### Session

## Deep Learning Theory 2

Moderator: Rene Vidal

**On Energy-Based Models with Overparametrized Shallow Neural Networks**

Carles Domingo-Enrich · Alberto Bietti · Eric Vanden-Eijnden · Joan Bruna

Energy-based models (EBMs) are a simple yet powerful framework for generative modeling. They are based on a trainable energy function which defines an associated Gibbs measure, and they can be trained and sampled from via well-established statistical tools, such as MCMC. Neural networks may be used as energy function approximators, providing both a rich class of expressive models as well as a flexible device to incorporate data structure. In this work we focus on shallow neural networks. Building from the incipient theory of overparametrized neural networks, we show that models trained in the so-called 'active' regime provide a statistical advantage over their associated 'lazy' or kernel regime, leading to improved adaptivity to hidden low-dimensional structure in the data distribution, as already observed in supervised learning. Our study covers both the maximum likelihood and Stein Discrepancy estimators, and we validate our theoretical results with numerical experiments on synthetic data.

**Uncertainty Principles of Encoding GANs**

Ruili Feng · Zhouchen Lin · Jiapeng Zhu · Deli Zhao · Jingren Zhou · Zheng-Jun Zha

The compelling synthesis results of Generative Adversarial Networks (GANs) demonstrate rich semantic knowledge in their latent codes. To obtain this knowledge for downstream applications, encoding GANs has been proposed to learn encoders, such that real world data can be encoded to latent codes, which can be fed to generators to reconstruct those data.
However, despite the theoretical guarantees of precise reconstruction in previous works, current algorithms generally reconstruct inputs with non-negligible deviations from inputs. In this paper we study this predicament of encoding GANs, which is indispensable research for the GAN community. We prove three uncertainty principles of encoding GANs in practice: a) the `perfect' encoder and generator cannot be continuous at the same time, which implies that current framework of encoding GANs is ill-posed and needs rethinking; b) neural networks cannot approximate the underlying encoder and generator precisely at the same time, which explains why we cannot get`

perfect' encoders and generators as promised in previous theories; c) neural networks cannot be stable and accurate at the same time, which demonstrates the difficulty of training and trade-off between fidelity and disentanglement encountered in previous works. Our work may eliminate gaps between previous theories and empirical results, promote the understanding of GANs, and guide network designs for follow-up works.

**On the Proof of Global Convergence of Gradient Descent for Deep ReLU Networks with Linear Widths**

Quynh Nguyen

We give a simple proof for the global convergence of gradient descent in training deep ReLU networks with the standard square loss, and show some of its improvements over the state-of-the-art. In particular, while prior works require all the hidden layers to be wide with width at least $\Omega(N^8)$ ($N$ being the number of training samples), we require a single wide layer of linear, quadratic or cubic width depending on the type of initialization. Unlike many recent proofs based on the Neural Tangent Kernel (NTK), our proof need not track the evolution of the entire NTK matrix, or more generally, any quantities related to the changes of activation patterns during training. Instead, we only need to track the evolution of the output at the last hidden layer, which can be done much more easily thanks to the Lipschitz property of ReLU. Some highlights of our setting: (i) all the layers are trained with standard gradient descent, (ii) the network has standard parameterization as opposed to the NTK one, and (iii) the network has a single wide layer as opposed to having all wide hidden layers as in most of NTK-related results.

**Tight Bounds on the Smallest Eigenvalue of the Neural Tangent Kernel for Deep ReLU Networks**

Quynh Nguyen · Marco Mondelli · Guido Montufar

A recent line of work has analyzed the theoretical properties of deep neural networks via the Neural Tangent Kernel (NTK). In particular, the smallest eigenvalue of the NTK has been related to the memorization capacity, the global convergence of gradient descent algorithms and the generalization of deep nets. However, existing results either provide bounds in the two-layer setting or assume that the spectrum of the NTK matrices is bounded away from 0 for multi-layer networks. In this paper, we provide tight bounds on the smallest eigenvalue of NTK matrices for deep ReLU nets, both in the limiting case of infinite widths and for finite widths. In the finite-width setting, the network architectures we consider are fairly general: we require the existence of a wide layer with roughly order of $N$ neurons, $N$ being the number of data samples; and the scaling of the remaining layer widths is arbitrary (up to logarithmic factors). To obtain our results, we analyze various quantities of independent interest: we give lower bounds on the smallest singular value of hidden feature matrices, and upper bounds on the Lipschitz constant of input-output feature maps.

**Functional Space Analysis of Local GAN Convergence**

Valentin Khrulkov · Artem Babenko · Ivan Oseledets

Recent work demonstrated the benefits of studying continuous-time dynamics governing the GAN training. However, this dynamics is analyzed in the model parameter space, which results in finite-dimensional dynamical systems. We propose a novel perspective where we study the local dynamics of adversarial training in the general functional space and show how it can be represented as a system of partial differential equations. Thus, the convergence properties can be inferred from the eigenvalues of the resulting differential operator. We show that these eigenvalues can be efficiently estimated from the target dataset before training. Our perspective reveals several insights on the practical tricks commonly used to stabilize GANs, such as gradient penalty, data augmentation, and advanced integration schemes. As an immediate practical benefit, we demonstrate how one can a priori select an optimal data augmentation strategy for a particular generation task.

**Exact Gap between Generalization Error and Uniform Convergence in Random Feature Models**

Zitong Yang · Yu Bai · Song Mei

Recent work showed that there could be a large gap between the classical uniform convergence bound and the actual test error of zero-training-error predictors (interpolators) such as deep neural networks. To better understand this gap, we study the uniform convergence in the nonlinear random feature model and perform a precise theoretical analysis on how uniform convergence depends on the sample size and the number of parameters. We derive and prove analytical expressions for three quantities in this model: 1) classical uniform convergence over norm balls, 2) uniform convergence over interpolators in the norm ball (recently proposed by~\citet{zhou2021uniform}), and 3) the risk of minimum norm interpolator. We show that, in the setting where the classical uniform convergence bound is vacuous (diverges to $\infty$), uniform convergence over the interpolators still gives a non-trivial bound of the test error of interpolating solutions. We also showcase a different setting where classical uniform convergence bound is non-vacuous, but uniform convergence over interpolators can give an improved sample complexity guarantee. Our result provides a first exact comparison between the test errors and uniform convergence bounds for interpolators beyond simple linear models.

**Provable Generalization of SGD-trained Neural Networks of Any Width in the Presence of Adversarial Label Noise**

Spencer Frei · Yuan Cao · Quanquan Gu

We consider a one-hidden-layer leaky ReLU network of arbitrary width trained by stochastic gradient descent (SGD) following an arbitrary initialization. We prove that SGD produces neural networks that have classification accuracy competitive with that of the best halfspace over the distribution for a broad class of distributions that includes log-concave isotropic and hard margin distributions. Equivalently, such networks can generalize when the data distribution is linearly separable but corrupted with adversarial label noise, despite the capacity to overfit. To the best of our knowledge, this is the first work to show that overparameterized neural networks trained by SGD can generalize when the data is corrupted with adversarial label noise.