This paper studies the problem of simultaneously learning relevant features and minimising regret in contextual bandit problems. We introduce and analyse a new class of contextual bandit problems, called sparse nonparametric contextual bandits, in which the expected reward function lies in the linear span of a small unknown set of features that belongs to a known infinite set of candidate features. We consider two notions of sparsity, for which the set of candidate features is either countable or uncountable. Our contribution is two-fold. First, we provide lower bounds on the minimax regret, which show that polynomial dependence on the number of actions is generally unavoidable in this setting. Second, we show that a variant of the Feel-Good Thompson Sampling algorithm enjoys regret bounds that match our lower bounds up to logarithmic factors of the horizon, and have logarithmic dependence on the effective number of candidate features. When we apply our results to kernelised and neural contextual bandits, we find that sparsity always enables better regret bounds, as long as the horizon is large enough relative to the sparsity and the number of actions.

Reinforcement learning utilizing kernel ridge regression to predict the expected value function represents a powerful method with great representational capacity. This setting is a highly versatile framework amenable to analytical results. We consider kernel-based function approximation for RL in the infinite horizon average reward setting, also referred to as the undiscounted setting. We propose an optimistic algorithm, similar to acquisition function based algorithms in the special case of bandits. We establish novel no-regret performance guarantees for our algorithm, under kernel-based modelling assumptions. Additionally, we derive a novel confidence interval for the kernel-based prediction of the expected value function, applicable across various RL problems.

In this research note, we revisit the bandits with expert advice problem. Under a restricted feedback model, we prove a lower bound of order $\sqrt{K T \ln(N/K)}$ for the worst-case regret, where $K$ is the number of actions, $N>K$ the number of experts, and $T$ the time horizon. This matches a previously known upper bound of the same order and improves upon the best available lower bound of $\sqrt{K T (\ln N) / (\ln K)}$. For the standard feedback model, we prove a new instance-based upper bound that depends on the agreement between the experts and provides a logarithmic improvement compared to prior results.

Reinforcement learning (RL) has shown empirical success in various real world settings with complex models and large state-action spaces. The existing analytical results, however, typically focus on settings with a small number of state-actions or simple models such as linearly modeled state-action value functions. To derive RL policies that efficiently handle large state-action spaces with more general value functions, some recent works have considered nonlinear function approximation using kernel ridge regression. We propose $\pi$-KRVI, an optimistic modification of least-squares value iteration, when the state-action value function is represented by a reproducing kernel Hilbert space (RKHS). We prove the first order-optimal regret guarantees under a general setting. Our results show a significant polynomial in the number of episodes improvement over the state of the art. In particular, with highly non-smooth kernels (such as Neural Tangent kernel or some Mat\'ern kernels) the existing results lead to trivial (superlinear in the number of episodes) regret bounds. We show a sublinear regret bound that is order optimal in the case of Mat\'ern kernels where a lower bound on regret is known.

We consider the adversarial linear contextual bandit setting, which allows for the loss functions associated with each of $K$ arms to change over time without restriction. Assuming the $d$-dimensional contexts are drawn from a fixed known distribution, the worst-case expected regret over the course of $T$ rounds is known to scale as $\tilde O(\sqrt{Kd T})$. Under the additional assumption that the density of the contexts is log-concave, we obtain a second-order bound of order $\tilde O(K\sqrt{d V_T})$ in terms of the cumulative second moment of the learner's losses $V_T$, and a closely related first-order bound of order $\tilde O(K\sqrt{d L_T^*})$ in terms of the cumulative loss of the best policy $L_T^*$. Since $V_T$ or $L_T^*$ may be significantly smaller than $T$, these improve over the worst-case regret whenever the environment is relatively benign. Our results are obtained using a truncated version of the continuous exponential weights algorithm over the probability simplex, which we analyse by exploiting a novel connection to the linear bandit setting without contexts.

We study the Bayesian regret of the renowned Thompson Sampling algorithm in contextual bandits with binary losses and adversarially-selected contexts. We adapt the information-theoretic perspective of \cite{RvR16} to the contextual setting by considering a lifted version of the information ratio defined in terms of the unknown model parameter instead of the optimal action or optimal policy as done in previous works on the same setting. This allows us to bound the regret in terms of the entropy of the prior distribution through a remarkably simple proof, and with no structural assumptions on the likelihood or the prior. The extension to priors with infinite entropy only requires a Lipschitz assumption on the log-likelihood. An interesting special case is that of logistic bandits with $d$-dimensional parameters, $K$ actions, and Lipschitz logits, for which we provide a $\widetilde{O}(\sqrt{dKT})$ regret upper-bound that does not depend on the smallest slope of the sigmoid link function.

We consider an online learning problem where the learner interacts with a Markov decision process in a sequence of episodes, where the reward function is allowed to change between episodes in an adversarial manner and the learner only gets to observe the rewards associated with its actions. We allow the state space to be arbitrarily large, but we assume that all action-value functions can be represented as linear functions in terms of a known low-dimensional feature map, and that the learner has access to a simulator of the environment that allows generating trajectories from the true MDP dynamics. Our main contribution is developing a computationally efficient algorithm that we call MDP-LinExp3, and prove that its regret is bounded by $\widetilde{\mathcal{O}}\big(H^2 T^{2/3} (dK)^{1/3}\big)$, where $T$ is the number of episodes, $H$ is the number of steps in each episode, $K$ is the number of actions, and $d$ is the dimension of the feature map. We also show that the regret can be improved to $\widetilde{\mathcal{O}}\big(H^2 \sqrt{TdK}\big)$ under much stronger assumptions on the MDP dynamics. To our knowledge, MDP-LinExp3 is the first provably efficient algorithm for this problem setting.

We consider an online influence maximization problem in which a decision maker selects a node among a large number of possibilities and places a piece of information at the node. The node transmits the information to some others that are in the same connected component in a random graph. The goal of the decision maker is to reach as many nodes as possible, with the added complication that feedback is only available about the degree of the selected node. Our main result shows that such local observations can be sufficient for maximizing global influence in two broadly studied families of random graph models: stochastic block models and Chung--Lu models. With this insight, we propose a bandit algorithm that aims at maximizing local (and thus global) influence, and provide its theoretical analysis in both the subcritical and supercritical regimes of both considered models. Notably, our performance guarantees show no explicit dependence on the total number of nodes in the network, making our approach well-suited for large-scale applications.