Optimal SQ Lower Bounds for Learning Halfspaces with Massart Noise
This work addresses a fundamental limitation in robust machine learning for researchers, providing tight hardness results that match known algorithms and extend to related noise models, though it is incremental in refining prior lower bounds.
The paper tackles the problem of learning halfspaces with Massart noise by establishing tight statistical query (SQ) lower bounds, showing that achieving misclassification error better than the noise rate η requires superpolynomial queries or accuracy, even when the optimal error is extremely small and most examples are noiseless.
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 $η$. We show that for arbitrary $η\in [0,1/2]$ every SQ algorithm achieving misclassification error better than $η$ 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 $\mathrm{OPT} + ε$ or error better than $Ω(η)$ or, if $η$ is close to $1/2$, error $η- o_η(1)$, where the term $o_η(1)$ is constant in $d$ but going to 0 for $η$ approaching $1/2$. As a consequence, we also show that achieving misclassification error better than $1/2$ in the $(A,α)$-Tsybakov model is SQ-hard for $A$ constant and $α$ bounded away from 1.