Tight Hardness Results for Training Depth-2 ReLU Networks

TL;DR

This paper establishes strong computational hardness results for training depth-2 ReLU neural networks, showing NP-hardness even for simple cases and deriving lower bounds on training time based on complexity hypotheses.

Contribution

It proves NP-hardness for training depth-2 ReLU networks with a single ReLU and provides lower bounds on training time under complexity assumptions, highlighting fundamental computational limits.

Findings

Training depth-2 ReLU networks is NP-hard even for a single ReLU.

Proper learning algorithms require exponential time in 1/ε² under certain hypotheses.

Upper bounds on training time match lower bounds in terms of ε dependency.

Abstract

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 network that minimizes the square loss with respect to a given training set. We prove that this problem is NP-hard already for a network with a single ReLU. We also prove NP-hardness for outputting a weighted sum of $k$ ReLUs minimizing the squared error (for $k>1$) even in the realizable setting (i.e., when the labels are consistent with an unknown depth-2 ReLU network). We are also able to obtain lower bounds on the running time in terms of the desired additive error $\epsilon$. To obtain our lower bounds, we use the Gap Exponential Time Hypothesis (Gap-ETH) as well as a new hypothesis regarding the hardness of approximating the well known Densest $\kappa$-Subgraph problem in subexponential time (these hypotheses are used separately in proving different lower bounds). For example, we prove that under reasonable hardness assumptions, any proper learning algorithm for finding the best fitting ReLU must run in time exponential in $1/\epsilon^2$. Together with a previous work regarding improperly learning a ReLU (Goel et al., COLT'17), this implies the first separation between proper and improper algorithms for learning a ReLU. We also study the problem of properly learning a depth-2 network of ReLUs with bounded weights giving new (worst-case) upper bounds on the running time needed to learn such networks both in the realizable and agnostic settings. Our upper bounds on the running time essentially matches our lower bounds in terms of the dependency on $\epsilon$.