We develop a unifying framework for information-theoretic lower bound in statistical estimation and interactive decision making. Classical lower bound techniques---such as Fano's method, Le Cam's method, and Assouad's lemma---are central to the study of minimax risk in statistical estimation, yet are insufficient to provide tight lower bounds for \emph{interactive decision making} algorithms that collect data interactively (e.g., algorithms for bandits and reinforcement learning). Recent work of Foster et al. provides minimax lower bounds for interactive decision making using seemingly different analysis techniques from the classical methods. These results---which are proven using a complexity measure known as the \emph{Decision-Estimation Coefficient} (DEC)---capture difficulties unique to interactive learning, yet do not recover the tightest known lower bounds for passive estimation. We propose a unified view of these distinct methodologies through a new lower bound approach called \emph{interactive Fano method}. As an application, we introduce a novel complexity measure, the \emph{Fractional Covering Number}, which facilitates the new lower bounds for interactive decision making that extend the DEC methodology by incorporating the complexity of estimation. Using the fractional covering number, we (i) provide a unified characterization of learnability for \emph{any} stochastic bandit problem, (ii) close the remaining gap between the upper and lower bounds in Foster et al. (up to polynomial factors) for any interactive decision making problem in which the underlying model class is convex.

We develop a unifying framework for information-theoretic lower bound in statistical estimation and interactive decision making. Classical lower bound techniques---such as Fano's method, Le Cam's method, and Assouad's lemma---are central to the study of minimax risk in statistical estimation, yet are insufficient to provide tight lower bounds for \emph{interactive decision making} algorithms that collect data interactively (e.g., algorithms for bandits and reinforcement learning). Recent work of Foster et al. provides minimax lower bounds for interactive decision making using seemingly different analysis techniques from the classical methods. These results---which are proven using a complexity measure known as the \emph{Decision-Estimation Coefficient} (DEC)---capture difficulties unique to interactive learning, yet do not recover the tightest known lower bounds for passive estimation. We propose a unified view of these distinct methodologies through a new lower bound approach called \emph{interactive Fano method}.

We develop a unifying framework for information-theoretic lower bound in statistical estimation and interactive decision making. Classical lower bound techniques -- such as Fano's method, Le Cam's method, and Assouad's lemma -- are central to the study of minimax risk in statistical estimation, yet are insufficient to provide tight lower bounds for \emph{interactive decision making} algorithms that collect data interactively (e.g., algorithms for bandits and reinforcement learning). Recent work of Foster et al. (2021, 2023) provides minimax lower bounds for interactive decision making using seemingly different analysis techniques from the classical methods. These results -- which are proven using a complexity measure known as the \emph{Decision-Estimation Coefficient} (DEC) -- capture difficulties unique to interactive learning, yet do not recover the tightest known lower bounds for passive estimation. We propose a unified view of these distinct methodologies through a new lower bound approach called \emph{interactive Fano method}. As an application, we introduce a novel complexity measure, the \emph{Fractional Covering Number}, which facilitates the new lower bounds for interactive decision making that extend the DEC methodology by incorporating the complexity of estimation. Using the fractional covering number, we (i) provide a unified characterization of learnability for \emph{any} stochastic bandit problem, (ii) close the remaining gap between the upper and lower bounds in Foster et al. (2021, 2023) (up to polynomial factors) for any interactive decision making problem in which the underlying model class is convex.

We study sample efficient reinforcement learning (RL) under the general framework of interactive decision making, which includes Markov decision process (MDP), partially observable Markov decision process (POMDP), and predictive state representation (PSR) as special cases. Toward finding the minimum assumption that empowers sample efficient learning, we propose a novel complexity measure, generalized eluder coefficient (GEC), which characterizes the fundamental tradeoff between exploration and exploitation in online interactive decision making. In specific, GEC captures the hardness of exploration by comparing the error of predicting the performance of the updated policy with the in-sample training error evaluated on the historical data. We show that RL problems with low GEC form a remarkably rich class, which subsumes low Bellman eluder dimension problems, bilinear class, low witness rank problems, PO-bilinear class, and generalized regular PSR, where generalized regular PSR, a new tractable PSR class identified by us, includes nearly all known tractable POMDPs and PSRs. Furthermore, in terms of algorithm design, we propose a generic posterior sampling algorithm, which can be implemented in both model-free and model-based fashion, under both fully observable and partially observable settings. The proposed algorithm modifies the standard posterior sampling algorithm in two aspects: (i) we use an optimistic prior distribution that biases towards hypotheses with higher values and (ii) a loglikelihood function is set to be the empirical loss evaluated on the historical data, where the choice of loss function supports both model-free and model-based learning. We prove that the proposed algorithm is sample efficient by establishing a sublinear regret upper bound in terms of GEC. In summary, we provide a new and unified understanding of both fully observable and partially observable RL.

Bandit and reinforcement learning (RL) algorithms demonstrated a wide range of successful real-life applications [Silver et al., 2016, 2017, Mnih et al., 2013, Berner et al., 2019, Vinyals et al., 2019, Mnih et al., 2015, Degrave et al., 2022]. Past works have theoretically studied the regret or sample complexity of various interactive decision making problems, such as contextual bandits, reinforcement learning (RL), partially observable Markov decision process (see Azar et al. [2017], Jin et al. [2018], Dong et al. [2021], Li et al. [2019], Agarwal et al. [2014], Foster and Rakhlin [2020], Jin et al. [2020], and references therein). Recently, Foster et al. [2021] present a unified algorithmic principle for achieving the minimax regret--the optimal regret for the worst-case problem instances. However, minimax regret bounds do not necessarily always present a full picture of the statistical complexity of the problem. They characterize the complexity of the most difficult instances, but potentially many other instances are much easier. An ideal algorithm should adapt to the complexity of a particular instance and incur smaller regrets on easy instances than the worst-case instances. Thus, an ideal regret bound should be instance-dependent, that is, depending on some properties of each instance. Prior works designed algorithms with instance-dependent regret bounds that are stronger than minimax regret bounds, but they are often not directly comparable because they depend on different properties of the instances, such as the gap conditions and the variance of the value function [Zanette and Brunskill, 2019, Xu et al., 2021, Foster et al., 2020, Tirinzoni et al., 2021]. A more ambitious and challenging goal is to design instance-optimal algorithms that outperform, on every instance, all consistent algorithms (those achieving non-trivial regrets on all instances).

Interactive Decision Making for Autonomous V ehicles in Dense Traffic David Isele 1 Abstract -- Dense urban traffic environments can produce situations where accurate prediction and dynamic models are insufficient for successful autonomous vehicle motion planning. We investigate how an autonomous agent can safely negotiate with other traffic participants, enabling the agent to handle potential deadlocks. Specifically we consider merges where the gap between cars is smaller than the size of the ego vehicle. We propose a game theoretic framework capable of generating and responding to interactive behaviors. Our main contribution is to show how game-tree decision making can be executed by an autonomous vehicle, including approximations and reasoning that make the tree-search computationally tractable. Additionally, to test our model we develop a stochastic rule-based traffic agent capable of generating interactive behaviors that can be used as a benchmark for simulating traffic participants in a crowded merge setting. I NTRODUCTION Much of the long tail around autonomous driving behavior relates to complex interactions between self-interested agents. Since other traffic participants exhibit a great deal of variety and are often neither purely adversarial, nor purely cooperative, it can be difficult to reason about their behavior. However this type of reasoning is essential to numerous traffic situations in congested traffic such as overcrowded merge scenarios depicted in Figure 1.