Quantitative CLTs in Deep Neural Networks

We study the distribution of a fully connected neural network with random Gaussian weights and biases in which the hidden layer widths are proportional to a large constant $n$. Under mild assumptions on the non-linearity, we obtain quantitative bounds on normal approximations valid at large but finite $n$ and any fixed network depth. Our theorems show both for the finite-dimensional distributions and the entire process, that the distance between a random fully connected network (and its derivatives) to the corresponding infinite width Gaussian process scales like $n^{-\gamma}$ for $\gamma>0$, with the exponent depending on the metric used to measure discrepancy. Our bounds are strictly stronger in terms of their dependence on network width than any previously available in the literature; in the one-dimensional case, we also prove that they are optimal, i.e., we establish matching lower bounds.

Principles for Initialization and Architecture Selection in Graph Neural Networks with ReLU Activations

This article derives and validates three principles for initialization and architecture selection in finite width graph neural networks (GNNs) with ReLU activations. First, we theoretically derive what is essentially the unique generalization to ReLU GNNs of the well-known He-initialization. Our initialization scheme guarantees that the average scale of network outputs and gradients remains order one at initialization. Second, we prove in finite width vanilla ReLU GNNs that oversmoothing is unavoidable at large depth when using fixed aggregation operator, regardless of initialization. We then prove that using residual aggregation operators, obtained by interpolating a fixed aggregation operator with the identity, provably alleviates oversmoothing at initialization. Finally, we show that the common practice of using residual connections with a fixup-type initialization provably avoids correlation collapse in final layer features at initialization. Through ablation studies we find that using the correct initialization, residual aggregation operators, and residual connections in the forward pass significantly and reliably speeds up early training dynamics in deep ReLU GNNs on a variety of tasks.

Depth Dependence of $μ$P Learning Rates in ReLU MLPs

In this short note we consider random fully connected ReLU networks of width $n$ and depth $L$ equipped with a mean-field weight initialization. Our purpose is to study the dependence on $n$ and $L$ of the maximal update ($\mu$P) learning rate, the largest learning rate for which the mean squared change in pre-activations after one step of gradient descent remains uniformly bounded at large $n,L$. As in prior work on $\mu$P of Yang et. al., we find that this maximal update learning rate is independent of $n$ for all but the first and last layer weights. However, we find that it has a non-trivial dependence of $L$, scaling like $L^{-3/2}.$

Bayesian Interpolation with Deep Linear Networks

This article concerns Bayesian inference using deep linear networks with output dimension one. In the interpolating (zero noise) regime we show that with Gaussian weight priors and MSE negative log-likelihood loss both the predictive posterior and the Bayesian model evidence can be written in closed form in terms of a class of meromorphic special functions called Meijer-G functions. These results are non-asymptotic and hold for any training dataset, network depth, and hidden layer widths, giving exact solutions to Bayesian interpolation using a deep Gaussian process with a Euclidean covariance at each layer. Through novel asymptotic expansions of Meijer-G functions, a rich new picture of the role of depth emerges. Specifically, we find that the posteriors in deep linear networks with data-independent priors are the same as in shallow networks with evidence maximizing data-dependent priors. In this sense, deep linear networks make provably optimal predictions. We also prove that, starting from data-agnostic priors, Bayesian model evidence in wide networks is only maximized at infinite depth. This gives a principled reason to prefer deeper networks (at least in the linear case). Finally, our results show that with data-agnostic priors a novel notion of effective depth given by \[\#\text{hidden layers}\times\frac{\#\text{training data}}{\text{network width}}\] determines the Bayesian posterior in wide linear networks, giving rigorous new scaling laws for generalization error.

Maximal Initial Learning Rates in Deep ReLU Networks

Training a neural network requires choosing a suitable learning rate, involving 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 $\eta^{\ast}$ - 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 $\eta^{\ast}$, we observe that in constant-width fully-connected ReLU networks, $\eta^{\ast}$ demonstrates different behavior to the maximum learning rate later in training. Specifically, we find that $\eta^{\ast}$ is well predicted as a power of $(\text{depth} \times \text{width})$, provided that (i) the width of the network is sufficiently large compared to the depth, and (ii) the input layer of the network is trained at a relatively small learning rate. We further analyze the relationship between $\eta^{\ast}$ and the sharpness $\lambda_{1}$ of the network at initialization, indicating that they are closely though not inversely related. We formally prove bounds for $\lambda_{1}$ in terms of $(\text{depth} \times \text{width})$ that align with our empirical results.

Deep Architecture Connectivity Matters for Its Convergence: A Fine-Grained Analysis

Advanced deep neural networks (DNNs), designed by either human or AutoML algorithms, are growing increasingly complex. Diverse operations are connected by complicated connectivity patterns, e.g., various types of skip connections. Those topological compositions are empirically effective and observed to smooth the loss landscape and facilitate the gradient flow in general. However, it remains elusive to derive any principled understanding of their effects on the DNN capacity or trainability, and to understand why or in which aspect one specific connectivity pattern is better than another. In this work, we theoretically characterize the impact of connectivity patterns on the convergence of DNNs under gradient descent training in fine granularity. By analyzing a wide network's Neural Network Gaussian Process (NNGP), we are able to depict how the spectrum of an NNGP kernel propagates through a particular connectivity pattern, and how that affects the bound of convergence rates. As one practical implication of our results, we show that by a simple filtration on "unpromising" connectivity patterns, we can trim down the number of models to evaluate, and significantly accelerate the large-scale neural architecture search without any overhead. Codes will be released at https://github.com/chenwydj/architecture_convergence.

Correlation Functions in Random Fully Connected Neural Networks at Finite Width

This article considers fully connected neural networks with Gaussian random weights and biases and $L$ hidden layers, each of width proportional to a large parameter $n$. For polynomially bounded non-linearities we give sharp estimates in powers of $1/n$ for the joint correlation functions of the network output and its derivatives. Moreover, we obtain exact layerwise recursions for these correlation functions and solve a number of special cases for classes of non-linearities including $\mathrm{ReLU}$ and $\tanh$. We find in both cases that the depth-to-width ratio $L/n$ plays the role of an effective network depth, controlling both the scale of fluctuations at individual neurons and the size of inter-neuron correlations. We use this to study a somewhat simplified version of the so-called exploding and vanishing gradient problem, proving that this particular variant occurs if and only if $L/n$ is large. Several of the key ideas in this article were first developed at a physics level of rigor in a recent monograph with Roberts and Yaida.

Ridgeless Interpolation with Shallow ReLU Networks in $1D$ is Nearest Neighbor Curvature Extrapolation and Provably Generalizes on Lipschitz Functions

We prove a precise geometric description of all one layer ReLU networks $z(x;\theta)$ with a single linear unit and input/output dimensions equal to one that interpolate a given dataset $\mathcal D=\{(x_i,f(x_i))\}$ and, among all such interpolants, minimize the $\ell_2$-norm of the neuron weights. Such networks can intuitively be thought of as those that minimize the mean-squared error over $\mathcal D$ plus an infinitesimal weight decay penalty. We therefore refer to them as ridgeless ReLU interpolants. Our description proves that, to extrapolate values $z(x;\theta)$ for inputs $x\in (x_i,x_{i+1})$ lying between two consecutive datapoints, a ridgeless ReLU interpolant simply compares the signs of the discrete estimates for the curvature of $f$ at $x_i$ and $x_{i+1}$ derived from the dataset $\mathcal D$. If the curvature estimates at $x_i$ and $x_{i+1}$ have different signs, then $z(x;\theta)$ must be linear on $(x_i,x_{i+1})$. If in contrast the curvature estimates at $x_i$ and $x_{i+1}$ are both positive (resp. negative), then $z(x;\theta)$ is convex (resp. concave) on $(x_i,x_{i+1})$. Our results show that ridgeless ReLU interpolants achieve the best possible generalization for learning $1d$ Lipschitz functions, up to universal constants.

Random Neural Networks in the Infinite Width Limit as Gaussian Processes

This article gives a new proof that fully connected neural networks with random weights and biases converge to Gaussian processes in the regime where the input dimension, output dimension, and depth are kept fixed, while the hidden layer widths tend to infinity. Unlike prior work, convergence is shown assuming only moment conditions for the distribution of weights and for quite general non-linearities.

The Principles of Deep Learning Theory

This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.