Related papers: Complexity, Statistical Risk, and Metric Entropy o…

We prove several hardness results for training depth-2 neural networks with the ReLU activation function; these networks are simply weighted sums (that may include negative coefficients) of ReLUs. Our goal is to output a depth-2 neural…

Generalised linear models for multi-class classification problems are one of the fundamental building blocks of modern machine learning tasks. In this manuscript, we characterise the learning of a mixture of $K$ Gaussians with generic means…

This study explores the number of neurons required for a Rectified Linear Unit (ReLU) neural network to approximate multivariate monomials. We establish an exponential lower bound on the complexity of any shallow network approximating the…

Weight initialization governs signal propagation and gradient flow at the start of training. This paper offers a theory-grounded and empirically validated study across two regimes: compact ReLU multilayer perceptrons and GPT-2-style…

We investigate the sample complexity of networks with bounds on the magnitude of its weights. In particular, we consider the class \[ H=\left\{W_t\circ\rho\circ \ldots\circ\rho\circ W_{1} :W_1,\ldots,W_{t-1}\in M_{d, d}, W_t\in…

Quantifying the relations (e.g., similarity) between complex networks paves the way for studying the latent information shared across networks. However, fundamental relation metrics are not well-defined between networks. As a compromise,…

We obtained convergence rates of the collocation approximation by deep ReLU neural networks of solutions to elliptic PDEs with lognormal inputs, parametrized by $\boldsymbol{y}$ from the non-compact set $\mathbb{R}^\infty$. The…

Recent theoretical work has guaranteed that overparameterized networks trained by gradient descent achieve arbitrarily low training error, and sometimes even low test error. The required width, however, is always polynomial in at least one…

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…

Whereas recovery of the manifold from data is a well-studied topic, approximation rates for functions defined on manifolds are less known. In this work, we study a regression problem with inputs on a $d^*$-dimensional manifold that is…

Algorithm- and data-dependent generalization bounds are required to explain the generalization behavior of modern machine learning algorithms. In this context, there exists information theoretic generalization bounds that involve (various…

Over-parameterized neural networks generalize well in practice without any explicit regularization. Although it has not been proven yet, empirical evidence suggests that implicit regularization plays a crucial role in deep learning and…

The generalization mystery of overparametrized deep nets has motivated efforts to understand how gradient descent (GD) converges to low-loss solutions that generalize well. Real-life neural networks are initialized from small random values…

We study products of random matrices in the regime where the number of terms and the size of the matrices simultaneously tend to infinity. Our main theorem is that the logarithm of the $\ell_2$ norm of such a product applied to any fixed…

This paper presents a general framework for norm-based capacity control for $L_{p,q}$ weight normalized deep neural networks. We establish the upper bound on the Rademacher complexities of this family. With an $L_{p,q}$ normalization where…

This paper presents a new mathematical framework to analyze the loss functions of deep neural networks with ReLU functions. Furthermore, as as application of this theory, we prove that the loss functions can reconstruct the inputs of the…

Existing Rademacher complexity bounds for neural networks rely only on norm control of the weight matrices and depend exponentially on depth via a product of the matrix norms. Lower bounds show that this exponential dependence on depth is…

It is well-known that modern neural networks are vulnerable to adversarial examples. To mitigate this problem, a series of robust learning algorithms have been proposed. However, although the robust training error can be near zero via some…

This paper establishes the (nearly) optimal approximation error characterization of deep rectified linear unit (ReLU) networks for smooth functions in terms of both width and depth simultaneously. To that end, we first prove that…

This theoretical paper is devoted to developing a rigorous theory for demystifying the global convergence phenomenon in a challenging scenario: learning over-parameterized Rectified Linear Unit (ReLU) nets for very high dimensional dataset…