Can stable and accurate neural networks be computed? -- On the barriers of deep learning and Smale's 18th problem

TL;DR

This paper investigates the fundamental barriers in training stable and accurate neural networks, demonstrating theoretical limitations and proposing FIRENETs as a stable, efficient solution for inverse problems.

Contribution

The paper reveals intrinsic computational barriers in neural network training and introduces FIRENETs, a new stable network architecture with provable efficiency for inverse problems.

Findings

Certain stable NNs cannot be computed by any algorithm with high probability.

Deterministic algorithms can approximate NNs with fewer training samples under specific conditions.

FIRENETs are stable and require logarithmic layers for epsilon-accuracy in inverse problems.

Abstract

Deep learning (DL) has had unprecedented success and is now entering scientific computing with full force. However, current DL methods typically suffer from instability, even when universal approximation properties guarantee the existence of stable neural networks (NNs). We address this paradox by demonstrating basic well-conditioned problems in scientific computing where one can prove the existence of NNs with great approximation qualities, however, there does not exist any algorithm, even randomised, that can train (or compute) such a NN. For any positive integers $K > 2$ and $L$, there are cases where simultaneously: (a) no randomised training algorithm can compute a NN correct to $K$ digits with probability greater than $1/2$, (b) there exists a deterministic training algorithm that computes a NN with $K-1$ correct digits, but any such (even randomised) algorithm needs arbitrarily many training data, (c) there exists a deterministic training algorithm that computes a NN with $K-2$ correct digits using no more than $L$ training samples. These results imply a classification theory describing conditions under which (stable) NNs with a given accuracy can be computed by an algorithm. We begin this theory by establishing sufficient conditions for the existence of algorithms that compute stable NNs in inverse problems. We introduce Fast Iterative REstarted NETworks (FIRENETs), which we both prove and numerically verify are stable. Moreover, we prove that only $\mathcal{O}(|\log(\epsilon)|)$ layers are needed for an $\epsilon$-accurate solution to the inverse problem.