In this paper, we develop a unified framework for lower bound methods in statistical estimation and interactive decision making. Classical lower bound techniques -- such as Fano's inequality, Le Cam's method, and Assouad's lemma -- have been central to the study of minimax risk in statistical estimation, yet they are insufficient for the analysis of methods that collect data in an interactive manner. The recent minimax lower bounds for interactive decision making via the Decision-Estimation Coefficient (DEC) appear to be genuinely different from the classical methods. We propose a unified view of these distinct methodologies through a general algorithmic lower bound method. We further introduce a novel complexity measure, decision dimension, which facilitates the derivation of new lower bounds for interactive decision making. In particular, decision dimension provides a characterization of bandit learnability for any structured bandit model class. Further, we characterize the sample complexity of learning convex model class up to a polynomial gap with the decision dimension, addressing the remaining gap between upper and lower bounds in Foster et al. (2021, 2023).

翻译：本文提出了一个适用于统计估计与交互式决策中下界方法的统一框架。经典下界技术——如Fano不等式、Le Cam方法与Assouad引理——在统计估计的极小极大风险研究中占据核心地位，然而它们不足以分析以交互方式收集数据的方法。近期通过决策-估计系数（DEC）建立的交互式决策极小极大下界，似乎与经典方法存在本质差异。我们通过一种通用的算法下界方法，为这些不同方法论提供了统一视角。进一步，我们引入了一种新颖的复杂度度量——决策维度，该度量促进了交互式决策中新下界的推导。特别地，决策维度为任意结构化赌博机模型类提供了赌博机可学习性的完整刻画。此外，我们以决策维度为度量（至多相差多项式间隙），刻画了凸模型类学习的样本复杂度，从而解决了Foster等人（2021, 2023）工作中遗留的上下界间隙问题。