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  1. Training a neural network requires choosing a suitable learning rate, which involves a trade-off between speed and effectiveness of convergence. While there has been considerable theoretical and empirical analysis of how large the learning rate can be, most prior work focuses only on late-stage training. In this work, we introduce the maximal initial learning rate - the largest learning rate at which a randomly initialized neural network can successfully begin training and achieve (at least) a given threshold accuracy. Using a simple approach to estimate the maximal initial learning rate, we observe that in constant-width fully-connected ReLU networks, the maximal initial learning rate behaves differently from the maximum learning rate later in training. Specifically, we find that the maximal initial learning rate is well predicted as a power of depth times width, provided that (i) the width of the network is sufficiently large compared to the depth, and (ii) the input layer is trained at a relatively small learning rate. We further analyze the relationship between the maximal initial learning rate and the sharpness of the network at initialization, indicating they are closely though not inversely related. We formally prove bounds for the maximal initial learning rate in terms of depth times width that align with our empirical results. 
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  2. Characterizing how neural network depth, width, and dataset size jointly impact model quality is a central problem in deep learning theory. We give here a complete solution in the special case of linear networks with output dimension one trained using zero noise Bayesian inference with Gaussian weight priors and mean squared error as a negative log-likelihood. For any training dataset, network depth, and hidden layer widths, we find non-asymptotic expressions for the predictive posterior and Bayesian model evidence in terms of Meijer-G functions, a class of meromorphic special functions of a single complex variable. Through novel asymptotic expansions of these Meijer-G functions, a rich new picture of the joint role of depth, width, and dataset size emerges. We show that linear networks make provably optimal predictions at infinite depth: the posterior of infinitely deep linear networks with data-agnostic priors is the same as that of shallow networks with evidence-maximizing data-dependent priors. This yields a principled reason to prefer deeper networks when priors are forced to be data-agnostic. Moreover, we show that with data-agnostic priors, Bayesian model evidence in wide linear networks is maximized at infinite depth, elucidating the salutary role of increased depth for model selection. Underpinning our results is a novel emergent notion of effective depth, given by the number of hidden layers times the number of data points divided by the network width; this determines the structure of the posterior in the large-data limit. 
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    Free, publicly-accessible full text available June 6, 2024
  3. Though data augmentation has rapidly emerged as a key tool for optimization in modern machine learning, a clear picture of how augmentation schedules affect optimization and interact with optimization hyperparameters such as learning rate is nascent. In the spirit of classical convex optimization and recent work on implicit bias, the present work analyzes the effect of augmentation on optimization in the simple convex setting of linear regression with MSE loss.We find joint schedules for learning rate and data augmentation scheme under which augmented gradient descent provably converges and characterize the resulting minimum. Our results apply to arbitrary augmentation schemes, revealing complex interactions between learning rates and augmentations even in the convex setting. Our approach interprets augmented (S)GD as a stochastic optimization method for a time-varying sequence of proxy losses. This gives a unified way to analyze learning rate, batch size, and augmentations ranging from additive noise to random projections. From this perspective, our results, which also give rates of convergence, can be viewed as Monro-Robbins type conditions for augmented (S)GD. 
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