Tor Lattimore

TMLR Journal 2024 Journal Article

Sequential Best-Arm Identification with Application to P300 Speller

• Xin Zhou
• Botao Hao
• Tor Lattimore
• Jian Kang
• Lexin Li

A brain-computer interface (BCI) is an advanced technology that facilitates direct communication between the human brain and a computer system, by enabling individuals to interact with devices using only their thoughts. The P300 speller is a primary type of BCI system, which allows users to spell words without using a physical keyboard, but instead by capturing and interpreting brain electroencephalogram (EEG) signals under different stimulus presentation paradigms. Traditional non-adaptive presentation paradigms, however, treat each word selection as an isolated event, resulting in a lengthy learning process. To enhance efficiency, we cast the problem as a sequence of best-arm identification tasks within the context of multi-armed bandits, where each task corresponds to the interaction between the user and the system for a single character or word. Leveraging large language models, we utilize the prior knowledge learned from previous tasks to inform and facilitate subsequent tasks. We propose a sequential top-two Thompson sampling algorithm under two scenarios: the fixed-confidence setting and the fixed-budget setting. We study the theoretical property of the proposed algorithm, and demonstrate its substantial empirical improvement through both simulations as well as the data generated from a P300 speller simulator that was built upon the real BCI experiments.

NeurIPS Conference 2023 Conference Paper

Context-lumpable stochastic bandits

• Chung-Wei Lee
• Qinghua Liu
• Yasin Abbasi Yadkori
• Chi Jin
• Tor Lattimore
• Csaba Szepesvari

We consider a contextual bandit problem with $S $ contexts and $K $ actions. In each round $t=1, 2, \dots$ the learnerobserves 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.

ICML Conference 2023 Conference Paper

Distributed Contextual Linear Bandits with Minimax Optimal Communication Cost

• Sanae Amani
• Tor Lattimore
• András György 0001
• Lin F. Yang

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.

ICML Conference 2023 Conference Paper

Leveraging Demonstrations to Improve Online Learning: Quality Matters

• Botao Hao
• Rahul Jain 0002
• Tor Lattimore
• Benjamin Van Roy
• Zheng Wen 0002

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.

JMLR Journal 2023 Journal Article

Linear Partial Monitoring for Sequential Decision Making: Algorithms, Regret Bounds and Applications

• Johannes Kirschner
• Tor Lattimore
• Andreas Krause

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. [abs] [ pdf ][ bib ] &copy JMLR 2023. ( edit, beta )

NeurIPS Conference 2023 Conference Paper

Probabilistic Inference in Reinforcement Learning Done Right

• Jean Tarbouriech
• Tor Lattimore
• Brendan O'Donoghue

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.

EWRL Workshop 2023 Workshop Paper

Probabilistic Inference in Reinforcement Learning Done Right

• Jean Tarbouriech
• Tor Lattimore
• Brendan O'Donoghue

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 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.

ICML Conference 2022 Conference Paper

Contextual Information-Directed Sampling

• Botao Hao
• Tor Lattimore
• Chao Qin

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.

NeurIPS Conference 2022 Conference Paper

Regret Bounds for Information-Directed Reinforcement Learning

• Botao Hao
• Tor Lattimore

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.

NeurIPS Conference 2021 Conference Paper

Bandit Phase Retrieval

• Tor Lattimore
• Botao Hao

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.

AAAI Conference 2021 Conference Paper

Gated Linear Networks

• Joel Veness
• Tor Lattimore
• David Budden
• Avishkar Bhoopchand
• Christopher Mattern
• Agnieszka Grabska-Barwinska
• Eren Sezener
• Jianan Wang

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.

NeurIPS Conference 2021 Conference Paper

Information Directed Sampling for Sparse Linear Bandits

• Botao Hao
• Tor Lattimore
• Wei Deng

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.

UAI Conference 2021 Conference Paper

Matrix games with bandit feedback

• Brendan O'Donoghue
• Tor Lattimore
• Ian Osband

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.

ICML Conference 2021 Conference Paper

On the Optimality of Batch Policy Optimization Algorithms

• Chenjun Xiao
• Yifan Wu
• Jincheng Mei
• Bo Dai 0001
• Tor Lattimore
• Lihong Li 0001
• Csaba Szepesvári
• Dale Schuurmans

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.

ICML Conference 2021 Conference Paper

Sparse Feature Selection Makes Batch Reinforcement Learning More Sample Efficient

• Botao Hao
• Yaqi Duan
• Tor Lattimore
• Csaba Szepesvári
• Mengdi Wang 0001

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.

NeurIPS Conference 2021 Conference Paper

Variational Bayesian Optimistic Sampling

• Brendan O'Donoghue
• Tor Lattimore

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.

ICLR Conference 2020 Conference Paper

Behaviour Suite for Reinforcement Learning

• Ian Osband
• Yotam Doron
• Matteo Hessel
• John Aslanides
• Eren Sezener
• Andre Saraiva 0001
• Katrina McKinney
• Tor Lattimore

This paper introduces the Behaviour Suite for Reinforcement Learning, or bsuite for short. bsuite is a collection of carefully-designed experiments that investigate core capabilities of reinforcement learning (RL) agents with two objectives. First, to collect clear, informative and scalable problems that capture key issues in the design of general and efficient learning algorithms. Second, to study agent behaviour through their performance on these shared benchmarks. To complement this effort, we open source this http URL, which automates evaluation and analysis of any agent on bsuite. This library facilitates reproducible and accessible research on the core issues in RL, and ultimately the design of superior learning algorithms. Our code is Python, and easy to use within existing projects. We include examples with OpenAI Baselines, Dopamine as well as new reference implementations. Going forward, we hope to incorporate more excellent experiments from the research community, and commit to a periodic review of bsuite from a committee of prominent researchers.

NeurIPS Conference 2020 Conference Paper

Gaussian Gated Linear Networks

• David Budden
• Adam Marblestone
• Eren Sezener
• Tor Lattimore
• Gregory Wayne
• Joel Veness

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.

NeurIPS Conference 2020 Conference Paper

High-Dimensional Sparse Linear Bandits

• Botao Hao
• Tor Lattimore
• Mengdi Wang

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.

ICML Conference 2020 Conference Paper

Learning with Good Feature Representations in Bandits and in RL with a Generative Model

• Tor Lattimore
• Csaba Szepesvári
• Gellért Weisz

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.

ICML Conference 2020 Conference Paper

Linear bandits with Stochastic Delayed Feedback

• Claire Vernade
• Alexandra Carpentier
• Tor Lattimore
• Giovanni Zappella
• Beyza Ermis
• Michael Brückner

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.

NeurIPS Conference 2020 Conference Paper

Model Selection in Contextual Stochastic Bandit Problems

• Aldo Pacchiano
• My Phan
• Yasin Abbasi Yadkori
• Anup Rao
• Julian Zimmert
• Tor Lattimore
• Csaba Szepesvari

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.

NeurIPS Conference 2019 Conference Paper

A Geometric Perspective on Optimal Representations for Reinforcement Learning

• Marc Bellemare
• Will Dabney
• Robert Dadashi
• Adrien Ali Taiga
• Pablo Samuel Castro
• Nicolas Le Roux
• Dale Schuurmans
• Tor Lattimore

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.

UAI Conference 2019 Conference Paper

BubbleRank: Safe Online Learning to Re-Rank via Implicit Click Feedback

• Chang Li 0003
• Branislav Kveton
• Tor Lattimore
• Ilya Markov
• Maarten de Rijke
• Csaba Szepesvári
• Masrour Zoghi

In this paper, we study the problem of safe online learning to re-rank, where user feedback is used to improve the quality of displayed lists. Learning to rank has traditionally been studied in two settings. In the offline setting, rankers are typically learned from relevance labels created by judges. This approach has generally become standard in industrial applications of ranking, such as search. However, this approach lacks exploration and thus is limited by the information content of the offline training data. In the online setting, an algorithm can experiment with lists and learn from feedback on them in a sequential fashion. Bandit algorithms are well-suited for this setting but they tend to learn user preferences from scratch, which results in a high initial cost of exploration. This poses an additional challenge of safe exploration in ranked lists. We propose BubbleRank, a bandit algorithm for safe re-ranking that combines the strengths of both the offline and online settings. The algorithm starts with an initial base list and improves it online by gradually exchanging higher-ranked less attractive items for lower-ranked more attractive items. We prove an upper bound on the n-step regret of BubbleRank that degrades gracefully with the quality of the initial base list. Our theoretical findings are supported by extensive experiments on a large-scale real-world click dataset.

NeurIPS Conference 2019 Conference Paper

Connections Between Mirror Descent, Thompson Sampling and the Information Ratio

• Julian Zimmert
• Tor Lattimore

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.

ICML Conference 2019 Conference Paper

Garbage In, Reward Out: Bootstrapping Exploration in Multi-Armed Bandits

• Branislav Kveton
• Csaba Szepesvári
• Sharan Vaswani
• Zheng Wen 0002
• Tor Lattimore
• Mohammad Ghavamzadeh

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.

IJCAI Conference 2019 Conference Paper

Iterative Budgeted Exponential Search

• Malte Helmert
• Tor Lattimore
• Levi H. S. Lelis
• Laurent Orseau
• Nathan R. Sturtevant

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*.

UAI Conference 2019 Conference Paper

On First-Order Bounds, Variance and Gap-Dependent Bounds for Adversarial Bandits

• Roman Pogodin
• Tor Lattimore

We make three contributions to the theory of k-armed adversarial bandits. First, we prove a first-order bound for a modified variant of the INF strategy by Audibert and Bubeck [2009], without sacrificing worst case optimality or modifying the loss estimators. Second, we provide a variance analysis for algorithms based on follow the regularised leader, showing that without adaptation the variance of the regret is typically {\Omega}(n^2) where n is the horizon. Finally, we study bounds that depend on the degree of separation of the arms, generalising the results by Cowan and Katehakis [2015] from the stochastic setting to the adversarial and improving the result of Seldin and Slivkins [2014] by a factor of log(n)/log(log(n)).

ICML Conference 2019 Conference Paper

Online Learning to Rank with Features

• Shuai Li 0010
• Tor Lattimore
• Csaba Szepesvári

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.

JMLR Journal 2018 Journal Article

Refining the Confidence Level for Optimistic Bandit Strategies

• Tor Lattimore

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. [abs] [ pdf ][ bib ] &copy JMLR 2018. ( edit, beta )

NeurIPS Conference 2018 Conference Paper

Single-Agent Policy Tree Search With Guarantees

• Laurent Orseau
• Levi Lelis
• Tor Lattimore
• Theophane Weber

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.

NeurIPS Conference 2018 Conference Paper

TopRank: A practical algorithm for online stochastic ranking

• Tor Lattimore
• Branislav Kveton
• Shuai Li
• Csaba Szepesvari

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.

NeurIPS Conference 2017 Conference Paper

A Scale Free Algorithm for Stochastic Bandits with Bounded Kurtosis

• Tor Lattimore

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.

JMLR Journal 2017 Journal Article

Following the Leader and Fast Rates in Online Linear Prediction: Curved Constraint Sets and Other Regularities

• Ruitong Huang
• Tor Lattimore
• András György
• Csaba Szepesvári

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. [abs] [ pdf ][ bib ] &copy JMLR 2017. ( edit, beta )

IJCAI Conference 2017 Conference Paper

On Thompson Sampling and Asymptotic Optimality

• Jan Leike
• Tor Lattimore
• Laurent Orseau
• Marcus Hutter

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.

RLDM Conference 2017 Conference Abstract

UBEV - A More Practical Algorithm for Episodic RL with Near-Optimal PAC and Regret Guar- antees

• Christoph Dann
• Tor Lattimore
• Emma Brunskill

We present UBEV, a simple and efficient reinforcement learning algorithm for fixed-horizon episodic Markov decision processes. The main contribution is a proof that UBEV enjoys a sample-complexity bound that holds for all accuracy levels simultaneously with high probability, and matches the lower bound except for logarithmic terms and one factor of the horizon. A consequence of the fact that our sample- complexity bound holds for all accuracy levels is that the new algorithm achieves a sub-linear regret of O(sqrt(SAT)), which is the first time the dependence on the size of the state space has provably appeared inside the square root. A brief empirical evaluation shows that UBEV is practically superior to existing algorithms with known sample-complexity guarantees.

NeurIPS Conference 2017 Conference Paper

Unifying PAC and Regret: Uniform PAC Bounds for Episodic Reinforcement Learning

• Christoph Dann
• Tor Lattimore
• Emma Brunskill

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.

NeurIPS Conference 2016 Conference Paper

Causal Bandits: Learning Good Interventions via Causal Inference

• Finnian Lattimore
• Tor Lattimore
• Mark Reid

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.

ICML Conference 2016 Conference Paper

Conservative Bandits

• Yifan Wu
• Roshan Shariff
• Tor Lattimore
• Csaba Szepesvári

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.

NeurIPS Conference 2016 Conference Paper

Following the Leader and Fast Rates in Linear Prediction: Curved Constraint Sets and Other Regularities

• Ruitong Huang
• Tor Lattimore
• András György
• Csaba Szepesvari

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.

NeurIPS Conference 2016 Conference Paper

On Explore-Then-Commit strategies

• Aurelien Garivier
• Tor Lattimore
• Emilie Kaufmann

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.

NeurIPS Conference 2016 Conference Paper

Refined Lower Bounds for Adversarial Bandits

• Sébastien Gerchinovitz
• Tor Lattimore

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.

UAI Conference 2016 Conference Paper

Thompson Sampling is Asymptotically Optimal in General Environments

• Jan Leike
• Tor Lattimore
• Laurent Orseau
• Marcus Hutter

We discuss two different notions of optimality: asymptotic optimality and worst-case regret. We discuss a variant of Thompson sampling for nonparametric reinforcement learning in countable classes of general stochastic environments. These environments can be non-Markov, nonergodic, and partially observable. We show that Thompson sampling learns the environment class in the sense that (1) asymptotically its value converges to the optimal value in mean and (2) given a recoverability assumption regret is sublinear. Asymptotic optimality requires that asymptotically the agent learns to act optimally, i. e. , that the discounted value of the agent’s policy π converges to the optimal discounted value, Vµ∗ − Vµπ → 0 for all environments µ from the environment class. This convergence is impossible for deterministic policies since the agent has to explore infinitely often and for long stretches of time, but there are policies that converge almost surely in Cesàro average [LH11]. Bayes-optimal agents are generally not asymptotically optimal [Ors13]. However, asymptotic optimality can be achieved through an exploration component on top of a Bayes-optimal agent [Lat13, Ch. 5] or through optimism [SH15].

NeurIPS Conference 2015 Conference Paper

Linear Multi-Resource Allocation with Semi-Bandit Feedback

• Tor Lattimore
• Koby Crammer
• Csaba Szepesvari

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.

TCS Journal 2015 Journal Article

On Martin-Löf (non-)convergence of Solomonoff's universal mixture

• Tor Lattimore
• Marcus Hutter

We study the convergence of Solomonoff's universal mixture on individual Martin-Löf random sequences. A new result is presented extending the work of Hutter and Muchnik [3] by showing that there does not exist a universal mixture that converges on all Martin-Löf random sequences. We show that this is not an artifact of the fact that the universal mixture is not a proper measure and that the normalised universal mixture also fails to converge on all Martin-Löf random sequences.

NeurIPS Conference 2015 Conference Paper

The Pareto Regret Frontier for Bandits

• Tor Lattimore

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.

NeurIPS Conference 2014 Conference Paper

Bounded Regret for Finite-Armed Structured Bandits

• Tor Lattimore
• Remi Munos

We study a new type of K-armed bandit problem where the expected return of one arm may depend on the returns of other arms. We present a new algorithm for this general class of problems and show that under certain circumstances it is possible to achieve finite expected cumulative regret. We also give problem-dependent lower bounds on the cumulative regret showing that at least in special cases the new algorithm is nearly optimal.

TCS Journal 2014 Journal Article

General time consistent discounting

• Tor Lattimore
• Marcus Hutter

Modeling inter-temporal choice is a key problem in both computer science and economic theory. The discounted utility model of Samuelson is currently the most popular model for measuring the global utility of a time-series of local utilities. The model is limited by not allowing the discount function to change with the age of the agent. This is despite the fact that many agents, in particular humans, are best modelled with age-dependent discount functions. It is well known that discounting can lead to time-inconsistent behaviour where agents change their preferences over time. In this paper we generalise the discounted utility model to allow age-dependent discount functions. We then extend previous work in time-inconsistency to our new setting, including a complete characterisation of time-(in)consistent discount functions, the existence of sub-game perfect equilibrium policies where the discount function is time-inconsistent and a continuity result showing that “nearly” time-consistent discount rates lead to “nearly” time-consistent behaviour.

TCS Journal 2014 Journal Article

Near-optimal PAC bounds for discounted MDPs

• Tor Lattimore
• Marcus Hutter

We study upper and lower bounds on the sample-complexity of learning near-optimal behaviour in finite-state discounted Markov Decision Processes (mdps). We prove a new bound for a modified version of Upper Confidence Reinforcement Learning (ucrl) with only cubic dependence on the horizon. The bound is unimprovable in all parameters except the size of the state/action space, where it depends linearly on the number of non-zero transition probabilities. The lower bound strengthens previous work by being both more general (it applies to all policies) and tighter. The upper and lower bounds match up to logarithmic factors provided the transition matrix is not too dense.

UAI Conference 2014 Conference Paper

Optimal Resource Allocation with Semi-Bandit Feedback

• Tor Lattimore
• Koby Crammer
• Csaba Szepesvári

We study a sequential resource allocation problem involving a fixed number of recurring jobs. At each time-step the manager should distribute available resources among the jobs in order to maximise the expected number of completed jobs. Allocating more resources to a given job increases the probability that it completes, but with a cut-off. Specifically, we assume a linear model where the probability increases linearly until it equals one, after which allocating additional resources is wasteful. We assume the difficulty of each job is unknown and present the first algorithm for this problem and prove upper and lower bounds on its regret. Despite its apparent simplicity, the problem has a rich structure: we show that an appropriate optimistic algorithm can improve its learning speed dramatically beyond the results one normally expects for similar problems as the problem becomes resource-laden.

EWRL Workshop 2013 Workshop Paper

Bayesian Reinforcement Learning + Exploration

• Tor Lattimore
• Marcus Hutter

EWRL Workshop 2013 Workshop Paper

Optimal Universal Explorative Agents

• Laurent Orseau
• Tor Lattimore
• Marcus Hutter

ICML Conference 2013 Conference Paper

The Sample-Complexity of General Reinforcement Learning

• Tor Lattimore
• Marcus Hutter
• Peter Sunehag

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.