Neural networks with a large number of parameters admit a mean-field description, which has recently served as a theoretical explanation for the favorable training properties of "overparameterized" models. In this regime, gradient descent obeys a deterministic partial differential equation (PDE) that converges to a globally optimal solution for networks with a single hidden layer under appropriate assumptions. In this work, we propose a non-local mass transport dynamics that leads to a modified PDE with the same minimizer. We implement this non-local dynamics as a stochastic neuronal birth-death process and we prove that it accelerates the rate of convergence in the mean-field limit. We subsequently realize this PDE with two classes of numerical schemes that converge to the mean-field equation, each of which can easily be implemented for neural networks with finite numbers of parameters. We illustrate our algorithms with two models to provide intuition for the mechanism through which convergence is accelerated.
Designing an incentive compatible auction that maximizes expected revenue is a central problem in Auction Design. Theoretical approaches to the problem have hit some limits in the past decades and analytical solutions are known for only a few simple settings. Computational approaches to the problem through the use of LPs have their own set of limitations. Building on the success of deep learning, a new approach was recently proposed by Dütting et al. (2017) in which the auction is modeled by a feed-forward neural network and the design problem is framed as a learning problem. The neural architectures used in that work are general purpose and do not take advantage of any of the symmetries the problem could present, such as permutation equivariance. In this work, we consider auction design problems that have permutation-equivariant symmetry and construct a neural architecture that is capable of perfectly recovering the permutationequivariant optimal mechanism, which we show is not possible with the previous architecture. We demonstrate that permutation-equivariant architectures are not only capable of recovering previous results, they also have better generalization properties.
Designing an incentive compatible auction that maximizes expected revenue is a central problem in Auction Design. While theoretical approaches to the problem have hit some limits, a recent research direction initiated by Duetting et al. (2019) consists in building neural network architectures to find optimal auctions. We propose two conceptual deviations from their approach which result in enhanced performance. First, we use recent results in theoretical auction design (Rubinstein and Weinberg, 2018) to introduce a time-independent Lagrangian. This not only circumvents the need for an expensive hyper-parameter search (as in prior work), but also provides a principled metric to compare the performance of two auctions (absent from prior work). Second,the optimization procedure in previous work uses an inner maximization loop to compute optimal misreports. We amortize this process through the introduction of an additional neural network. We demonstrate the effectiveness of our approach by learning competitive or strictly improved auctions compared to prior work. Both results together further imply a novel formulation of Auction Design as a two-player game with stationary utility functions.
We consider semidefinite programs (SDPs) of size n with equality constraints. In order to overcome scalability issues, Burer and Monteiro proposed a factorized approach based on optimizing over a matrix Y of size n × k such that X = Y Y * is the SDP variable. The advantages of such formulation are twofold: the dimension of the optimization variable is reduced, and positive semidefiniteness is naturally enforced. However, optimization in Y is non-convex. In prior work, it has been shown that, when the constraints on the factorized variable regularly define a smooth manifold, provided k is large enough, for almost all cost matrices, all second-order stationary points (SOSPs) are optimal. Importantly, in practice, one can only compute points which approximately satisfy necessary optimality conditions, leading to the question: are such points also approximately optimal? To answer it, under similar assumptions, we use smoothed analysis to show that approximate SOSPs for a randomly perturbed objective function are approximate global optima, with k scaling like the square root of the number of constraints (up to log factors). Moreover, we bound the optimality gap at the approximate solution of the perturbed problem with respect to the original problem. We particularize our results to an SDP relaxation of phase retrieval.
High-dimensional depth separation results for neural networks show that certain functions can be efficiently approximated by two-hidden-layer networks but not by one-hidden-layer ones in high-dimensions d. Existing results of this type mainly focus on functions with an underlying radial or one-dimensional structure, which are usually not encountered in practice. The first contribution of this paper is to extend such results to a more general class of functions, namely functions with piece-wise oscillatory structure, by building on the proof strategy of (Eldan and Shamir, 2016). We complement these results by showing that, if the domain radius and the rate of oscillation of the objective function are constant, then approximation by one-hiddenlayer networks holds at a poly(d) rate for any fixed error threshold.A common theme in the proofs of depth-separation results is the fact that one-hidden-layer networks fail to approximate high-energy functions whose Fourier representation is spread in the domain. On the other hand, existing approximation results of a function by one-hidden-layer neural networks rely on the function having a sparse Fourier representation. The choice of the domain also represents a source of gaps between upper and lower approximation bounds. Focusing on a fixed approximation domain, namely the sphere S d−1 in dimension d, we provide a characterization of both functions which are efficiently approximable by one-hidden-layer networks and of functions which are provably not, in terms of their Fourier expansion.
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