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Optimization · Everything Else

Dayal Singh Kalra, Tianyu He, Maissam Barkeshli

In gradient descent dynamics of neural networks, the top eigenvalue of the Hessian of the loss (sharpness) displays a variety of robust phenomena throughout training. This includes early time regimes where the sharpness may decrease during early periods of training (sharpness reduction), and later time behavior such as progressive sharpening and edge of stability. We demonstrate that a simple -layer linear network (UV model) trained on a single training example exhibits all of the essential sharpness phenomenology observed in real-world scenarios. By analyzing the structure of dynamical fixed points in function space and the vector field of function updates, we uncover the underlying mechanisms behind these sharpness trends. Our analysis reveals (i) the mechanism behind early sharpness reduction and progressive sharpening, (ii) the required conditions for edge of stability, and (iii) a period-doubling route to chaos on the edge of stability manifold as learning rate is increased. Finally, we demonstrate that various predictions from this simplified model generalize to real-world scenarios and discuss its limitations.

Applications · Computer Vision

Margret Keuper, Julia Grabinski, Janis Keuper

Recent work in neural networks for image classification has seen a strong tendency towards increasing the spatial context during encoding. Whether achieved through large convolution kernels or self-attention, models scale poorly with the increased spatial context, such that the improved model accuracy often comes at significant costs. In this paper, we propose a module for studying the effective filter size of convolutional neural networks (CNNs). To facilitate such a study, several challenges need to be addressed: (i) we need an effective means to train models with large filters (potentially as large as the input data) without increasing the number of learnable parameters, (ii) the employed convolution operation should be a plug-and-play module that can replace conventional convolutions in a CNN and allow for an efficient implementation in current frameworks, (iii) the study of filter sizes has to be decoupled from other aspects such as the network width or the number of learnable parameters, and (iv) the cost of the convolution operation itself has to remain manageable i.e.~we can not naïvely increase the size of the convolution kernel. To address these challenges, we propose to learn the frequency representations of filter weights as neural implicit functions, such that the better scalability of the convolution in the frequency domain can be leveraged. Additionally, due to the implementation of the proposed neural implicit function, even large and expressive spatial filters can be parameterized by only a few learnable weights. Interestingly, our analysis shows that, although the proposed networks could learn very large convolution kernels, the learned filters are well localized and relatively small in practice when transformed from the frequency to the spatial domain. We anticipate that our analysis of individually optimized filter sizes will allow for more efficient, yet effective, models in the future. Our code is available at https://github.com/GeJulia/NIFF .

Deep Learning · Graph Neural Networks

Olga Solodova, Nick Richardson, Deniz Oktay, Ryan P Adams

Graph neural networks (GNNs) appear to be powerful tools to learn state representations for agents in distributed, decentralized multi-agent systems, but generate catastrophically incorrect predictions when nodes update asynchronously during inference. This failure under asynchrony effectively excludes these architectures from many potential applications where synchrony is difficult or impossible to enforce, e.g., robotic swarms or sensor networks. In this work we identify ''implicitly-defined'' GNNs as a class of architectures which is provably robust to asynchronous ''hogwild'' inference, adapting convergence guarantees from work in asynchronous and distributed optimization. We then propose a novel implicitly-defined GNN architecture, which we call an energy GNN. We show that this architecture outperforms other GNNs from this class on a variety of synthetic tasks inspired by multi-agent systems.

Theory · Probabilistic Methods

Megh Shukla, Aziz Shameem, Mathieu Salzmann, Alexandre Alahi

Deep heteroscedastic regression models the mean and covariance of the target distribution through neural networks. The challenge arises from heteroscedasticity, which implies that the covariance is sample dependent and is often unknown. Consequently, recent methods learn the covariance through unsupervised frameworks, which unfortunately yield a trade-off between computational complexity and accuracy. While this trade-off could be alleviated through supervision, obtaining labels for the covariance is non-trivial.Here, we study self-supervised covariance estimation in deep heteroscedastic regression. We address two questions: (1) How should we supervise the covariance assuming ground truth is available? (2) How can we obtain pseudo labels in the absence of the ground-truth? We address (1) by analysing two popular measures: the KL Divergence and the 2-Wasserstein distance. Subsequently, we derive an upper bound on the 2-Wasserstein distance between normal distributions with non-commutative covariances that is stable to optimize. We address (2) through a simple neighborhood based heuristic algorithm which results in surprisingly effective pseudo labels for the covariance. Our experiments over a wide range of synthetic and real datasets demonstrate that the proposed 2-Wasserstein bound coupled with pseudo label annotations results in a computationally cheaper yet accurate deep heteroscedastic regression.

Deep Learning · Algorithms

Satoki Ishikawa, Rio Yokota, Ryo Karakida

Local learning, which trains a network through layer-wise local targets and losses, has been studied as an alternative to backpropagation (BP) in neural computation. However, its algorithms often become more complex or require additional hyperparameters due to the locality, making it challenging to identify desirable settings where the algorithm progresses in a stable manner.To provide theoretical and quantitative insights, we introduce maximal update parameterization ($\mu$P) in the infinite-width limit for two representative designs of local targets: predictive coding (PC) and target propagation (TP). We verify that $\mu$P enables hyperparameter transfer across models of different widths.Furthermore, our analysis reveals unique and intriguing properties of $\mu$P that are not present in conventional BP. By analyzing deep linear networks, we find that PC's gradients interpolate between first-order and Gauss-Newton-like gradients, depending on the parameterization. We demonstrate that, in specific standard settings, PC in the infinite-width limit behaves more similarly to the first-order gradient.For TP, even with the standard scaling of the last layer differing from classical $\mu$P, its local loss optimization favors the feature learning regime over the kernel regime.

Binghui Li, Zhixuan Pan, Kaifeng Lyu, Jian Li

In this work, we investigate a particular implicit bias in gradient descent training, which we term “Feature Averaging,” and argue that it is one of the principal factors contributing to the non-robustness of deep neural networks. We show that, even when multiple discriminative features are present in the input data, neural networks trained by gradient descent tend to rely on an average (or a certain combination) of these features for classification, rather than distinguishing and leveraging each feature individually. Specifically, we provide a detailed theoretical analysis of the training dynamics of two-layer ReLU networks on a binary classification task, where the data distribution consists of multiple clusters with mutually orthogonal centers. We rigorously prove that gradient descent biases the network towards feature averaging, where the weights of each hidden neuron represent an average of the cluster centers (each corresponding to a distinct feature), thereby making the network vulnerable to input perturbations aligned with the negative direction of the averaged features. On the positive side, we demonstrate that this vulnerability can be mitigated through more granular supervision. In particular, we prove that a two-layer ReLU network can achieve optimal robustness when trained to classify individual features rather than merely the original binary classes. Finally, we validate our theoretical findings with experiments on synthetic datasets, MNIST, and CIFAR-10, and confirm the prevalence of feature averaging and its impact on adversarial robustness. We hope these theoretical and empirical insights deepen the understanding of how gradient descent shapes feature learning and adversarial robustness, and how more detailed supervision can enhance robustness.

Deep Learning · Theory

Zhanpeng Zhou, Mingze Wang, Yuchen Mao, Bingrui Li, Junchi Yan

Sharpness-Aware Minimization (SAM) has substantially improved the generalization of neural networks under various settings. Despite the success, its effectiveness remains poorly understood. In this work, we discover an intriguing phenomenon in the training dynamics of SAM, shedding light on understanding its implicit bias towards flatter minima over Stochastic Gradient Descent (SGD). Specifically, we find that *SAM efficiently selects flatter minima late in training*. Remarkably, even a few epochs of SAM applied at the end of training yield nearly the same generalization and solution sharpness as full SAM training. Subsequently, we delve deeper into the underlying mechanism behind this phenomenon. Theoretically, we identify two phases in the learning dynamics after applying SAM late in training: i) SAM first escapes the minimum found by SGD exponentially fast; and ii) then rapidly converges to a flatter minimum within the same valley. Furthermore, we empirically investigate the role of SAM during the early training phase. We conjecture that the optimization method chosen in the late phase is more crucial in shaping the final solution's properties. Based on this viewpoint, we extend our findings from SAM to Adversarial Training.

Deep Learning · Graph Neural Networks

Hongliang Chi, Qiong Wu, Zhengyi Zhou, Yao Ma

Graph Neural Networks (GNNs) have demonstrated remarkable performance in various graph-based machine learning tasks, yet evaluating the importance of neighbors of testing nodes remains largely unexplored due to the challenge of assessing data importance without test labels. To address this gap, we propose Shapley-Guided Utility Learning (SGUL), a novel framework for graph inference data valuation. SGUL innovatively combines transferable data-specific and modelspecific features to approximate test accuracy without relying on ground truth labels. By incorporating Shapley values as a preprocessing step and using feature Shapley values as input, our method enables direct optimization of Shapley value prediction while reducing computational demands. SGUL overcomes key limitations of existing methods, including poor generalization to unseen test-time structures and indirect optimization. Experiments on diverse graph datasets demonstrate that SGUL consistently outperforms existing baselines in both inductive and transductive settings. SGUL offers an effective, efficient, and interpretable approach for quantifying the value of test-time neighbors.

Applications · Computer Vision

Yizhi Song, Liu He, Zhifei Zhang, Soo Ye Kim, HE Zhang, Wei Xiong, Zhe Lin, Brian Price, Scott Cohen, Jianming Zhang 等

Personalized image generation has emerged from the recent advancements in generative models. However, these generated personalized images often suffer from localized artifacts such as incorrect logos, reducing fidelity and fine-grained identity details of the generated results. Furthermore, there is little prior work tackling this problem. To help improve these identity details in the personalized image generation, we introduce a new task: reference-guided artifacts refinement. We present Refine-by-Align, a first-of-its-kind model that employs a diffusion-based framework to address this challenge. Our model consists of two stages: Alignment Stage and Refinement Stage, which share weights of a unified neural network model. Given a generated image, a masked artifact region, and a reference image, the alignment stage identifies and extracts the corresponding regional features in the reference, which are then used by the refinement stage to fix the artifacts. Our model-agnostic pipeline requires no test-time tuning or optimization. It automatically enhances image fidelity and reference identity in the generated image, generalizing well to existing models on various tasks including but not limited to customization, generative compositing, view synthesis, and virtual try-on. Extensive experiments and comparisons demonstrate that our pipeline greatly pushes the boundary of fine details in the image synthesis models.

Miscellaneous Aspects of Machine Learning · Online Learning, Active Learning and Bandits

Jaehyun Park, Dongmin Park, Jae-Gil Lee

*Continual learning (CL)* enables deep neural networks to adapt to ever-changing data distributions. In practice, there may be scenarios where annotation is costly, leading to *active continual learning (ACL)*, which performs *active learning (AL)* for the CL scenarios when reducing the labeling cost by selecting the most informative subset is preferable. However, conventional AL strategies are not suitable for ACL, as they focus solely on learning the new knowledge, leading to *catastrophic forgetting* of previously learned tasks. Therefore, ACL requires a new AL strategy that can balance the prevention of catastrophic forgetting and the ability to quickly learn new tasks. In this paper, we propose **AccuACL**, **Accu**mulated informativeness-based **A**ctive **C**ontinual **L**earning, by the novel use of the Fisher information matrix as a criterion for sample selection, derived from a theoretical analysis of the Fisher-optimality preservation properties within the framework of ACL, while also addressing the scalability issue of Fisher information-based AL. Extensive experiments demonstrate that AccuACL significantly outperforms AL baselines across various CL algorithms, increasing the average accuracy and forgetting by 23.8% and 17.0%, respectively, on average.

Theory · Learning Theory

Hossein Taheri, Christos Thrampoulidis, Arya Mazumdar

In this paper, we study the data-dependent convergence and generalization behavior of gradient methods for neural networks with smooth activation. Our first result is a novel bound on the excess risk of deep networks trained by the logistic loss via an alogirthmic stability analysis. Compared to previous works, our results improve upon the shortcomings of the well-established Rademacher complexity-based bounds. Importantly, the bounds we derive in this paper are tighter, hold even for neural networks of small width, do not scale unfavorably with width, are algorithm-dependent, and consequently capture the role of initialization on the sample complexity of gradient descent for deep nets. Specialized to noiseless data separable with margin $\gamma$ by neural tangent kernel (NTK) features of a network of width $\Omega(poly(\log(n)))$, we show the test-error rate $e^{O(L)}/{\gamma^2 n}$, where $n$ is the training set size and $L$ denotes the number of hidden layers. This results in an improvement in the test loss bound compared to previous works while maintaining the poly-logarithmic width conditions. We further investigate excess risk bounds for deep nets trained with noisy data, establishing that under a polynomial condition on the network width, gradient descent can achieve the optimal excess risk. Finally, we show that a large step-size significantly improves upon the NTK regime's results in classifying the XOR distribution. In particular, we show for a one-hidden layer neural network of constant width $m$ with quadratic activation and standard Gaussian initialization that SGD with linear sample complexity and with a large step-size $\eta=m$ reaches the perfect test accuracy after only $\lceil\log(d)\rceil$ iterations, where $d$ is the data dimension.

Deep Learning · Theory

Siyu Chen, Beining Wu, Miao Lu, Zhuoran Yang, Tianhao Wang

In this work, we tackle the following question: Can neural networks trained with gradient-based methods achieve the optimal statistical-computational tradeoff in learning Gaussian single-index models? Prior research has shown that any polynomial-time algorithm under the statistical query (SQ) framework requires $\Omega(d^{s^\star/2}\lor d)$ samples, where $s^\star$ is the generative exponent representing the intrinsic difficulty of learning the underlying model.However, it remains unknown whether neural networks can achieve this sample complexity. Inspired by prior techniques such as label transformation and landscape smoothing for learning single-index models, we propose a unified gradient-based algorithm for training a two-layer neural network in polynomial time.Our method is adaptable to a variety of loss and activation functions, covering a broad class of existing approaches.We show that our algorithm learns a feature representation that strongly aligns with the unknown signal $\theta^\star$, with sample complexity $\tilde O (d^{s^\star/2} \lor d)$, matching the SQ lower bound up to a polylogarithmic factor for all generative exponents $s^\star\geq 1$.Furthermore, we extend our approach to the setting where $\theta^\star$ is $k$-sparse for $k = o(\sqrt{d})$ by introducing a novel weight perturbation technique that leverages the sparsity structure. We derive a corresponding SQ lower bound of order $\tilde\Omega(k^{s^\star})$, matched by our method up to a polylogarithmic factor.Our framework, especially the weight perturbation technique, is of independent interest, and suggests potential gradient-based solutions to other problems such as sparse tensor PCA.

Siyu Chen, Beining Wu, Miao Lu, Zhuoran Yang, Tianhao Wang

In this work, we tackle the following question: Can neural networks trained with gradient-based methods achieve the optimal statistical-computational tradeoff in learning Gaussian single-index models? Prior research has shown that any polynomial-time algorithm under the statistical query (SQ) framework requires $\Omega(d^{s^\star/2}\lor d)$ samples, where $s^\star$ is the generative exponent representing the intrinsic difficulty of learning the underlying model.However, it remains unknown whether neural networks can achieve this sample complexity. Inspired by prior techniques such as label transformation and landscape smoothing for learning single-index models, we propose a unified gradient-based algorithm for training a two-layer neural network in polynomial time.Our method is adaptable to a variety of loss and activation functions, covering a broad class of existing approaches.We show that our algorithm learns a feature representation that strongly aligns with the unknown signal $\theta^\star$, with sample complexity $\tilde O (d^{s^\star/2} \lor d)$, matching the SQ lower bound up to a polylogarithmic factor for all generative exponents $s^\star\geq 1$.Furthermore, we extend our approach to the setting where $\theta^\star$ is $k$-sparse for $k = o(\sqrt{d})$ by introducing a novel weight perturbation technique that leverages the sparsity structure. We derive a corresponding SQ lower bound of order $\tilde\Omega(k^{s^\star})$, matched by our method up to a polylogarithmic factor.Our framework, especially the weight perturbation technique, is of independent interest, and suggests potential gradient-based solutions to other problems such as sparse tensor PCA.

Optimization · Everything Else

Joshua Russell, Ignacio Gavier, Devdhar Patel, Edward Rietman, Hava Siegelmann

Neural networks are known to be universal computers for Boolean functions. Recent advancements in hardware have significantly reduced matrix multiplication times, making neural network simulation both fast and efficient. Consequently, functions defined by complex Boolean networks are increasingly viable candidates for simulation through their neural network representation. Prior research has introduced a general method for deriving neural network representations of Boolean networks. However, the resulting neural networks are often suboptimal in terms of the number of neurons and connections, leading to slower simulation performance. Optimizing them while preserving functional equivalence --lossless optimization-- is an NP-hard problem, and current methods only provide lossy solutions. In this paper, we present a deterministic algorithm to optimize such neural networks in terms of neurons and connections while preserving functional equivalence. Moreover, to accelerate the compression of the neural network, we introduce an objective-aware algorithm that exploits representations that are shared among subproblems of the overall optimization. We demonstrate experimentally that we are able to reduce connections and neurons by up to 70% and 60%, respectively, in comparison to state-of-the-art. We also find that our objective-aware algorithm results in consistent speedups in optimization time, achieving up to 34.3x and 5.9x speedup relative to naive and caching solutions, respectively. Our methods are of practical relevance to applications such as high-throughput circuit simulation and placing neurosymbolic systems on the same hardware architecture.

Optimization · Learning for Optimization

Boris Knyazev, Abhinav Moudgil, Guillaume Lajoie, Eugene Belilovsky, Simon Lacoste-Julien

Neural network training can be accelerated when a learnable update rule is used in lieu of classic adaptive optimizers (e.g. Adam). However, learnable update rules can be costly and unstable to train and use. Recently, Jang et al. (2023) proposed a simpler approach to accelerate training based on weight nowcaster networks (WNNs). In their approach, Adam is used for most of the optimization steps and periodically, only every few steps, a WNN nowcasts (predicts near future) parameters. We improve WNNs by proposing neuron interaction and nowcasting (NiNo) networks. In contrast to WNNs, NiNo leverages neuron connectivity and graph neural networks to more accurately nowcast parameters. We further show that in some networks, such as Transformers, modeling neuron connectivity accurately is challenging. We address this and other limitations, which allows NiNo to accelerate Adam training by up to 50% in vision and language tasks.

Deep Learning · Algorithms

Nikos Dimitriadis, Pascal Frossard, François Fleuret

Multi-task trade-offs in machine learning can be addressed via Pareto Front Learning (PFL) methods that parameterize the Pareto Front (PF) with a single model. PFL permits to select the desired operational point during inference, contrary to traditional Multi-Task Learning (MTL) that optimizes for a single trade-off decided prior to training. However, recent PFL methodologies suffer from limited scalability, slow convergence, and excessive memory requirements, while exhibiting inconsistent mappings from preference to objective space. We introduce PaLoRA, a novel parameter-efficient method that addresses these limitations in two ways. First, we augment any neural network architecture with task-specific low-rank adapters and continuously parameterize the Pareto Front in their convex hull. Our approach steers the original model and the adapters towards learning general and task-specific features, respectively. Second, we propose a deterministic sampling schedule of preference vectors that reinforces this division of labor, enabling faster convergence and strengthening the validity of the mapping from preference to objective space throughout training. Our experiments show that PaLoRA outperforms state-of-the-art MTL and PFL baselines across various datasets, scales to large networks, reducing the memory overhead $23.8-31.7$ times compared with competing PFL baselines in scene understanding benchmarks.

Optimization · Non-Convex

Minhak Song, Kwangjun Ahn, Chulhee Yun

Understanding the training dynamics of deep neural networks is challenging due to their high-dimensional nature and intricate loss landscapes. Recent studies have revealed that, along the training trajectory, the gradient approximately aligns with a low-rank top eigenspace of the training loss Hessian, referred to as the dominant subspace. Given this alignment, this paper explores whether neural networks can be trained within the dominant subspace, which, if feasible, could lead to more efficient training methods. Our primary observation is that when the SGD update is projected onto the dominant subspace, the training loss does not decrease further.This suggests that the observed alignment between the gradient and the dominant subspace is spurious. Surprisingly, projecting out the dominant subspace proves to be just as effective as the original update, despite removing the majority of the original update component. We observe similar behavior across practical setups, including the large learning rate regime (also known as Edge of Stability), Sharpness-Aware Minimization, momentum, and adaptive optimizers. We discuss the main causes and implications of this spurious alignment, shedding light on the dynamics of neural network training.

Optimization · Non-Convex

Enea Monzio Compagnoni, Tianlin Liu, Rustem Islamov, Frank Proske, Antonio Orvieto, Aurelien Lucchi

Despite the vast empirical evidence supporting the efficacy of adaptive optimization methods in deep learning, their theoretical understanding is far from complete. This work introduces novel SDEs for commonly used adaptive optimizers: SignSGD, RMSprop(W), and Adam(W). These SDEs offer a quantitatively accurate description of these optimizers and help illuminate an intricate relationship between adaptivity, gradient noise, and curvature. Our novel analysis of SignSGD highlights a noteworthy and precise contrast to SGD in terms of convergence speed, stationary distribution, and robustness to heavy-tail noise. We extend this analysis to AdamW and RMSpropW, for which we observe that the role of noise is much more complex. Crucially, we support our theoretical analysis with experimental evidence by verifying our insights: this includes numerically integrating our SDEs using Euler-Maruyama discretization on various neural network architectures such as MLPs, CNNs, ResNets, and Transformers. Our SDEs accurately track the behavior of the respective optimizers, especially when compared to previous SDEs derived for Adam and RMSprop. We believe our approach can provide valuable insights into best training practices and novel scaling rules.

Deep Learning · Theory

Arthur Jacot, Peter Súkeník, Zihan Wang, Marco Mondelli

Deep neural networks (DNNs) at convergence consistently represent the training data in the last layer via a geometric structure referred to as neural collapse. This empirical evidence has spurred a line of theoretical research aimed at proving the emergence of neural collapse, mostly focusing on the unconstrained features model. Here, the features of the penultimate layer are free variables, which makes the model data-agnostic and puts into question its ability to capture DNN training. Our work addresses the issue, moving away from unconstrained features and studying DNNs that end with at least two linear layers. We first prove generic guarantees on neural collapse that assume \emph{(i)} low training error and balancedness of linear layers (for within-class variability collapse), and \emph{(ii)} bounded conditioning of the features before the linear part (for orthogonality of class-means, and their alignment with weight matrices). The balancedness refers to the fact that $W_{\ell+1}^\top W_{\ell+1}\approx W_\ell W_\ell ^\top$ for any pair ofconsecutive weight matricesof the linear part, and the bounded conditioning requires a well-behaved ratio between largest and smallest non-zero singular values of the features. We then show that such assumptions hold for gradient descent training with weight decay: \emph{(i)} for networks with a wide first layer, we prove low training error and balancedness, and \emph{(ii)} for solutions that are either nearly optimal or stable under large learning rates, we additionally prove the bounded conditioning. Taken together, our results are the first to show neural collapse in the end-to-end training of DNNs.

Deep Learning · Theory

Arthur Jacot, Peter Súkeník, Zihan Wang, Marco Mondelli

Deep neural networks (DNNs) at convergence consistently represent the training data in the last layer via a geometric structure referred to as neural collapse. This empirical evidence has spurred a line of theoretical research aimed at proving the emergence of neural collapse, mostly focusing on the unconstrained features model. Here, the features of the penultimate layer are free variables, which makes the model data-agnostic and puts into question its ability to capture DNN training. Our work addresses the issue, moving away from unconstrained features and studying DNNs that end with at least two linear layers. We first prove generic guarantees on neural collapse that assume \emph{(i)} low training error and balancedness of linear layers (for within-class variability collapse), and \emph{(ii)} bounded conditioning of the features before the linear part (for orthogonality of class-means, and their alignment with weight matrices). The balancedness refers to the fact that $W_{\ell+1}^\top W_{\ell+1}\approx W_\ell W_\ell ^\top$ for any pair ofconsecutive weight matricesof the linear part, and the bounded conditioning requires a well-behaved ratio between largest and smallest non-zero singular values of the features. We then show that such assumptions hold for gradient descent training with weight decay: \emph{(i)} for networks with a wide first layer, we prove low training error and balancedness, and \emph{(ii)} for solutions that are either nearly optimal or stable under large learning rates, we additionally prove the bounded conditioning. Taken together, our results are the first to show neural collapse in the end-to-end training of DNNs.