Various algorithms in reinforcement learning exhibit dramatic variability in their convergence rates and ultimate accuracy as a function of the problem structure. Such instance-specific behavior is not captured by existing global minimax bounds, which are worst-case in nature. We analyze the problem of estimating optimalQ-state-action value functions for a discounted Markov decision process with discrete states and actions; our main result is to identify an instance-dependent functional that controls the difficulty of estimation in the ℓ∞-norm. Using a local minimax framework, we show that this functional arises in lower bounds on the accuracy on any estimation procedure. We establish the sharpness of these lower bounds, up to factors logarithmic in the state and action spaces, by analyzing a variance-reduced version ofQ-learning. Our theory provides a precise way of distinguishing “easy” problems from “hard” ones in the context ofQ-learning, as illustrated by an ensemble with a continuum of difficulty.

The recent success of transformer language models owes much to their conversational fluency, which includes linguistic and morphological proficiency. An affine Taylor approximation has been found to be a good approximation for transformer computations over certain factual and encyclopedic relations. We show that the truly linear approximation W s, where s is a early layer representation of the base form and W is a local model derivative, is necessary and sufficient to approximate morphological derivation, achieving above 80% top-1 accuracy across most morphological tasks in the Bigger Analogy Test Set. We argue that many morphological forms in transformer models are likely linearly encoded.

We present and analyze the Krylov–Bellman Boosting algorithm for policy evaluation in general state spaces. It alternates between fitting the Bellman residual using non-parametric regression (as in boosting), and estimating the value function via the least-squares temporal difference (LSTD) procedure applied with a feature set that grows adaptively over time. By exploiting the connection to Krylov methods, we equip this method with two attractive guarantees. First, we provide a general convergence bound that allows for separate estimation errors in residual fitting and LSTD computation. Consistent with our numerical experiments, this bound shows that convergence rates depend on the restricted spectral structure, and are typically super-linear. Second, by combining this meta-result with sample-size dependent guarantees for residual fitting and LTSD computation, we obtain concrete statistical guarantees that depend on the sample size along with the complexity of the function class used to fit the residuals. We illustrate the behavior of the KBB algorithm for various types of policy evaluation problems, and typically find large reductions in sample complexity relative to the standard approach of fitted value iteration.

Reinforcement learning algorithms are known to exhibit a variety of convergence rates depending on the problem structure. Recent years have witnessed considerable progress in developing theory that is instance-dependent, along with algorithms that achieve such instance-optimal guarantees. However, important questions remain in how to utilize such notions for inferential purposes, or for early stopping, so that data and computational resources can be saved for “easy” problems. This paper develops data-dependent procedures that output instance-dependent confidence regions for evaluating and optimizing policies in a Markov decision process. Notably, our procedures require only black-box access to an instance-optimal algorithm, and re-use the samples used in the estimation algorithm itself. The resulting data-dependent stopping rule adapts instance-specific difficulty of the problem and allows for early termination for problems with favorable structure. We highlight benefit of such early stopping rules via some numerical studies.