Small Covers for Near-Zero Sets of Polynomials and Learning Latent Variable Models

TL;DR

This paper develops optimal covers for near-zero polynomial sets and leverages these to create quasi-polynomial algorithms for learning complex high-dimensional models with hidden variables.

Contribution

It introduces a new structural bound on polynomial near-zero sets and applies it to improve learning algorithms for various probabilistic models.

Findings

Established an optimal upper bound on epsilon-covers for near-zero polynomial sets.

Developed a constructive algorithm to compute epsilon-covers efficiently.

Designed quasi-polynomial time algorithms for learning high-dimensional models with hidden variables.

Abstract

Let $V$ be any vector space of multivariate degree-$d$ homogeneous polynomials with co-dimension at most $k$, and $S$ be the set of points where all polynomials in $V$ {\em nearly} vanish. We establish a qualitatively optimal upper bound on the size of $\epsilon$-covers for $S$, in the $\ell_2$-norm. Roughly speaking, we show that there exists an $\epsilon$-cover for $S$ of cardinality $M = (k/\epsilon)^{O_d(k^{1/d})}$. Our result is constructive yielding an algorithm to compute such an $\epsilon$-cover that runs in time $\mathrm{poly}(M)$. Building on our structural result, we obtain significantly improved learning algorithms for several fundamental high-dimensional probabilistic models with hidden variables. These include density and parameter estimation for $k$-mixtures of spherical Gaussians (with known common covariance), PAC learning one-hidden-layer ReLU networks with $k$ hidden units (under the Gaussian distribution), density and parameter estimation for $k$-mixtures of linear regressions (with Gaussian covariates), and parameter estimation for $k$-mixtures of hyperplanes. Our algorithms run in time {\em quasi-polynomial} in the parameter $k$. Previous algorithms for these problems had running times exponential in $k^{\Omega(1)}$. At a high-level our algorithms for all these learning problems work as follows: By computing the low-degree moments of the hidden parameters, we are able to find a vector space of polynomials that nearly vanish on the unknown parameters. Our structural result allows us to compute a quasi-polynomial sized cover for the set of hidden parameters, which we exploit in our learning algorithms.