Author name cluster

Michal Valko

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83 papers

2 author rows

ICML Conference 2025 Conference Paper

The Harder Path: Last Iterate Convergence for Uncoupled Learning in Zero-Sum Games with Bandit Feedback

Côme Fiegel
Pierre Ménard
Tadashi Kozuno
Michal Valko
Vianney Perchet

We study the problem of learning in zero-sum matrix games with repeated play and bandit feedback. Specifically, we focus on developing uncoupled algorithms that guarantee, without communication between players, convergence of the last-iterate to a Nash equilibrium. Although the non-bandit case has been studied extensively, this setting has only been explored recently, with a bound of $\mathcal{O}(T^{-1/8})$ on the exploitability gap. We show that, for uncoupled algorithms, guaranteeing convergence of the policy profiles to a Nash equilibrium is detrimental to the performances, with the best attainable rate being $\mathcal{O}(T^{-1/4})$ in contrast to the usual $\mathcal{O}(T^{-1/2})$ rate for convergence of the average iterates. We then propose two algorithms that achieve this optimal rate. The first algorithm leverages a straightforward tradeoff between exploration and exploitation, while the second employs a regularization technique based on a two-step mirror descent approach.

ICML Conference 2024 Conference Paper

Decoding-time Realignment of Language Models

Tianlin Liu
Shangmin Guo
Leonardo Bianco
Daniele Calandriello
Quentin Berthet
Felipe Llinares-López
Jessica Hoffmann
Lucas Dixon

Aligning language models with human preferences is crucial for reducing errors and biases in these models. Alignment techniques, such as reinforcement learning from human feedback (RLHF), are typically cast as optimizing a tradeoff between human preference rewards and a proximity regularization term that encourages staying close to the unaligned model. Selecting an appropriate level of regularization is critical: insufficient regularization can lead to reduced model capabilities due to reward hacking, whereas excessive regularization hinders alignment. Traditional methods for finding the optimal regularization level require retraining multiple models with varying regularization strengths. This process, however, is resource-intensive, especially for large models. To address this challenge, we propose decoding-time realignment (DeRa), a simple method to explore and evaluate different regularization strengths in aligned models without retraining. DeRa enables control over the degree of alignment, allowing users to smoothly transition between unaligned and aligned models. It also enhances the efficiency of hyperparameter tuning by enabling the identification of effective regularization strengths using a validation dataset.

ICLR Conference 2024 Conference Paper

Demonstration-Regularized RL

Daniil Tiapkin
Denis Belomestny
Daniele Calandriello
Eric Moulines
Alexey Naumov
Pierre Perrault
Michal Valko
Pierre Ménard

Incorporating expert demonstrations has empirically helped to improve the sample efficiency of reinforcement learning (RL). This paper quantifies theoretically to what extent this extra information reduces RL's sample complexity. In particular, we study the demonstration-regularized reinforcement learning framework that leverages the expert demonstrations by $\mathrm{KL}$-regularization for a policy learned by behavior cloning. Our findings reveal that using $N^{\mathrm{E}}$ expert demonstrations enables the identification of an optimal policy at a sample complexity of order $\widetilde{\mathcal{O}}(\mathrm{Poly}(S,A,H)/(\varepsilon^2 N^{\mathrm{E}}))$ in finite and $\widetilde{\mathcal{O}}(\mathrm{Poly}(d,H)/(\varepsilon^2 N^{\mathrm{E}}))$ in linear Markov decision processes, where $\varepsilon$is the target precision, $H$ the horizon, $A$ the number of action, $S$ the number of states in the finite case and $d$ the dimension of the feature space in the linear case. As a by-product, we provide tight convergence guarantees for the behavior cloning procedure under general assumptions on the policy classes. Additionally, we establish that demonstration-regularized methods are provably efficient for reinforcement learning from human feedback (RLHF). In this respect, we provide theoretical evidence showing the benefits of KL-regularization for RLHF in tabular and linear MDPs. Interestingly, we avoid pessimism injection by employing computationally feasible regularization to handle reward estimation uncertainty, thus setting our approach apart from the prior works.

ICML Conference 2024 Conference Paper

Generalized Preference Optimization: A Unified Approach to Offline Alignment

Yunhao Tang
Daniel Guo 0001
Zeyu Zheng
Daniele Calandriello
Rémi Munos
Mark Rowland 0001
Pierre Harvey Richemond
Michal Valko

Offline preference optimization allows fine-tuning large models directly from offline data, and has proved effective in recent alignment practices. We propose generalized preference optimization (GPO), a family of offline losses parameterized by a general class of convex functions. GPO enables a unified view over preference optimization, encompassing existing algorithms such as DPO, IPO and SLiC as special cases, while naturally introducing new variants. The GPO framework also sheds light on how offline algorithms enforce regularization, through the design of the convex function that defines the loss. Our analysis and experiments reveal the connections and subtle differences between the offline regularization and the KL divergence regularization intended by the canonical RLHF formulation. In a controlled setting akin to Gao et al 2023, we also show that different GPO variants achieve similar trade-offs between regularization and performance, though the optimal values of hyper-parameter might differ as predicted by theory. In all, our results present new algorithmic toolkits and empirical insights to alignment practitioners.

ICML Conference 2024 Conference Paper

Human Alignment of Large Language Models through Online Preference Optimisation

Daniele Calandriello
Daniel Guo 0001
Rémi Munos
Mark Rowland 0001
Yunhao Tang
Bernardo Ávila Pires
Pierre Harvey Richemond
Charline Le Lan

Ensuring alignment of language model’s outputs with human preferences is critical to guarantee a useful, safe, and pleasant user experience. Thus, human alignment has been extensively studied recently and several methods such as Reinforcement Learning from Human Feedback (RLHF), Direct Policy Optimisation (DPO) and Sequence Likelihood Calibration (SLiC) have emerged. In this paper, our contribution is two-fold. First, we show the equivalence between two recent alignment methods, namely Identity Policy Optimisation (IPO) and Nash Mirror Descent (Nash-MD). Second, we introduce a generalisation of IPO, named IPO-MD, that leverages the regularised sampling approach proposed by Nash-MD. This equivalence may seem surprising at first sight, since IPO is an offline method whereas Nash-MD is an online method using a preference model. However, this equivalence can be proven when we consider the online version of IPO, that is when both generations are sampled by the online policy and annotated by a trained preference model. Optimising the IPO loss with such a stream of data becomes then equivalent to finding the Nash equilibrium of the preference model through self-play. Building on this equivalence, we introduce the IPO-MD algorithm that generates data with a mixture policy (between the online and reference policy) similarly as the general Nash-MD algorithm. We compare online-IPO and IPO-MD to different online versions of existing losses on preference data such as DPO and SLiC on a summarisation task.

NeurIPS Conference 2024 Conference Paper

Local and Adaptive Mirror Descents in Extensive-Form Games

Côme Fiegel
Pierre Ménard
Tadashi Kozuno
Rémi Munos
Vianney Perchet
Michal Valko

We study how to learn $\epsilon$-optimal strategies in zero-sum imperfect information games (IIG) with *trajectory feedback*. In this setting, players update their policies sequentially, based on their observations over a fixed number of episodes denoted by $T$. Most existing procedures suffer from high variance due to the use of importance sampling over sequences of actions. To reduce this variance, we consider a *fixed sampling* approach, where players still update their policies over time, but with observations obtained through a given fixed sampling policy. Our approach is based on an adaptive Online Mirror Descent (OMD) algorithm that applies OMD locally to each information set, using individually decreasing learning rates and a *regularized loss*. We show that this approach guarantees a convergence rate of $\tilde{\mathcal{O}}(T^{-1/2})$ with high probability and has a near-optimal dependence on the game parameters when applied with the best theoretical choices of learning rates and sampling policies. To achieve these results, we generalize the notion of OMD stabilization, allowing for time-varying regularization with convex increments.

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NeurIPS Conference 2024 Conference Paper

Metacognitive Capabilities of LLMs: An Exploration in Mathematical Problem Solving

Aniket Didolkar
Anirudh Goyal
Nan R. Ke
Siyuan Guo
Michal Valko
Timothy Lillicrap
Danilo Rezende
Yoshua Bengio

\emph{Metacognitive knowledge} refers to humans' intuitive knowledge of their own thinking and reasoning processes. Today's best LLMs clearly possess some reasoning processes. The paper gives evidence that they also have metacognitive knowledge, including ability to name skills and procedures to apply given a task. We explore this primarily in context of math reasoning, developing a prompt-guided interaction procedure to get a powerful LLM to assign sensible skill labels to math questions, followed by having it perform semantic clustering to obtain coarser families of skill labels. These coarse skill labels look interpretable to humans. To validate that these skill labels are meaningful and relevant to the LLM's reasoning processes we perform the following experiments. (a) We ask GPT-4 to assign skill labels to training questions in math datasets GSM8K and MATH. (b) When using an LLM to solve the test questions, we present it with the full list of skill labels and ask it to identify the skill needed. Then it is presented with randomly selected exemplar solved questions associated with that skill label. This improves accuracy on GSM8k and MATH for several strong LLMs, including code-assisted models. The methodology presented is domain-agnostic, even though this article applies it to math problems.

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ICML Conference 2024 Conference Paper

Nash Learning from Human Feedback

Rémi Munos
Michal Valko
Daniele Calandriello
Mohammad Gheshlaghi Azar
Mark Rowland 0001
Daniel Guo 0001
Yunhao Tang
Matthieu Geist

Reinforcement learning from human feedback (RLHF) has emerged as the main paradigm for aligning large language models (LLMs) with human preferences. Traditionally, RLHF involves the initial step of learning a reward model from pairwise human feedback, i. e. , expressed as preferences between pairs of text generations. Subsequently, the LLM’s policy is fine-tuned to maximize the reward through a reinforcement learning algorithm. In this study, we introduce an alternative pipeline for the fine-tuning of LLMs using pairwise human feedback. Our approach entails the initial learning of a pairwise preference model, which is conditioned on two inputs (instead of a single input in the case of a reward model) given a prompt, followed by the pursuit of a policy that consistently generates responses preferred over those generated by any competing policy, thus defining the Nash equilibrium of this preference model. We term this approach Nash learning from human feedback (NLHF). In the context of a tabular policy representation, we present a novel algorithmic solution, Nash-MD, founded on the principles of mirror descent. This algorithm produces a sequence of policies, with the last iteration converging to the regularized Nash equilibrium. Additionally, we explore parametric representations of policies and introduce gradient descent algorithms for deep-learning architectures. We illustrate the effectiveness of our approach by presenting experimental results on a text summarization task. We believe NLHF offers a compelling avenue for fine-tuning LLMs and enhancing the alignment of LLMs with human preferences.

ICLR Conference 2024 Conference Paper

Unlocking the Power of Representations in Long-term Novelty-based Exploration

Alaa Saade
Steven Kapturowski
Daniele Calandriello
Charles Blundell
Pablo Sprechmann
Leopoldo Sarra
Oliver Groth
Michal Valko

We introduce Robust Exploration via Clustering-based Online Density Estimation (RECODE), a non-parametric method for novelty-based exploration that estimates visitation counts for clusters of states based on their similarity in a chosen embedding space. By adapting classical clustering to the nonstationary setting of Deep RL, RECODE can efficiently track state visitation counts over thousands of episodes. We further propose a novel generalization of the inverse dynamics loss, which leverages masked transformer architectures for multi-step prediction; which in conjunction with \DETOCS achieves a new state-of-the-art in a suite of challenging 3D-exploration tasks in DM-Hard-8. RECODE also sets new state-of-the-art in hard exploration Atari games, and is the first agent to reach the end screen in "Pitfall!"

ICML Conference 2023 Conference Paper

Adapting to game trees in zero-sum imperfect information games

Côme Fiegel
Pierre Ménard
Tadashi Kozuno
Rémi Munos
Vianney Perchet
Michal Valko

Imperfect information games (IIG) are games in which each player only partially observes the current game state. We study how to learn $\epsilon$-optimal strategies in a zero-sum IIG through self-play with trajectory feedback. We give a problem-independent lower bound $\widetilde{\mathcal{O}}(H(A_{\mathcal{X}}+B_{\mathcal{Y}})/\epsilon^2)$ on the required number of realizations to learn these strategies with high probability, where $H$ is the length of the game, $A_{\mathcal{X}}$ and $B_{\mathcal{Y}}$ are the total number of actions for the two players. We also propose two Follow the Regularized leader (FTRL) algorithms for this setting: Balanced FTRL which matches this lower bound, but requires the knowledge of the information set structure beforehand to define the regularization; and Adaptive FTRL which needs $\widetilde{\mathcal{O}}(H^2(A_{\mathcal{X}}+B_{\mathcal{Y}})/\epsilon^2)$ realizations without this requirement by progressively adapting the regularization to the observations.

ICML Conference 2023 Conference Paper

Curiosity in Hindsight: Intrinsic Exploration in Stochastic Environments

Daniel Jarrett
Corentin Tallec
Florent Altché
Thomas Mesnard
Rémi Munos
Michal Valko

Consider the problem of exploration in sparse-reward or reward-free environments, such as in Montezuma’s Revenge. In the curiosity-driven paradigm, the agent is rewarded for how much each realized outcome differs from their predicted outcome. But using predictive error as intrinsic motivation is fragile in stochastic environments, as the agent may become trapped by high-entropy areas of the state-action space, such as a "noisy TV". In this work, we study a natural solution derived from structural causal models of the world: Our key idea is to learn representations of the future that capture precisely the unpredictable aspects of each outcome—which we use as additional input for predictions, such that intrinsic rewards only reflect the predictable aspects of world dynamics. First, we propose incorporating such hindsight representations into models to disentangle "noise" from "novelty", yielding Curiosity in Hindsight: a simple and scalable generalization of curiosity that is robust to stochasticity. Second, we instantiate this framework for the recently introduced BYOL-Explore algorithm as our prime example, resulting in the noise-robust BYOL-Hindsight. Third, we illustrate its behavior under a variety of different stochasticities in a grid world, and find improvements over BYOL-Explore in hard-exploration Atari games with sticky actions. Notably, we show state-of-the-art results in exploring Montezuma’s Revenge with sticky actions, while preserving performance in the non-sticky setting.

ICML Conference 2023 Conference Paper

DoMo-AC: Doubly Multi-step Off-policy Actor-Critic Algorithm

Yunhao Tang
Tadashi Kozuno
Mark Rowland 0001
Anna Harutyunyan
Rémi Munos
Bernardo Ávila Pires
Michal Valko

Multi-step learning applies lookahead over multiple time steps and has proved valuable in policy evaluation settings. However, in the optimal control case, the impact of multi-step learning has been relatively limited despite a number of prior efforts. Fundamentally, this might be because multi-step policy improvements require operations that cannot be approximated by stochastic samples, hence hindering the widespread adoption of such methods in practice. To address such limitations, we introduce doubly multi-step off-policy VI (DoMo-VI), a novel oracle algorithm that combines multi-step policy improvements and policy evaluations. DoMo-VI enjoys guaranteed convergence speed-up to the optimal policy and is applicable in general off-policy learning settings. We then propose doubly multi-step off-policy actor-critic (DoMo-AC), a practical instantiation of the DoMo-VI algorithm. DoMo-AC introduces a bias-variance trade-off that ensures improved policy gradient estimates. When combined with the IMPALA architecture, DoMo-AC has showed improvements over the baseline algorithm on Atari-57 game benchmarks.

ICML Conference 2023 Conference Paper

Fast Rates for Maximum Entropy Exploration

Daniil Tiapkin
Denis Belomestny
Daniele Calandriello
Eric Moulines
Rémi Munos
Alexey Naumov
Pierre Perrault
Yunhao Tang

We address the challenge of exploration in reinforcement learning (RL) when the agent operates in an unknown environment with sparse or no rewards. In this work, we study the maximum entropy exploration problem of two different types. The first type is visitation entropy maximization previously considered by Hazan et al. (2019) in the discounted setting. For this type of exploration, we propose a game-theoretic algorithm that has $\widetilde{\mathcal{O}}(H^3S^2A/\varepsilon^2)$ sample complexity thus improving the $\varepsilon$-dependence upon existing results, where $S$ is a number of states, $A$ is a number of actions, $H$ is an episode length, and $\varepsilon$ is a desired accuracy. The second type of entropy we study is the trajectory entropy. This objective function is closely related to the entropy-regularized MDPs, and we propose a simple algorithm that has a sample complexity of order $\widetilde{\mathcal{O}}(\mathrm{poly}(S, A, H)/\varepsilon)$. Interestingly, it is the first theoretical result in RL literature that establishes the potential statistical advantage of regularized MDPs for exploration. Finally, we apply developed regularization techniques to reduce sample complexity of visitation entropy maximization to $\widetilde{\mathcal{O}}(H^2SA/\varepsilon^2)$, yielding a statistical separation between maximum entropy exploration and reward-free exploration.

ICML Conference 2023 Conference Paper

Half-Hop: A graph upsampling approach for slowing down message passing

Mehdi Azabou
Venkataramana Ganesh
Shantanu Thakoor
Chi-Heng Lin
Lakshmi Sathidevi
Ran Liu
Michal Valko
Petar Velickovic

Message passing neural networks have shown a lot of success on graph-structured data. However, there are many instances where message passing can lead to over-smoothing or fail when neighboring nodes belong to different classes. In this work, we introduce a simple yet general framework for improving learning in message passing neural networks. Our approach essentially upsamples edges in the original graph by adding "slow nodes" at each edge that can mediate communication between a source and a target node. Our method only modifies the input graph, making it plug-and-play and easy to use with existing models. To understand the benefits of slowing down message passing, we provide theoretical and empirical analyses. We report results on several supervised and self-supervised benchmarks, and show improvements across the board, notably in heterophilic conditions where adjacent nodes are more likely to have different labels. Finally, we show how our approach can be used to generate augmentations for self-supervised learning, where slow nodes are randomly introduced into different edges in the graph to generate multi-scale views with variable path lengths.

NeurIPS Conference 2023 Conference Paper

Model-free Posterior Sampling via Learning Rate Randomization

Daniil Tiapkin
Denis Belomestny
Daniele Calandriello
Eric Moulines
Remi Munos
Alexey Naumov
Pierre Perrault
Michal Valko

In this paper, we introduce Randomized Q-learning (RandQL), a novel randomized model-free algorithm for regret minimization in episodic Markov Decision Processes (MDPs). To the best of our knowledge, RandQL is the first tractable model-free posterior sampling-based algorithm. We analyze the performance of RandQL in both tabular and non-tabular metric space settings. In tabular MDPs, RandQL achieves a regret bound of order $\widetilde{\mathcal{O}}(\sqrt{H^{5}SAT})$, where $H$ is the planning horizon, $S$ is the number of states, $A$ is the number of actions, and $T$ is the number of episodes. For a metric state-action space, RandQL enjoys a regret bound of order $\widetilde{\mathcal{O}}(H^{5/2} T^{(d_z+1)/(d_z+2)})$, where $d_z$ denotes the zooming dimension. Notably, RandQL achieves optimistic exploration without using bonuses, relying instead on a novel idea of learning rate randomization. Our empirical study shows that RandQL outperforms existing approaches on baseline exploration environments.

ICML Conference 2023 Conference Paper

Quantile Credit Assignment

Thomas Mesnard
Wenqi Chen
Alaa Saade
Yunhao Tang
Mark Rowland 0001
Theophane Weber
Clare Lyle
Audrunas Gruslys

In reinforcement learning, the credit assignment problem is to distinguish luck from skill, that is, separate the inherent randomness in the environment from the controllable effects of the agent’s actions. This paper proposes two novel algorithms, Quantile Credit Assignment (QCA) and Hindsight QCA (HQCA), which incorporate distributional value estimation to perform credit assignment. QCA uses a network that predicts the quantiles of the return distribution, whereas HQCA additionally incorporates information about the future. Both QCA and HQCA have the appealing interpretation of leveraging an estimate of the quantile level of the return (interpreted as the level of "luck") in order to derive a "luck-dependent" baseline for policy gradient methods. We show theoretically that this approach gives an unbiased policy gradient estimate that can yield significant variance reductions over a standard value estimate baseline. QCA and HQCA significantly outperform prior state-of-the-art methods on a range of extremely difficult credit assignment problems.

ICML Conference 2023 Conference Paper

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

Toshinori Kitamura
Tadashi Kozuno
Yunhao Tang
Nino Vieillard
Michal Valko
Wenhao Yang
Jincheng Mei
Pierre Ménard

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

ICML Conference 2023 Conference Paper

Understanding Self-Predictive Learning for Reinforcement Learning

Yunhao Tang
Daniel Guo 0001
Pierre Harvey Richemond
Bernardo Ávila Pires
Yash Chandak
Rémi Munos
Mark Rowland 0001
Mohammad Gheshlaghi Azar

We study the learning dynamics of self-predictive learning for reinforcement learning, a family of algorithms that learn representations by minimizing the prediction error of their own future latent representations. Despite its recent empirical success, such algorithms have an apparent defect: trivial representations (such as constants) minimize the prediction error, yet it is obviously undesirable to converge to such solutions. Our central insight is that careful designs of the optimization dynamics are critical to learning meaningful representations. We identify that a faster paced optimization of the predictor and semi-gradient updates on the representation, are crucial to preventing the representation collapse. Then in an idealized setup, we show self-predictive learning dynamics carries out spectral decomposition on the state transition matrix, effectively capturing information of the transition dynamics. Building on the theoretical insights, we propose bidirectional self-predictive learning, a novel self-predictive algorithm that learns two representations simultaneously. We examine the robustness of our theoretical insights with a number of small-scale experiments and showcase the promise of the novel representation learning algorithm with large-scale experiments.

ICML Conference 2023 Conference Paper

VA-learning as a more efficient alternative to Q-learning

Yunhao Tang
Rémi Munos
Mark Rowland 0001
Michal Valko

In reinforcement learning, the advantage function is critical for policy improvement, but is often extracted from a learned Q-function. A natural question is: Why not learn the advantage function directly? In this work, we introduce VA-learning, which directly learns advantage function and value function using bootstrapping, without explicit reference to Q-functions. VA-learning learns off-policy and enjoys similar theoretical guarantees as Q-learning. Thanks to the direct learning of advantage function and value function, VA-learning improves the sample efficiency over Q-learning both in tabular implementations and deep RL agents on Atari-57 games. We also identify a close connection between VA-learning and the dueling architecture, which partially explains why a simple architectural change to DQN agents tends to improve performance.

NeurIPS Conference 2022 Conference Paper

BYOL-Explore: Exploration by Bootstrapped Prediction

Zhaohan Guo
Shantanu Thakoor
Miruna Pislar
Bernardo Avila Pires
Florent Altché
Corentin Tallec
Alaa Saade
Daniele Calandriello

We present BYOL-Explore, a conceptually simple yet general approach for curiosity-driven exploration in visually complex environments. BYOL-Explore learns the world representation, the world dynamics and the exploration policy all-together by optimizing a single prediction loss in the latent space with no additional auxiliary objective. We show that BYOL-Explore is effective in DM-HARD-8, a challenging partially-observable continuous-action hard-exploration benchmark with visually rich 3-D environment. On this benchmark, we solve the majority of the tasks purely through augmenting the extrinsic reward with BYOL-Explore intrinsic reward, whereas prior work could only get off the ground with human demonstrations. As further evidence of the generality of BYOL-Explore, we show that it achieves superhuman performance on the ten hardest exploration games in Atari while having a much simpler design than other competitive agents.

ICML Conference 2022 Conference Paper

From Dirichlet to Rubin: Optimistic Exploration in RL without Bonuses

Daniil Tiapkin
Denis Belomestny
Eric Moulines
Alexey Naumov
Sergey Samsonov
Yunhao Tang
Michal Valko
Pierre Ménard

We propose the Bayes-UCBVI algorithm for reinforcement learning in tabular, stage-dependent, episodic Markov decision process: a natural extension of the Bayes-UCB algorithm by Kaufmann et al. 2012 for multi-armed bandits. Our method uses the quantile of a Q-value function posterior as upper confidence bound on the optimal Q-value function. For Bayes-UCBVI, we prove a regret bound of order $\widetilde{\mathcal{O}}(\sqrt{H^3SAT})$ where $H$ is the length of one episode, $S$ is the number of states, $A$ the number of actions, $T$ the number of episodes, that matches the lower-bound of $\Omega(\sqrt{H^3SAT})$ up to poly-$\log$ terms in $H, S, A, T$ for a large enough $T$. To the best of our knowledge, this is the first algorithm that obtains an optimal dependence on the horizon $H$ (and $S$) without the need of an involved Bernstein-like bonus or noise. Crucial to our analysis is a new fine-grained anti-concentration bound for a weighted Dirichlet sum that can be of independent interest. We then explain how Bayes-UCBVI can be easily extended beyond the tabular setting, exhibiting a strong link between our algorithm and Bayesian bootstrap (Rubin, 1981).

ICLR Conference 2022 Conference Paper

Large-Scale Representation Learning on Graphs via Bootstrapping

Shantanu Thakoor
Corentin Tallec
Mohammad Gheshlaghi Azar
Mehdi Azabou
Eva L. Dyer
Rémi Munos
Petar Velickovic
Michal Valko

Self-supervised learning provides a promising path towards eliminating the need for costly label information in representation learning on graphs. However, to achieve state-of-the-art performance, methods often need large numbers of negative examples and rely on complex augmentations. This can be prohibitively expensive, especially for large graphs. To address these challenges, we introduce Bootstrapped Graph Latents (BGRL) - a graph representation learning method that learns by predicting alternative augmentations of the input. BGRL uses only simple augmentations and alleviates the need for contrasting with negative examples, and thus is scalable by design. BGRL outperforms or matches prior methods on several established benchmarks, while achieving a 2-10x reduction in memory costs. Furthermore, we show that BGRL can be scaled up to extremely large graphs with hundreds of millions of nodes in the semi-supervised regime, achieving state-of-the-art performance and improving over supervised baselines where representations are shaped only through label information. In particular, our solution centered on BGRL constituted one of the winning entries to the Open Graph Benchmark -Large Scale Challenge at KDD Cup 2021, on a graph orders of magnitudes larger than all previously available benchmarks, thus demonstrating the scalability and effectiveness of our approach.

NeurIPS Conference 2022 Conference Paper

Optimistic Posterior Sampling for Reinforcement Learning with Few Samples and Tight Guarantees

Daniil Tiapkin
Denis Belomestny
Daniele Calandriello
Eric Moulines
Remi Munos
Alexey Naumov
Mark Rowland
Michal Valko

We consider reinforcement learning in an environment modeled by an episodic, tabular, step-dependent Markov decision process of horizon $H$ with $S$ states, and $A$ actions. The performance of an agent is measured by the regret after interacting with the environment for $T$ episodes. We propose an optimistic posterior sampling algorithm for reinforcement learning (OPSRL), a simple variant of posterior sampling that only needs a number of posterior samples logarithmic in $H$, $S$, $A$, and $T$ per state-action pair. For OPSRL we guarantee a high-probability regret bound of order at most $O(\sqrt{H^3SAT})$ ignoring $\text{poly}\log(HSAT)$ terms. The key novel technical ingredient is a new sharp anti-concentration inequality for linear forms of a Dirichlet random vector which may be of independent interest. Specifically, we extend the normal approximation-based lower bound for Beta distributions by Alfers and Dinges (1984) to Dirichlet distributions. Our bound matches the lower bound of order $\Omega(\sqrt{H^3SAT})$, thereby answering the open problems raised by Agrawal and Jia (2017) for the episodic setting.

ICML Conference 2022 Conference Paper

Retrieval-Augmented Reinforcement Learning

Anirudh Goyal
Abram L. Friesen
Andrea Banino
Theophane Weber
Nan Rosemary Ke
Adrià Puigdomènech Badia
Arthur Guez
Mehdi Mirza

Most deep reinforcement learning (RL) algorithms distill experience into parametric behavior policies or value functions via gradient updates. While effective, this approach has several disadvantages: (1) it is computationally expensive, (2) it can take many updates to integrate experiences into the parametric model, (3) experiences that are not fully integrated do not appropriately influence the agent’s behavior, and (4) behavior is limited by the capacity of the model. In this paper we explore an alternative paradigm in which we train a network to map a dataset of past experiences to optimal behavior. Specifically, we augment an RL agent with a retrieval process (parameterized as a neural network) that has direct access to a dataset of experiences. This dataset can come from the agent’s past experiences, expert demonstrations, or any other relevant source. The retrieval process is trained to retrieve information from the dataset that may be useful in the current context, to help the agent achieve its goal faster and more efficiently. The proposed method facilitates learning agents that at test time can condition their behavior on the entire dataset and not only the current state, or current trajectory. We integrate our method into two different RL agents: an offline DQN agent and an online R2D2 agent. In offline multi-task problems, we show that the retrieval-augmented DQN agent avoids task interference and learns faster than the baseline DQN agent. On Atari, we show that retrieval-augmented R2D2 learns significantly faster than the baseline R2D2 agent and achieves higher scores. We run extensive ablations to measure the contributions of the components of our proposed method.

ICML Conference 2022 Conference Paper

Scaling Gaussian Process Optimization by Evaluating a Few Unique Candidates Multiple Times

Daniele Calandriello
Luigi Carratino
Alessandro Lazaric
Michal Valko
Lorenzo Rosasco

Computing a Gaussian process (GP) posterior has a computational cost cubical in the number of historical points. A reformulation of the same GP posterior highlights that this complexity mainly depends on how many unique historical points are considered. This can have important implication in active learning settings, where the set of historical points is constructed sequentially by the learner. We show that sequential black-box optimization based on GPs (GP-Opt) can be made efficient by sticking to a candidate solution for multiple evaluation steps and switch only when necessary. Limiting the number of switches also limits the number of unique points in the history of the GP. Thus, the efficient GP reformulation can be used to exactly and cheaply compute the posteriors required to run the GP-Opt algorithms. This approach is especially useful in real-world applications of GP-Opt with high switch costs (e. g. switching chemicals in wet labs, data/model loading in hyperparameter optimization). As examples of this meta-approach, we modify two well-established GP-Opt algorithms, GP-UCB and GP-EI, to switch candidates as infrequently as possible adapting rules from batched GP-Opt. These versions preserve all the theoretical no-regret guarantees while improving practical aspects of the algorithms such as runtime, memory complexity, and the ability of batching candidates and evaluating them in parallel.

NeurIPS Conference 2021 Conference Paper

A Provably Efficient Sample Collection Strategy for Reinforcement Learning

Jean Tarbouriech
Matteo Pirotta
Michal Valko
Alessandro Lazaric

One of the challenges in online reinforcement learning (RL) is that the agent needs to trade off the exploration of the environment and the exploitation of the samples to optimize its behavior. Whether we optimize for regret, sample complexity, state-space coverage or model estimation, we need to strike a different exploration-exploitation trade-off. In this paper, we propose to tackle the exploration-exploitation problem following a decoupled approach composed of: 1) An "objective-specific" algorithm that (adaptively) prescribes how many samples to collect at which states, as if it has access to a generative model (i. e. , a simulator of the environment); 2) An "objective-agnostic" sample collection exploration strategy responsible for generating the prescribed samples as fast as possible. Building on recent methods for exploration in the stochastic shortest path problem, we first provide an algorithm that, given as input the number of samples $b(s, a)$ needed in each state-action pair, requires $\widetilde{O}(B D + D^{3/2} S^2 A)$ time steps to collect the $B=\sum_{s, a} b(s, a)$ desired samples, in any unknown communicating MDP with $S$ states, $A$ actions and diameter $D$. Then we show how this general-purpose exploration algorithm can be paired with "objective-specific" strategies that prescribe the sample requirements to tackle a variety of settings — e. g. , model estimation, sparse reward discovery, goal-free cost-free exploration in communicating MDPs — for which we obtain improved or novel sample complexity guarantees.

NeurIPS Conference 2021 Conference Paper

Drop, Swap, and Generate: A Self-Supervised Approach for Generating Neural Activity

Ran Liu
Mehdi Azabou
Max Dabagia
Chi-Heng Lin
Mohammad Gheshlaghi Azar
Keith Hengen
Michal Valko
Eva Dyer

Meaningful and simplified representations of neural activity can yield insights into how and what information is being processed within a neural circuit. However, without labels, finding representations that reveal the link between the brain and behavior can be challenging. Here, we introduce a novel unsupervised approach for learning disentangled representations of neural activity called Swap-VAE. Our approach combines a generative modeling framework with an instance-specific alignment loss that tries to maximize the representational similarity between transformed views of the input (brain state). These transformed (or augmented) views are created by dropping out neurons and jittering samples in time, which intuitively should lead the network to a representation that maintains both temporal consistency and invariance to the specific neurons used to represent the neural state. Through evaluations on both synthetic data and neural recordings from hundreds of neurons in different primate brains, we show that it is possible to build representations that disentangle neural datasets along relevant latent dimensions linked to behavior.

ICML Conference 2021 Conference Paper

Fast active learning for pure exploration in reinforcement learning

Pierre Ménard
Omar Darwiche Domingues
Anders Jonsson 0001
Emilie Kaufmann
Edouard Leurent
Michal Valko

Realistic environments often provide agents with very limited feedback. When the environment is initially unknown, the feedback, in the beginning, can be completely absent, and the agents may first choose to devote all their effort on \emph{exploring efficiently. } The exploration remains a challenge while it has been addressed with many hand-tuned heuristics with different levels of generality on one side, and a few theoretically-backed exploration strategies on the other. Many of them are incarnated by \emph{intrinsic motivation} and in particular \emph{explorations bonuses}. A common choice is to use $1/\sqrt{n}$ bonus, where $n$ is a number of times this particular state-action pair was visited. We show that, surprisingly, for a pure-exploration objective of \emph{reward-free exploration}, bonuses that scale with $1/n$ bring faster learning rates, improving the known upper bounds with respect to the dependence on the horizon $H$. Furthermore, we show that with an improved analysis of the stopping time, we can improve by a factor $H$ the sample complexity in the \emph{best-policy identification} setting, which is another pure-exploration objective, where the environment provides rewards but the agent is not penalized for its behavior during the exploration phase.

JAIR Journal 2021 Journal Article

Game Plan: What AI can do for Football, and What Football can do for AI

Karl Tuyls
Shayegan Omidshafiei
Paul Muller
Zhe Wang
Jerome Connor
Daniel Hennes
Ian Graham
William Spearman

The rapid progress in artificial intelligence (AI) and machine learning has opened unprecedented analytics possibilities in various team and individual sports, including baseball, basketball, and tennis. More recently, AI techniques have been applied to football, due to a huge increase in data collection by professional teams, increased computational power, and advances in machine learning, with the goal of better addressing new scientific challenges involved in the analysis of both individual players’ and coordinated teams’ behaviors. The research challenges associated with predictive and prescriptive football analytics require new developments and progress at the intersection of statistical learning, game theory, and computer vision. In this paper, we provide an overarching perspective highlighting how the combination of these fields, in particular, forms a unique microcosm for AI research, while offering mutual benefits for professional teams, spectators, and broadcasters in the years to come. We illustrate that this duality makes football analytics a game changer of tremendous value, in terms of not only changing the game of football itself, but also in terms of what this domain can mean for the field of AI. We review the state-of-the-art and exemplify the types of analysis enabled by combining the aforementioned fields, including illustrative examples of counterfactual analysis using predictive models, and the combination of game-theoretic analysis of penalty kicks with statistical learning of player attributes. We conclude by highlighting envisioned downstream impacts, including possibilities for extensions to other sports (real and virtual).

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ICML Conference 2021 Conference Paper

Kernel-Based Reinforcement Learning: A Finite-Time Analysis

Omar Darwiche Domingues
Pierre Ménard
Matteo Pirotta
Emilie Kaufmann
Michal Valko

We consider the exploration-exploitation dilemma in finite-horizon reinforcement learning problems whose state-action space is endowed with a metric. We introduce Kernel-UCBVI, a model-based optimistic algorithm that leverages the smoothness of the MDP and a non-parametric kernel estimator of the rewards and transitions to efficiently balance exploration and exploitation. For problems with $K$ episodes and horizon $H$, we provide a regret bound of $\widetilde{O}\left( H^3 K^{\frac{2d}{2d+1}}\right)$, where $d$ is the covering dimension of the joint state-action space. This is the first regret bound for kernel-based RL using smoothing kernels, which requires very weak assumptions on the MDP and applies to a wide range of tasks. We empirically validate our approach in continuous MDPs with sparse rewards.

NeurIPS Conference 2021 Conference Paper

Learning in two-player zero-sum partially observable Markov games with perfect recall

Tadashi Kozuno
Pierre Ménard
Remi Munos
Michal Valko

We study the problem of learning a Nash equilibrium (NE) in an extensive game with imperfect information (EGII) through self-play. Precisely, we focus on two-player, zero-sum, episodic, tabular EGII under the \textit{perfect-recall} assumption where the only feedback is realizations of the game (bandit feedback). In particular the \textit{dynamics of the EGII is not known}---we can only access it by sampling or interacting with a game simulator. For this learning setting, we provide the Implicit Exploration Online Mirror Descent (IXOMD) algorithm. It is a model-free algorithm with a high-probability bound on convergence rate to the NE of order $1/\sqrt{T}$ where~$T$ is the number of played games. Moreover IXOMD is computationally efficient as it needs to perform the updates only along the sampled trajectory.

ICML Conference 2021 Conference Paper

Online A-Optimal Design and Active Linear Regression

Xavier Fontaine
Pierre Perrault
Michal Valko
Vianney Perchet

We consider in this paper the problem of optimal experiment design where a decision maker can choose which points to sample to obtain an estimate $\hat{\beta}$ of the hidden parameter $\beta^{\star}$ of an underlying linear model. The key challenge of this work lies in the heteroscedasticity assumption that we make, meaning that each covariate has a different and unknown variance. The goal of the decision maker is then to figure out on the fly the optimal way to allocate the total budget of $T$ samples between covariates, as sampling several times a specific one will reduce the variance of the estimated model around it (but at the cost of a possible higher variance elsewhere). By trying to minimize the $\ell^2$-loss $\mathbb{E} [\lVert\hat{\beta}-\beta^{\star}\rVert^2]$ the decision maker is actually minimizing the trace of the covariance matrix of the problem, which corresponds then to online A-optimal design. Combining techniques from bandit and convex optimization we propose a new active sampling algorithm and we compare it with existing ones. We provide theoretical guarantees of this algorithm in different settings, including a $\mathcal{O}(T^{-2})$ regret bound in the case where the covariates form a basis of the feature space, generalizing and improving existing results. Numerical experiments validate our theoretical findings.

ICML Conference 2021 Conference Paper

Revisiting Peng's Q(λ) for Modern Reinforcement Learning

Tadashi Kozuno
Yunhao Tang
Mark Rowland 0001
Rémi Munos
Steven Kapturowski
Will Dabney
Michal Valko
David Abel

Off-policy multi-step reinforcement learning algorithms consist of conservative and non-conservative algorithms: the former actively cut traces, whereas the latter do not. Recently, Munos et al. (2016) proved the convergence of conservative algorithms to an optimal Q-function. In contrast, non-conservative algorithms are thought to be unsafe and have a limited or no theoretical guarantee. Nonetheless, recent studies have shown that non-conservative algorithms empirically outperform conservative ones. Motivated by the empirical results and the lack of theory, we carry out theoretical analyses of Peng’s Q($\lambda$), a representative example of non-conservative algorithms. We prove that \emph{it also converges to an optimal policy} provided that the behavior policy slowly tracks a greedy policy in a way similar to conservative policy iteration. Such a result has been conjectured to be true but has not been proven. We also experiment with Peng’s Q($\lambda$) in complex continuous control tasks, confirming that Peng’s Q($\lambda$) often outperforms conservative algorithms despite its simplicity. These results indicate that Peng’s Q($\lambda$), which was thought to be unsafe, is a theoretically-sound and practically effective algorithm.

NeurIPS Conference 2021 Conference Paper

Stochastic Shortest Path: Minimax, Parameter-Free and Towards Horizon-Free Regret

Jean Tarbouriech
Runlong Zhou
Simon S. Du
Matteo Pirotta
Michal Valko
Alessandro Lazaric

We study the problem of learning in the stochastic shortest path (SSP) setting, where an agent seeks to minimize the expected cost accumulated before reaching a goal state. We design a novel model-based algorithm EB-SSP that carefully skews the empirical transitions and perturbs the empirical costs with an exploration bonus to induce an optimistic SSP problem whose associated value iteration scheme is guaranteed to converge. We prove that EB-SSP achieves the minimax regret rate $\widetilde{O}(B_{\star} \sqrt{S A K})$, where $K$ is the number of episodes, $S$ is the number of states, $A$ is the number of actions and $B_{\star}$ bounds the expected cumulative cost of the optimal policy from any state, thus closing the gap with the lower bound. Interestingly, EB-SSP obtains this result while being parameter-free, i. e. , it does not require any prior knowledge of $B_{\star}$, nor of $T_{\star}$, which bounds the expected time-to-goal of the optimal policy from any state. Furthermore, we illustrate various cases (e. g. , positive costs, or general costs when an order-accurate estimate of $T_{\star}$ is available) where the regret only contains a logarithmic dependence on $T_{\star}$, thus yielding the first (nearly) horizon-free regret bound beyond the finite-horizon MDP setting.

ICML Conference 2021 Conference Paper

Taylor Expansion of Discount Factors

Yunhao Tang
Mark Rowland 0001
Rémi Munos
Michal Valko

In practical reinforcement learning (RL), the discount factor used for estimating value functions often differs from that used for defining the evaluation objective. In this work, we study the effect that this discrepancy of discount factors has during learning, and discover a family of objectives that interpolate value functions of two distinct discount factors. Our analysis suggests new ways for estimating value functions and performing policy optimization updates, which demonstrate empirical performance gains. This framework also leads to new insights on commonly-used deep RL heuristic modifications to policy optimization algorithms.

ICML Conference 2021 Conference Paper

UCB Momentum Q-learning: Correcting the bias without forgetting

Pierre Ménard
Omar Darwiche Domingues
Xuedong Shang
Michal Valko

We propose UCBMQ, Upper Confidence Bound Momentum Q-learning, a new algorithm for reinforcement learning in tabular and possibly stage-dependent, episodic Markov decision process. UCBMQ is based on Q-learning where we add a momentum term and rely on the principle of optimism in face of uncertainty to deal with exploration. Our new technical ingredient of UCBMQ is the use of momentum to correct the bias that Q-learning suffers while, \emph{at the same time}, limiting the impact it has on the second-order term of the regret. For UCBMQ, we are able to guarantee a regret of at most $\tilde{O}(\sqrt{H^3SAT}+ H^4 S A)$ where $H$ is the length of an episode, $S$ the number of states, $A$ the number of actions, $T$ the number of episodes and ignoring terms in poly$\log(SAHT)$. Notably, UCBMQ is the first algorithm that simultaneously matches the lower bound of $\Omega(\sqrt{H^3SAT})$ for large enough $T$ and has a second-order term (with respect to $T$) that scales \emph{only linearly} with the number of states $S$.

NeurIPS Conference 2021 Conference Paper

Unifying Gradient Estimators for Meta-Reinforcement Learning via Off-Policy Evaluation

Yunhao Tang
Tadashi Kozuno
Mark Rowland
Remi Munos
Michal Valko

Model-agnostic meta-reinforcement learning requires estimating the Hessian matrix of value functions. This is challenging from an implementation perspective, as repeatedly differentiating policy gradient estimates may lead to biased Hessian estimates. In this work, we provide a unifying framework for estimating higher-order derivatives of value functions, based on off-policy evaluation. Our framework interprets a number of prior approaches as special cases and elucidates the bias and variance trade-off of Hessian estimates. This framework also opens the door to a new family of estimates, which can be easily implemented with auto-differentiation libraries, and lead to performance gains in practice.

NeurIPS Conference 2020 Conference Paper

Bootstrap Your Own Latent - A New Approach to Self-Supervised Learning

Jean-Bastien Grill
Florian Strub
Florent Altché
Corentin Tallec
Pierre Richemond
Elena Buchatskaya
Carl Doersch
Bernardo Avila Pires

We introduce Bootstrap Your Own Latent (BYOL), a new approach to self-supervised image representation learning. BYOL relies on two neural networks, referred to as online and target networks, that interact and learn from each other. From an augmented view of an image, we train the online network to predict the target network representation of the same image under a different augmented view. At the same time, we update the target network with a slow-moving average of the online network. While state-of-the art methods intrinsically rely on negative pairs, BYOL achieves a new state of the art without them. BYOL reaches 74. 3% top-1 classification accuracy on ImageNet using the standard linear evaluation protocol with a standard ResNet-50 architecture and 79. 6% with a larger ResNet. We also show that BYOL performs on par or better than the current state of the art on both transfer and semi-supervised benchmarks.

ICML Conference 2020 Conference Paper

Budgeted Online Influence Maximization

Pierre Perrault
Jennifer Healey
Zheng Wen
Michal Valko

We introduce a new budgeted framework for online influence maximization, considering the total cost of an advertising campaign instead of the common cardinality constraint on a chosen influencer set. Our approach models better the real-world setting where the cost of influencers varies and advertizers want to find the best value for their overall social advertising budget. We propose an algorithm assuming an independent cascade diffusion model and edge-level semi-bandit feedback, and provide both theoretical and experimental results. Our analysis is also valid for the cardinality-constraint setting and improves the state of the art regret bound in this case.

ICML Conference 2020 Conference Paper

Gamification of Pure Exploration for Linear Bandits

Rémy Degenne
Pierre Ménard
Xuedong Shang
Michal Valko

We investigate an active \emph{pure-exploration} setting, that includes \emph{best-arm identification}, in the context of \emph{linear stochastic bandits}. While asymptotically optimal algorithms exist for standard \emph{multi-armed bandits}, the existence of such algorithms for the best-arm identification in linear bandits has been elusive despite several attempts to address it. First, we provide a thorough comparison and new insight over different notions of optimality in the linear case, including G-optimality, transductive optimality from optimal experimental design and asymptotic optimality. Second, we design the first asymptotically optimal algorithm for fixed-confidence pure exploration in linear bandits. As a consequence, our algorithm naturally bypasses the pitfall caused by a simple but difficult instance, that most prior algorithms had to be engineered to deal with explicitly. Finally, we avoid the need to fully solve an optimal design problem by providing an approach that entails an efficient implementation.

NeurIPS Conference 2020 Conference Paper

Improved Sample Complexity for Incremental Autonomous Exploration in MDPs

Jean Tarbouriech
Matteo Pirotta
Michal Valko
Alessandro Lazaric

We study the problem of exploring an unknown environment when no reward function is provided to the agent. Building on the incremental exploration setting introduced by Lim and Auer (2012), we define the objective of learning the set of $\epsilon$-optimal goal-conditioned policies attaining all states that are incrementally reachable within $L$ steps (in expectation) from a reference state $s_0$. In this paper, we introduce a novel model-based approach that interleaves discovering new states from $s_0$ and improving the accuracy of a model estimate that is used to compute goal-conditioned policies. The resulting algorithm, DisCo, achieves a sample complexity scaling as $\widetilde{O}_{\epsilon}(L^5 S_{L+\epsilon} \Gamma_{L+\epsilon} A \epsilon^{-2})$, where $A$ is the number of actions, $S_{L+\epsilon}$ is the number of states that are incrementally reachable from $s_0$ in $L+\epsilon$ steps, and $\Gamma_{L+\epsilon}$ is the branching factor of the dynamics over such states. This improves over the algorithm proposed in (Lim and Auer, 2012) in both $\epsilon$ and $L$ at the cost of an extra $\Gamma_{L+\epsilon}$ factor, which is small in most environments of interest. Furthermore, DisCo is the first algorithm that can return an $\epsilon/c_{\min}$-optimal policy for any cost-sensitive shortest-path problem defined on the $L$-reachable states with minimum cost $c_{\min}$. Finally, we report preliminary empirical results confirming our theoretical findings.

ICML Conference 2020 Conference Paper

Improved Sleeping Bandits with Stochastic Action Sets and Adversarial Rewards

Aadirupa Saha
Pierre Gaillard
Michal Valko

In this paper, we consider the problem of sleeping bandits with stochastic action sets and adversarial rewards. In this setting, in contrast to most work in bandits, the actions may not be available at all times. For instance, some products might be out of stock in item recommendation. The best existing efficient (i. e. , polynomial-time) algorithms for this problem only guarantee an $O(T^{2/3})$ upper-bound on the regret. Yet, inefficient algorithms based on EXP4 can achieve $O(\sqrt{T})$. In this paper, we provide a new computationally efficient algorithm inspired by EXP3 satisfying a regret of order $O(\sqrt{T})$ when the availabilities of each action $i \in \cA$ are independent. We then study the most general version of the problem where at each round available sets are generated from some unknown arbitrary distribution (i. e. , without the independence assumption) and propose an efficient algorithm with $O(\sqrt {2^K T})$ regret guarantee. Our theoretical results are corroborated with experimental evaluations.

ICML Conference 2020 Conference Paper

Monte-Carlo Tree Search as Regularized Policy Optimization

Jean-Bastien Grill
Florent Altché
Yunhao Tang
Thomas Hubert
Michal Valko
Ioannis Antonoglou
Rémi Munos

The combination of Monte-Carlo tree search (MCTS) with deep reinforcement learning has led to groundbreaking results in artificial intelligence. However, AlphaZero, the current state-of-the-art MCTS algorithm still relies on handcrafted heuristics that are only partially understood. In this paper, we show that AlphaZero’s search heuristic, along with other common ones, can be interpreted as an approximation to the solution of a specific regularized policy optimization problem. With this insight, we propose a variant of AlphaZero which uses the exact solution to this policy optimization problem, and show experimentally that it reliably outperforms the original algorithm in multiple domains.

ICML Conference 2020 Conference Paper

Near-linear time Gaussian process optimization with adaptive batching and resparsification

Daniele Calandriello
Luigi Carratino
Alessandro Lazaric
Michal Valko
Lorenzo Rosasco

Gaussian processes (GP) are one of the most successful frameworks to model uncertainty. However, GP optimization (e. g. , GP-UCB) suffers from major scalability issues. Experimental time grows linearly with the number of evaluations, unless candidates are selected in batches (e. g. , using GP-BUCB) and evaluated in parallel. Furthermore, computational cost is often prohibitive since algorithms such as GP-BUCB require a time at least quadratic in the number of dimensions and iterations to select each batch. In this paper, we introduce BBKB (Batch Budgeted Kernel Bandits), the first no-regret GP optimization algorithm that provably runs in near-linear time and selects candidates in batches. This is obtained with a new guarantee for the tracking of the posterior variances that allows BBKB to choose increasingly larger batches, improving over GP-BUCB. Moreover, we show that the same bound can be used to adaptively delay costly updates to the sparse GP approximation used by BBKB, achieving a near-constant per-step amortized cost. These findings are then confirmed in several experiments, where BBKB is much faster than state-of-the-art methods.

ICML Conference 2020 Conference Paper

No-Regret Exploration in Goal-Oriented Reinforcement Learning

Jean Tarbouriech
Evrard Garcelon
Michal Valko
Matteo Pirotta
Alessandro Lazaric

Many popular reinforcement learning problems (e. g. , navigation in a maze, some Atari games, mountain car) are instances of the episodic setting under its stochastic shortest path (SSP) formulation, where an agent has to achieve a goal state while minimizing the cumulative cost. Despite the popularity of this setting, the exploration-exploitation dilemma has been sparsely studied in general SSP problems, with most of the theoretical literature focusing on different problems (i. e. , fixed-horizon and infinite-horizon) or making the restrictive loop-free SSP assumption (i. e. , no state can be visited twice during an episode). In this paper, we study the general SSP problem with no assumption on its dynamics (some policies may actually never reach the goal). We introduce UC-SSP, the first no-regret algorithm in this setting, and prove a regret bound scaling as $\widetilde{\mathcal{O}}( D S \sqrt{ A D K})$ after $K$ episodes for any unknown SSP with $S$ states, $A$ actions, positive costs and SSP-diameter $D$, defined as the smallest expected hitting time from any starting state to the goal. We achieve this result by crafting a novel stopping rule, such that UC-SSP may interrupt the current policy if it is taking too long to achieve the goal and switch to alternative policies that are designed to rapidly terminate the episode.

NeurIPS Conference 2020 Conference Paper

Planning in Markov Decision Processes with Gap-Dependent Sample Complexity

Anders Jonsson
Emilie Kaufmann
Pierre Menard
Omar Darwiche Domingues
Edouard Leurent
Michal Valko

We propose MDP-GapE, a new trajectory-based Monte-Carlo Tree Search algorithm for planning in a Markov Decision Process in which transitions have a finite support. We prove an upper bound on the number of sampled trajectories needed for MDP-GapE to identify a near-optimal action with high probability. This problem-dependent result is expressed in terms of the sub-optimality gaps of the state-action pairs that are visited during exploration. Our experiments reveal that MDP-GapE is also effective in practice, in contrast with other algorithms with sample complexity guarantees in the fixed-confidence setting, that are mostly theoretical.

NeurIPS Conference 2020 Conference Paper

Sampling from a k-DPP without looking at all items

Daniele Calandriello
Michal Derezinski
Michal Valko

Determinantal point processes (DPPs) are a useful probabilistic model for selecting a small diverse subset out of a large collection of items, with applications in summarization, recommendation, stochastic optimization, experimental design and more. Given a kernel function and a subset size k, our goal is to sample k out of n items with probability proportional to the determinant of the kernel matrix induced by the subset (a. k. a. k-DPP). Existing k-DPP sampling algorithms require an expensive preprocessing step which involves multiple passes over all n items, making it infeasible for large datasets. A naïve heuristic addressing this problem is to uniformly subsample a fraction of the data and perform k-DPP sampling only on those items, however this method offers no guarantee that the produced sample will even approximately resemble the target distribution over the original dataset. In this paper, we develop alpha-DPP, an algorithm which adaptively builds a sufficiently large uniform sample of data that is then used to efficiently generate a smaller set of k items, while ensuring that this set is drawn exactly from the target distribution defined on all n items. We show empirically that our algorithm produces a k-DPP sample after observing only a small fraction of all elements, leading to several orders of magnitude faster performance compared to the state-of-the-art. Our implementation of alpha-DPP is provided at https: //github. com/guilgautier/DPPy/.

JMLR Journal 2020 Journal Article

Spectral bandits

Tomáš Kocák
Rémi Munos
Branislav Kveton
Shipra Agrawal
Michal Valko

Smooth functions on graphs have wide applications in manifold and semi-supervised learning. In this work, we study a bandit problem where the payoffs of arms are smooth on a graph. This framework is suitable for solving online learning problems that involve graphs, such as content-based recommendation. In this problem, each item we can recommend is a node of an undirected graph and its expected rating is similar to the one of its neighbors. The goal is to recommend items that have high expected ratings. We aim for the algorithms where the cumulative regret with respect to the optimal policy would not scale poorly with the number of nodes. In particular, we introduce the notion of an effective dimension, which is small in real-world graphs, and propose three algorithms for solving our problem that scale linearly and sublinearly in this dimension. Our experiments on content recommendation problem show that a good estimator of user preferences for thousands of items can be learned from just tens of node evaluations. [abs] [ pdf ][ bib ] &copy JMLR 2020. ( edit, beta )

NeurIPS Conference 2020 Conference Paper

Statistical Efficiency of Thompson Sampling for Combinatorial Semi-Bandits

Pierre Perrault
Etienne Boursier
Michal Valko
Vianney Perchet

We investigate stochastic combinatorial multi-armed bandit with semi-bandit feedback (CMAB). In CMAB, the question of the existence of an efficient policy with an optimal asymptotic regret (up to a factor poly-logarithmic with the action size) is still open for many families of distributions, including mutually independent outcomes, and more generally the multivariate \emph{sub-Gaussian} family. We propose to answer the above question for these two families by analyzing variants of the Combinatorial Thompson Sampling policy (CTS). For mutually independent outcomes in $[0, 1]$, we propose a tight analysis of CTS using Beta priors. We then look at the more general setting of multivariate sub-Gaussian outcomes and propose a tight analysis of CTS using Gaussian priors. This last result gives us an alternative to the Efficient Sampling for Combinatorial Bandit policy (ESCB), which, although optimal, is not computationally efficient.

ICML Conference 2020 Conference Paper

Stochastic bandits with arm-dependent delays

Anne Gael Manegueu
Claire Vernade
Alexandra Carpentier
Michal Valko

Significant work has been recently dedicated to the stochastic delayed bandits because of its relevance in applications. The applicability of existing algorithms is however restricted by the fact that strong assumptions are often made on the delay distributions, such as full observability, restrictive shape constraints, or uniformity over arms. In this work, we weaken them significantly and only assume that there is a bound on the tail of the delay. In particular, we cover the important case where the delay distributions vary across arms, and the case where the delays are heavy-tailed. Addressing these difficulties, we propose a simple but efficient UCB-based algorithm called the PatientBandits. We provide both problemsdependent and problems-independent bounds on the regret as well as performance lower bounds.

ICML Conference 2020 Conference Paper

Taylor Expansion Policy Optimization

Yunhao Tang
Michal Valko
Rémi Munos

In this work, we investigate the application of Taylor expansions in reinforcement learning. In particular, we propose Taylor Expansion Policy Optimization, a policy optimization formalism that generalizes prior work as a first-order special case. We also show that Taylor expansions intimately relate to off-policy evaluation. Finally, we show that this new formulation entails modifications which improve the performance of several state-of-the-art distributed algorithms.

JMLR Journal 2019 Journal Article

DPPy: DPP Sampling with Python

Guillaume Gautier
Guillermo Polito
Rémi Bardenet
Michal Valko

Determinantal point processes (DPPs) are specific probability distributions over clouds of points that are used as models and computational tools across physics, probability, statistics, and more recently machine learning. Sampling from DPPs is a challenge and therefore we present DPPy, a Python toolbox that gathers known exact and approximate sampling algorithms for both finite and continuous DPPs. The project is hosted on GitHub, and equipped with an extensive documentation. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2019. ( edit, beta )

NeurIPS Conference 2019 Conference Paper

Exact sampling of determinantal point processes with sublinear time preprocessing

Michal Derezinski
Daniele Calandriello
Michal Valko

We study the complexity of sampling from a distribution over all index subsets of the set {1, .. ., n} with the probability of a subset S proportional to the determinant of the submatrix L S of some n x n positive semidefinite matrix L, where L S corresponds to the entries of L indexed by S. Known as a determinantal point process (DPP), this distribution is used in machine learning to induce diversity in subset selection. When sampling from DDPs, we often wish to sample multiple subsets S with small expected size k = E[|S|] << n from a very large matrix L, so it is important to minimize the preprocessing cost of the procedure (performed once) as well as the sampling cost (performed repeatedly). For this purpose we provide DPP-VFX, a new algorithm which, given access only to L, samples exactly from a determinantal point process while satisfying the following two properties: (1) its preprocessing cost is n poly(k), i. e. , sublinear in the size of L, and (2) its sampling cost is poly(k), i. e. , independent of the size of L. Prior to our results, state-of-the-art exact samplers required O(n^3) preprocessing time and sampling time linear in n or dependent on the spectral properties of L. We furthermore give a reduction which allows using our algorithm for exact sampling from cardinality constrained determinantal point processes with n poly(k) time preprocessing. Our implementation of DPP-VFX is provided at https: //github. com/guilgautier/DPPy/.

ICML Conference 2019 Conference Paper

Exploiting structure of uncertainty for efficient matroid semi-bandits

Pierre Perrault
Vianney Perchet
Michal Valko

We improve the efficiency of algorithms for stochastic combinatorial semi-bandits. In most interesting problems, state-of-the-art algorithms take advantage of structural properties of rewards, such as independence. However, while being minimax optimal in terms of regret, these algorithms are intractable. In our paper, we first reduce their implementation to a specific submodular maximization. Then, in case of matroid constraints, we design adapted approximation routines, thereby providing the first efficient algorithms that exploit the reward structure. In particular, we improve the state-of-the-art efficient gap-free regret bound by a factor sqrt(k), where k is the maximum action size. Finally, we show how our improvement translates to more general budgeted combinatorial semi-bandits.

NeurIPS Conference 2019 Conference Paper

Multiagent Evaluation under Incomplete Information

Mark Rowland
Shayegan Omidshafiei
Karl Tuyls
Julien Perolat
Michal Valko
Georgios Piliouras
Remi Munos

This paper investigates the evaluation of learned multiagent strategies in the incomplete information setting, which plays a critical role in ranking and training of agents. Traditionally, researchers have relied on Elo ratings for this purpose, with recent works also using methods based on Nash equilibria. Unfortunately, Elo is unable to handle intransitive agent interactions, and other techniques are restricted to zero-sum, two-player settings or are limited by the fact that the Nash equilibrium is intractable to compute. Recently, a ranking method called $\alpha$-Rank, relying on a new graph-based game-theoretic solution concept, was shown to tractably apply to general games. However, evaluations based on Elo or $\alpha$-Rank typically assume noise-free game outcomes, despite the data often being collected from noisy simulations, making this assumption unrealistic in practice. This paper investigates multiagent evaluation in the incomplete information regime, involving general-sum many-player games with noisy outcomes. We derive sample complexity guarantees required to confidently rank agents in this setting. We propose adaptive algorithms for accurate ranking, provide correctness and sample complexity guarantees, then introduce a means of connecting uncertainties in noisy match outcomes to uncertainties in rankings. We evaluate the performance of these approaches in several domains, including Bernoulli games, a soccer meta-game, and Kuhn poker.

NeurIPS Conference 2019 Conference Paper

On two ways to use determinantal point processes for Monte Carlo integration

Guillaume Gautier
Rémi Bardenet
Michal Valko

When approximating an integral by a weighted sum of function evaluations, determinantal point processes (DPPs) provide a way to enforce repulsion between the evaluation points. This negative dependence is encoded by a kernel. Fifteen years before the discovery of DPPs, Ermakov & Zolotukhin (EZ, 1960) had the intuition of sampling a DPP and solving a linear system to compute an unbiased Monte Carlo estimator of the integral. In the absence of DPP machinery to derive an efficient sampler and analyze their estimator, the idea of Monte Carlo integration with DPPs was stored in the cellar of numerical integration. Recently, Bardenet & Hardy (BH, 2019) came up with a more natural estimator with a fast central limit theorem (CLT). In this paper, we first take the EZ estimator out of the cellar, and analyze it using modern arguments. Second, we provide an efficient implementation to sample exactly a particular multidimensional DPP called multivariate Jacobi ensemble. The latter satisfies the assumptions of the aforementioned CLT. Third, our new implementation lets us investigate the behavior of the two unbiased Monte Carlo estimators in yet unexplored regimes. We demonstrate experimentally good properties when the kernel is adapted to basis of functions in which the integrand is sparse or has fast-decaying coefficients. If such a basis and the level of sparsity are known (e. g. , we integrate a linear combination of kernel eigenfunctions), the EZ estimator can be the right choice, but otherwise it can display an erratic behavior.

NeurIPS Conference 2019 Conference Paper

Planning in entropy-regularized Markov decision processes and games

Jean-Bastien Grill
Omar Darwiche Domingues
Pierre Menard
Remi Munos
Michal Valko

We propose SmoothCruiser, a new planning algorithm for estimating the value function in entropy-regularized Markov decision processes and two-player games, given a generative model of the SmoothCruiser. SmoothCruiser makes use of the smoothness of the Bellman operator promoted by the regularization to achieve problem-independent sample complexity of order $\tilde{\mathcal{O}}(1/\epsilon^4)$ for a desired accuracy $\epsilon$, whereas for non-regularized settings there are no known algorithms with guaranteed polynomial sample complexity in the worst case.

ICML Conference 2019 Conference Paper

Scale-free adaptive planning for deterministic dynamics & discounted rewards

Peter L. Bartlett
Victor Gabillon
Jennifer Healey
Michal Valko

We address the problem of planning in an environment with deterministic dynamics and stochastic discounted rewards under a limited numerical budget where the ranges of both rewards and noise are unknown. We introduce PlaTypOOS, an adaptive, robust, and efficient alternative to the OLOP (open-loop optimistic planning) algorithm. Whereas OLOP requires a priori knowledge of the ranges of both rewards and noise, PlaTypOOS dynamically adapts its behavior to both. This allows PlaTypOOS to be immune to two vulnerabilities of OLOP: failure when given underestimated ranges of noise and rewards and inefficiency when these are overestimated. PlaTypOOS additionally adapts to the global smoothness of the value function. PlaTypOOS acts in a provably more efficient manner vs. OLOP when OLOP is given an overestimated reward and show that in the case of no noise, PlaTypOOS learns exponentially faster.

EWRL Workshop 2018 Workshop Paper

Adaptive black-box optimization got easier: HCT needs only local smoothness

Xuedong Shang
Emilie Kaufmann
Michal Valko

Hierarchical bandits is an approach for global optimization of extremely irregular functions. This paper provides new elements regarding POO, an adaptive meta-algorithm that does not require the knowledge of local smoothness of the target function. We first highlight the fact that the subroutine algorithm used in POO should have a small regret under the assumption of local smoothness with respect to the chosen partitioning, which is unknown if it is satisfied by the standard subroutine HOO. In this work, we establish such regret guarantee for HCT, which is another hierarchical optimistic optimization algorithm that needs to know the smoothness. This confirms the validity of POO. We show that POO can be used with HCT as a subroutine with a regret upper bound that matches the one of best-known algorithms using the knowledge of smoothness up to a √ log n factor.

ICML Conference 2018 Conference Paper

Improved Large-Scale Graph Learning through Ridge Spectral Sparsification

Daniele Calandriello
Ioannis Koutis
Alessandro Lazaric
Michal Valko

The representation and learning benefits of methods based on graph Laplacians, such as Laplacian smoothing or harmonic function solution for semi-supervised learning (SSL), are empirically and theoretically well supported. Nonetheless, the exact versions of these methods scale poorly with the number of nodes $n$ of the graph. In this paper, we combine a spectral sparsification routine with Laplacian learning. Given a graph $G$ as input, our algorithm computes a sparsifier in a distributed way in $O(n\log^3(n))$ time, $O(m\log^3(n))$ work and $O(n\log(n))$ memory, using only $\log(n)$ rounds of communication. Furthermore, motivated by the regularization often employed in learning algorithms, we show that constructing sparsifiers that preserve the spectrum of the Laplacian only up to the regularization level may drastically reduce the size of the final graph. By constructing a spectrally-similar graph, we are able to bound the error induced by the sparsification for a variety of downstream tasks (e. g. , SSL). We empirically validate the theoretical guarantees on Amazon co-purchase graph and compare to the state-of-the-art heuristics.

NeurIPS Conference 2018 Conference Paper

Optimistic optimization of a Brownian

Jean-Bastien Grill
Michal Valko
Remi Munos

We address the problem of optimizing a Brownian motion. We consider a (random) realization $W$ of a Brownian motion with input space in $[0, 1]$. Given $W$, our goal is to return an $\epsilon$-approximation of its maximum using the smallest possible number of function evaluations, the sample complexity of the algorithm. We provide an algorithm with sample complexity of order $\log^2(1/\epsilon)$. This improves over previous results of Al-Mharmah and Calvin (1996) and Calvin et al. (2017) which provided only polynomial rates. Our algorithm is adaptive---each query depends on previous values---and is an instance of the optimism-in-the-face-of-uncertainty principle.

NeurIPS Conference 2017 Conference Paper

Efficient Second-Order Online Kernel Learning with Adaptive Embedding

Daniele Calandriello
Alessandro Lazaric
Michal Valko

Online kernel learning (OKL) is a flexible framework to approach prediction problems, since the large approximation space provided by reproducing kernel Hilbert spaces can contain an accurate function for the problem. Nonetheless, optimizing over this space is computationally expensive. Not only first order methods accumulate $\O(\sqrt{T})$ more loss than the optimal function, but the curse of kernelization results in a $\O(t)$ per step complexity. Second-order methods get closer to the optimum much faster, suffering only $\O(\log(T))$ regret, but second-order updates are even more expensive, with a $\O(t^2)$ per-step cost. Existing approximate OKL methods try to reduce this complexity either by limiting the Support Vectors (SV) introduced in the predictor, or by avoiding the kernelization process altogether using embedding. Nonetheless, as long as the size of the approximation space or the number of SV does not grow over time, an adversary can always exploit the approximation process. In this paper, we propose PROS-N-KONS, a method that combines Nystrom sketching to project the input point in a small, accurate embedded space, and performs efficient second-order updates in this space. The embedded space is continuously updated to guarantee that the embedding remains accurate, and we show that the per-step cost only grows with the effective dimension of the problem and not with $T$. Moreover, the second-order updated allows us to achieve the logarithmic regret. We empirically compare our algorithm on recent large-scales benchmarks and show it performs favorably.

NeurIPS Conference 2017 Conference Paper

Online Influence Maximization under Independent Cascade Model with Semi-Bandit Feedback

Zheng Wen
Branislav Kveton
Michal Valko
Sharan Vaswani

We study the online influence maximization problem in social networks under the independent cascade model. Specifically, we aim to learn the set of "best influencers" in a social network online while repeatedly interacting with it. We address the challenges of (i) combinatorial action space, since the number of feasible influencer sets grows exponentially with the maximum number of influencers, and (ii) limited feedback, since only the influenced portion of the network is observed. Under a stochastic semi-bandit feedback, we propose and analyze IMLinUCB, a computationally efficient UCB-based algorithm. Our bounds on the cumulative regret are polynomial in all quantities of interest, achieve near-optimal dependence on the number of interactions and reflect the topology of the network and the activation probabilities of its edges, thereby giving insights on the problem complexity. To the best of our knowledge, these are the first such results. Our experiments show that in several representative graph topologies, the regret of IMLinUCB scales as suggested by our upper bounds. IMLinUCB permits linear generalization and thus is both statistically and computationally suitable for large-scale problems. Our experiments also show that IMLinUCB with linear generalization can lead to low regret in real-world online influence maximization.

ICML Conference 2017 Conference Paper

Second-Order Kernel Online Convex Optimization with Adaptive Sketching

Daniele Calandriello
Alessandro Lazaric
Michal Valko

Kernel online convex optimization (KOCO) is a framework combining the expressiveness of non-parametric kernel models with the regret guarantees of online learning. First-order KOCO methods such as functional gradient descent require only $O(t)$ time and space per iteration, and, when the only information on the losses is their convexity, achieve a minimax optimal $O(\sqrt{T})$ regret. Nonetheless, many common losses in kernel problems, such as squared loss, logistic loss, and squared hinge loss posses stronger curvature that can be exploited. In this case, second-order KOCO methods achieve $O(\log(\mathrm{Det}(K)))$ regret, which we show scales as $O(deff \log T)$, where $deff$ is the effective dimension of the problem and is usually much smaller than $O(\sqrt{T})$. The main drawback of second-order methods is their much higher $O(t^2)$ space and time complexity. In this paper, we introduce kernel online Newton step (KONS), a new second-order KOCO method that also achieves $O(deff\log T)$ regret. To address the computational complexity of second-order methods, we introduce a new matrix sketching algorithm for the kernel matrix~$K$, and show that for a chosen parameter $\gamma \leq 1$ our Sketched-KONS reduces the space and time complexity by a factor of $\gamma^2$ to $O(t^2\gamma^2)$ space and time per iteration, while incurring only $1/\gamma$ times more regret.

ICML Conference 2017 Conference Paper

Zonotope Hit-and-run for Efficient Sampling from Projection DPPs

Guillaume Gautier
Rémi Bardenet
Michal Valko

Determinantal point processes (DPPs) are distributions over sets of items that model diversity using kernels. Their applications in machine learning include summary extraction and recommendation systems. Yet, the cost of sampling from a DPP is prohibitive in large-scale applications, which has triggered an effort towards efficient approximate samplers. We build a novel MCMC sampler that combines ideas from combinatorial geometry, linear programming, and Monte Carlo methods to sample from DPPs with a fixed sample cardinality, also called projection DPPs. Our sampler leverages the ability of the hit-and-run MCMC kernel to efficiently move across convex bodies. Previous theoretical results yield a fast mixing time of our chain when targeting a distribution that is close to a projection DPP, but not a DPP in general. Our empirical results demonstrate that this extends to sampling projection DPPs, i. e. , our sampler is more sample-efficient than previous approaches which in turn translates to faster convergence when dealing with costly-to-evaluate functions, such as summary extraction in our experiments.

UAI Conference 2016 Conference Paper

Analysis of Nyström method with sequential ridge leverage scores

Daniele Calandriello
Alessandro Lazaric
Michal Valko

Large-scale kernel ridge regression (KRR) is limited by the need to store a large kernel matrix Kt. To avoid storing the entire matrix Kt, Nyström methods subsample a subset of columns of the kernel matrix, and efficiently find an approximate e t. The choKRR solution on the reconstructed K sen subsampling distribution in turn affects the statistical and computational tradeoffs. For KRR problems, [16, 1] show that a sampling distribution proportional to the ridge leverage scores (RLSs) provides strong reconstruction guarane t. While exact RLSs are as difficult to tees for K compute as a KRR solution, we may be able to approximate them well enough. In this paper, we study KRR problems in a sequential setting and introduce the INK-E STIMATE algorithm, that incrementally computes the RLSs estimates. INKE STIMATE maintains a small sketch of Kt, that at each step is used to compute an intermediate estimate of the RLSs. First, our sketch update does not require access to previously seen columns, and therefore a single pass over the kernel matrix is sufficient. Second, the algorithm requires a fixed, small space budget to run dependent only on the effective dimension of the kernel matrix. Finally, our sketch provides strong approximae t k2, and tion guarantees on the distance kKt − K on the statistical risk of the approximate KRR solution at any time, because all our guarantees hold at any intermediate step.

JMLR Journal 2016 Journal Article

Bayesian Policy Gradient and Actor-Critic Algorithms

Mohammad Ghavamzadeh
Yaakov Engel
Michal Valko

Policy gradient methods are reinforcement learning algorithms that adapt a parameterized policy by following a performance gradient estimate. Many conventional policy gradient methods use Monte-Carlo techniques to estimate this gradient. The policy is improved by adjusting the parameters in the direction of the gradient estimate. Since Monte-Carlo methods tend to have high variance, a large number of samples is required to attain accurate estimates, resulting in slow convergence. In this paper, we first propose a Bayesian framework for policy gradient, based on modeling the policy gradient as a Gaussian process. This reduces the number of samples needed to obtain accurate gradient estimates. Moreover, estimates of the natural gradient as well as a measure of the uncertainty in the gradient estimates, namely, the gradient covariance, are provided at little extra cost. Since the proposed Bayesian framework considers system trajectories as its basic observable unit, it does not require the dynamics within trajectories to be of any particular form, and thus, can be easily extended to partially observable problems. On the downside, it cannot take advantage of the Markov property when the system is Markovian. To address this issue, we proceed to supplement our Bayesian policy gradient framework with a new actor-critic learning model in which a Bayesian class of non- parametric critics, based on Gaussian process temporal difference learning, is used. Such critics model the action- value function as a Gaussian process, allowing Bayes' rule to be used in computing the posterior distribution over action-value functions, conditioned on the observed data. Appropriate choices of the policy parameterization and of the prior covariance (kernel) between action-values allow us to obtain closed-form expressions for the posterior distribution of the gradient of the expected return with respect to the policy parameters. We perform detailed experimental comparisons of the proposed Bayesian policy gradient and actor-critic algorithms with classic Monte-Carlo based policy gradient methods, as well as with each other, on a number of reinforcement learning problems. [abs] [ pdf ][ bib ] &copy JMLR 2016. ( edit, beta )

NeurIPS Conference 2016 Conference Paper

Blazing the trails before beating the path: Sample-efficient Monte-Carlo planning

Jean-Bastien Grill
Michal Valko
Remi Munos

We study the sampling-based planning problem in Markov decision processes (MDPs) that we can access only through a generative model, usually referred to as Monte-Carlo planning. Our objective is to return a good estimate of the optimal value function at any state while minimizing the number of calls to the generative model, i. e. the sample complexity. We propose a new algorithm, TrailBlazer, able to handle MDPs with a finite or an infinite number of transitions from state-action to next states. TrailBlazer is an adaptive algorithm that exploits possible structures of the MDP by exploring only a subset of states reachable by following near-optimal policies. We provide bounds on its sample complexity that depend on a measure of the quantity of near-optimal states. The algorithm behavior can be considered as an extension of Monte-Carlo sampling (for estimating an expectation) to problems that alternate maximization (over actions) and expectation (over next states). Finally, another appealing feature of TrailBlazer is that it is simple to implement and computationally efficient.

UAI Conference 2016 Conference Paper

Online learning with Erdos-Renyi side-observation graphs

Tomás Kocák
Gergely Neu
Michal Valko

We consider adversarial multi-armed bandit problems where the learner is allowed to observe losses of a number of arms beside the arm that it actually chose. We study the case where all non-chosen arms reveal their loss with an unknown probability rt, independently of each other and the action of the learner. Moreover, we allow rt to change in every round t, which rules out the possibility of estimating rt by a well-concentrated sample average. We propose an algorithm which operates under the assumption that rt is large enough to warrant at least one side observation with high probability. We show that after T rounds in a bandit problem with N arms, the expected regret of our algorithm is qP T (1/r ) log N, given that of order O t t=1 rt ≥ log T /(2N − 2) for all t. All our bounds are within logarithmic factors of the best achievable performance of any algorithm that is even allowed to know exact values of rt.

ICML Conference 2016 Conference Paper

Pliable Rejection Sampling

Akram Erraqabi
Michal Valko
Alexandra Carpentier
Odalric-Ambrym Maillard

Rejection sampling is a technique for sampling from difficult distributions. However, its use is limited due to a high rejection rate. Common adaptive rejection sampling methods either work only for very specific distributions or without performance guarantees. In this paper, we present pliable rejection sampling (PRS), a new approach to rejection sampling, where we learn the sampling proposal using a kernel estimator. Since our method builds on rejection sampling, the samples obtained are with high probability i. i. d. and distributed according to f. Moreover, PRS comes with a guarantee on the number of accepted samples.

NeurIPS Conference 2015 Conference Paper

Black-box optimization of noisy functions with unknown smoothness

Jean-Bastien Grill
Michal Valko
Remi Munos

We study the problem of black-box optimization of a function $f$ of any dimension, given function evaluations perturbed by noise. The function is assumed to be locally smooth around one of its global optima, but this smoothness is unknown. Our contribution is an adaptive optimization algorithm, POO or parallel optimistic optimization, that is able to deal with this setting. POO performs almost as well as the best known algorithms requiring the knowledge of the smoothness. Furthermore, POO works for a larger class of functions than what was previously considered, especially for functions that are difficult to optimize, in a very precise sense. We provide a finite-time analysis of POO's performance, which shows that its error after $n$ evaluations is at most a factor of $\sqrt{\ln n}$ away from the error of the best known optimization algorithms using the knowledge of the smoothness.

ICML Conference 2015 Conference Paper

Cheap Bandits

Manjesh Kumar Hanawal
Venkatesh Saligrama
Michal Valko
Rémi Munos

We consider stochastic sequential learning problems where the learner can observe the average reward of several actions. Such a setting is interesting in many applications involving monitoring and surveillance, where the set of the actions to observe represent some (geographical) area. The importance of this setting is that in these applications, it is actually cheaper to observe average reward of a group of actions rather than the reward of a single action. We show that when the reward is smooth over a given graph representing the neighboring actions, we can maximize the cumulative reward of learning while minimizing the sensing cost. In this paper we propose CheapUCB, an algorithm that matches the regret guarantees of the known algorithms for this setting and at the same time guarantees a linear cost again over them. As a by-product of our analysis, we establish a Ω(\sqrt(dT)) lower bound on the cumulative regret of spectral bandits for a class of graphs with effective dimension d.

IJCAI Conference 2015 Conference Paper

Maximum Entropy Semi-Supervised Inverse Reinforcement Learning

Julien Audiffren
Michal Valko
Alessandro Lazaric
Mohammad Ghavamzadeh

A popular approach to apprenticeship learning (AL) is to formulate it as an inverse reinforcement learning (IRL) problem. The MaxEnt-IRL algorithm successfully integrates the maximum entropy principle into IRL and unlike its predecessors, it resolves the ambiguity arising from the fact that a possibly large number of policies could match the expert’s behavior. In this paper, we study an AL setting in which in addition to the expert’s trajectories, a number of unsupervised trajectories is available. We introduce MESSI, a novel algorithm that combines MaxEnt-IRL with principles coming from semi-supervised learning. In particular, MESSI integrates the unsupervised data into the MaxEnt-IRL framework using a pairwise penalty on trajectories. Empirical results in a highway driving and grid-world problems indicate that MESSI is able to take advantage of the unsupervised trajectories and improve the performance of MaxEnt-IRL.

ICML Conference 2015 Conference Paper

Simple regret for infinitely many armed bandits

Alexandra Carpentier
Michal Valko

We consider a stochastic bandit problem with infinitely many arms. In this setting, the learner has no chance of trying all the arms even once and has to dedicate its limited number of samples only to a certain number of arms. All previous algorithms for this setting were designed for minimizing the cumulative regret of the learner. In this paper, we propose an algorithm aiming at minimizing the simple regret. As in the cumulative regret setting of infinitely many armed bandits, the rate of the simple regret will depend on a parameter βcharacterizing the distribution of the near-optimal arms. We prove that depending on β, our algorithm is minimax optimal either up to a multiplicative constant or up to a \log(n) factor. We also provide extensions to several important cases: when βis unknown, in a natural setting where the near-optimal arms have a small variance, and in the case of unknown time horizon.

NeurIPS Conference 2014 Conference Paper

Efficient learning by implicit exploration in bandit problems with side observations

Tomáš Kocák
Gergely Neu
Michal Valko
Remi Munos

We consider online learning problems under a a partial observability model capturing situations where the information conveyed to the learner is between full information and bandit feedback. In the simplest variant, we assume that in addition to its own loss, the learner also gets to observe losses of some other actions. The revealed losses depend on the learner's action and a directed observation system chosen by the environment. For this setting, we propose the first algorithm that enjoys near-optimal regret guarantees without having to know the observation system before selecting its actions. Along similar lines, we also define a new partial information setting that models online combinatorial optimization problems where the feedback received by the learner is between semi-bandit and full feedback. As the predictions of our first algorithm cannot be always computed efficiently in this setting, we propose another algorithm with similar properties and with the benefit of always being computationally efficient, at the price of a slightly more complicated tuning mechanism. Both algorithms rely on a novel exploration strategy called implicit exploration, which is shown to be more efficient both computationally and information-theoretically than previously studied exploration strategies for the problem.

NeurIPS Conference 2014 Conference Paper

Extreme bandits

Alexandra Carpentier
Michal Valko

In many areas of medicine, security, and life sciences, we want to allocate limited resources to different sources in order to detect extreme values. In this paper, we study an efficient way to allocate these resources sequentially under limited feedback. While sequential design of experiments is well studied in bandit theory, the most commonly optimized property is the regret with respect to the maximum mean reward. However, in other problems such as network intrusion detection, we are interested in detecting the most extreme value output by the sources. Therefore, in our work we study extreme regret which measures the efficiency of an algorithm compared to the oracle policy selecting the source with the heaviest tail. We propose the ExtremeHunter algorithm, provide its analysis, and evaluate it empirically on synthetic and real-world experiments.

NeurIPS Conference 2014 Conference Paper

Online combinatorial optimization with stochastic decision sets and adversarial losses

Gergely Neu
Michal Valko

Most work on sequential learning assumes a fixed set of actions that are available all the time. However, in practice, actions can consist of picking subsets of readings from sensors that may break from time to time, road segments that can be blocked or goods that are out of stock. In this paper we study learning algorithms that are able to deal with stochastic availability of such unreliable composite actions. We propose and analyze algorithms based on the Follow-The-Perturbed-Leader prediction method for several learning settings differing in the feedback provided to the learner. Our algorithms rely on a novel loss estimation technique that we call Counting Asleep Times. We deliver regret bounds for our algorithms for the previously studied full information and (semi-)bandit settings, as well as a natural middle point between the two that we call the restricted information setting. A special consequence of our results is a significant improvement of the best known performance guarantees achieved by an efficient algorithm for the sleeping bandit problem with stochastic availability. Finally, we evaluate our algorithms empirically and show their improvement over the known approaches.

ICML Conference 2014 Conference Paper

Spectral Bandits for Smooth Graph Functions

Michal Valko
Rémi Munos
Branislav Kveton
Tomás Kocák

Smooth functions on graphs have wide applications in manifold and semi-supervised learning. In this paper, we study a bandit problem where the payoffs of arms are smooth on a graph. This framework is suitable for solving online learning problems that involve graphs, such as content-based recommendation. In this problem, each item we can recommend is a node and its expected rating is similar to its neighbors. The goal is to recommend items that have high expected ratings. We aim for the algorithms where the cumulative regret with respect to the optimal policy would not scale poorly with the number of nodes. In particular, we introduce the notion of an effective dimension, which is small in real-world graphs, and propose two algorithms for solving our problem that scale linearly and sublinearly in this dimension. Our experiments on real-world content recommendation problem show that a good estimator of user preferences for thousands of items can be learned from just tens of nodes evaluations.

AAAI Conference 2014 Conference Paper

Spectral Thompson Sampling

Tomáš Kocák
Michal Valko
Rémi Munos
Shipra Agrawal

Thompson Sampling (TS) has surged a lot of interest due to its good empirical performance, in particular in the computational advertising. Though successful, the tools for its performance analysis appeared only recently. In this paper, we describe and analyze SpectralTS algorithm for a bandit problem, where the payoffs of the choices are smooth given an underlying graph. In this setting, each choice is a node of a graph and the expected payoffs of the neighboring nodes are assumed to be similar. Although the setting has application both in recommender systems and advertising, the traditional algorithms would scale poorly with the number of choices. For that purpose we consider an effective dimension d, which is small in real-world graphs. We deliver the analysis showing that the regret of SpectralTS scales as d √ T ln N with high probability, where T is the time horizon and N is the number of choices. Since a d √ T ln N regret is comparable to the known results, SpectralTS offers a computationally more efficient alternative. We also show that our algorithm is competitive on both synthetic and real-world data.

UAI Conference 2013 Conference Paper

Finite-Time Analysis of Kernelised Contextual Bandits

Michal Valko
Nathaniel Korda
Rémi Munos
Ilias N. Flaounas
Nello Cristianini

We tackle the problem of online reward maximisation over a large finite set of actions described by their contexts. We focus on the case when the number of actions is too big to sample all of them even once. However we assume that we have access to the similarities between actions’ contexts and that the expected reward is an arbitrary linear function of the contexts’ images in the related reproducing kernel Hilbert space (RKHS). We propose KernelUCB, a kernelised UCB algorithm, and give a cumulative regret bound through a frequentist analysis. For contextual bandits, the related algorithm GP-UCB turns out to be a special case of our algorithm, and our finite-time analysis improves the regret bound of GP-UCB for the agnostic case, both in the terms of the kerneldependent quantity and the RKHS norm of the reward function. Moreover, for the linear kernel, our regret bound matches the lower bound for contextual linear bandits.

ICML Conference 2013 Conference Paper

Stochastic Simultaneous Optimistic Optimization

Michal Valko
Alexandra Carpentier
Rémi Munos

We study the problem of global maximization of a function f given a finite number of evaluations perturbed by noise. We consider a very weak assumption on the function, namely that it is locally smooth (in some precise sense) with respect to some semi-metric, around one of its global maxima. Compared to previous works on bandits in general spaces (Kleinberg et al. , 2008; Bubeck et al. , 2011a) our algorithm does not require the knowledge of this semi-metric. Our algorithm, StoSOO, follows an optimistic strategy to iteratively construct upper confidence bounds over the hierarchical partitions of the function domain to decide which point to sample next. A finite-time analysis of StoSOO shows that it performs almost as well as the best specifically-tuned algorithms even though the local smoothness of the function is not known.

EWRL Workshop 2012 Conference Paper

Semi-Supervised Apprenticeship Learning

Michal Valko
Mohammad Ghavamzadeh
Alessandro Lazaric

In apprenticeship learning we aim to learn a good policy by observing the behavior of an expert or a set of experts. In particular, we consider the case where the expert acts so as to maximize an unknown reward function defined as a linear combination of a set of state features. In this paper, we consider the setting where we observe many sample trajectories (i. e. , sequences of states) but only one or a few of them are labeled as experts' trajectories. We investigate the conditions under which the remaining unlabeled trajectories can help in learning a policy with a good performance. In particular, we define an extension to the max-margin inverse reinforcement learning proposed by Abbeel and Ng [2004] where, at each iteration, the max-margin optimization step is replaced by a semi-supervised optimiza- tion problem which favors classifiers separating clusters of trajectories. Finally, we report empirical results on two grid-world domains showing that the semi-supervised algorithm is able to output a better policy in fewer iterations than the related algorithm that does not take the unlabeled trajectories into account.

UAI Conference 2010 Conference Paper

Online Semi-Supervised Learning on Quantized Graphs

Michal Valko
Branislav Kveton
Ling Huang
Daniel Ting

In this paper, we tackle the problem of online semi-supervised learning (SSL). When data arrive in a stream, the dual problems of computation and data storage arise for any SSL method. We propose a fast approximate online SSL algorithm that solves for the harmonic solution on an approximate graph. We show, both empirically and theoretically, that good behavior can be achieved by collapsing nearby points into a set of local “representative points” that minimize distortion. Moreover, we regularize the harmonic solution to achieve better stability properties. We apply our algorithm to face recognition and optical character recognition applications to show that we can take advantage of the manifold structure to outperform the previous methods. Unlike previous heuristic approaches, we show that our method yields provable performance bounds.