Dynamically Learning to Integrate in Recurrent Neural Networks

Learning to remember over long timescales is fundamentally challenging for recurrent neural networks (RNNs). While much prior work has explored why RNNs struggle to learn long timescales and how to mitigate this, we still lack a clear understanding of the dynamics involved when RNNs learn long timescales via gradient descent. Here we build a mathematical theory of the learning dynamics of linear RNNs trained to integrate white noise. We show that when the initial recurrent weights are small, the dynamics of learning are described by a low-dimensional system that tracks a single outlier eigenvalue of the recurrent weights. This reveals the precise manner in which the long timescale associated with white noise integration is learned. We extend our analyses to RNNs learning a damped oscillatory filter, and find rich dynamical equations for the evolution of a conjugate pair of outlier eigenvalues. Taken together, our analyses build a rich mathematical framework for studying dynamical learning problems salient for both machine learning and neuroscience.

Learning richness modulates equality reasoning in neural networks

Equality reasoning is ubiquitous and purely abstract: sameness or difference may be evaluated no matter the nature of the underlying objects. As a result, same-different tasks (SD) have been extensively studied as a starting point for understanding abstract reasoning in humans and across animal species. With the rise of neural networks (NN) that exhibit striking apparent proficiency for abstractions, equality reasoning in NNs has also gained interest. Yet despite extensive study, conclusions about equality reasoning vary widely and with little consensus. To clarify the underlying principles in learning SD, we develop a theory of equality reasoning in multi-layer perceptrons (MLP). Following observations in comparative psychology, we propose a spectrum of behavior that ranges from conceptual to perceptual outcomes. Conceptual behavior is characterized by task-specific representations, efficient learning, and insensitivity to spurious perceptual details. Perceptual behavior is characterized by strong sensitivity to spurious perceptual details, accompanied by the need for exhaustive training to learn the task. We develop a mathematical theory to show that an MLP's behavior is driven by learning richness. Rich-regime MLPs exhibit conceptual behavior, whereas lazy-regime MLPs exhibit perceptual behavior. We validate our theoretical findings in vision SD experiments, showing that rich feature learning promotes success by encouraging hallmarks of conceptual behavior. Overall, our work identifies feature learning richness as a key parameter modulating equality reasoning, and suggests that equality reasoning in humans and animals may similarly depend on learning richness in neural circuits.

Adaptive kernel predictors from feature-learning infinite limits of neural networks

Previous influential work showed that infinite width limits of neural networks in the lazy training regime are described by kernel machines. Here, we show that neural networks trained in the rich, feature learning infinite-width regime in two different settings are also described by kernel machines, but with data-dependent kernels. For both cases, we provide explicit expressions for the kernel predictors and prescriptions to numerically calculate them. To derive the first predictor, we study the large-width limit of feature-learning Bayesian networks, showing how feature learning leads to task-relevant adaptation of layer kernels and preactivation densities. The saddle point equations governing this limit result in a min-max optimization problem that defines the kernel predictor. To derive the second predictor, we study gradient flow training of randomly initialized networks trained with weight decay in the infinite-width limit using dynamical mean field theory (DMFT). The fixed point equations of the arising DMFT defines the task-adapted internal representations and the kernel predictor. We compare our kernel predictors to kernels derived from lazy regime and demonstrate that our adaptive kernels achieve lower test loss on benchmark datasets.

Two-Point Deterministic Equivalence for Stochastic Gradient Dynamics in Linear Models

We derive a novel deterministic equivalence for the two-point function of a random matrix resolvent. Using this result, we give a unified derivation of the performance of a wide variety of high-dimensional linear models trained with stochastic gradient descent. This includes high-dimensional linear regression, kernel regression, and random feature models. Our results include previously known asymptotics as well as novel ones.

Deep Linear Network Training Dynamics from Random Initialization: Data, Width, Depth, and Hyperparameter Transfer

We theoretically characterize gradient descent dynamics in deep linear networks trained at large width from random initialization and on large quantities of random data. Our theory captures the ``wider is better" effect of mean-field/maximum-update parameterized networks as well as hyperparameter transfer effects, which can be contrasted with the neural-tangent parameterization where optimal learning rates shift with model width. We provide asymptotic descriptions of both non-residual and residual neural networks, the latter of which enables an infinite depth limit when branches are scaled as $1/\sqrt{\text{depth}}$. We also compare training with one-pass stochastic gradient descent to the dynamics when training data are repeated at each iteration. Lastly, we show that this model recovers the accelerated power law training dynamics for power law structured data in the rich regime observed in recent works.

A precise asymptotic analysis of learning diffusion models: theory and insights

In this manuscript, we consider the problem of learning a flow or diffusion-based generative model parametrized by a two-layer auto-encoder, trained with online stochastic gradient descent, on a high-dimensional target density with an underlying low-dimensional manifold structure. We derive a tight asymptotic characterization of low-dimensional projections of the distribution of samples generated by the learned model, ascertaining in particular its dependence on the number of training samples. Building on this analysis, we discuss how mode collapse can arise, and lead to model collapse when the generative model is re-trained on generated synthetic data.

No Free Lunch From Random Feature Ensembles

Given a budget on total model size, one must decide whether to train a single, large neural network or to combine the predictions of many smaller networks. We study this trade-off for ensembles of random-feature ridge regression models. We prove that when a fixed number of trainable parameters are partitioned among $K$ independently trained models, $K=1$ achieves optimal performance, provided the ridge parameter is optimally tuned. We then derive scaling laws which describe how the test risk of an ensemble of regression models decays with its total size. We identify conditions on the kernel and task eigenstructure under which ensembles can achieve near-optimal scaling laws. Training ensembles of deep convolutional neural networks on CIFAR-10 and a transformer architecture on C4, we find that a single large network outperforms any ensemble of networks with the same total number of parameters, provided the weight decay and feature-learning strength are tuned to their optimal values.

Scaling Laws for Precision

Low precision training and inference affect both the quality and cost of language models, but current scaling laws do not account for this. In this work, we devise "precision-aware" scaling laws for both training and inference. We propose that training in lower precision reduces the model's "effective parameter count," allowing us to predict the additional loss incurred from training in low precision and post-train quantization. For inference, we find that the degradation introduced by post-training quantization increases as models are trained on more data, eventually making additional pretraining data actively harmful. For training, our scaling laws allow us to predict the loss of a model with different parts in different precisions, and suggest that training larger models in lower precision may be compute optimal. We unify the scaling laws for post and pretraining quantization to arrive at a single functional form that predicts degradation from training and inference in varied precisions. We fit on over 465 pretraining runs and validate our predictions on model sizes up to 1.7B parameters trained on up to 26B tokens.

Do Mice Grok? Glimpses of Hidden Progress During Overtraining in Sensory Cortex

Does learning of task-relevant representations stop when behavior stops changing? Motivated by recent theoretical advances in machine learning and the intuitive observation that human experts continue to learn from practice even after mastery, we hypothesize that task-specific representation learning can continue, even when behavior plateaus. In a novel reanalysis of recently published neural data, we find evidence for such learning in posterior piriform cortex of mice following continued training on a task, long after behavior saturates at near-ceiling performance ("overtraining"). This learning is marked by an increase in decoding accuracy from piriform neural populations and improved performance on held-out generalization tests. We demonstrate that class representations in cortex continue to separate during overtraining, so that examples that were incorrectly classified at the beginning of overtraining can abruptly be correctly classified later on, despite no changes in behavior during that time. We hypothesize this hidden yet rich learning takes the form of approximate margin maximization; we validate this and other predictions in the neural data, as well as build and interpret a simple synthetic model that recapitulates these phenomena. We conclude by showing how this model of late-time feature learning implies an explanation for the empirical puzzle of overtraining reversal in animal learning, where task-specific representations are more robust to particular task changes because the learned features can be reused.

A Brain-Inspired Regularizer for Adversarial Robustness

Convolutional Neural Networks (CNNs) excel in many visual tasks, but they tend to be sensitive to slight input perturbations that are imperceptible to the human eye, often resulting in task failures. Recent studies indicate that training CNNs with regularizers that promote brain-like representations, using neural recordings, can improve model robustness. However, the requirement to use neural data severely restricts the utility of these methods. Is it possible to develop regularizers that mimic the computational function of neural regularizers without the need for neural recordings, thereby expanding the usability and effectiveness of these techniques? In this work, we inspect a neural regularizer introduced in Li et al. (2019) to extract its underlying strength. The regularizer uses neural representational similarities, which we find also correlate with pixel similarities. Motivated by this finding, we introduce a new regularizer that retains the essence of the original but is computed using image pixel similarities, eliminating the need for neural recordings. We show that our regularization method 1) significantly increases model robustness to a range of black box attacks on various datasets and 2) is computationally inexpensive and relies only on original datasets. Our work explores how biologically motivated loss functions can be used to drive the performance of artificial neural networks.