On the Complexity of Learning Neural Networks

TL;DR

This paper proves that certain neural network functions are computationally hard to learn efficiently with current algorithms, even under ideal conditions, highlighting fundamental limitations in understanding neural network training.

Contribution

It establishes a comprehensive lower bound showing that a wide class of neural network functions are hard to learn with statistical query algorithms, including stochastic gradient descent.

Findings

Exponential lower bounds for learning certain neural network functions.

Hard functions can be realized with a small number of hidden units.

Experiments show a phase transition in training error as predicted by the theory.

Abstract

The stunning empirical successes of neural networks currently lack rigorous theoretical explanation. What form would such an explanation take, in the face of existing complexity-theoretic lower bounds? A first step might be to show that data generated by neural networks with a single hidden layer, smooth activation functions and benign input distributions can be learned efficiently. We demonstrate here a comprehensive lower bound ruling out this possibility: for a wide class of activation functions (including all currently used), and inputs drawn from any logconcave distribution, there is a family of one-hidden-layer functions whose output is a sum gate, that are hard to learn in a precise sense: any statistical query algorithm (which includes all known variants of stochastic gradient descent with any loss function) needs an exponential number of queries even using tolerance inversely proportional to the input dimensionality. Moreover, this hard family of functions is realizable with a small (sublinear in dimension) number of activation units in the single hidden layer. The lower bound is also robust to small perturbations of the true weights. Systematic experiments illustrate a phase transition in the training error as predicted by the analysis.