On Agnostic PAC Learning in the Small Error Regime

Binary classification in the classic PAC model exhibits a curious phenomenon: Empirical Risk Minimization (ERM) learners are suboptimal in the realizable case yet optimal in the agnostic case. Roughly speaking, this owes itself to the fact that non-realizable distributions $\mathcal{D}$ are simply more difficult to learn than realizable distributions -- even when one discounts a learner's error by $\mathrm{err}(h^*_{\mathcal{D}})$, the error of the best hypothesis in $\mathcal{H}$ for $\mathcal{D}$. Thus, optimal agnostic learners are permitted to incur excess error on (easier-to-learn) distributions $\mathcal{D}$ for which $τ= \mathrm{err}(h^*_{\mathcal{D}})$ is small. Recent work of Hanneke, Larsen, and Zhivotovskiy (FOCS `24) addresses this shortcoming by including $τ$ itself as a parameter in the agnostic error term. In this more fine-grained model, they demonstrate tightness of the error lower bound $τ+ Ω\left(\sqrt{\frac{τ(d + \log(1 / δ))}{m}} + \frac{d + \log(1 / δ)}{m} \right)$ in a regime where $τ> d/m$, and leave open the question of whether there may be a higher lower bound when $τ\approx d/m$, with $d$ denoting $\mathrm{VC}(\mathcal{H})$. In this work, we resolve this question by exhibiting a learner which achieves error $c \cdot τ+ O \left(\sqrt{\frac{τ(d + \log(1 / δ))}{m}} + \frac{d + \log(1 / δ)}{m} \right)$ for a constant $c \leq 2.1$, thus matching the lower bound when $τ\approx d/m$. Further, our learner is computationally efficient and is based upon careful aggregations of ERM classifiers, making progress on two other questions of Hanneke, Larsen, and Zhivotovskiy (FOCS `24). We leave open the interesting question of whether our approach can be refined to lower the constant from 2.1 to 1, which would completely settle the complexity of agnostic learning.