誤指定回帰に対する符号ベッティング：分散適応性と最適近似
Sign Betting for Misspecified Regression: Variance Adaptivity and Optimal Approximation
モデル誤指定下の有界回帰において、L1距離に基づくmin-max推定量「符号ベッティング回帰」を解析し、分散適応的なオラクル不等式と最適な近似係数3を導出した研究。
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著者: Yinan Li, Han Han, Kwang-Sung Jun
分類: cs.LG, stat.ML
原文アブストラクト
We study bounded regression under model misspecification, evaluating a function $f$ by its $L_1(P_X)$ distance to the conditional mean $η(x)=\mathbb{E}[Y\mid X=x]$, rather than the standard $L_2(P_X)$. This $L_1$ metric is natural for downstream decision problems such as bandits and reinforcement learning. We analyze the min--max estimator of Baraud & Maillard (2025), which we call \emph{sign-betting regression}. Our main results show that sign-betting regression achieves high-probability $L_1$ oracle inequalities under misspecification. Under realizability, our bound adapts to the average conditional variance and recovers the variance-adaptive rate of Li et al. (2026). We further characterize the optimal universal approximation factor for proper learning by showing that sign betting achieves factor 3, while no proper learner can satisfy a distribution-free, high-probability $L_1$ oracle inequality with a universal approximation factor strictly below 3. We extend the upper-bound analysis beyond finite classes and complement the theory with experiments in synthetic and real-world model-selection tasks.