In this work, we showthat this "lazy training" phenomenon isnot specific tooverparameterized neural networks, and is due to a choice of scaling, often implicit, that makes the model behave as its linearization around the initialization, thus yielding amodel equivalenttolearning withpositive-definite kernels.

The fundamental learning theory behind neural networks remains largely open. What classes of functions can neural networks actually learn?

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The fundamental learning theory behind neural networks remains largely open. What classes of functions can neural networks actually learn? Why doesn't the trained network overfit when it is overparameterized? In this work, we prove that overparameterized neural networks can learn some notable concept classes, including two and three-layer networks with fewer parameters and smooth activations. Moreover, the learning can be simply done by SGD (stochastic gradient descent) or its variants in polynomial time using polynomially many samples. The sample complexity can also be almost independent of the number of parameters in the network. On the technique side, our analysis goes beyond the so-called NTK (neural tangent kernel) linearization of neural networks in prior works. We establish a new notion of quadratic approximation of the neural network, and connect it to the SGD theory of escaping saddle points.

This paper proposes a novel algorithm, Transformative Bayesian Learning (TansBL), which bridges the gap between empirical risk minimization (ERM) and Bayesian learning for neural networks. We compare ERM, which uses gradient descent to optimize, and Bayesian learning with importance sampling for their generalization and computational complexity. We derive the first algorithm-dependent PAC-Bayesian generalization bound for infinitely wide networks based on an exact KL divergence between the trained posterior distribution obtained by infinitesimal step size gradient descent and a Gaussian prior. Moreover, we show how to transform gradient-based optimization into importance sampling by incorporating a weight. While Bayesian learning has better generalization, it suffers from low sampling efficiency. Optimization methods, on the other hand, have good sampling efficiency but poor generalization. Our proposed algorithm TansBL enables a trade-off between generalization and sampling efficiency.

We analytically investigate how over-parameterization of models in randomized machine learning algorithms impacts the information leakage about their training data. Specifically, we prove a privacy bound for the KL divergence between model distributions on worst-case neighboring datasets, and explore its dependence on the initialization, width, and depth of fully connected neural networks. We find that this KL privacy bound is largely determined by the expected squared gradient norm relative to model parameters during training. Notably, for the special setting of linearized network, our analysis indicates that the squared gradient norm (and therefore the escalation of privacy loss) is tied directly to the per-layer variance of the initialization distribution. By using this analysis, we demonstrate that privacy bound improves with increasing depth under certain initializations (LeCun and Xavier), while degrades with increasing depth under other initializations (He and NTK). Our work reveals a complex interplay between privacy and depth that depends on the chosen initialization distribution. We further prove excess empirical risk bounds under a fixed KL privacy budget, and show that the interplay between privacy utility trade-off and depth is similarly affected by the initialization.

The classical theory of reinforcement learning (RL) has focused on tabular and linear representations of value functions.

Overparameterized fully-connected neural networks have been shown to behave like kernel models when trained with gradient descent, under mild conditions on the width, the learning rate, and the parameter initialization. In the limit of infinitely large widths and small learning rate, the kernel that is obtained allows to represent the output of the learned model with a closed-form solution. This closed-form solution hinges on the invertibility of the limiting kernel, a property that often holds on real-world datasets. In this work, we analyze the sensitivity of large ReLU networks to increasing depths by characterizing the corresponding limiting kernel. Our theoretical results demonstrate that the normalized limiting kernel approaches the matrix of ones. In contrast, they show the corresponding closed-form solution approaches a fixed limit on the sphere. We empirically evaluate the order of magnitude in network depth required to observe this convergent behavior, and we describe the essential properties that enable the generalization of our results to other kernels.

The classical theory of reinforcement learning (RL) has focused on tabular and linear representations of value functions.