the weights throughout the diffusion process, reporting superior results but without providing any rationale or analysis. By conducting comprehensive experiments, this paper provides insights into CFG weight schedulers. Our findings suggest that simple, monotonically increasing weight schedulers consistently lead to improved performances, requiring merely a single line of code. In addition, more complex parametrized schedulers can be optimized for further improvement, but do not generalize across different models and tasks.

We present a novel adaptive optimization algorithm for large-scale machine learning problems. Equipped with a low-cost estimate of local curvature and Lipschitz smoothness, our method dynamically adapts the search direction and step-size. The search direction contains gradient information preconditioned by a well-scaled diagonal preconditioning matrix that captures the local curvature information. Our methodology does not require the tedious task of learning rate tuning, as the learning rate is updated automatically without adding an extra hyperparameter. We provide convergence guarantees on a comprehensive collection of optimization problems, including convex, strongly convex, and nonconvex problems, in both deterministic and stochastic regimes. We also conduct an extensive empirical evaluation on standard machine learning problems, justifying our algorithm's versatility and demonstrating its strong performance compared to other start-of-the-art first-order and second-order methods.

This paper presents a comprehensive study on the role of Classifier-Free Guidance (CFG) in text-conditioned diffusion models from the perspective of inference efficiency. In particular, we relax the default choice of applying CFG in all diffusion steps and instead search for efficient guidance policies. We formulate the discovery of such policies in the differentiable Neural Architecture Search framework. Our findings suggest that the denoising steps proposed by CFG become increasingly aligned with simple conditional steps, which renders the extra neural network evaluation of CFG redundant, especially in the second half of the denoising process. Building upon this insight, we propose "Adaptive Guidance" (AG), an efficient variant of CFG, that adaptively omits network evaluations when the denoising process displays convergence. Our experiments demonstrate that AG preserves CFG's image quality while reducing computation by 25%. Thus, AG constitutes a plug-and-play alternative to Guidance Distillation, achieving 50% of the speed-ups of the latter while being training-free and retaining the capacity to handle negative prompts. Finally, we uncover further redundancies of CFG in the first half of the diffusion process, showing that entire neural function evaluations can be replaced by simple affine transformations of past score estimates. This method, termed LinearAG, offers even cheaper inference at the cost of deviating from the baseline model. Our findings provide insights into the efficiency of the conditional denoising process that contribute to more practical and swift deployment of text-conditioned diffusion models.

Adaptive gradient methods, e.g. Adam, have achieved tremendous success in machine learning. Scaling the learning rate element-wisely by a certain form of second moment estimate of gradients, such methods are able to attain rapid training of modern deep neural networks. Nevertheless, they are observed to suffer from compromised generalization ability compared with stochastic gradient descent (SGD) and tend to be trapped in local minima at an early stage during training. Intriguingly, we discover that substituting the gradient in the second raw moment estimate term with its momentumized version in Adam can resolve the issue. The intuition is that gradient with momentum contains more accurate directional information and therefore its second moment estimation is a more favorable option for learning rate scaling than that of the raw gradient. Thereby we propose AdaMomentum as a new optimizer reaching the goal of training fast while generalizing much better. We further develop a theory to back up the improvement in generalization and provide convergence guarantees under both convex and nonconvex settings. Extensive experiments on a wide range of tasks and models demonstrate that AdaMomentum exhibits state-of-the-art performance and superior training stability consistently.

Gradient-based meta-learning has proven to be highly effective at learning model initializations, representations, and update rules that allow fast adaptation from a few samples. The core idea behind these approaches is to use fast adaptation and generalization -- two second-order metrics -- as training signals on a meta-training dataset. However, little attention has been given to other possible second-order metrics. In this paper, we investigate a different training signal -- robustness to catastrophic interference -- and demonstrate that representations learned by directing minimizing interference are more conducive to incremental learning than those learned by just maximizing fast adaptation.

In distributed computing, slower nodes (stragglers) usually become a bottleneck. Gradient Coding (GC), introduced by Tandon et al., is an efficient technique that uses principles of error-correcting codes to distribute gradient computation in the presence of stragglers. In this paper, we consider the distributed computation of a sequence of gradients {g(1),g(2),ldots,g(J)}, where processing of each gradient g(t) starts in round-t and finishes by round-(t+T). Here Tgeq 0 denotes a delay parameter. For the GC scheme, coding is only across computing nodes and this results in a solution where T=0. On the other hand, having T>0 allows for designing schemes which exploit the temporal dimension as well. In this work, we propose two schemes that demonstrate improved performance compared to GC. Our first scheme combines GC with selective repetition of previously unfinished tasks and achieves improved straggler mitigation. In our second scheme, which constitutes our main contribution, we apply GC to a subset of the tasks and repetition for the remainder of the tasks. We then multiplex these two classes of tasks across workers and rounds in an adaptive manner, based on past straggler patterns. Using theoretical analysis, we demonstrate that our second scheme achieves significant reduction in the computational load. In our experiments, we study a practical setting of concurrently training multiple neural networks over an AWS Lambda cluster involving 256 worker nodes, where our framework naturally applies. We demonstrate that the latter scheme can yield a 16\% improvement in runtime over the baseline GC scheme, in the presence of naturally occurring, non-simulated stragglers.

Many evaluation metrics can be used to assess the performance of models in binary classification tasks. However, most of them are derived from a confusion matrix in a non-differentiable form, making it very difficult to generate a differentiable loss function that could directly optimize them. The lack of solutions to bridge this challenge not only hinders our ability to solve difficult tasks, such as imbalanced learning, but also requires the deployment of computationally expensive hyperparameter search processes in model selection. In this paper, we propose a general-purpose approach that transforms any confusion matrix-based metric into a loss function, AnyLoss, that is available in optimization processes. To this end, we use an approximation function to make a confusion matrix represented in a differentiable form, and this approach enables any confusion matrix-based metric to be directly used as a loss function. The mechanism of the approximation function is provided to ensure its operability and the differentiability of our loss functions is proved by suggesting their derivatives. We conduct extensive experiments under diverse neural networks with many datasets, and we demonstrate their general availability to target any confusion matrix-based metrics. Our method, especially, shows outstanding achievements in dealing with imbalanced datasets, and its competitive learning speed, compared to multiple baseline models, underscores its efficiency.

This work proposes aesthetic gradients, a method to personalize a CLIP-conditioned diffusion model by guiding the generative process towards custom aesthetics defined by the user from a set of images. The approach is validated with qualitative and quantitative experiments, using the recent stable diffusion model and several aesthetically-filtered datasets. Code is released at https://github.com/vicgalle/stable-diffusion-aesthetic-gradients

A novel gradient boosting framework is proposed where shallow neural networks are employed as ``weak learners''. General loss functions are considered under this unified framework with specific examples presented for classification, regression, and learning to rank. A fully corrective step is incorporated to remedy the pitfall of greedy function approximation of classic gradient boosting decision tree. The proposed model rendered outperforming results against state-of-the-art boosting methods in all three tasks on multiple datasets. An ablation study is performed to shed light on the effect of each model components and model hyperparameters.

Beyond minimizing a single training loss, many deep learning estimation pipelines rely on an auxiliary objective to quantify and encourage desirable properties of the model (e.g. performance on another dataset, robustness, agreement with a prior). Although the simplest approach to incorporating an auxiliary loss is to sum it with the training loss as a regularizer, recent works have shown that one can improve performance by blending the gradients beyond a simple sum; this is known as gradient surgery. We cast the problem as a constrained minimization problem where the auxiliary objective is minimized among the set of minimizers of the training loss. To solve this bilevel problem, we follow a parameter update direction that combines the training loss gradient and the orthogonal projection of the auxiliary gradient to the training gradient. In a setting where gradients come from mini-batches, we explain how, using a moving average of the training loss gradients, we can carefully maintain this critical orthogonality property. We demonstrate that our method, Bloop, can lead to much better performances on NLP and vision experiments than other gradient surgery methods without EMA.

This paper investigates the distinctions between gradient methods applied to non-differentiable functions (NGDMs) and classical gradient descents (GDs) designed for differentiable functions. First, we demonstrate significant differences in the convergence properties of NGDMs compared to GDs, challenging the applicability of the extensive neural network convergence literature based on L-smoothness to non-smooth neural networks. Next, we demonstrate the paradoxical nature of NGDM solutions for L_{1}-regularized problems, showing that increasing the regularization penalty leads to an increase in the L_{1} norm of optimal solutions in NGDMs. Consequently, we show that widely adopted L_{1} penalization-based techniques for network pruning do not yield expected results. Finally, we explore the Edge of Stability phenomenon, indicating its inapplicability even to Lipschitz continuous convex differentiable functions, leaving its relevance to non-convex non-differentiable neural networks inconclusive. Our analysis exposes misguided interpretations of NGDMs in widely referenced papers and texts due to an overreliance on strong smoothness assumptions, emphasizing the necessity for a nuanced understanding of foundational assumptions in the analysis of these systems.

While the Transformer architecture has achieved remarkable success across various domains, a thorough theoretical foundation explaining its optimization dynamics is yet to be fully developed. In this study, we aim to bridge this understanding gap by answering the following two core questions: (1) Which types of Transformer architectures allow Gradient Descent (GD) to achieve guaranteed convergence? and (2) Under what initial conditions and architectural specifics does the Transformer achieve rapid convergence during training? By analyzing the loss landscape of a single Transformer layer using Softmax and Gaussian attention kernels, our work provides concrete answers to these questions. Our findings demonstrate that, with appropriate weight initialization, GD can train a Transformer model (with either kernel type) to achieve a global optimal solution, especially when the input embedding dimension is large. Nonetheless, certain scenarios highlight potential pitfalls: training a Transformer using the Softmax attention kernel may sometimes lead to suboptimal local solutions. In contrast, the Gaussian attention kernel exhibits a much favorable behavior. Our empirical study further validate the theoretical findings.

Text-to-image diffusion models have recently emerged at the forefront of image generation, powered by very large-scale unsupervised or weakly supervised text-to-image training datasets. Due to their unsupervised training, controlling their behavior in downstream tasks, such as maximizing human-perceived image quality, image-text alignment, or ethical image generation, is difficult. Recent works finetune diffusion models to downstream reward functions using vanilla reinforcement learning, notorious for the high variance of the gradient estimators. In this paper, we propose AlignProp, a method that aligns diffusion models to downstream reward functions using end-to-end backpropagation of the reward gradient through the denoising process. While naive implementation of such backpropagation would require prohibitive memory resources for storing the partial derivatives of modern text-to-image models, AlignProp finetunes low-rank adapter weight modules and uses gradient checkpointing, to render its memory usage viable. We test AlignProp in finetuning diffusion models to various objectives, such as image-text semantic alignment, aesthetics, compressibility and controllability of the number of objects present, as well as their combinations. We show AlignProp achieves higher rewards in fewer training steps than alternatives, while being conceptually simpler, making it a straightforward choice for optimizing diffusion models for differentiable reward functions of interest. Code and Visualization results are available at https://align-prop.github.io/.

Large Language Models (LLMs) have shown impressive performance as general purpose agents, but their abilities remain highly dependent on prompts which are hand written with onerous trial-and-error effort. We propose a simple and nonparametric solution to this problem, Automatic Prompt Optimization (APO), which is inspired by numerical gradient descent to automatically improve prompts, assuming access to training data and an LLM API. The algorithm uses minibatches of data to form natural language ``gradients'' that criticize the current prompt. The gradients are then ``propagated'' into the prompt by editing the prompt in the opposite semantic direction of the gradient. These gradient descent steps are guided by a beam search and bandit selection procedure which significantly improves algorithmic efficiency. Preliminary results across three benchmark NLP tasks and the novel problem of LLM jailbreak detection suggest that Automatic Prompt Optimization can outperform prior prompt editing techniques and improve an initial prompt's performance by up to 31\%, by using data to rewrite vague task descriptions into more precise annotation instructions.

Diffusion models have shown impressive results in generating high-quality conditional samples using guidance techniques such as Classifier-Free Guidance (CFG). However, existing methods often require additional training or neural function evaluations (NFEs), making them incompatible with guidance-distilled models. Also, they rely on heuristic approaches that need identifying target layers. In this work, we propose a novel and efficient method, termed PLADIS, which boosts pre-trained models (U-Net/Transformer) by leveraging sparse attention. Specifically, we extrapolate query-key correlations using softmax and its sparse counterpart in the cross-attention layer during inference, without requiring extra training or NFEs. By leveraging the noise robustness of sparse attention, our PLADIS unleashes the latent potential of text-to-image diffusion models, enabling them to excel in areas where they once struggled with newfound effectiveness. It integrates seamlessly with guidance techniques, including guidance-distilled models. Extensive experiments show notable improvements in text alignment and human preference, offering a highly efficient and universally applicable solution.

When trying to gain better visibility into a machine learning model in order to understand and mitigate the associated risks, a potentially valuable source of evidence is: which training examples most contribute to a given behavior? Influence functions aim to answer a counterfactual: how would the model's parameters (and hence its outputs) change if a given sequence were added to the training set? While influence functions have produced insights for small models, they are difficult to scale to large language models (LLMs) due to the difficulty of computing an inverse-Hessian-vector product (IHVP). We use the Eigenvalue-corrected Kronecker-Factored Approximate Curvature (EK-FAC) approximation to scale influence functions up to LLMs with up to 52 billion parameters. In our experiments, EK-FAC achieves similar accuracy to traditional influence function estimators despite the IHVP computation being orders of magnitude faster. We investigate two algorithmic techniques to reduce the cost of computing gradients of candidate training sequences: TF-IDF filtering and query batching. We use influence functions to investigate the generalization patterns of LLMs, including the sparsity of the influence patterns, increasing abstraction with scale, math and programming abilities, cross-lingual generalization, and role-playing behavior. Despite many apparently sophisticated forms of generalization, we identify a surprising limitation: influences decay to near-zero when the order of key phrases is flipped. Overall, influence functions give us a powerful new tool for studying the generalization properties of LLMs.

In this work, we study the evolution of the loss Hessian across many classification tasks in order to understand the effect the curvature of the loss has on the training dynamics. Whereas prior work has focused on how different learning rates affect the loss Hessian observed during training, we also analyze the effects of model initialization, architectural choices, and common training heuristics such as gradient clipping and learning rate warmup. Our results demonstrate that successful model and hyperparameter choices allow the early optimization trajectory to either avoid -- or navigate out of -- regions of high curvature and into flatter regions that tolerate a higher learning rate. Our results suggest a unifying perspective on how disparate mitigation strategies for training instability ultimately address the same underlying failure mode of neural network optimization, namely poor conditioning. Inspired by the conditioning perspective, we show that learning rate warmup can improve training stability just as much as batch normalization, layer normalization, MetaInit, GradInit, and Fixup initialization.

Many optimization problems are inherently multi-objective. To address them, we formalize Jacobian descent (JD), a direct generalization of gradient descent for vector-valued functions. Each step of this algorithm relies on a Jacobian matrix consisting of one gradient per objective. The aggregator, responsible for reducing this matrix into an update vector, characterizes JD. While the multi-task learning literature already contains a variety of aggregators, they often lack some natural properties. In particular, the update should not conflict with any objective and should scale proportionally to the norm of each gradient. We propose a new aggregator specifically designed to satisfy this. Emphasizing conflict between objectives, we then highlight direct applications for our methods. Most notably, we introduce instance-wise risk minimization (IWRM), a learning paradigm in which the loss of each training example is considered a separate objective. On simple image classification tasks, IWRM exhibits promising results compared to the direct minimization of the average loss. The performance of our aggregator in those experiments also corroborates our theoretical findings. Lastly, as speed is the main limitation of JD, we provide a path towards a more efficient implementation.

Particle gradient descent, which uses particles to represent a probability measure and performs gradient descent on particles in parallel, is widely used to optimize functions of probability measures. This paper considers particle gradient descent with a finite number of particles and establishes its theoretical guarantees to optimize functions that are displacement convex in measures. Concretely, for Lipschitz displacement convex functions defined on probability over R^d, we prove that O(1/epsilon^2) particles and O(d/epsilon^4) computations are sufficient to find the epsilon-optimal solutions. We further provide improved complexity bounds for optimizing smooth displacement convex functions. We demonstrate the application of our results for function approximation with specific neural architectures with two-dimensional inputs.

Image generation using diffusion models have demonstrated outstanding learning capabilities, effectively capturing the full distribution of the training dataset. They are known to generate wide variations in sampled images, albeit with a trade-off in image fidelity. Guided sampling methods, such as classifier guidance (CG) and classifier-free guidance (CFG), focus sampling in well-learned high-probability regions to generate images of high fidelity, but each has its limitations. CG is computationally expensive due to the use of back-propagation for classifier gradient descent, while CFG, being gradient-free, is more efficient but compromises class label alignment compared to CG. In this work, we propose an efficient guidance method that fully utilizes a pre-trained classifier without using gradient descent. By using the classifier solely in inference mode, a time-adaptive reference class label and corresponding guidance scale are determined at each time step for guided sampling. Experiments on both class-conditioned and text-to-image generation diffusion models demonstrate that the proposed Gradient-free Classifier Guidance (GFCG) method consistently improves class prediction accuracy. We also show GFCG to be complementary to other guided sampling methods like CFG. When combined with the state-of-the-art Autoguidance (ATG), without additional computational overhead, it enhances image fidelity while preserving diversity. For ImageNet 512times512, we achieve a record FD_{DINOv2} of 23.09, while simultaneously attaining a higher classification Precision (94.3%) compared to ATG (90.2%)

Out-of-distribution (OOD) generalization is a critical ability for deep learning models in many real-world scenarios including healthcare and autonomous vehicles. Recently, different techniques have been proposed to improve OOD generalization. Among these methods, gradient-based regularizers have shown promising performance compared with other competitors. Despite this success, our understanding of the role of Hessian and gradient alignment in domain generalization is still limited. To address this shortcoming, we analyze the role of the classifier's head Hessian matrix and gradient in domain generalization using recent OOD theory of transferability. Theoretically, we show that spectral norm between the classifier's head Hessian matrices across domains is an upper bound of the transfer measure, a notion of distance between target and source domains. Furthermore, we analyze all the attributes that get aligned when we encourage similarity between Hessians and gradients. Our analysis explains the success of many regularizers like CORAL, IRM, V-REx, Fish, IGA, and Fishr as they regularize part of the classifier's head Hessian and/or gradient. Finally, we propose two simple yet effective methods to match the classifier's head Hessians and gradients in an efficient way, based on the Hessian Gradient Product (HGP) and Hutchinson's method (Hutchinson), and without directly calculating Hessians. We validate the OOD generalization ability of proposed methods in different scenarios, including transferability, severe correlation shift, label shift and diversity shift. Our results show that Hessian alignment methods achieve promising performance on various OOD benchmarks. The code is available at https://github.com/huawei-noah/Federated-Learning/tree/main/HessianAlignment.

Recent research has demonstrated that feature attribution methods for deep networks can themselves be incorporated into training; these attribution priors optimize for a model whose attributions have certain desirable properties -- most frequently, that particular features are important or unimportant. These attribution priors are often based on attribution methods that are not guaranteed to satisfy desirable interpretability axioms, such as completeness and implementation invariance. Here, we introduce attribution priors to optimize for higher-level properties of explanations, such as smoothness and sparsity, enabled by a fast new attribution method formulation called expected gradients that satisfies many important interpretability axioms. This improves model performance on many real-world tasks where previous attribution priors fail. Our experiments show that the gains from combining higher-level attribution priors with expected gradients attributions are consistent across image, gene expression, and health care data sets. We believe this work motivates and provides the necessary tools to support the widespread adoption of axiomatic attribution priors in many areas of applied machine learning. The implementations and our results have been made freely available to academic communities.

Our work presents extensive empirical evidence that layer rotation, i.e. the evolution across training of the cosine distance between each layer's weight vector and its initialization, constitutes an impressively consistent indicator of generalization performance. In particular, larger cosine distances between final and initial weights of each layer consistently translate into better generalization performance of the final model. Interestingly, this relation admits a network independent optimum: training procedures during which all layers' weights reach a cosine distance of 1 from their initialization consistently outperform other configurations -by up to 30% test accuracy. Moreover, we show that layer rotations are easily monitored and controlled (helpful for hyperparameter tuning) and potentially provide a unified framework to explain the impact of learning rate tuning, weight decay, learning rate warmups and adaptive gradient methods on generalization and training speed. In an attempt to explain the surprising properties of layer rotation, we show on a 1-layer MLP trained on MNIST that layer rotation correlates with the degree to which features of intermediate layers have been trained.

This paper presents a novel technique based on gradient boosting to train the final layers of a neural network (NN). Gradient boosting is an additive expansion algorithm in which a series of models are trained sequentially to approximate a given function. A neural network can also be seen as an additive expansion where the scalar product of the responses of the last hidden layer and its weights provide the final output of the network. Instead of training the network as a whole, the proposed algorithm trains the network sequentially in T steps. First, the bias term of the network is initialized with a constant approximation that minimizes the average loss of the data. Then, at each step, a portion of the network, composed of J neurons, is trained to approximate the pseudo-residuals on the training data computed from the previous iterations. Finally, the T partial models and bias are integrated as a single NN with T times J neurons in the hidden layer. Extensive experiments in classification and regression tasks, as well as in combination with deep neural networks, are carried out showing a competitive generalization performance with respect to neural networks trained with different standard solvers, such as Adam, L-BFGS, SGD and deep models. Furthermore, we show that the proposed method design permits to switch off a number of hidden units during test (the units that were last trained) without a significant reduction of its generalization ability. This permits the adaptation of the model to different classification speed requirements on the fly.

Guidance in conditional diffusion generation is of great importance for sample quality and controllability. However, existing guidance schemes are to be desired. On one hand, mainstream methods such as classifier guidance and classifier-free guidance both require extra training with labeled data, which is time-consuming and unable to adapt to new conditions. On the other hand, training-free methods such as universal guidance, though more flexible, have yet to demonstrate comparable performance. In this work, through a comprehensive investigation into the design space, we show that it is possible to achieve significant performance improvements over existing guidance schemes by leveraging off-the-shelf classifiers in a training-free fashion, enjoying the best of both worlds. Employing calibration as a general guideline, we propose several pre-conditioning techniques to better exploit pretrained off-the-shelf classifiers for guiding diffusion generation. Extensive experiments on ImageNet validate our proposed method, showing that state-of-the-art diffusion models (DDPM, EDM, DiT) can be further improved (up to 20%) using off-the-shelf classifiers with barely any extra computational cost. With the proliferation of publicly available pretrained classifiers, our proposed approach has great potential and can be readily scaled up to text-to-image generation tasks. The code is available at https://github.com/AlexMaOLS/EluCD/tree/main.