Regret Minimization via Saddle Point Optimization

A long line of works characterizes the sample complexity of regret minimization in sequential decision-making by min-max programs. In the corresponding saddle-point game, the min-player optimizes the sampling distribution against an adversarial max-player that chooses confusing models leading to large regret. The most recent instantiation of this idea is the decision-estimation coefficient (DEC), which was shown to provide nearly tight lower and upper bounds on the worst-case expected regret in structured bandits and reinforcement learning. By re-parametrizing the offset DEC with the confidence radius and solving the corresponding min-max program, we derive an anytime variant of the Estimation-To-Decisions (E2D) algorithm. Importantly, the algorithm optimizes the exploration-exploitation trade-off online instead of via the analysis. Our formulation leads to a practical algorithm for finite model classes and linear feedback models. We further point out connections to the information ratio, decoupling coefficient and PAC-DEC, and numerically evaluate the performance of E2D on simple examples.

Switching the Loss Reduces the Cost in Batch Reinforcement Learning

We propose training fitted Q-iteration with log-loss (FQI-LOG) for batch reinforcement learning (RL). We show that the number of samples needed to learn a near-optimal policy with FQI-LOG scales with the accumulated cost of the optimal policy, which is zero in problems where acting optimally achieves the goal and incurs no cost. In doing so, we provide a general framework for proving $\textit{small-cost}$ bounds, i.e. bounds that scale with the optimal achievable cost, in batch RL. Moreover, we empirically verify that FQI-LOG uses fewer samples than FQI trained with squared loss on problems where the optimal policy reliably achieves the goal.

Ensemble sampling for linear bandits: small ensembles suffice

We provide the first useful, rigorous analysis of ensemble sampling for the stochastic linear bandit setting. In particular, we show that, under standard assumptions, for a $d$-dimensional stochastic linear bandit with an interaction horizon $T$, ensemble sampling with an ensemble of size $m$ on the order of $d \log T$ incurs regret bounded by order $(d \log T)^{5/2} \sqrt{T}$. Ours is the first result in any structured setting not to require the size of the ensemble to scale linearly with $T$ -- which defeats the purpose of ensemble sampling -- while obtaining near $\sqrt{T}$ order regret. Ours is also the first result that allows infinite action sets.

Exploration via linearly perturbed loss minimisation

We introduce exploration via linear loss perturbations (EVILL), a randomised exploration method for structured stochastic bandit problems that works by solving for the minimiser of a linearly perturbed regularised negative log-likelihood function. We show that, for the case of generalised linear bandits, EVILL reduces to perturbed history exploration (PHE), a method where exploration is done by training on randomly perturbed rewards. In doing so, we provide a simple and clean explanation of when and why random reward perturbations give rise to good bandit algorithms. With the data-dependent perturbations we propose, not present in previous PHE-type methods, EVILL is shown to match the performance of Thompson-sampling-style parameter-perturbation methods, both in theory and in practice. Moreover, we show an example outside of generalised linear bandits where PHE leads to inconsistent estimates, and thus linear regret, while EVILL remains performant. Like PHE, EVILL can be implemented in just a few lines of code.

Stochastic Gradient Descent for Gaussian Processes Done Right

We study the optimisation problem associated with Gaussian process regression using squared loss. The most common approach to this problem is to apply an exact solver, such as conjugate gradient descent, either directly, or to a reduced-order version of the problem. Recently, driven by successes in deep learning, stochastic gradient descent has gained traction as an alternative. In this paper, we show that when done right$\unicode{x2014}$by which we mean using specific insights from the optimisation and kernel communities$\unicode{x2014}$this approach is highly effective. We thus introduce a particular stochastic dual gradient descent algorithm, that may be implemented with a few lines of code using any deep learning framework. We explain our design decisions by illustrating their advantage against alternatives with ablation studies and show that the new method is highly competitive. Our evaluations on standard regression benchmarks and a Bayesian optimisation task set our approach apart from preconditioned conjugate gradients, variational Gaussian process approximations, and a previous version of stochastic gradient descent for Gaussian processes. On a molecular binding affinity prediction task, our method places Gaussian process regression on par in terms of performance with state-of-the-art graph neural networks.

Online RL in Linearly $q^π$-Realizable MDPs Is as Easy as in Linear MDPs If You Learn What to Ignore

We consider online reinforcement learning (RL) in episodic Markov decision processes (MDPs) under the linear $q^\pi$-realizability assumption, where it is assumed that the action-values of all policies can be expressed as linear functions of state-action features. This class is known to be more general than linear MDPs, where the transition kernel and the reward function are assumed to be linear functions of the feature vectors. As our first contribution, we show that the difference between the two classes is the presence of states in linearly $q^\pi$-realizable MDPs where for any policy, all the actions have approximately equal values, and skipping over these states by following an arbitrarily fixed policy in those states transforms the problem to a linear MDP. Based on this observation, we derive a novel (computationally inefficient) learning algorithm for linearly $q^\pi$-realizable MDPs that simultaneously learns what states should be skipped over and runs another learning algorithm on the linear MDP hidden in the problem. The method returns an $\epsilon$-optimal policy after $\text{polylog}(H, d)/\epsilon^2$ interactions with the MDP, where $H$ is the time horizon and $d$ is the dimension of the feature vectors, giving the first polynomial-sample-complexity online RL algorithm for this setting. The results are proved for the misspecified case, where the sample complexity is shown to degrade gracefully with the misspecification error.

The Optimal Approximation Factors in Misspecified Off-Policy Value Function Estimation

Theoretical guarantees in reinforcement learning (RL) are known to suffer multiplicative blow-up factors with respect to the misspecification error of function approximation. Yet, the nature of such \emph{approximation factors} -- especially their optimal form in a given learning problem -- is poorly understood. In this paper we study this question in linear off-policy value function estimation, where many open questions remain. We study the approximation factor in a broad spectrum of settings, such as with the weighted $L_2$-norm (where the weighting is the offline state distribution), the $L_\infty$ norm, the presence vs. absence of state aliasing, and full vs. partial coverage of the state space. We establish the optimal asymptotic approximation factors (up to constants) for all of these settings. In particular, our bounds identify two instance-dependent factors for the $L_2(\mu)$ norm and only one for the $L_\infty$ norm, which are shown to dictate the hardness of off-policy evaluation under misspecification.

Context-lumpable stochastic bandits

We consider a contextual bandit problem with $S $ contexts and $A $ actions. In each round $t=1,2,\dots$ the learner observes a random context and chooses an action based on its past experience. The learner then observes a random reward whose mean is a function of the context and the action for the round. Under the assumption that the contexts can be lumped into $r\le \min\{S ,A \}$ groups such that the mean reward for the various actions is the same for any two contexts that are in the same group, we give an algorithm that outputs an $\epsilon$-optimal policy after using at most $\widetilde O(r (S +A )/\epsilon^2)$ samples with high probability and provide a matching $\widetilde\Omega(r (S +A )/\epsilon^2)$ lower bound. In the regret minimization setting, we give an algorithm whose cumulative regret up to time $T$ is bounded by $\widetilde O(\sqrt{r^3(S +A )T})$. To the best of our knowledge, we are the first to show the near-optimal sample complexity in the PAC setting and $\widetilde O(\sqrt{{poly}(r)(S+K)T})$ minimax regret in the online setting for this problem. We also show our algorithms can be applied to more general low-rank bandits and get improved regret bounds in some scenarios.

Regularization and Variance-Weighted Regression Achieves Minimax Optimality in Linear MDPs: Theory and Practice

Mirror descent value iteration (MDVI), an abstraction of Kullback-Leibler (KL) and entropy-regularized reinforcement learning (RL), has served as the basis for recent high-performing practical RL algorithms. However, despite the use of function approximation in practice, the theoretical understanding of MDVI has been limited to tabular Markov decision processes (MDPs). We study MDVI with linear function approximation through its sample complexity required to identify an $\varepsilon$-optimal policy with probability $1-\delta$ under the settings of an infinite-horizon linear MDP, generative model, and G-optimal design. We demonstrate that least-squares regression weighted by the variance of an estimated optimal value function of the next state is crucial to achieving minimax optimality. Based on this observation, we present Variance-Weighted Least-Squares MDVI (VWLS-MDVI), the first theoretical algorithm that achieves nearly minimax optimal sample complexity for infinite-horizon linear MDPs. Furthermore, we propose a practical VWLS algorithm for value-based deep RL, Deep Variance Weighting (DVW). Our experiments demonstrate that DVW improves the performance of popular value-based deep RL algorithms on a set of MinAtar benchmarks.