Optimal SQ Lower Bounds for Learning Halfspaces with Massart Noise

TL;DR

This paper establishes tight statistical query lower bounds for learning halfspaces with Massart noise, demonstrating the computational difficulty of achieving low error under certain noise conditions, matching existing algorithms.

Contribution

It provides the first tight SQ lower bounds for learning halfspaces with Massart noise, even when the optimal error is exponentially small, confirming the computational limits of such learning tasks.

Findings

SQ algorithms require superpolynomial accuracy or queries for error below η.

Lower bounds hold even when the optimal error is exponentially small.

Achieving error less than 1/2 in the Tsybakov model is SQ-hard.

Abstract

We give tight statistical query (SQ) lower bounds for learnining halfspaces in the presence of Massart noise. In particular, suppose that all labels are corrupted with probability at most $\eta$. We show that for arbitrary $\eta \in [0,1/2]$ every SQ algorithm achieving misclassification error better than $\eta$ requires queries of superpolynomial accuracy or at least a superpolynomial number of queries. Further, this continues to hold even if the information-theoretically optimal error $\mathrm{OPT}$ is as small as $\exp\left(-\log^c(d)\right)$, where $d$ is the dimension and $0 < c < 1$ is an arbitrary absolute constant, and an overwhelming fraction of examples are noiseless. Our lower bound matches known polynomial time algorithms, which are also implementable in the SQ framework. Previously, such lower bounds only ruled out algorithms achieving error $\mathrm{OPT} + \epsilon$ or error better than $\Omega(\eta)$ or, if $\eta$ is close to $1/2$, error $\eta - o_\eta(1)$, where the term $o_\eta(1)$ is constant in $d$ but going to 0 for $\eta$ approaching $1/2$. As a consequence, we also show that achieving misclassification error better than $1/2$ in the $(A,\alpha)$-Tsybakov model is SQ-hard for $A$ constant and $\alpha$ bounded away from 1.