Learning (Very) Simple Generative Models Is Hard

Sitan Chen, Jerry Li, Yuanzhi Li

Keywords: generative models, statistical query, distribution learning, computational hardness, computational-statistical tradeoffs

TL;DR: We prove the first computational hardness result for learning pushforwards of Gaussians under one hidden layer ReLU networks of logarithmic size.

Abstract:

Motivated by the recent empirical successes of deep generative models, we study the computational complexity of the following unsupervised learning problem. For an unknown neural network $F:\mathbb{R}^d\to\mathbb{R}^{d'}$, let $D$ be the distribution over $\mathbb{R}^{d'}$ given by pushing the standard Gaussian $\mathcal{N}(0,\textrm{Id}_d)$ through $F$. Given i.i.d. samples from $D$, the goal is to output *any* distribution close to $D$ in statistical distance.

We show under the statistical query (SQ) model that no polynomial-time algorithm can solve this problem even when the output coordinates of $F$ are one-hidden-layer ReLU networks with $\log(d)$ neurons. Previously, the best lower bounds for this problem simply followed from lower bounds for *supervised learning* and required at least two hidden layers and $\textrm{poly}(d)$ neurons [Daniely-Vardi '21, Chen-Gollakota-Klivans-Meka '22].

The key ingredient in our proof is an ODE-based construction of a compactly supported, piecewise-linear function $f$ with polynomially-bounded slopes such that the pushforward of $\mathcal{N}(0,1)$ under $f$ matches all low-degree moments of $\mathcal{N}(0,1)$.