We design a simple reinforcement learning (RL) agent that implements an optimistic version of $Q$-learning and establish through regret analysis that this agent can operate with some level of competence in any environment. While we leverage concepts from the literature on provably efficient RL, we consider a general agent-environment interface and provide a novel agent design and analysis. This level of generality positions our results to inform the design of future agents for operation in complex real environments. We establish that, as time progresses, our agent performs competitively relative to policies that require longer times to evaluate. The time it takes to approach asymptotic performance is polynomial in the complexity of the agent’s state representation and the time required to evaluate the best policy that the agent can represent. Notably, there is no dependence on the complexity of the environment. The ultimate per-period performance loss of the agent is bounded by a constant multiple of a measure of distortion introduced by the agent’s state representation. This work is the first to establish that an algorithm approaches this asymptotic condition within a tractable time frame.

We study the logistic bandit, in which rewards are binary with success probability $\exp(\beta a^\top \theta) / (1 + \exp(\beta a^\top \theta))$ and actions $a$ and coefficients $\theta$ are within the $d$-dimensional unit ball. While prior regret bounds for algorithms that address the logistic bandit exhibit exponential dependence on the slope parameter $\beta$, we establish a regret bound for Thompson sampling that is independent of $\beta$. Specifically, we establish that, when the set of feasible actions is identical to the set of possible coefficient vectors, the Bayesian regret of Thompson sampling is $\tilde{O}(d\sqrt{T})$. We also establish a $\tilde{O}(\sqrt{d\eta T}/\Delta)$ bound that applies more broadly, where $\Delta$ is the worst-case optimal log-odds and $\eta$ is the “fragility dimension,” a new statistic we define to capture the degree to which an optimal action for one model fails to satisfice for others. We demonstrate that the fragility dimension plays an essential role by showing that, for any $\epsilon > 0$, no algorithm can achieve $\mathrm{poly}(d, 1/\Delta)\cdot T^{1-\epsilon}$ regret.

Information-theoretic Bayesian regret bounds of Russo and Van Roy capture the dependence of regret on prior uncertainty. However, this dependence is through entropy, which can become arbitrarily large as the number of actions increases. We establish new bounds that depend instead on a notion of rate-distortion. Among other things, this allows us to recover through information-theoretic arguments a near-optimal bound for the linear bandit. We also offer a bound for the logistic bandit that dramatically improves on the best previously available, though this bound depends on an information-theoretic statistic that we have only been able to quantify via computation.