We show that Thompson sampling has a Bayesian regret of at most $\tilde O(\sqrt{n})$ for $1$-dimensional bandit convex optimisation where $n$ is the time horizon and no assumptions are made on the loss function beyond convexity, boundedness and a mild Lipschitz assumption. For general high-dimensional problems we show that Thompson sampling can fail catastrophically. More positively, we show that Thompson sampling has Bayesian regret of $\tilde O(d^{2.5} \sqrt{n})$ for generalised linear bandits with an unknown convex monotone link function. Lastly, we prove that the standard information-theoretic machinery can never give a bound on the regret in the general case that improveson the best known bound of $\tilde O(d^{1.5} \sqrt{n})$.

We introduce a computationally efficient algorithm for zeroth-order bandit convex optimisation and prove that in the adversarial setting its regret is at most $d^{3.5} \sqrt{n} \mathrm{polylog}(n, d)$ with high probability where $d$ is the dimension and $n$ is the time horizon. In the stochastic setting the bound improves to $M d^{2} \sqrt{n} \mathrm{polylog}(n, d)$ where $M \in [d^{-1/2}, d^{-1/4}]$ is a constant that depends on the geometry of the constraint set and the desired computational properties.

We introduce a simple and efficient algorithm for unconstrained zeroth-order stochastic convex bandits and prove its regret is at most (1 + r/d)[d^1.5 sqrt(n) + d^3] polylog(n, d, r) where n is the horizon, d the dimension and r is the radius of a known ball containing the minimiser of the loss.

We prove that the continuous time version of the concentration bounds by Abbasi-Yadkori et al. (2011) for adaptive linear regression cannot be improved in general, showing that there can be a significant price for sequential design. This resolves the continuous time version of the COLT open problem by Vakili et al. (2021b) on confidence intervals for kernel regression with sequential designs. Experimental evidence suggests that improved confidence bounds are also not possible in discrete time.

We study distributed contextual linear bandits with stochastic contexts, where $N$ agents/learners act cooperatively to solve a linear bandit-optimization problem with $d$-dimensional features over the course of $T$ rounds. For this problem, we derive the first ever information-theoretic lower bound $\Omega(dN)$ on the communication cost of any algorithm that performs optimally in a regret minimization setup. We then propose a distributed batch elimination version of the LinUCB algorithm, DisBE-LUCB, where the agents share information among each other through a central server. We prove that the communication cost of DisBE-LUCB, matches our lower bound up to logarithmic factors. In particular, for scenarios with known context distribution, the communication cost of DisBE-LUCB is only $\tilde{\mathcal{O}}(dN)$ and its regret is $\tilde{\mathcal{O}}(\sqrt{dNT})$, which is of the same order as that incurred by an optimal single-agent algorithm for $NT$ rounds. We also provide similar bounds for practical settings where the context distribution can only be estimated. Therefore, our proposed algorithm is nearly minimax optimal in terms of both regret and communication cost. Finally, we propose DecBE-LUCB, a fully decentralized version of DisBE-LUCB, which operates without a central server, where agents share information with their immediate neighbors through a carefully designed consensus procedure.

We investigate the extent to which offline demonstration data can improve online learning. It is natural to expect some improvement, but *the question is how, and by how much?* We show that the degree of improvement must depend on the *quality* of the demonstration data. To generate portable insights, we focus on Thompson sampling (TS) applied to a multi-armed bandit as a prototypical online learning algorithm and model. The demonstration data is generated by an expert with a given *competence* level, a notion we introduce. We propose an informed TS algorithm that utilizes the demonstration data in a coherent way through Bayes' rule and derive a prior-dependent Bayesian regret bound. This offers insight into how pretraining can greatly improve online performance and how the degree of improvement increases with the expert's competence level. We also develop a practical, approximate informed TS algorithm through Bayesian bootstrapping and show substantial empirical regret reduction through experiments.

We consider a contextual bandit problem with $S $ contexts and $K $ 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 $r\le \min(S ,K)$ 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 $\widetilde O(r (S +K )/\epsilon^2)$ samples with high probability and provide a matching $\widetilde\Omega(r (S +K )/\epsilon^2)$ lower bound. In the regret minimization setting, we give an algorithm whose cumulative regret up to time $T$ is bounded by $\widetilde O(\sqrt{r ^3(S +K )T})$. To the best of our knowledge, we are the first to show the near-optimal sample complexity in the PAC setting and $\widetilde O{\sqrt{\text{poly}(r)(S+K)T}}$ 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.

A popular perspective in Reinforcement learning (RL) casts the problem as probabilistic inference on a graphical model of the Markov decision process (MDP). The core object of study is the probability of each state-action pair being visited under the optimal policy. Previous approaches to approximate this quantity can be arbitrarily poor, leading to algorithms that do not implement genuine statistical inference and consequently do not perform well in challenging problems. In this work, we undertake a rigorous Bayesian treatment of the posterior probability of state-action optimality and clarify how it flows through the MDP. We first reveal that this quantity can indeed be used to generate a policy that explores efficiently, as measured by regret. Unfortunately, computing it is intractable, so we derive a new variational Bayesian approximation yielding a tractable convex optimization problem and establish that the resulting policy also explores efficiently. We call our approach VAPOR and show that it has strong connections to Thompson sampling, K-learning, and maximum entropy exploration. We conclude with some experiments demonstrating the performance advantage of a deep RL version of VAPOR.

Partial monitoring is an expressive framework for sequential decision-making with an abundance of applications, including graph-structured and dueling bandits, dynamic pricing and transductive feedback models. We survey and extend recent results on the linear formulation of partial monitoring that naturally generalizes the standard linear bandit setting. The main result is that a single algorithm, information-directed sampling (IDS), is (nearly) worst-case rate optimal in all finite-action games. We present a simple and unified analysis of stochastic partial monitoring, and further extend the model to the contextual and kernelized setting.

We show that a version of the generalised information ratio of Lattimore and Gyorgy (2020) determines the asymptotic minimax regret for all finite-action partial monitoring games provided that (a) the standard definition of regret is used but the latent space where the adversary plays is potentially infinite; or (b) the regret introduced by Rustichini (1999) is used and the latent space is finite. Our results are complemented by a number of examples. For any p ∈ [1/2, 1] there exists an infinite partial monitoring game for which the minimax regret over n rounds is n^p up to subpolynomial factors and there exist finite games for which the minimax Rustichini regret is n^(4/7) up to subpolynomial factors.

We introduce a modification of follow the regularised leader and combine it with the log determinant potential and suitable loss estimators to prove that the minimax regret for adaptive adversarial linear bandits is at most $O(d \sqrt{T \log(T)})$ where $d$ is the dimension and $T$ is the number of rounds. By using exponential weights, we improve this bound to $O(\sqrt{dT\log(kT)})$ when the action set has size $k$. These results confirms an old conjecture. We also show that follow the regularized leader with the entropic barrier and suitable loss estimators has regret against an adaptive adversary of at most $O(d^2 \sqrt{T} \log(T))$ and can be implement in polynomial time, which improves on the best known bound for an efficient algorithm of $O(d^{7/2} \sqrt{T} \poly(\log(T)))$ by Lee et al 2020.

Information-directed sampling (IDS) has recently demonstrated its potential as a data-efficient reinforcement learning algorithm. However, it is still unclear what is the right form of information ratio to optimize when contextual information is available. We investigate the IDS design through two contextual bandit problems: contextual bandits with graph feedback and sparse linear contextual bandits. We provably demonstrate the advantage of contextual IDS over conditional IDS and emphasize the importance of considering the context distribution. The main message is that an intelligent agent should invest more on the actions that are beneficial for the future unseen contexts while the conditional IDS can be myopic. We further propose a computationally-efficient version of contextual IDS based on Actor-Critic and evaluate it empirically on a neural network contextual bandit.

Information-directed sampling (IDS) has revealed its potential as a data-efficient algorithm for reinforcement learning (RL). However, theoretical understanding of IDS for Markov Decision Processes (MDPs) is still limited. We develop novel information-theoretic tools to bound the information ratio and cumulative information gain about the learning target. Our theoretical results shed light on the importance of choosing the learning target such that the practitioners can balance the computation and regret bounds. As a consequence, we derive prior-free Bayesian regret bounds for vanilla-IDS which learns the whole environment under tabular finite-horizon MDPs. In addition, we propose a computationally-efficient regularized-IDS that maximizes an additive form rather than the ratio form and show that it enjoys the same regret bound as vanilla-IDS. With the aid of rate-distortion theory, we improve the regret bound by learning a surrogate, less informative environment. Furthermore, we extend our analysis to linear MDPs and prove similar regret bounds for Thompson sampling as a by-product.

This paper presents a new family of backpropagation-free neural architectures, Gated Linear Networks (GLNs). What distinguishes GLNs from contemporary neural networks is the distributed and local nature of their credit assignment mechanism; each neuron directly predicts the target, forgoing the ability to learn feature representations in favor of rapid online learning. Individual neurons are able to model nonlinear functions via the use of data-dependent gating in conjunction with online convex optimization. We show that this architecture gives rise to universal learning capabilities in the limit, with effective model capacity increasing as a function of network size in a manner comparable with deep ReLU networks. Furthermore, we demonstrate that the GLN learning mechanism possesses extraordinary resilience to catastrophic forgetting, performing almost on par to an MLP with dropout and Elastic Weight Consolidation on standard benchmarks.

We investigate the hardness of online reinforcement learning in sparse linear Markov decision process (MDP), with a special focus on the high-dimensional regime where the ambient dimension is larger than the number of episodes. Our contribution is two-fold. First, we provide a lower bound showing that linear regret is generally unavoidable, even if there exists a policy that collects well-conditioned data. Second, we show that if the learner has oracle access to a policy that collects well-conditioned data, then a variant of Lasso fitted Q-iteration enjoys a regret of $O(N^{2/3})$ where $N$ is the number of episodes.

We introduce a simple and efficient algorithm for stochastic linear bandits with finitely many actions that is asymptotically optimal and (nearly) worst-case optimal in finite time. The approach is based on the frequentist information-directed sampling (IDS) framework, with a surrogate for the information gain that is informed by the optimization problem that defines the asymptotic lower bound. Our analysis sheds light on how IDS balances the trade-off between regret and information and uncovers a surprising connection between the recently proposed primal-dual methods and the IDS algorithm. We demonstrate empirically that IDS is competitive with UCB in finite-time, and can be significantly better in the asymptotic regime.

We present an efficient algorithm for stochastic bandit convex optimisation with no assumptions on smoothness or strong convexity and for which the regret is bounded by O(d^(4.5) sqrt(n) polylog(n)), where n is the number of interactions and d is the dimension.

We establish a connection between the stability of mirror descent and the information ratio by Russo and Van Roy (2014). Our analysis shows that mirror descent with suitable loss estimators and exploratory distributions enjoys the same bound on the adversarial regret as the bounds on the Bayesian regret for information-directed sampling. Along the way, we develop the theory for information-directed sampling and provide an efficient algorithm for adversarial bandits for which the regret upper bound matches exactly the best known information-theoretic upper bound. Keywords: Bandits, partial monitoring, mirror descent, information theory.

This paper provides a statistical analysis of high-dimensional batch reinforcement learning (RL) using sparse linear function approximation. When there is a large number of candidate features, our result sheds light on the fact that sparsity-aware methods can make batch RL more sample efficient. We first consider the off-policy policy evaluation problem. To evaluate a new target policy, we analyze a Lasso fitted Q-evaluation method and establish a finite-sample error bound that has no polynomial dependence on the ambient dimension. To reduce the Lasso bias, we further propose a post model-selection estimator that applies fitted Q-evaluation to the features selected via group Lasso. Under an additional signal strength assumption, we derive a sharper instance-dependent error bound that depends on a divergence function measuring the distribution mismatch between the data distribution and occupancy measure of the target policy. Further, we study the Lasso fitted Q-iteration for batch policy optimization and establish a finite-sample error bound depending on the ratio between the number of relevant features and restricted minimal eigenvalue of the data’s covariance. In the end, we complement the results with minimax lower bounds for batch-data policy evaluation/optimization that nearly match our upper bounds. The results suggest that having well-conditioned data is crucial for sparse batch policy learning.

Batch policy optimization considers leveraging existing data for policy construction before interacting with an environment. Although interest in this problem has grown significantly in recent years, its theoretical foundations remain under-developed. To advance the understanding of this problem, we provide three results that characterize the limits and possibilities of batch policy optimization in the finite-armed stochastic bandit setting. First, we introduce a class of confidence-adjusted index algorithms that unifies optimistic and pessimistic principles in a common framework, which enables a general analysis. For this family, we show that any confidence-adjusted index algorithm is minimax optimal, whether it be optimistic, pessimistic or neutral. Our analysis reveals that instance-dependent optimality, commonly used to establish optimality of on-line stochastic bandit algorithms, cannot be achieved by any algorithm in the batch setting. In particular, for any algorithm that performs optimally in some environment, there exists another environment where the same algorithm suffers arbitrarily larger regret. Therefore, to establish a framework for distinguishing algorithms, we introduce a new weighted-minimax criterion that considers the inherent difficulty of optimal value prediction. We demonstrate how this criterion can be used to justify commonly used pessimistic principles for batch policy optimization.

Stochastic sparse linear bandits offer a practical model for high-dimensional online decision-making problems and have a rich information-regret structure. In this work we explore the use of information-directed sampling (IDS), which naturally balances the information-regret trade-off. We develop a class of information-theoretic Bayesian regret bounds that nearly match existing lower bounds on a variety of problem instances, demonstrating the adaptivity of IDS. To efficiently implement sparse IDS, we propose an empirical Bayesian approach for sparse posterior sampling using a spike-and-slab Gaussian-Laplace prior. Numerical results demonstrate significant regret reductions by sparse IDS relative to several baselines.

We study a bandit version of phase retrieval where the learner chooses actions $(A_t)_{t=1}^n$ in the $d$-dimensional unit ball and the expected reward is $\langle A_t, \theta_\star \rangle^2$ with $\theta_\star \in \mathbb R^d$ an unknown parameter vector. We prove an upper bound on the minimax cumulative regret in this problem of $\smash{\tilde \Theta(d \sqrt{n})}$, which matches known lower bounds up to logarithmic factors and improves on the best known upper bound by a factor of $\smash{\sqrt{d}}$. We also show that the minimax simple regret is $\smash{\tilde \Theta(d / \sqrt{n})}$ and that this is only achievable by an adaptive algorithm. Our analysis shows that an apparently convincing heuristic for guessing lower bounds can be misleading and that uniform bounds on the information ratio for information-directed sampling (Russo and Van Roy, 2014) are not sufficient for optimal regret.

We consider online sequential decision problems where an agent must balance exploration and exploitation. We derive a set of Bayesian `optimistic' policies which, in the stochastic multi-armed bandit case, includes the Thompson sampling policy. We provide a new analysis showing that any algorithm producing policies in the optimistic set enjoys $\tilde O(\sqrt{AT})$ Bayesian regret for a problem with $A$ actions after $T$ rounds. We extend the regret analysis for optimistic policies to bilinear saddle-point problems which include zero-sum matrix games and constrained bandits as special cases. In this case we show that Thompson sampling can produce policies outside of the optimistic set and suffer linear regret in some instances. Finding a policy inside the optimistic set amounts to solving a convex optimization problem and we call the resulting algorithm `variational Bayesian optimistic sampling' (VBOS). The procedure works for any posteriors, \ie, it does not require the posterior to have any special properties, such as log-concavity, unimodality, or smoothness. The variational view of the problem has many useful properties, including the ability to tune the exploration-exploitation tradeoff, add regularization, incorporate constraints, and linearly parameterize the policy.

We study a version of the classical zero-sum matrix game with unknown payoff matrix and bandit feedback, where the players only observe each others actions and a noisy payoff. This generalizes the usual matrix game, where the payoff matrix is known to the players. Despite numerous applications, this problem has received relatively little attention. Although adversarial bandit algorithms achieve low regret, they do not exploit the matrix structure and perform poorly relative to the new algorithms. The main contributions are regret analyses of variants of UCB and K-learning that hold for any opponent, e.g., even when the opponent adversarially plays the best-response to the learner’s mixed strategy. Along the way, we show that Thompson fails catastrophically in this setting and provide empirical comparison to existing algorithms.

Contextual bandits serve as a fundamental model for many sequential decision making tasks. The most popular theoretically justified approaches are based on the optimism principle and Thompson sampling. While these algorithms can be practical, they are known to be suboptimal asymptotically. On the other hand, existing asymptotically optimal algorithms for this problem do not exploit the linear structure in an optimal way and suffer from lower-order terms that dominate the regret in all practically interesting regimes. We start to bridge the gap by designing an algorithm that is asymptotically optimal and has good finite-time empirical performance. At the same time, we make connections to the recent literature on when exploration-free methods are effective. Indeed, if the distribution of contexts is well behaved, then our algorithm acts mostly greedily and enjoys sub-logarithmic regret. Furthermore, our approach is adaptive in the sense that it automatically detects the nice case. Numerical results demonstrate significant regret reductions by our method relative to several baselines.

Partial monitoring is a rich framework for sequential decision making under uncertainty that generalizes many well known bandit models, including linear, combinatorial and dueling bandits. We introduce {\em information directed sampling} (IDS) for stochastic partial monitoring with a linear reward and observation structure. IDS achieves adaptive worst-case regret rates that depend on precise observability conditions of the game. Moreover, we prove lower bounds that classify the minimax regret of all finite games into four possible regimes. IDS achieves the optimal rate in all cases up to logarithmic factors, without tuning any hyper-parameters. We further extend our results to the contextual and the kernelized setting, which significantly increases the range of possible applications.

We provide a novel algorithm for adversarial k-action d-outcome partial monitoring that is adaptive, intuitive and efficient. The highlight is that for the non-degenerate locally observable games, the n-round minimax regret is bounded by 6m k^(3/2) sqrt(n log(k)), where m is the number of signals. This matches the best known information-theoretic upper bound derived via Bayesian minimax duality. The same algorithm also achieves near-optimal regret for full information, bandit and globally observable games. High probability bounds and simple experiments are also provided.

The construction in the recent paper by Du et al. [2019] implies that searching for a near-optimal action in a bandit sometimes requires examining essentially all the actions, even if the learner is given linear features in R^d that approximate the rewards with a small uniform error. We use the Kiefer-Wolfowitz theorem to prove a positive result that by checking only a few actions, a learner can always find an action that is suboptimal with an error of at most O($\epsilon$$\sqrt{}$d) where $\epsilon$ is the approximation error of the features. Thus, features are useful when the approximation error is small relative to the dimensionality of the features. The idea is applied to stochastic bandits and reinforcement learning with a generative model where the learner has access to d-dimensional linear features that approximate the action-value functions for all policies to an accuracy of $\epsilon$. For linear bandits, we prove a bound on the regret of order d$\sqrt{}$(n log(k)) + $\epsilon$n$\sqrt{}$d log(n) with k the number of actions and n the horizon. For RL we show that approximate policy iteration can learn a policy that is optimal up to an additive error of order $\epsilon$$\sqrt{}$d/(1 − $\gamma$)^2 and using about d/($\epsilon$^2(1 − $\gamma$)^4) samples from the generative model. These bounds are independent of the finer details of the features. We also investigate how the structure of the feature set impacts the tradeoff between sample complexity and estimation error.

Stochastic linear bandits are a natural and well-studied model for structured exploration/exploitation problems and are widely used in applications such as on-line marketing and recommendation. One of the main challenges faced by practitioners hoping to apply existing algorithms is that usually the feedback is randomly delayed and delays are only partially observable. For example, while a purchase is usually observable some time after the display, the decision of not buying is never explicitly sent to the system. In other words, the learner only observes delayed positive events. We formalize this problem as a novel stochastic delayed linear bandit and propose OTFLinUCB and OTFLinTS, two computationally efficient algorithms able to integrate new information as it becomes available and to deal with the permanently censored feedback. We prove optimal O(d\sqrt{T}) bounds on the regret of the first algorithm and study the dependency on delay-dependent parameters. Our model, assumptions and results are validated by experiments on simulated and real data.

We propose the Gaussian Gated Linear Network (G-GLN), an extension to the recently proposed GLN family of deep neural networks. Instead of using backpropagation to learn features, GLNs have a distributed and local credit assignment mechanism based on optimizing a convex objective. This gives rise to many desirable properties including universality, data-efficient online learning, trivial interpretability and robustness to catastrophic forgetting. We extend the GLN framework from classification to multiple regression and density modelling by generalizing geometric mixing to a product of Gaussian densities. The G-GLN achieves competitive or state-of-the-art performance on several univariate and multivariate regression benchmarks, and we demonstrate its applicability to practical tasks including online contextual bandits and density estimation via denoising.

Stochastic linear bandits with high-dimensional sparse features are a practical model for a variety of domains, such as personalized medicine and online advertising. We derive a novel O(n^{2/3}) dimension-free minimax regret lower bound for sparse linear bandits in the data-poor regime where the horizon is larger than the ambient dimension and where the feature vectors admit a well-conditioned exploration distribution. This is complemented by a nearly matching upper bound for an explore-then-commit algorithm showing that that O(n^{2/3}) is the optimal rate in the data-poor regime. The results complement existing bounds for the data-rich regime and also provide another example where carefully balancing the trade-off between information and regret is necessary. Finally, we prove a dimension-free O(\sqrt{n}) regret upper bound under an additional assumption on the magnitude of the signal for relevant features.

We study bandit model selection in stochastic environments. Our approach relies on a master algorithm that selects between candidate base algorithms. We develop a master-base algorithm abstraction that can work with general classes of base algorithms and different type of adversarial master algorithms. Our methods rely on a novel and generic smoothing transformation for bandit algorithms that permits us to obtain optimal $O(\sqrt{T})$ model selection guarantees for stochastic contextual bandit problems as long as the optimal base algorithm satisfies a high probability regret guarantee. We show through a lower bound that even when one of the base algorithms has $O(\log T)$ regret, in general it is impossible to get better than $\Omega(\sqrt{T})$ regret in model selection, even asymptotically. Using our techniques, we address model selection in a variety of problems such as misspecified linear contextual bandits \citep{lattimore2019learning}, linear bandit with unknown dimension \citep{Foster-Krishnamurthy-Luo-2019} and reinforcement learning with unknown feature maps. Our algorithm requires the knowledge of the optimal base regret to adjust the master learning rate. We show that without such prior knowledge any master can suffer a regret larger than the optimal base regret.

Machine learning is used extensively in recommender systems deployed in products. The decisions made by these systems can influence user beliefs and preferences which in turn affect the feedback the learning system receives - thus creating a feedback loop. This phenomenon can give rise to the so-called "echo chambers" or "filter bubbles" that have user and societal implications. In this paper, we provide a novel theoretical analysis that examines both the role of user dynamics and the behavior of recommender systems, disentangling the echo chamber from the filter bubble effect. In addition, we offer practical solutions to slow down system degeneracy. Our study contributes toward understanding and developing solutions to commonly cited issues in the complex temporal scenario, an area that is still largely unexplored.

Partial monitoring is a generalization of the well-known multi-armed bandit framework where the loss is not directly observed by the learner. We complete the classification of finite adversarial partial monitoring to include all games, solving an open problem posed by Bartok et al. (2014). Along the way we simplify and improve existing algorithms and correct errors in previous analyses. Our second contribution is a new algorithm for the class of games studied by Bartok (2013) where we prove upper and lower regret bounds that shed more light on the dependence of the regret on the game structure.

We prove a new minimax theorem connecting the worst-case Bayesian regret and minimax regret under finite-action partial monitoring with no assumptions on the space of signals or decisions of the adversary. We then generalise the information-theoretic tools of Russo and Van Roy (2016) for proving Bayesian regret bounds and combine them with the minimax theorem to derive minimax regret bounds for various partial monitoring settings. The highlight is a clean analysis of ‘easy’ and ‘hard’ finite partial monitoring, with new regret bounds that are independent of arbitrarily large game-dependent constants and eliminate the logarithmic dependence on the horizon for easy games that appeared in earlier work. The power of the generalised machinery is further demonstrated by proving that the minimax regret for $k$-armed adversarial bandits is at most $\sqrt{2kn}$, improving on existing results by a factor of 2. Finally, we provide a simple analysis of the cops and robbers game, also improving best known constants.

We propose a bandit algorithm that explores by randomizing its history of rewards. Specifically, it pulls the arm with the highest mean reward in a non-parametric bootstrap sample of its history with pseudo rewards. We design the pseudo rewards such that the bootstrap mean is optimistic with a sufficiently high probability. We call our algorithm Giro, which stands for garbage in, reward out. We analyze Giro in a Bernoulli bandit and derive a $O(K \Delta^{-1} \log n)$ bound on its $n$-round regret, where $\Delta$ is the difference in the expected rewards of the optimal and the best suboptimal arms, and $K$ is the number of arms. The main advantage of our exploration design is that it easily generalizes to structured problems. To show this, we propose contextual Giro with an arbitrary reward generalization model. We evaluate Giro and its contextual variant on multiple synthetic and real-world problems, and observe that it performs well.

We introduce a new model for online ranking in which the click probability factors into an examination and attractiveness function and the attractiveness function is a linear function of a feature vector and an unknown parameter. Only relatively mild assumptions are made on the examination function. A novel algorithm for this setup is analysed, showing that the dependence on the number of items is replaced by a dependence on the dimension, allowing the new algorithm to handle a large number of items. When reduced to the orthogonal case, the regret of the algorithm improves on the state-of-the-art.

We tackle two long-standing problems related to re-expansions in heuristic search algorithms. For graph search, A* can require Ω(2ⁿ) expansions, where n is the number of states within the final f bound. Existing algorithms that address this problem like B and B’ improve this bound to Ω(n²). For tree search, IDA* can also require Ω(n²) expansions. We describe a new algorithmic framework that iteratively controls an expansion budget and solution cost limit, giving rise to new graph and tree search algorithms for which the number of expansions is O(n log C*), where C* is the optimal solution cost. Our experiments show that the new algorithms are robust in scenarios where existing algorithms fail. In the case of tree search, our new algorithms have no overhead over IDA* in scenarios to which IDA* is well suited and can therefore be recommended as a general replacement for IDA*.

We propose a new perspective on representation learning in reinforcement learning based on geometric properties of the space of value functions. From there, we provide formal evidence regarding the usefulness of value functions as auxiliary tasks in reinforcement learning. Our formulation considers adapting the representation to minimize the (linear) approximation of the value function of all stationary policies for a given environment. We show that this optimization reduces to making accurate predictions regarding a special class of value functions which we call adversarial value functions (AVFs). We demonstrate that using value functions as auxiliary tasks corresponds to an expected-error relaxation of our formulation, with AVFs a natural candidate, and identify a close relationship with proto-value functions (Mahadevan, 2005). We highlight characteristics of AVFs and their usefulness as auxiliary tasks in a series of experiments on the four-room domain.

The information-theoretic analysis by Russo and Van Roy [2014] in combination with minimax duality has proved a powerful tool for the analysis of online learning algorithms in full and partial information settings. In most applications there is a tantalising similarity to the classical analysis based on mirror descent. We make a formal connection, showing that the information-theoretic bounds in most applications are derived from existing techniques from online convex optimisation. Besides this, we improve best known regret guarantees for $k$-armed adversarial bandits, online linear optimisation on $\ell_p$-balls and bandits with graph feedback.

Online learning to rank is a sequential decision-making problem where in each round the learning agent chooses a list of items and receives feedback in the form of clicks from the user. Many sample-efficient algorithms have been proposed for this problem that assume a specific click model connecting rankings and user behavior. We propose a generalized click model that encompasses many existing models, including the position-based and cascade models. Our generalization motivates a novel online learning algorithm based on topological sort, which we call TopRank. TopRank is (a) more natural than existing algorithms, (b) has stronger regret guarantees than existing algorithms with comparable generality, (c) has a more insightful proof that leaves the door open to many generalizations, (d) outperforms existing algorithms empirically.

We introduce two novel tree search algorithms that use a policy to guide search. The first algorithm is a best-first enumeration that uses a cost function that allows us to provide an upper bound on the number of nodes to be expanded before reaching a goal state. We show that this best-first algorithm is particularly well suited for ``needle-in-a-haystack'' problems. The second algorithm, which is based on sampling, provides an upper bound on the expected number of nodes to be expanded before reaching a set of goal states. We show that this algorithm is better suited for problems where many paths lead to a goal. We validate these tree search algorithms on 1,000 computer-generated levels of Sokoban, where the policy used to guide search comes from a neural network trained using A3C. Our results show that the policy tree search algorithms we introduce are competitive with a state-of-the-art domain-independent planner that uses heuristic search.

This paper introduces the first strategy for stochastic bandits with unit variance Gaussian noise that is simultaneously minimax optimal up to constant factors, asymptotically optimal, and never worse than the classical upper confidence bound strategy up to universal constant factors. Preliminary empirical evidence is also promising. Besides this, a conjecture on the optimal form of the regret is shown to be false and a finite-time lower bound on the regret of any strategy is presented that very nearly matches the finite-time upper bound of the newly proposed strategy.

Stochastic linear bandits are a natural and simple generalisation of finite-armed bandits with numerous practical applications. Current approaches focus on generalising existing techniques for finite-armed bandits, notably the otimism principle and Thompson sampling. Prior analysis has mostly focussed on the worst-case setting. We analyse the asymptotic regret and show matching upper and lower bounds on what is achievable. Surprisingly, our results show that no algorithm based on optimism or Thompson sampling will ever achieve the optimal rate. In fact, they can be arbitrarily far from optimal, even in very simple cases. This is a disturbing result because these techniques are standard tools that are widely used for sequential optimisation, for example, generalised linear bandits and reinforcement learning.

We consider prediction with expert advice under the log-loss with the goal of deriving efficient and robust algorithms. We argue that existing algorithms such as exponentiated gradient, online gradient descent and online Newton step do not adequately satisfy both requirements. Our main contribution is an analysis of the Prod algorithm that is robust to any data sequence and runs in linear time relative to the number of experts in each round. Despite the unbounded nature of the log-loss, we derive a bound that is independent of the largest loss and of the largest gradient, and depends only on the number of experts and the time horizon. Furthermore we give a Bayesian interpretation of Prod and adapt the algorithm to derive a tracking regret.

We discuss some recent results on Thompson sampling for nonparametric reinforcement learning in countable classes of general stochastic environments. These environments can be non-Markovian, non-ergodic, and partially observable. We show that Thompson sampling learns the environment class in the sense that (1) asymptotically its value converges in mean to the optimal value and (2) given a recoverability assumption regret is sublinear. We conclude with a discussion about optimality in reinforcement learning.

Statistical performance bounds for reinforcement learning (RL) algorithms can be critical for high-stakes applications like healthcare. This paper introduces a new framework for theoretically measuring the performance of such algorithms called Uniform-PAC, which is a strengthening of the classical Probably Approximately Correct (PAC) framework. In contrast to the PAC framework, the uniform version may be used to derive high probability regret guarantees and so forms a bridge between the two setups that has been missing in the literature. We demonstrate the benefits of the new framework for finite-state episodic MDPs with a new algorithm that is Uniform-PAC and simultaneously achieves optimal regret and PAC guarantees except for a factor of the horizon.

Existing strategies for finite-armed stochastic bandits mostly depend on a parameter of scale that must be known in advance. Sometimes this is in the form of a bound on the payoffs, or the knowledge of a variance or subgaussian parameter. The notable exceptions are the analysis of Gaussian bandits with unknown mean and variance by Cowan and Katehakis [2015a] and of uniform distributions with unknown support [Cowan and Katehakis, 2015b]. The results derived in these specialised cases are generalised here to the non-parametric setup, where the learner knows only a bound on the kurtosis of the noise, which is a scale free measure of the extremity of outliers.

Follow the leader (FTL) is a simple online learning algorithm that is known to perform well when the loss functions are convex and positively curved. In this paper we ask whether there are other settings when FTL achieves low regret. In particular, we study the fundamental problem of linear prediction over a convex, compact domain with non-empty interior. Amongst other results, we prove that the curvature of the boundary of the domain can act as if the losses were curved: In this case, we prove that as long as the mean of the loss vectors have positive lengths bounded away from zero, FTL enjoys logarithmic regret, while for polytope domains and stochastic data it enjoys finite expected regret. The former result is also extended to strongly convex domains by establishing an equivalence between the strong convexity of sets and the minimum curvature of their boundary, which may be of independent interest. Building on a previously known meta-algorithm, we also get an algorithm that simultaneously enjoys the worst-case guarantees and the smaller regret of FTL when the data is `easy'. Finally, we show that such guarantees are achievable directly (e.g., by the follow the regularized leader algorithm or by a shrinkage-based variant of FTL) when the constraint set is an ellipsoid.

I prove near-optimal frequentist regret guarantees for the finite-horizon Gittins index strategy for multi-armed bandits with Gaussian noise and prior. Along the way I derive finite-time bounds on the Gittins index that are asymptotically exact and may be of independent interest. I also discuss computational issues and present experimental results suggesting that a particular version of the Gittins index strategy is an improvement on existing algorithms with finite-time regret guarantees such as UCB and Thompson sampling.

We study a novel multi-armed bandit problem that models the challenge faced by a company wishing to explore new strategies to maximize revenue whilst simultaneously maintaining their revenue above a fixed baseline, uniformly over time. While previous work addressed the problem under the weaker requirement of maintaining the revenue constraint only at a given fixed time in the future, the design of those algorithms makes them unsuitable under the more stringent constraints. We consider both the stochastic and the adversarial settings, where we propose natural yet novel strategies and analyze the price for maintaining the constraints. Amongst other things, we prove both high probability and expectation bounds on the regret, while we also consider both the problem of maintaining the constraints with high probability or expectation. For the adversarial setting the price of maintaining the constraint appears to be higher, at least for the algorithm considered. A lower bound is given showing that the algorithm for the stochastic setting is almost optimal. Empirical results obtained in synthetic environments complement our theoretical findings.

We study the problem of minimising regret in two-armed bandit problems with Gaussian rewards. Our objective is to use this simple setting to illustrate that strategies based on an exploration phase (up to a stopping time) followed by exploitation are necessarily suboptimal. The results hold regardless of whether or not the difference in means between the two arms is known. Besides the main message, we also refine existing deviation inequalities, which allow us to design fully sequential strategies with finite-time regret guarantees that are (a) asymptotically optimal as the horizon grows and (b) order-optimal in the minimax sense. Furthermore we provide empirical evidence that the theory also holds in practice and discuss extensions to non-gaussian and multiple-armed case.

We provide new lower bounds on the regret that must be suffered by adversarial bandit algorithms. The new results show that recent upper bounds that either (a) hold with high-probability or (b) depend on the total loss of the best arm or (c) depend on the quadratic variation of the losses, are close to tight. Besides this we prove two impossibility results. First, the existence of a single arm that is optimal in every round cannot improve the regret in the worst case. Second, the regret cannot scale with the effective range of the losses. In contrast, both results are possible in the full-information setting.

The follow the leader (FTL) algorithm, perhaps the simplest of all online learning algorithms, is known to perform well when the loss functions it is used on are positively curved. In this paper we ask whether there are other "lucky" settings when FTL achieves sublinear, "small" regret. In particular, we study the fundamental problem of linear prediction over a non-empty convex, compact domain. Amongst other results, we prove that the curvature of the boundary of the domain can act as if the losses were curved: In this case, we prove that as long as the mean of the loss vectors have positive lengths bounded away from zero, FTL enjoys a logarithmic growth rate of regret, while, e.g., for polyhedral domains and stochastic data it enjoys finite expected regret. Building on a previously known meta-algorithm, we also get an algorithm that simultaneously enjoys the worst-case guarantees and the bound available for FTL.

We study the problem of using causal models to improve the rate at which good interventions can be learned online in a stochastic environment. Our formalism combines multi-arm bandits and causal inference to model a novel type of bandit feedback that is not exploited by existing approaches. We propose a new algorithm that exploits the causal feedback and prove a bound on its simple regret that is strictly better (in all quantities) than algorithms that do not use the additional causal information.

Given a multi-armed bandit problem it may be desirable to achieve a smaller-than-usual worst-case regret for some special actions. I show that the price for such unbalanced worst-case regret guarantees is rather high. Specifically, if an algorithm enjoys a worst-case regret of B with respect to some action, then there must exist another action for which the worst-case regret is at least Ω(nK/B), where n is the horizon and K the number of actions. I also give upper bounds in both the stochastic and adversarial settings showing that this result cannot be improved. For the stochastic case the pareto regret frontier is characterised exactly up to constant factors.

We study an idealised sequential resource allocation problem. In each time step the learner chooses an allocation of several resource types between a number of tasks. Assigning more resources to a task increases the probability that it is completed. The problem is challenging because the alignment of the tasks to the resource types is unknown and the feedback is noisy. Our main contribution is the new setting and an algorithm with nearly-optimal regret analysis. Along the way we draw connections to the problem of minimising regret for stochastic linear bandits with heteroscedastic noise. We also present some new results for stochastic linear bandits on the hypercube that significantly out-performs existing work, especially in the sparse case.

Function optimisation is a major challenge in computer science. The No Free Lunch theorems state that if all functions with the same histogram are assumed to be equally probable then no algorithm outperforms any other in expectation. We argue against the uniform assumption and suggest a universal prior exists for which there is a free lunch, but where no particular class of functions is favoured over another. We also prove upper and lower bounds on the size of the free lunch.

We study the sample-complexity of reinforcement learning in a general setting without assuming ergodicity or finiteness of the environment. Instead, we define a topology on the space of environments and show that if an environment class is compact with respect to this topology then finite sample-complexity bounds are possible and give an algorithm achieving these bounds. We also show the existence of environment classes that are non-compact where finite sample-complexity bounds are not achievable. A lower bound is presented that matches the upper bound except for logarithmic factors.