15 accepted papers
In overparameterized logistic regression, gradient descent (GD) iterates diverge in norm while converging in direction to the maximum $\ell_2$-margin solution---a phenomenon known as the implicit bias of GD. This work investigates additional regularization effects induced by early stopping in well-s…
We show the effectiveness of automatic differentiation in efficiently and correctly computing and controlling the spectrum of implicitly linear operators, a rich family of layer types including all standard convolutional and dense layers. We provide the first clipping method which is correct for gen…
We show that a constant number of self-attention layers can efficiently simulate—and be simulated by—a constant number of communication rounds of *Massively Parallel Computation*. As a consequence, we show that logarithmic-depth is sufficient for transformers to solve basic computational tasks that…
This work establishes low test error of gradient flow (GF) and stochastic gradient descent (SGD) on two-layer ReLU networks with standard initialization scale, in three regimes where key sets of weights rotate little (either naturally due to GF and SGD, or due to an artificial constraint), and makin…
We first elucidate various fundamental properties of optimal adversarial predictors: the structure of optimal adversarial convex predictors in terms of optimal adversarial zero-one predictors, bounds relating the adversarial convex loss to the adversarial zero-one loss, and the fact that continuous…
Attention layers, as commonly used in transformers, form the backbone of modern deep learning, yet there is no mathematical description of their benefits and deficiencies as compared with other architectures. In this work we establish both positive and negative results on the representation power of…
We show that the simplest actor-critic method — a linear softmax policy updated with TD through interaction with a linear MDP, but featuring no explicit regularization or exploration — does not merely find an optimal policy, but moreover prefers high entropy optimal policies. To demonstrate the stre…
This work studies the behavior of shallow ReLU networks trained with the logistic loss via gradient descent on binary classification data where the underlying data distribution is general, and the (optimal) Bayes risk is not necessarily zero. In this setting, it is shown that gradient descent with…
We present and analyze a momentum-based gradient method for training linear classifiers with an exponentially-tailed loss (e.g., the exponential or logistic loss), which maximizes the classification margin on separable data at a rate of O(1/t^2). This contrasts with a rate of O(1/log(t)) for standar…
This paper theoretically investigates the following empirical phenomenon: given a high-complexity network with poor generalization bounds, one can distill it into a network with nearly identical predictions but low complexity and vastly smaller generalization bounds. The main contribution is an ana…
This paper establishes rates of universal approximation for the shallow neural tangent kernel (NTK): network weights are only allowed microscopic changes from random initialization, which entails that activations are mostly unchanged, and the network is nearly equivalent to its linearization. Concre…
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 of the sample size $n$, the (inverse) target error $1/\epsil…
This paper provides a simple procedure to fit generative networks to target distributions, with the goal of a small Wasserstein distance (or other optimal transport costs). The approach is based on two principles: (a) if the source randomness of the network is a continuous distribution (the "semi-di…
This paper establishes risk convergence and asymptotic weight matrix alignment --- a form of implicit regularization --- of gradient flow and gradient descent when applied to deep linear networks on linearly separable data. In more detail, for gradient flow applied to strictly decreasing loss functi…
Neural networks and rational functions efficiently approximate each other. In more detail, it is shown here that for any ReLU network, there exists a rational function of degree $O(polylog(1/\epsilon))$ which is $\epsilon$-close, and similarly for any rational function there exists a ReLU network of…