A General Characterization of the Statistical Query Complexity

TL;DR

This paper introduces a simple statistical dimension to characterize the complexity of solving distribution-based problems with statistical query algorithms, unifying various approaches and addressing open problems in learning theory.

Contribution

It provides the first precise characterization of SQ complexity and query tolerance using a linear-algebraic parameter, extending prior work beyond classification.

Findings

Introduces a statistical dimension capturing SQ complexity.

Characterizes the necessary tolerance of SQ queries.

Applies to open problems in learning theory and constrained algorithms.

Abstract

Statistical query (SQ) algorithms are algorithms that have access to an {\em SQ oracle} for the input distribution $D$ instead of i.i.d.~ samples from $D$. Given a query function $\phi:X \rightarrow [-1,1]$, the oracle returns an estimate of ${\bf E}_{ x\sim D}[\phi(x)]$ within some tolerance $\tau_\phi$ that roughly corresponds to the number of samples. In this work we demonstrate that the complexity of solving general problems over distributions using SQ algorithms can be captured by a relatively simple notion of statistical dimension that we introduce. SQ algorithms capture a broad spectrum of algorithmic approaches used in theory and practice, most notably, convex optimization techniques. Hence our statistical dimension allows to investigate the power of a variety of algorithmic approaches by analyzing a single linear-algebraic parameter. Such characterizations were investigated over the past 20 years in learning theory but prior characterizations are restricted to the much simpler setting of classification problems relative to a fixed distribution on the domain (Blum et al., 1994; Bshouty and Feldman, 2002; Yang, 2001; Simon, 2007; Feldman, 2012; Szorenyi, 2009). Our characterization is also the first to precisely characterize the necessary tolerance of queries. We give applications of our techniques to two open problems in learning theory and to algorithms that are subject to memory and communication constraints.