David J. Schwab

Vudtiwat Ngampruetikorn, David J. Schwab

Avoiding overfitting is a central challenge in machine learning, yet many large neural networks readily achieve zero training loss. This puzzling contradiction necessitates new approaches to the study of overfitting. Here we quantify overfitting via residual information, defined as the bits in fitted models that encode noise in training data. Information efficient learning algorithms minimize residual information while maximizing the relevant bits, which are predictive of the unknown generative models. We solve this optimization to obtain the information content of optimal algorithms for a linear regression problem and compare it to that of randomized ridge regression. Our results demonstrate the fundamental trade-off between residual and relevant information and characterize the relative information efficiency of randomized regression with respect to optimal algorithms. Finally, using results from random matrix theory, we reveal the information complexity of learning a linear map in high dimensions and unveil information-theoretic analogs of double and multiple descent phenomena.

Vudtiwat Ngampruetikorn, David J. Schwab

The information bottleneck (IB) method offers an attractive framework for understanding representation learning, however its applications are often limited by its computational intractability. Analytical characterization of the IB method is not only of practical interest, but it can also lead to new insights into learning phenomena. Here we consider a generalized IB problem, in which the mutual information in the original IB method is replaced by correlation measures based on Renyi and Jeffreys divergences. We derive an exact analytical IB solution for the case of Gaussian correlated variables. Our analysis reveals a series of structural transitions, similar to those previously observed in the original IB case. We find further that although solving the original, Renyi and Jeffreys IB problems yields different representations in general, the structural transitions occur at the same critical tradeoff parameters, and the Renyi and Jeffreys IB solutions perform well under the original IB objective. Our results suggest that formulating the IB method with alternative correlation measures could offer a strategy for obtaining an approximate solution to the original IB problem.

Jello Zhou, Vudtiwat Ngampruetikorn, David J. Schwab

Stochastic resetting, where a dynamical process is intermittently returned to a fixed reference state, has emerged as a powerful mechanism for optimizing first-passage properties. Existing theory largely treats static, non-learning processes. Here we ask how stochastic resetting interacts with reinforcement learning, where the underlying dynamics adapt through experience. In tabular grid environments, we find that resetting accelerates policy convergence even when it does not reduce the search time of a purely diffusive agent, indicating a novel mechanism beyond classical first-passage optimization. In a continuous control task with neural-network-based value approximation, we show that random resetting improves deep reinforcement learning when exploration is difficult and rewards are sparse. Unlike temporal discounting, resetting preserves the optimal policy while accelerating convergence by truncating long, uninformative trajectories to enhance value propagation. Our results establish stochastic resetting as a simple, tunable mechanism for accelerating learning, translating a canonical phenomenon of statistical mechanics into an optimization principle for reinforcement learning.

Alex Nguyen, David J. Schwab, Vudtiwat Ngampruetikorn

Machine learning models are often brittle under distribution shift, i.e., when data distributions at test time differ from those during training. Understanding this failure mode is central to identifying and mitigating safety risks of mass adoption of machine learning. Here we analyze ridge regression under concept shift -- a form of distribution shift in which the input-label relationship changes at test time. We derive an exact expression for prediction risk in the thermodynamic limit. Our results reveal nontrivial effects of concept shift on generalization performance, including a phase transition between weak and strong concept shift regimes and nonmonotonic data dependence of test performance even when double descent is absent. Our theoretical results are in good agreement with experiments based on transformers pretrained to solve linear regression; under concept shift, too long context length can be detrimental to generalization performance of next token prediction. Finally, our experiments on MNIST and FashionMNIST suggest that this intriguing behavior is present also in classification problems.

Luca Di Carlo, Chase Goddard, David J. Schwab

Modern neural networks exhibit a striking property: basins of attraction in the loss landscape are often connected by low-loss paths, yet optimization dynamics generally remain confined to a single convex basin and rarely explore intermediate points. We resolve this paradox by identifying entropic barriers arising from the interplay between curvature variations along these paths and noise in optimization dynamics. Empirically, we find that curvature systematically rises away from minima, producing effective forces that bias noisy dynamics back toward the endpoints - even when the loss remains nearly flat. These barriers persist longer than energetic barriers, shaping the late-time localization of solutions in parameter space. Our results highlight the role of curvature-induced entropic forces in governing both connectivity and confinement in deep learning landscapes.

Chase Goddard, Lindsay M. Smith, Vudtiwat Ngampruetikorn et al. · princeton

In-context learning (ICL) is a remarkable capability of pretrained transformers that allows models to generalize to unseen tasks after seeing only a few examples. We investigate empirically the conditions necessary on the pretraining distribution for ICL to emerge and generalize \emph{out-of-distribution}. Previous work has focused on the number of distinct tasks necessary in the pretraining dataset. Here, we use a different notion of task diversity to study the emergence of ICL in transformers trained on linear functions. We find that as task diversity increases, transformers undergo a transition from a specialized solution, which exhibits ICL only within the pretraining task distribution, to a solution which generalizes out of distribution to the entire task space. We also investigate the nature of the solutions learned by the transformer on both sides of the transition, and observe similar transitions in nonlinear regression problems. We construct a phase diagram to characterize how our concept of task diversity interacts with the number of pretraining tasks. In addition, we explore how factors such as the depth of the model and the dimensionality of the regression problem influence the transition.

Alex Nguyen, David J. Schwab, Vudtiwat Ngampruetikorn

Lossy data transformations by definition lose information. Yet, in modern machine learning, methods like data pruning and lossy data augmentation can help improve generalization performance. We study this paradox using a solvable model of high-dimensional, ridge-regularized linear regression under 'data coarse graining.' Inspired by the renormalization group in statistical physics, we analyze coarse-graining schemes that systematically discard features based on their relevance to the learning task. Our results reveal a nonmonotonic dependence of the prediction risk on the degree of coarse graining. A 'high-pass' scheme--which filters out less relevant, lower-signal features--can help models generalize better. By contrast, a 'low-pass' scheme that integrates out more relevant, higher-signal features is purely detrimental. Crucially, using optimal regularization, we demonstrate that this nonmonotonicity is a distinct effect of data coarse graining and not an artifact of double descent. Our framework offers a clear, analytical explanation for why careful data augmentation works: it strips away less relevant degrees of freedom and isolates more predictive signals. Our results highlight a complex, nonmonotonic risk landscape shaped by the structure of the data, and illustrate how ideas from statistical physics provide a principled lens for understanding modern machine learning phenomena.

Vudtiwat Ngampruetikorn, David J. Schwab

Extracting relevant information from data is crucial for all forms of learning. The information bottleneck (IB) method formalizes this, offering a mathematically precise and conceptually appealing framework for understanding learning phenomena. However the nonlinearity of the IB problem makes it computationally expensive and analytically intractable in general. Here we derive a perturbation theory for the IB method and report the first complete characterization of the learning onset, the limit of maximum relevant information per bit extracted from data. We test our results on synthetic probability distributions, finding good agreement with the exact numerical solution near the onset of learning. We explore the difference and subtleties in our derivation and previous attempts at deriving a perturbation theory for the learning onset and attribute the discrepancy to a flawed assumption. Our work also provides a fresh perspective on the intimate relationship between the IB method and the strong data processing inequality.

Chaitanya K. Ryali, David J. Schwab, Ari S. Morcos

Recent progress in self-supervised learning has demonstrated promising results in multiple visual tasks. An important ingredient in high-performing self-supervised methods is the use of data augmentation by training models to place different augmented views of the same image nearby in embedding space. However, commonly used augmentation pipelines treat images holistically, ignoring the semantic relevance of parts of an image-e.g. a subject vs. a background-which can lead to the learning of spurious correlations. Our work addresses this problem by investigating a class of simple, yet highly effective "background augmentations", which encourage models to focus on semantically-relevant content by discouraging them from focusing on image backgrounds. Through a systematic investigation, we show that background augmentations lead to substantial improvements in performance across a spectrum of state-of-the-art self-supervised methods (MoCo-v2, BYOL, SwAV) on a variety of tasks, e.g. $\sim$+1-2% gains on ImageNet, enabling performance on par with the supervised baseline. Further, we find the improvement in limited-labels settings is even larger (up to 4.2%). Background augmentations also improve robustness to a number of distribution shifts, including natural adversarial examples, ImageNet-9, adversarial attacks, ImageNet-Renditions. We also make progress in completely unsupervised saliency detection, in the process of generating saliency masks used for background augmentations.

Tiffany Tianhui Cai, Jonathan Frankle, David J. Schwab et al.

Self-supervised learning has recently begun to rival supervised learning on computer vision tasks. Many of the recent approaches have been based on contrastive instance discrimination (CID), in which the network is trained to recognize two augmented versions of the same instance (a query and positive) while discriminating against a pool of other instances (negatives). The learned representation is then used on downstream tasks such as image classification. Using methodology from MoCo v2 (Chen et al., 2020), we divided negatives by their difficulty for a given query and studied which difficulty ranges were most important for learning useful representations. We found a minority of negatives -- the hardest 5% -- were both necessary and sufficient for the downstream task to reach nearly full accuracy. Conversely, the easiest 95% of negatives were unnecessary and insufficient. Moreover, the very hardest 0.1% of negatives were unnecessary and sometimes detrimental. Finally, we studied the properties of negatives that affect their hardness, and found that hard negatives were more semantically similar to the query, and that some negatives were more consistently easy or hard than we would expect by chance. Together, our results indicate that negatives vary in importance and that CID may benefit from more intelligent negative treatment.

Yann Dubois, Douwe Kiela, David J. Schwab et al.

We address the question of characterizing and finding optimal representations for supervised learning. Traditionally, this question has been tackled using the Information Bottleneck, which compresses the inputs while retaining information about the targets, in a decoder-agnostic fashion. In machine learning, however, our goal is not compression but rather generalization, which is intimately linked to the predictive family or decoder of interest (e.g. linear classifier). We propose the Decodable Information Bottleneck (DIB) that considers information retention and compression from the perspective of the desired predictive family. As a result, DIB gives rise to representations that are optimal in terms of expected test performance and can be estimated with guarantees. Empirically, we show that the framework can be used to enforce a small generalization gap on downstream classifiers and to predict the generalization ability of neural networks.

Kamesh Krishnamurthy, Tankut Can, David J. Schwab

Recurrent neural networks (RNNs) are powerful dynamical models, widely used in machine learning (ML) and neuroscience. Prior theoretical work has focused on RNNs with additive interactions. However, gating - i.e. multiplicative - interactions are ubiquitous in real neurons and also the central feature of the best-performing RNNs in ML. Here, we show that gating offers flexible control of two salient features of the collective dynamics: i) timescales and ii) dimensionality. The gate controlling timescales leads to a novel, marginally stable state, where the network functions as a flexible integrator. Unlike previous approaches, gating permits this important function without parameter fine-tuning or special symmetries. Gates also provide a flexible, context-dependent mechanism to reset the memory trace, thus complementing the memory function. The gate modulating the dimensionality can induce a novel, discontinuous chaotic transition, where inputs push a stable system to strong chaotic activity, in contrast to the typically stabilizing effect of inputs. At this transition, unlike additive RNNs, the proliferation of critical points (topological complexity) is decoupled from the appearance of chaotic dynamics (dynamical complexity). The rich dynamics are summarized in phase diagrams, thus providing a map for principled parameter initialization choices to ML practitioners.

Jonathan Frankle, David J. Schwab, Ari S. Morcos

A wide variety of deep learning techniques from style transfer to multitask learning rely on training affine transformations of features. Most prominent among these is the popular feature normalization technique BatchNorm, which normalizes activations and then subsequently applies a learned affine transform. In this paper, we aim to understand the role and expressive power of affine parameters used to transform features in this way. To isolate the contribution of these parameters from that of the learned features they transform, we investigate the performance achieved when training only these parameters in BatchNorm and freezing all weights at their random initializations. Doing so leads to surprisingly high performance considering the significant limitations that this style of training imposes. For example, sufficiently deep ResNets reach 82% (CIFAR-10) and 32% (ImageNet, top-5) accuracy in this configuration, far higher than when training an equivalent number of randomly chosen parameters elsewhere in the network. BatchNorm achieves this performance in part by naturally learning to disable around a third of the random features. Not only do these results highlight the expressive power of affine parameters in deep learning, but - in a broader sense - they characterize the expressive power of neural networks constructed simply by shifting and rescaling random features.

Jonathan Frankle, David J. Schwab, Ari S. Morcos

Recent studies have shown that many important aspects of neural network learning take place within the very earliest iterations or epochs of training. For example, sparse, trainable sub-networks emerge (Frankle et al., 2019), gradient descent moves into a small subspace (Gur-Ari et al., 2018), and the network undergoes a critical period (Achille et al., 2019). Here, we examine the changes that deep neural networks undergo during this early phase of training. We perform extensive measurements of the network state during these early iterations of training and leverage the framework of Frankle et al. (2019) to quantitatively probe the weight distribution and its reliance on various aspects of the dataset. We find that, within this framework, deep networks are not robust to reinitializing with random weights while maintaining signs, and that weight distributions are highly non-independent even after only a few hundred iterations. Despite this behavior, pre-training with blurred inputs or an auxiliary self-supervised task can approximate the changes in supervised networks, suggesting that these changes are not inherently label-dependent, though labels significantly accelerate this process. Together, these results help to elucidate the network changes occurring during this pivotal initial period of learning.

Tankut Can, Kamesh Krishnamurthy, David J. Schwab

Recurrent neural networks (RNNs) are powerful dynamical models for data with complex temporal structure. However, training RNNs has traditionally proved challenging due to exploding or vanishing of gradients. RNN models such as LSTMs and GRUs (and their variants) significantly mitigate these issues associated with training by introducing various types of gating units into the architecture. While these gates empirically improve performance, how the addition of gates influences the dynamics and trainability of GRUs and LSTMs is not well understood. Here, we take the perspective of studying randomly initialized LSTMs and GRUs as dynamical systems, and ask how the salient dynamical properties are shaped by the gates. We leverage tools from random matrix theory and mean-field theory to study the state-to-state Jacobians of GRUs and LSTMs. We show that the update gate in the GRU and the forget gate in the LSTM can lead to an accumulation of slow modes in the dynamics. Moreover, the GRU update gate can poise the system at a marginally stable point. The reset gate in the GRU and the output and input gates in the LSTM control the spectral radius of the Jacobian, and the GRU reset gate also modulates the complexity of the landscape of fixed-points. Furthermore, for the GRU we obtain a phase diagram describing the statistical properties of fixed-points. We also provide a preliminary comparison of training performance to the various dynamical regimes realized by varying hyperparameters. Looking to the future, we have introduced a powerful set of techniques which can be adapted to a broad class of RNNs, to study the influence of various architectural choices on dynamics, and potentially motivate the principled discovery of novel architectures.

Pankaj Mehta, Marin Bukov, Ching-Hao Wang et al.

Machine Learning (ML) is one of the most exciting and dynamic areas of modern research and application. The purpose of this review is to provide an introduction to the core concepts and tools of machine learning in a manner easily understood and intuitive to physicists. The review begins by covering fundamental concepts in ML and modern statistics such as the bias-variance tradeoff, overfitting, regularization, generalization, and gradient descent before moving on to more advanced topics in both supervised and unsupervised learning. Topics covered in the review include ensemble models, deep learning and neural networks, clustering and data visualization, energy-based models (including MaxEnt models and Restricted Boltzmann Machines), and variational methods. Throughout, we emphasize the many natural connections between ML and statistical physics. A notable aspect of the review is the use of Python Jupyter notebooks to introduce modern ML/statistical packages to readers using physics-inspired datasets (the Ising Model and Monte-Carlo simulations of supersymmetric decays of proton-proton collisions). We conclude with an extended outlook discussing possible uses of machine learning for furthering our understanding of the physical world as well as open problems in ML where physicists may be able to contribute. (Notebooks are available at https://physics.bu.edu/~pankajm/MLnotebooks.html )

David J. Schwab, Pankaj Mehta

In a recent paper, "Why does deep and cheap learning work so well?", Lin and Tegmark claim to show that the mapping between deep belief networks and the variational renormalization group derived in [arXiv:1410.3831] is invalid, and present a "counterexample" that claims to show that this mapping does not hold. In this comment, we show that these claims are incorrect and stem from a misunderstanding of the variational RG procedure proposed by Kadanoff. We also explain why the "counterexample" of Lin and Tegmark is compatible with the mapping proposed in [arXiv:1410.3831].

E. Miles Stoudenmire, David J. Schwab

Tensor networks are efficient representations of high-dimensional tensors which have been very successful for physics and mathematics applications. We demonstrate how algorithms for optimizing such networks can be adapted to supervised learning tasks by using matrix product states (tensor trains) to parameterize models for classifying images. For the MNIST data set we obtain less than 1% test set classification error. We discuss how the tensor network form imparts additional structure to the learned model and suggest a possible generative interpretation.

Pankaj Mehta, David J. Schwab

Deep learning is a broad set of techniques that uses multiple layers of representation to automatically learn relevant features directly from structured data. Recently, such techniques have yielded record-breaking results on a diverse set of difficult machine learning tasks in computer vision, speech recognition, and natural language processing. Despite the enormous success of deep learning, relatively little is understood theoretically about why these techniques are so successful at feature learning and compression. Here, we show that deep learning is intimately related to one of the most important and successful techniques in theoretical physics, the renormalization group (RG). RG is an iterative coarse-graining scheme that allows for the extraction of relevant features (i.e. operators) as a physical system is examined at different length scales. We construct an exact mapping from the variational renormalization group, first introduced by Kadanoff, and deep learning architectures based on Restricted Boltzmann Machines (RBMs). We illustrate these ideas using the nearest-neighbor Ising Model in one and two-dimensions. Our results suggests that deep learning algorithms may be employing a generalized RG-like scheme to learn relevant features from data.