Revisiting Agnostic PAC Learning

• URL: http://arxiv.org/abs/2407.19777v1
• Date: Mon, 29 Jul 2024 08:20:49 GMT
• Title: Revisiting Agnostic PAC Learning
• Authors: Steve Hanneke, Kasper Green Larsen, Nikita Zhivotovskiy,
• Abstract summary: PAC learning, dating back to Valiant'84 and Vapnik and Chervonenkis'64,'74, is a classic model for studying supervised learning. Empirical Risk Minimization (ERM) is a natural learning algorithm, where one simply outputs the hypothesis from $mathcalH$ making the fewest mistakes on the training data. We revisit PAC learning and first show that ERM is in fact sub-optimal if we treat the performance of the best hypothesis, denoted $tau:=Pr_mathcalD[hstar_math
• Score: 30.67561230812141
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: PAC learning, dating back to Valiant'84 and Vapnik and Chervonenkis'64,'74, is a classic model for studying supervised learning. In the agnostic setting, we have access to a hypothesis set $\mathcal{H}$ and a training set of labeled samples $(x_1,y_1),\dots,(x_n,y_n) \in \mathcal{X} \times \{-1,1\}$ drawn i.i.d. from an unknown distribution $\mathcal{D}$. The goal is to produce a classifier $h : \mathcal{X} \to \{-1,1\}$ that is competitive with the hypothesis $h^\star_{\mathcal{D}} \in \mathcal{H}$ having the least probability of mispredicting the label $y$ of a new sample $(x,y)\sim \mathcal{D}$. Empirical Risk Minimization (ERM) is a natural learning algorithm, where one simply outputs the hypothesis from $\mathcal{H}$ making the fewest mistakes on the training data. This simple algorithm is known to have an optimal error in terms of the VC-dimension of $\mathcal{H}$ and the number of samples $n$. In this work, we revisit agnostic PAC learning and first show that ERM is in fact sub-optimal if we treat the performance of the best hypothesis, denoted $\tau:=\Pr_{\mathcal{D}}[h^\star_{\mathcal{D}}(x) \neq y]$, as a parameter. Concretely we show that ERM, and any other proper learning algorithm, is sub-optimal by a $\sqrt{\ln(1/\tau)}$ factor. We then complement this lower bound with the first learning algorithm achieving an optimal error for nearly the full range of $\tau$. Our algorithm introduces several new ideas that we hope may find further applications in learning theory.

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