We consider neural networks (NNs) where the final layer is down-scaled by a fixed hyperparameter $\gamma$. Recent work has identified $\gamma$ as controlling the strength of feature learning. As $\gamma$ increases, network evolution changes from "lazy" kernel dynamics to "rich" feature-learning dynamics, with a host of associated benefits including improved performance on common tasks. In this work, we conduct a thorough empirical investigation of the effect of scaling $\gamma$ across a variety of models and datasets in the online training setting. We first examine the interaction of $\gamma$ with the learning rate $\eta$, identifying several scaling regimes in the $\gamma$-$\eta$ plane which we explain theoretically using a simple model. We find that the optimal learning rate $\eta^*$ scales non-trivially with $\gamma$. In particular, $\eta^* \propto \gamma^2$ when $\gamma \ll 1$ and $\eta^* \propto \gamma^{2/L}$ when $\gamma \gg 1$ for a feed-forward network of depth $L$. Using this optimal learning rate scaling, we proceed with an empirical study of the under-explored ``ultra-rich'' $\gamma \gg 1$ regime. We find that networks in this regime display characteristic loss curves, starting with a long plateau followed by a drop-off, sometimes followed by one or more additional staircase steps. We find networks of different large $\gamma$ values optimize along similar trajectories up to a reparameterization of time. We further find that optimal online performance is often found at large $\gamma$ and could be missed if this hyperparameter is not tuned. Our findings indicate that analytical study of the large-$\gamma$ limit may yield useful insights into the dynamics of representation learning in performant models.

Self-supervised word embedding algorithms such as word2vec provide a minimal setting for studying representation learning in language modeling. We examine the quartic Taylor approximation of the word2vec loss around the origin, and we show that both the resulting training dynamics and the final performance on downstream tasks are empirically very similar to those of word2vec. Our main contribution is to analytically solve for both the gradient flow training dynamics and the final word embeddings in terms of only the corpus statistics and training hyperparameters. The solutions reveal that these models learn orthogonal linear subspaces one at a time, each one incrementing the effective rank of the embeddings until model capacity is saturated. Training on Wikipedia, we find that each of the top linear subspaces represents an interpretable topic-level concept. Finally, we apply our theory to describe how linear representations of more abstract semantic concepts emerge during training; these can be used to complete analogies via vector addition.

What features neural networks learn, and how, remains an open question. In this paper, we introduce Alternating Gradient Flows (AGF), an algorithmic framework that describes the dynamics of feature learning in two-layer networks trained from small initialization. Prior works have shown that gradient flow in this regime exhibits a staircase-like loss curve, alternating between plateaus where neurons slowly align to useful directions and sharp drops where neurons rapidly grow in norm. AGF approximates this behavior as an alternating two-step process: maximizing a utility function over dormant neurons and minimizing a cost function over active ones. AGF begins with all neurons dormant, corresponding to an initialization at the origin. At each iteration, a dormant neuron activates, triggering the acquisition of a feature and a drop in the loss. AGF quantifies the order, timing, and magnitude of these drops, matching experiments across several commonly studied architectures. We show that AGF unifies and extends existing saddle-to-saddle analyses in fully connected linear networks and attention-only linear transformers, where the learned features are singular modes and principal components, respectively. In diagonal linear networks, we prove AGF converges to gradient flow in the limit of vanishing initialization. Applying AGF to quadratic networks trained to perform modular addition, we give the first complete characterization of the training dynamics, revealing that networks learn Fourier features in decreasing order of coefficient magnitude. Altogether, AGF offers a promising step towards understanding feature learning in neural networks.

In our era of enormous neural networks, empirical progress has been driven by the philosophy that *more is better.* Recent deep learning practice has found repeatedly that larger model size, more data, and more computation (resulting in lower training loss) optimizing to near-interpolation improves performance. In this paper, we give theoretical backing to these empirical observations by showing that these three properties hold in random feature (RF) regression, a class of models equivalent to shallow networks with only the last layer trained. Concretely, we first show that the test risk of RF regression decreases monotonically with both the number of features and samples, provided the ridge penalty is tuned optimally. In particular, this implies that infinite width RF architectures are preferable to those of any finite width. We then proceed to demonstrate that, for a large class of tasks characterized by powerlaw eigenstructure, training to near-zero training loss is *obligatory:* near-optimal performance can *only* be achieved when the training error is much smaller than the test error. Grounding our theory in real-world data, we find empirically that standard computer vision tasks with convolutional neural kernels clearly fall into this class. Taken together, our results tell a simple, testable story of the benefits of overparameterization and overfitting in random feature models.

We study the cost of overfitting in noisy kernel ridge regression (KRR), which we define as the ratio between the test error of the interpolating ridgeless model and the test error of the optimally-tuned model. We take an ``agnostic'' view in the following sense: we consider the cost as a function of sample size for any target function, even if the sample size is not large enough for consistency or the target is outside the RKHS. We analyze the cost of overfitting under a Gaussian universality ansatz using recently derived (non-rigorous) risk estimates in terms of the task eigenstructure. Our analysis provides a more refined characterization of benign, tempered and catastrophic overfitting (cf. Mallinar et al. 2022).

We present a simple picture of the training process of self-supervised learning methods with dual deep networks. In our picture, these methods learn their high-dimensional embeddings one dimension at a time in a sequence of discrete, well-separated steps. We arrive at this picture via the study of a linear toy model of Barlow Twins, applicable to the case in which the trained network is infinitely wide. We solve the training dynamics of our toy model from small initialization, finding that the model learns the top eigenmodes of a certain contrastive kernel in a discrete, stepwise fashion, and find a closed-form expression for the final learned representations. Remarkably, we see the same stepwise learning phenomenon when training deep ResNets using the Barlow Twins, SimCLR, and VICReg losses. This stepwise picture partially demystifies the process of self-supervised training.

The development of methods to guide the design of neural networks is an important open challenge for deep learning theory. As a paradigm for principled neural architecture design, we propose the translation of high-performing kernels, which are better-understood and amenable to first-principles design, into equivalent network architectures, which have superior efficiency, flexibility, and feature learning. To this end, we constructively prove that, with just an appropriate choice of activation function, any positive-semidefinite dot-product kernel can be realized as either the NNGP or neural tangent kernel of a fully-connected neural network with only one hidden layer. We verify our construction numerically and demonstrate its utility as a design tool for finite fully-connected networks in several experiments.

Despite impressive successes, deep reinforcement learning (RL) systems still fall short of human performance on generalization to new tasks and environments that differ from their training. As a benchmark tailored for studying RL generalization, we introduce Avalon, a set of tasks in which embodied agents in highly diverse procedural 3D worlds must survive by navigating terrain, hunting or gathering food, and avoiding hazards. Avalon is unique among existing RL benchmarks in that the reward function, world dynamics, and action space are the same for every task, with tasks differentiated solely by altering the environment; its 20 tasks, ranging in complexity from eat and throw to hunt and navigate, each create worlds in which the agent must perform specific skills in order to survive. This setup enables investigations of generalization within tasks, between tasks, and to compositional tasks that require combining skills learned from previous tasks. Avalon includes a highly efficient simulator, a library of baselines, and a benchmark with scoring metrics evaluated against hundreds of hours of human performance, all of which are open-source and publicly available. We find that standard RL baselines make progress on most tasks but are still far from human performance, suggesting Avalon is challenging enough to advance the quest for generalizable RL.

The practical success of overparameterized neural networks has motivated the recent scientific study of \emph{interpolating methods}-- learning methods which are able fit their training data perfectly. Empirically, certain interpolating methods can fit noisy training data without catastrophically bad test performance, which defies standard intuitions from statistical learning theory. Aiming to explain this, a large body of recent work has studied \emph{benign overfitting}, a behavior seen in certain asymptotic settings under which interpolating methods approach Bayes-optimality, even in the presence of noise. In this work, we argue that, while benign overfitting has been instructive to study, real interpolating methods like deep networks do not fit benignly. That is, noise in the train set leads to suboptimal generalization, suggesting that these methods fall in an intermediate regime between benign and catastrophic overfitting, in which asymptotic risk is neither is neither Bayes-optimal nor unbounded, with the confounding effect of the noise being ``tempered" but non-negligible. We call this behavior \textit{tempered overfitting}. We first provide broad empirical evidence for our three-part taxonomy, demonstrating that deep neural networks and kernel machines fit to noisy data can be reasonably well classified as benign, tempered, or catastrophic. We then specialize to kernel (ridge) regression (KR), obtaining conditions on the ridge parameter and kernel eigenspectrum under which KR exhibits each of the three behaviors, demonstrating the consequences for KR with common kernels and trained neural networks of infinite width using experiments on natural and synthetic datasets.

Despite the fact that the loss functions of deep neural networks are highly nonconvex, gradient-based optimization algorithms converge to approximately the same performance from many random initial points. One thread of work has focused on explaining this phenomenon by numerically characterizing the local curvature near critical points of the loss function, where the gradients are near zero. Such studies have reported that neural network losses enjoy a no-bad-local-minima property, in disagreement with more recent theoretical results. We report here that the methods used to find these putative critical points suffer from a bad local minima problem of their own: they often converge to or pass through regions where the gradient norm has a stationary point. We call these gradient-flat regions, since they arise when the gradient is approximately in the kernel of the Hessian, such that the loss is locally approximately linear, or flat, in the direction of the gradient. We describe how the presence of these regions necessitates care in both interpreting past results that claimed to find critical points of neural network losses and in designing second-order methods for optimizing neural networks.