Neglected Hessian component explains mysteries in Sharpness regularization

Recent work has shown that methods like SAM which either explicitly or implicitly penalize second order information can improve generalization in deep learning. Seemingly similar methods like weight noise and gradient penalties often fail to provide such benefits. We show that these differences can be explained by the structure of the Hessian of the loss. First, we show that a common decomposition of the Hessian can be quantitatively interpreted as separating the feature exploitation from feature exploration. The feature exploration, which can be described by the Nonlinear Modeling Error matrix (NME), is commonly neglected in the literature since it vanishes at interpolation. Our work shows that the NME is in fact important as it can explain why gradient penalties are sensitive to the choice of activation function. Using this insight we design interventions to improve performance. We also provide evidence that challenges the long held equivalence of weight noise and gradient penalties. This equivalence relies on the assumption that the NME can be ignored, which we find does not hold for modern networks since they involve significant feature learning. We find that regularizing feature exploitation but not feature exploration yields performance similar to gradient penalties.

On the Foundations of Shortcut Learning

Deep-learning models can extract a rich assortment of features from data. Which features a model uses depends not only on predictivity-how reliably a feature indicates train-set labels-but also on availability-how easily the feature can be extracted, or leveraged, from inputs. The literature on shortcut learning has noted examples in which models privilege one feature over another, for example texture over shape and image backgrounds over foreground objects. Here, we test hypotheses about which input properties are more available to a model, and systematically study how predictivity and availability interact to shape models' feature use. We construct a minimal, explicit generative framework for synthesizing classification datasets with two latent features that vary in predictivity and in factors we hypothesize to relate to availability, and quantify a model's shortcut bias-its over-reliance on the shortcut (more available, less predictive) feature at the expense of the core (less available, more predictive) feature. We find that linear models are relatively unbiased, but introducing a single hidden layer with ReLU or Tanh units yields a bias. Our empirical findings are consistent with a theoretical account based on Neural Tangent Kernels. Finally, we study how models used in practice trade off predictivity and availability in naturalistic datasets, discovering availability manipulations which increase models' degree of shortcut bias. Taken together, these findings suggest that the propensity to learn shortcut features is a fundamental characteristic of deep nonlinear architectures warranting systematic study given its role in shaping how models solve tasks.

Sharpness-Aware Minimization Leads to Low-Rank Features

Sharpness-aware minimization (SAM) is a recently proposed method that minimizes the sharpness of the training loss of a neural network. While its generalization improvement is well-known and is the primary motivation, we uncover an additional intriguing effect of SAM: reduction of the feature rank which happens at different layers of a neural network. We show that this low-rank effect occurs very broadly: for different architectures such as fully-connected networks, convolutional networks, vision transformers and for different objectives such as regression, classification, language-image contrastive training. To better understand this phenomenon, we provide a mechanistic understanding of how low-rank features arise in a simple two-layer network. We observe that a significant number of activations gets entirely pruned by SAM which directly contributes to the rank reduction. We confirm this effect theoretically and check that it can also occur in deep networks, although the overall rank reduction mechanism can be more complex, especially for deep networks with pre-activation skip connections and self-attention layers. We make our code available at https://github.com/tml-epfl/sam-low-rank-features.

On student-teacher deviations in distillation: does it pay to disobey?

Knowledge distillation has been widely-used to improve the performance of a "student" network by hoping to mimic soft probabilities of a "teacher" network. Yet, for self-distillation to work, the student must somehow deviate from the teacher (Stanton et al., 2021). But what is the nature of these deviations, and how do they relate to gains in generalization? We investigate these questions through a series of experiments across image and language classification datasets. First, we observe that distillation consistently deviates in a characteristic way: on points where the teacher has low confidence, the student achieves even lower confidence than the teacher. Secondly, we find that deviations in the initial dynamics of training are not crucial -- simply switching to distillation loss in the middle of training can recover much of its gains. We then provide two parallel theoretical perspectives to understand the role of student-teacher deviations in our experiments, one casting distillation as a regularizer in eigenspace, and another as a gradient denoiser. Our analysis bridges several gaps between existing theory and practice by (a) focusing on gradient-descent training, (b) by avoiding label noise assumptions, and (c) by unifying several disjoint empirical and theoretical findings.

Sharpness-Aware Minimization Improves Language Model Generalization

The allure of superhuman-level capabilities has led to considerable interest in language models like GPT-3 and T5, wherein the research has, by and large, revolved around new model architectures, training tasks, and loss objectives, along with substantial engineering efforts to scale up model capacity and dataset size. Comparatively little work has been done to improve the generalization of these models through better optimization. In this work, we show that Sharpness-Aware Minimization (SAM), a recently proposed optimization procedure that encourages convergence to flatter minima, can substantially improve the generalization of language models without much computational overhead. We show that SAM is able to boost performance on SuperGLUE, GLUE, Web Questions, Natural Questions, Trivia QA, and TyDiQA, with particularly large gains when training data for these tasks is limited.

The Low-Rank Simplicity Bias in Deep Networks

Modern deep neural networks are highly over-parameterized compared to the data on which they are trained, yet they often generalize remarkably well. A flurry of recent work has asked: why do deep networks not overfit to their training data? We investigate the hypothesis that deeper nets are implicitly biased to find lower rank solutions and that these are the solutions that generalize well. We prove for the asymptotic case that the percent volume of low effective-rank solutions increases monotonically as linear neural networks are made deeper. We then show empirically that our claim holds true on finite width models. We further empirically find that a similar result holds for non-linear networks: deeper non-linear networks learn a feature space whose kernel has a lower rank. We further demonstrate how linear over-parameterization of deep non-linear models can be used to induce low-rank bias, improving generalization performance without changing the effective model capacity. We evaluate on various model architectures and demonstrate that linearly over-parameterized models outperform existing baselines on image classification tasks, including ImageNet.

NeurIPS 2020 Competition: Predicting Generalization in Deep Learning

Understanding generalization in deep learning is arguably one of the most important questions in deep learning. Deep learning has been successfully adopted to a large number of problems ranging from pattern recognition to complex decision making, but many recent researchers have raised many concerns about deep learning, among which the most important is generalization. Despite numerous attempts, conventional statistical learning approaches have yet been able to provide a satisfactory explanation on why deep learning works. A recent line of works aims to address the problem by trying to predict the generalization performance through complexity measures. In this competition, we invite the community to propose complexity measures that can accurately predict generalization of models. A robust and general complexity measure would potentially lead to a better understanding of deep learning's underlying mechanism and behavior of deep models on unseen data, or shed light on better generalization bounds. All these outcomes will be important for making deep learning more robust and reliable.

Data Augmentation via Structured Adversarial Perturbations

Data augmentation is a major component of many machine learning methods with state-of-the-art performance. Common augmentation strategies work by drawing random samples from a space of transformations. Unfortunately, such sampling approaches are limited in expressivity, as they are unable to scale to rich transformations that depend on numerous parameters due to the curse of dimensionality. Adversarial examples can be considered as an alternative scheme for data augmentation. By being trained on the most difficult modifications of the inputs, the resulting models are then hopefully able to handle other, presumably easier, modifications as well. The advantage of adversarial augmentation is that it replaces sampling with the use of a single, calculated perturbation that maximally increases the loss. The downside, however, is that these raw adversarial perturbations appear rather unstructured; applying them often does not produce a natural transformation, contrary to a desirable data augmentation technique. To address this, we propose a method to generate adversarial examples that maintain some desired natural structure. We first construct a subspace that only contains perturbations with the desired structure. We then project the raw adversarial gradient onto this space to select a structured transformation that would maximally increase the loss when applied. We demonstrate this approach through two types of image transformations: photometric and geometric. Furthermore, we show that training on such structured adversarial images improves generalization.

A Unifying View on Implicit Bias in Training Linear Neural Networks

We study the implicit bias of gradient flow (i.e., gradient descent with infinitesimal step size) on linear neural network training. We propose a tensor formulation of neural networks that includes fully-connected, diagonal, and convolutional networks as special cases, and investigate the linear version of the formulation called linear tensor networks. For $L$-layer linear tensor networks that are orthogonally decomposable, we show that gradient flow on separable classification finds a stationary point of the $\ell_{2/L}$ max-margin problem in a "transformed" input space defined by the network. For underdetermined regression, we prove that gradient flow finds a global minimum which minimizes a norm-like function that interpolates between weighted $\ell_1$ and $\ell_2$ norms in the transformed input space. Our theorems subsume existing results in the literature while removing most of the convergence assumptions. We also provide experiments that corroborate our analysis.

Sharpness-Aware Minimization for Efficiently Improving Generalization

In today's heavily overparameterized models, the value of the training loss provides few guarantees on model generalization ability. Indeed, optimizing only the training loss value, as is commonly done, can easily lead to suboptimal model quality. Motivated by the connection between geometry of the loss landscape and generalization---including a generalization bound that we prove here---we introduce a novel, effective procedure for instead simultaneously minimizing loss value and loss sharpness. In particular, our procedure, Sharpness-Aware Minimization (SAM), seeks parameters that lie in neighborhoods having uniformly low loss; this formulation results in a min-max optimization problem on which gradient descent can be performed efficiently. We present empirical results showing that SAM improves model generalization across a variety of benchmark datasets (e.g., CIFAR-{10, 100}, ImageNet, finetuning tasks) and models, yielding novel state-of-the-art performance for several. Additionally, we find that SAM natively provides robustness to label noise on par with that provided by state-of-the-art procedures that specifically target learning with noisy labels.