The field of learning-augmented algorithms has gained significant attention in recent years. Using potentially inaccurate predictions, these algorithms must exhibit three key properties: consistency, robustness, and smoothness. In scenarios with stochastic predictions, a strong average-case performance is required. Typically, the design of such algorithms involves a natural tradeoff between consistency and robustness, and previous works aimed to achieve Pareto-optimal tradeoffs for specific problems. However, in some settings, this comes at the expense of smoothness. In this paper, we explore the tradeoffs between all the mentioned criteria and show how they can be balanced.

Bilateral trade models the problem of facilitating trades between a seller and a buyer having private valuations for the item being sold. In the online version of the problem, the learner faces a new seller and buyer at each time step, and has to post a price for each of the two parties without any knowledge of their valuations. We consider a scenario where, at each time step, before posting prices the learner observes a context vector containing information about the features of the item for sale. The valuations of both the seller and the buyer follow an unknown linear function of the context. In this setting, the learner could leverage previous transactions in an attempt to estimate private valuations. We characterize the regret regimes of different settings, taking as a baseline the best context-dependent prices in hindsight. First, in the setting in which the learner has two-bit feedback and strong budget balance constraints, we propose an algorithm with $O(\log T)$ regret. Then, we study the same set-up with noisy valuations, providing a tight $\widetilde O(T^{2/3})$ regret upper bound. Finally, we show that loosening budget balance constraints allows the learner to operate under more restrictive feedback. Specifically, we show how to address the one-bit, global budget balance setting through a reduction from the two-bit, strong budget balance setup. This established a fundamental trade-off between the quality of the feedback and the strictness of the budget constraints.

One-max search is a classic problem in online decision-making, in which a trader acts on a sequence of revealed prices and accepts one of them irrevocably to maximise its profit. The problem has been studied both in probabilistic and in worst-case settings, notably through competitive analysis, and more recently in learning-augmented settings in which the trader has access to a prediction on the sequence. However, existing approaches either lack smoothness, or do not achieve optimal worst-case guarantees: they do not attain the best possible trade-off between the consistency and the robustness of the algorithm. We close this gap by presenting the first algorithm that simultaneously achieves both of these important objectives. Furthermore, we show how to leverage the obtained smoothness to provide an analysis of one-max search in stochastic learning-augmented settings which capture randomness in both the observed prices and the prediction.

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.

We consider matroid allocation problems under \textit{opportunity fairness} constraints: resources need to be allocated to a set of agents under matroid constraints (which includes classical problems such as bipartite matching). Agents are divided into $C$ groups according to a sensitive attribute, and an allocation is opportunity-fair if each group receives the same share proportional to the maximum feasible allocation it could achieve in isolation. We study the Price of Fairness (PoF), i.e., the ratio between maximum size allocations and maximum size opportunity-fair allocations. We first provide a characterization of the PoF leveraging the underlying polymatroid structure of the allocation problem. Based on this characterization, we prove bounds on the PoF in various settings from fully adversarial (worst-case) to fully random. Notably, one of our main results considers an arbitrary matroid structure with agents randomly divided into groups. In this setting, we prove a PoF bound as a function of the (relative) size of the largest group. Our result implies that, as long as there is no dominant group (i.e., the largest group is not too large), opportunity fairness constraints do not induce any loss of social welfare (defined as the allocation size). Overall, our results give insights into which aspects of the problem's structure affect the trade-off between opportunity fairness and social welfare.

We study matching markets with ties, where workers on one side of the market may have tied preferences over jobs, determined by their matching utilities. Unlike classical two-sided markets with strict preferences, no single stable matching exists that is utility-maximizing for all workers. To address this challenge, we introduce the \emph{Optimal Stable Share} (OSS)-ratio, which measures the ratio of a worker's maximum achievable utility in any stable matching to their utility in a given matching. We prove that distributions over only stable matchings can incur linear utility losses, i.e., an $\Omega (N)$ OSS-ratio, where $N$ is the number of workers. To overcome this, we design an algorithm that efficiently computes a distribution over (possibly non-stable) matchings, achieving an asymptotically tight $O (\log N)$ OSS-ratio. When exact utilities are unknown, our second algorithm guarantees workers a logarithmic approximation of their optimal utility under bounded instability. Finally, we extend our offline approximation results to a bandit learning setting where utilities are only observed for matched pairs. In this setting, we consider worker-optimal stable regret, design an adaptive algorithm that smoothly interpolates between markets with strict preferences and those with statistical ties, and establish a lower bound revealing the fundamental trade-off between strict and tied preference regimes.

Repeated multi-unit auctions, where a seller allocates multiple identical items over many rounds, are common mechanisms in electricity markets and treasury auctions. We compare the two predominant formats: uniform-price and discriminatory auctions, focusing on the perspective of a single bidder learning to bid against stochastic adversaries. We characterize the learning difficulty in each format, showing that the regret scales similarly for both auction formats under both full-information and bandit feedback, as $\tilde{\Theta} ( \sqrt{T} )$ and $\tilde{\Theta} ( T^{2/3} )$, respectively. However, analysis beyond worst-case regret reveals structural differences: uniform-price auctions may admit faster learning rates, with regret scaling as $\tilde{\Theta} ( \sqrt{T} )$ in settings where discriminatory auctions remain at $\tilde{\Theta} ( T^{2/3} )$. Finally, we provide a specific analysis for auctions in which the other participants are symmetric and have unit-demand, and show that in these instances a similar regret rate separation appears.

We explore a prophet inequality problem, where the values of a sequence of items are drawn i.i.d. from some distribution, and an online decision maker must select one item irrevocably. We establish that CRℓ the worst-case competitive ratio between the expected optimal performance of an online decision maker compared to that of a prophet who uses the average of the top ℓ items is exactly the solution to an integral equation. This quantity CRℓ is larger than 1 — e -ℓ. This implies that the bound converges exponentially fast to 1 as ℓ grows. In particular for ℓ = 2, CR2 ≈ 0.966 which is much closer to 1 than the classical bound of 0.745 for ℓ = 1. Additionally, we prove asymptotic lower bounds for the competitive ratio of a more general scenario, where the decision maker is permitted to select k items. This subsumes the k multi-unit i.i.d. prophet problem and provides the current best asymptotic guarantees, as well as enables broader understanding in the more general framework. Finally, we prove a tight asymptotic competitive ratio when only static threshold policies are allowed.

We consider a Multi-Armed Bandit problem with covering constraints, where the primary goal is to ensure that each arm receives a minimum expected reward while maximizing the total cumulative reward. In this scenario, the optimal policy then belongs to some unknown feasible set. Unlike much of the existing literature, we do not assume the presence of a safe policy or a feasibility margin, which hinders the exclusive use of conservative approaches. Consequently, we propose and analyze an algorithm that switches between pessimism and optimism in the face of uncertainty. We prove both precise problem-dependent and problem-independent bounds, demonstrating that our algorithm achieves the best of the two approaches—depending on the presence or absence of a feasibility margin—in terms of constraint violation guarantees. Furthermore, our results indicate that playing greedily on the constraints actually outperforms pessimism when considering long-term violations rather than violations on a per-round basis.

Multiplayer bandits have recently garnered significant attention due to their relevance in cognitive radio networks. While the existing body of literature predominantly focuses on synchronous players, real-world radio networks, such as those in IoT applications, often feature asynchronous (i.e., randomly activated) devices. This highlights the need for addressing the more challenging asynchronous multiplayer bandits problem. Our first result shows that a natural extension of UCB achieves a minimax regret of $\mathcal{O}(\sqrt{T\log(T)})$ in the centralized setting. More significantly, we introduce Cautious Greedy, which uses $\mathcal{O}(\log(T))$ communications and whose instance-dependent regret is constant if the optimal policy assigns at least one player to each arm (a situation proven to occur when arm means are sufficiently close). Otherwise, the regret is, as usual, $\log(T)$ times the sum of some inverse sub-optimality gaps. We substantiate the optimality of Cautious Greedy through lower-bound analysis based on data-dependent terms. Therefore, we establish a strong baseline for asynchronous multiplayer bandits, at least with $\mathcal{O}(\log(T))$ communications.

The combination of lightly supervised pre-training and online fine-tuning has played a key role in recent AI developments. These new learning pipelines call for new theoretical frameworks. In this paper, we formalize key aspects of weakly supervised and active learning with a simple problem: the estimation of the mode of a distribution with partial feedback. We showcase how entropy coding allows for optimal information acquisition from partial feedback, develop coarse sufficient statistics for mode identification, and adapt bandit algorithms to our new setting. Finally, we combine those contributions into a statistically and computationally efficient solution to our original problem.

In the strategic multi-armed bandit setting, when arms possess perfect information about the player’s behavior, they can establish an equilibrium where: 1. they retain almost all of their value, 2. they leave the player with a substantial (linear) regret. This study illustrates that, even if complete information is not publicly available to all arms but is shared among them, it is possible to achieve a similar equilibrium. The primary challenge lies in designing a communication protocol that incentivizes the arms to communicate truthfully.

The non-clairvoyant scheduling problem has gained new interest within learning-augmented algorithms, where the decision-maker is equipped with predictions without any quality guarantees. In practical settings, access to predictions may be reduced to specific instances, due to cost or data limitations. Our investigation focuses on scenarios where predictions for only $B$ job sizes out of $n$ are available to the algorithm. We first establish near-optimal lower bounds and algorithms in the case of perfect predictions. Subsequently, we present a learning-augmented algorithm satisfying the robustness, consistency, and smoothness criteria, and revealing a novel tradeoff between consistency and smoothness inherent in the scenario with a restricted number of predictions.

We address the challenge of actively ranking a set of items/players with varying values/strengths. The comparison outcomes are random, with a greater noise the closer the values. A crucial requirement is that, at each iteration of the algorithm, all items must be compared once, i.e., an iteration is a perfect matching. Furthermore, we presume that comparing two players with closely matched strengths incurs no cost and, in contrast, a unit cost is associated with comparing players whose strength difference is more substantial. Our secondary objective is to determine an optimal matching between players based on this cost function: we propose and analyze an algorithm that draws on concepts from both AKS sorting networks and bandit theory. Our algorithm achieves both objectives with high probability, and the total cost is optimal (up to logarithmic terms).

Motivated by online display advertising, this work considers repeated second-price auctions, where agents sample their value from an unknown distribution with cumulative distribution function $F$. In each auction $t$, a decision-maker bound by limited observations selects $n_t$ agents from a coalition of $N$ to compete for a prize with $p$ other agents, aiming to maximize the cumulative reward of the coalition across all auctions. The problem is framed as an $N$-armed structured bandit, each number of player sent being an arm $n$, with expected reward $r(n)$ fully characterized by $F$ and $p+n$. We present two algorithms, Local-Greedy (LG) and Greedy-Grid (GG), both achieving *constant* problem-dependent regret. This relies on three key ingredients: **1.** an estimator of $r(n)$ from feedback collected from any arm $k$, **2.** concentration bounds of these estimates for $k$ within an estimation neighborhood of $n$ and **3.** the unimodality property of $r$ under standard assumptions on $F$. Additionally, GG exhibits problem-independent guarantees on top of best problem-dependent guarantees. However, by avoiding to rely on confidence intervals, LG practically outperforms GG, as well as standard unimodal bandit algorithms such as OSUB or multi-armed bandit algorithms.

Prophet inequalities are fundamental optimal stopping problems, where a decision-maker observes sequentially items with values sampled independently from known distributions, and must decide at each new observation to either stop and gain the current value or reject it irrevocably and move to the next step. This model is often too pessimistic and does not adequately represent real-world online selection processes. Potentially, rejectesd items can be revisited and a fraction of their value can be recovered. To analyze this problem, we consider general decay functions $D_1,D_2,\ldots$, quantifying the value to be recovered from a rejected item, depending on how far it has been observed in the past. We analyze how lookback improves, or not, the competitive ratio in prophet inequalities in different order models. We show that, under mild monotonicity assumptions on the decay functions, the problem can be reduced to the case where all the decay functions are equal to the same function $x \mapsto \gamma x$, where $\gamma = \inf_{x>0} \inf_{j \geq 1} D_j(x)/x$. Consequently, we focus on this setting and refine the analyses of the competitive ratios, with upper and lower bounds expressed as increasing functions of $\gamma$.

Consider a hiring process with candidates coming from different universities. It is easy to order candidates with the same background, yet it can be challenging to compare them otherwise. The latter case requires additional costly assessments, leading to a potentially high total cost for the hiring organization. Given an assigned budget, what would be an optimal strategy to select the most qualified candidate? We model the above problem as a multicolor secretary problem, allowing comparisons between candidates from distinct groups at a fixed cost. Our study explores how the allocated budget enhances the success probability of online selection algorithms.

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.

We consider the dataset valuation problem, that is the problem of quantifying the incremental gain, to some relevant pre-defined utility of a machine learning task, of aggregating an individual dataset to others. The Shapley value is a natural tool to perform dataset valuation due to its formal axiomatic justification, which can be combined with Monte Carlo integration to overcome the computational tractability challenges. Such generic approximation methods, however, remain expensive in some cases. In this paper, we exploit the knowledge about the structure of the dataset valuation problem to devise more efficient Shapley value estimators. We propose a novel approximation, referred to as discrete uniform Shapley, which is expressed as an expectation under a discrete uniform distribution with support of reasonable size. We justify the relevancy of the proposed framework via asymptotic and non-asymptotic theoretical guarantees and illustrate its benefits via an extensive set of numerical experiments.

In reinforcement learning (RL), agents sequentially interact with changing environments while aiming to maximize the obtained rewards. Usually, rewards are observed only _after_ acting, and so the goal is to maximize the _expected_ cumulative reward. Yet, in many practical settings, reward information is observed in advance -- prices are observed before performing transactions; nearby traffic information is partially known; and goals are oftentimes given to agents prior to the interaction. In this work, we aim to quantifiably analyze the value of such future reward information through the lens of _competitive analysis. In particular, we measure the ratio between the value of standard RL agents and that of agents with partial future-reward lookahead. We characterize the worst-case reward distribution and derive exact ratios for the worst-case reward expectations. Surprisingly, the resulting ratios relate to known quantities in offline RL and reward-free exploration. We further provide tight bounds for the ratio given the worst-case dynamics. Our results cover the full spectrum between observing the immediate rewards before acting to observing all the rewards before the interaction starts.

Motivated by the strategic participation of electricity producers in electricity day-ahead market, we study the problem of online learning in repeated multi-unit uniform price auctions focusing on the adversarial opposing bid setting. The main contribution of this paper is the introduction of a new modeling of the bid space. Indeed, we prove that a learning algorithm leveraging the structure of this problem achieves a regret of $\tilde{O}(K^{4/3}T^{2/3})$ under bandit feedback, improving over the bound of $\tilde{O}(K^{7/4}T^{3/4})$ previously obtained in the literature. This improved regret rate is tight up to logarithmic terms. %by deducing a lower bound of $\Omega (T^{2/3})$ from the dynamic pricing literature, proving the optimality in $T$ of our algorithm up to log factors. Inspired by electricity reserve markets, we further introduce a different feedback model under which all winning bids are revealed. This feedback interpolates between the full-information and bandit scenarios depending on the auctions' results. We prove that, under this feedback, the algorithm that we propose achieves regret $\tilde{O}(K^{5/2}\sqrt{T})$.

In contextual dynamic pricing, a seller sequentially prices goods based on contextual information. Buyers will purchase products only if the prices are below their valuations. The goal of the seller is to design a pricing strategy that collects as much revenue as possible. We focus on two different valuation models. The first assumes that valuations linearly depend on the context and are further distorted by noise. Under minor regularity assumptions, our algorithm achieves an optimal regret bound of $\tilde{\mathcal{O}}(T^{2/3})$, improving the existing results. The second model removes the linearity assumption, requiring only that the expected buyer valuation is $\beta$-H\"older in the context. For this model, our algorithm obtains a regret $\tilde{\mathcal{O}}(T^{d+2\beta/d+3\beta})$, where $d$ is the dimension of the context space.

We examine multi-armed bandit problems featuring strategic arms under debt-free reporting. In this context, each arm is characterized by a bounded support reward distribution and strategically aims to maximize its own utility by retaining a portion of the observed reward, potentially disclosing only a fraction of it to the player. This scenario unfolds as a game over $T$ rounds, leading to a competition of objectives between the player, aiming to minimize regret, and the arms, motivated by the desire to maximize their individual utilities. To address these dynamics, we propose an algorithm that establishes an equilibrium wherein each arm behaves truthfully and discloses as much of its rewards as possible. Utilizing this algorithm, the player can attain the second-highest average (true) reward among arms, with a cumulative regret bounded by $O(\log(T)/\Delta)$ (problem-dependent) or $O(\sqrt{T\log(T)})$ (worst-case).

How should an expert send forecasts to maximize her utility subject to passing a calibration test? We consider a dynamic game where an expert sends probabilistic forecasts to a decision maker. The decision maker uses a calibration test based on past outcomes to verify the expert's forecasts. We characterize the optimal forecasting strategy by reducing the dynamic game to a static persuasion problem. A distribution of forecasts is implementable by a calibrated strategy if and only if it is a mean-preserving contraction of the distribution of conditionals (honest forecasts). We characterize the value of information by comparing what an informed and uninformed expert can attain. Moreover, we consider a decision maker who uses regret minimization, instead of the calibration test, to take actions. We show that the expert can always achieve the same payoff against a regret minimizer as under the calibration test, and in some instances, she can achieve strictly more. URL: https://atulya-jain.github.io/files/calibration.pdf

Due mostly to its application to cognitive radio networks, multiplayer bandits gained a lot of interest in the last decade. A considerable progress has been made on its theoretical aspect. However, the current algorithms are far from applicable and many obstacles remain between these theoretical results and a possible implementation of multiplayer bandits algorithms in real communication networks. This survey contextualizes and organizes the rich multiplayer bandits literature. In light of the existing works, some clear directions for future research appear. We believe that a further study of these different directions might lead to theoretical algorithms adapted to real-world situations.

We discuss an application of Stochastic Approximation to statistical estimation of high-dimensional sparse parameters. The proposed solution reduces to resolving a penalized stochastic optimization problem on each stage of a multistage algorithm; each problem being solved to a prescribed accuracy by the non-Euclidean Composite Stochastic Mirror Descent (CSMD) algorithm. Assuming that the problem objective is smooth and quadratically minorated and stochastic perturbations are sub-Gaussian, our analysis prescribes the method parameters which ensure fast convergence of the estimation error (the radius of a confidence ball of a given norm around the approximate solution). This convergence is linear during the first “preliminary” phase of the routine and is sublinear during the second “asymptotic” phase. We consider an application of the proposed approach to sparse Generalized Linear Regression problem. In this setting, we show that the proposed algorithm attains the optimal convergence of the estimation error under weak assumptions on the regressor distribution. We also present a numerical study illustrating the performance of the algorithm on high-dimensional simulation data.

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.

We study single-machine scheduling of jobs, each belonging to a job type that determines its duration distribution. We start by analyzing the scenario where the type characteristics are known and then move to two learning scenarios where the types are unknown: non-preemptive problems, where each started job must be completed before moving to another job; and preemptive problems, where job execution can be paused in the favor of moving to a different job. In both cases, we design algorithms that achieve sublinear excess cost, compared to the performance with known types, and prove lower bounds for the non-preemptive case. Notably, we demonstrate, both theoretically and through simulations, how preemptive algorithms can greatly outperform non-preemptive ones when the durations of different job types are far from one another, a phenomenon that does not occur when the type durations are known.

Several problems have been extensively studied in the learning-augmented setting, where the algorithm has access to some, possibly incorrect, predictions. However, it is assumed in most works that the predictions are provided to the algorithm as input, with no constraint on their size. In this paper, we consider algorithms with access to a limited number of predictions, that they can request at any time during their execution. We study three classical problems in competitive analysis, the ski rental problem, the secretary problem, and the non-clairvoyant job scheduling. We address the question of when to query predictions and how to use them.

We consider the problem of online allocation subject to a long-term fairness penalty. Contrary to existing works, however, we do not assume that the decision-maker observes the protected attributes---which is often unrealistic in practice. Instead they can purchase data that help estimate them from sources of different quality; and hence reduce the fairness penalty at some cost. We model this problem as a multi-armed bandit problem where each arm corresponds to the choice of a data source, coupled with the fair online allocation problem. We propose an algorithm that jointly solves both problems and show that it has a regret bounded by $\mathcal{O}(\sqrt{T})$. A key difficulty is that the rewards received by selecting a source are correlated by the fairness penalty, which leads to a need for randomization (despite a stochastic setting). Our algorithm takes into account contextual information available before the source selection, and can adapt to many different fairness notions.

Contextual bandit is a general framework for online learning in sequential decision-making problems that has found application in a wide range of domains, including recommendation systems, online advertising, and clinical trials. A critical aspect of bandit methods is that they require to observe the contexts –i.e., individual or group-level data– and rewards in order to solve the sequential problem. The large deployment in industrial applications has increased interest in methods that preserve the users’ privacy. In this paper, we introduce a privacy-preserving bandit framework based on homomorphic encryption which allows computations using encrypted data. The algorithm only observes encrypted information (contexts and rewards) and has no ability to decrypt it. Leveraging the properties of homomorphic encryption, we show that despite the complexity of the setting, it is possible to solve linear contextual bandits over encrypted data with a $\widetilde{O}(d\sqrt{T})$ regret bound in any linear contextual bandit problem, while keeping data encrypted.

Potential buyers of a product or service, before making their decisions, tend to read reviews written by previous consumers. We consider Bayesian consumers with heterogeneous preferences, who sequentially decide whether to buy an item of unknown quality, based on previous buyers’ reviews. The quality is multi-dimensional and may occasionally vary over time; the reviews are also multi-dimensional. In the simple uni-dimensional and static setting, beliefs about the quality are known to converge to its true value. Our paper extends this result in several ways. First, a multi-dimensional quality is considered, second, rates of convergence are provided, third, a dynamical Markovian model with varying quality is studied. In this dynamical setting the cost of learning is shown to be small.

Contextual bandit algorithms are widely used in domains where it is desirable to provide a personalized service by leveraging contextual information, that may contain sensitive information that needs to be protected. Inspired by this scenario, we study the contextual linear bandit problem with differential privacy (DP) constraints. While the literature has focused on either centralized (joint DP) or local (local DP) privacy, we consider the shuffle model of privacy and we show that it is possible to achieve a privacy/utility trade-off between JDP and LDP. By leveraging shuffling from privacy and batching from bandits, we present an algorithm with regret bound $\widetilde{\mathcal{O}}(T^{2/3}/\varepsilon^{1/3})$, while guaranteeing both central (joint) and local privacy. Our result shows that it is possible to obtain a trade-off between JDP and LDP by leveraging the shuffle model while preserving local privacy.

The workhorse of machine learning is stochastic gradient descent.To access stochastic gradients, it is common to consider iteratively input/output pairs of a training dataset.Interestingly, it appears that one does not need full supervision to access stochastic gradients, which is the main motivation of this paper.After formalizing the "active labeling" problem, which focuses on active learning with partial supervision, we provide a streaming technique that provably minimizes the ratio of generalization error over the number of samples.We illustrate our technique in depth for robust regression.

A century ago, Emile Borel published his seminal paper on the theory of play and integral equations with skew-symmetric kernels[1]. Borel describes what is now called the Blotto game: a resource-allocation game in which two players compete for over n different battlefields by simultaneously allocating resources to each battlefield. The following two additional characteristics are perhaps the most salient features of the Blotto game:

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.

Finding an optimal matching in a weighted graph is a standard combinatorial problem. We consider its semi-bandit version where either a pair or a full matching is sampled sequentially. We prove that it is possible to leverage a rank-1 assumption on the adjacency matrix to reduce the sample complexity and the regret of off-the-shelf algorithms up to reaching a linear dependency in the number of vertices (up to to poly-log terms).

We study online learning for optimal allocation when the resource to be allocated is time. An agent receives task proposals sequentially according to a Poisson process and can either accept or reject a proposed task. If she accepts the proposal, she is busy for the duration of the task and obtains a reward that depends on the task duration. If she rejects it, she remains on hold until a new task proposal arrives. We study the regret incurred by the agent first when she knows her reward function but does not know the distribution of the task duration, and then when she does not know her reward function, either. Faster rates are finally obtained by adding structural assumptions on the distribution of rides or on the reward function. This natural setting bears similarities with contextual (one-armed) bandits, but with the crucial difference that the normalized reward associated to a context depends on the whole distribution of contexts.

Reinforcement learning algorithms are widely used in domains where it is desirable to provide a personalized service. In these domains it is common that user data contains sensitive information that needs to be protected from third parties. Motivated by this, we study privacy in the context of finite-horizon Markov Decision Processes (MDPs) by requiring information to be obfuscated on the user side. We formulate this notion of privacy for RL by leveraging the local differential privacy (LDP) framework. We establish a lower bound for regret minimization in finite-horizon MDPs with LDP guarantees which shows that guaranteeing privacy has a multiplicative effect on the regret. This result shows that while LDP is an appealing notion of privacy, it makes the learning problem significantly more complex. Finally, we present an optimistic algorithm that simultaneously satisfies $\varepsilon$-LDP requirements, and achieves $\sqrt{K}/\varepsilon$ regret in any finite-horizon MDP after $K$ episodes, matching the lower bound dependency on the number of episodes $K$.

Motivated by sequential budgeted allocation problems, we investigate online matching problems where connections between vertices are not i.i.d., but they have fixed degree distributions -- the so-called configuration model. We estimate the competitive ratio of the simplest algorithm, GREEDY, by approximating some relevant stochastic discrete processes by their continuous counterparts, that are solutions of an explicit system of partial differential equations. This technique gives precise bounds on the estimation errors, with arbitrarily high probability as the problem size increases. In particular, it allows the formal comparison between different configuration models. We also prove that, quite surprisingly, GREEDY can have better performance guarantees than RANKING, another celebrated algorithm for online matching that usually outperforms the former.

We consider the problem of online linear regression in the stochastic setting. We derive high probability regret bounds for online $\textit{ridge}$ regression and the $\textit{forward}$ algorithm. This enables us to compare online regression algorithms more accurately and eliminate assumptions of bounded observations and predictions. Our study advocates for the use of the forward algorithm in lieu of ridge due to its enhanced bounds and robustness to the regularization parameter. Moreover, we explain how to integrate it in algorithms involving linear function approximation to remove a boundedness assumption without deteriorating theoretical bounds. We showcase this modification in linear bandit settings where it yields improved regret bounds. Last, we provide numerical experiments to illustrate our results and endorse our intuitions.

Motivated by packet routing in computer networks, online queuing systems are composed of queues receiving packets at different rates. Repeatedly, they send packets to servers, each of them treating only at most one packet at a time. In the centralized case, the number of accumulated packets remains bounded (i.e., the system is stable) as long as the ratio between service rates and arrival rates is larger than $1$. In the decentralized case, individual no-regret strategies ensures stability when this ratio is larger than $2$. Yet, myopically minimizing regret disregards the long term effects due to the carryover of packets to further rounds. On the other hand, minimizing long term costs leads to stable Nash equilibria as soon as the ratio exceeds $\frac{e}{e-1}$. Stability with decentralized learning strategies with a ratio below $2$ was a major remaining question. We first argue that for ratios up to $2$, cooperation is required for stability of learning strategies, as selfish minimization of policy regret, a patient notion of regret, might indeed still be unstable in this case. We therefore consider cooperative queues and propose the first learning decentralized algorithm guaranteeing stability of the system as long as the ratio of rates is larger than $1$, thus reaching performances comparable to centralized strategies.

Strategic information is valuable either by remaining private (for instance if it is sensitive) or, on the other hand, by being used publicly to increase some utility. These two objectives are antagonistic and leaking this information might be more rewarding than concealing it. Unlike classical solutions that focus on the first point, we consider instead agents that optimize a natural trade-off between both objectives.We formalize this as an optimization problem where the objective mapping is regularized by the amount of information revealed to the adversary (measured as a divergence between the prior and posterior on the private knowledge). Quite surprisingly, when combined with the entropic regularization, the Sinkhorn loss naturally emerges in the optimization objective, making it efficiently solvable. We apply these techniques to preserve some privacy in online repeated auctions.

We study a multiplayer stochastic multi-armed bandit problem in which players cannot communicate, and if two or more players pull the same arm, a collision occurs and the involved players receive zero reward. We consider the challenging heterogeneous setting, in which different arms may have different means for different players, and propose a new and efficient algorithm that combines the idea of leveraging forced collisions for implicit communication and that of performing matching eliminations. We present a finite-time analysis of our algorithm, giving the first sublinear minimax regret bound for this problem, and prove that if the optimal assignment of players to arms is unique, our algorithm attains the optimal O(ln(T)) regret, solving an open question raised at NeurIPS 2018 by Bistritz and Leshem (2018).

We consider the practical and classical setting where the seller is using an exploration stage to learn the value distributions of the bidders before running a revenue-maximizing auction in a exploitation phase. In this two-stage process, we exhibit practical, simple and robust strategies with large utility uplifts for the bidders. We quantify precisely the seller revenue against non-discounted buyers, complementing recent studies that had focused on impatient/heavily discounted buyers. We also prove the robustness of these shading strategies to sample approximation error of the seller, to bidder’s approximation error of the competition and to possible change of the mechanisms.

We consider the classical problem of sequential resource allocation where a decision maker must repeatedly divide a budget between several resources, each with diminishing returns. This can be recast as a specific stochastic optimization problem where the objective is to maximize the cumulative reward, or equivalently to minimize the regret. We construct an algorithm that is adaptive to the complexity of the problem, expressed in term of the regularity of the returns of the resources, measured by the exponent in the Łojasiewicz inequality (or by their universal concavity parameter). Our parameter-independent algorithm recovers the optimal rates for strongly-concave functions and the classical fast rates of multi-armed bandit (for linear reward functions). Moreover, the algorithm improves existing results on stochastic optimization in this regret minimization setting for intermediate cases.

When agents or decision maker have uncertainties of underlying parameters, they may want to take “robust” decisions that are optimal against the worst possible values of those parameters, leading to some max-min optimization problems. With several agents in competition, in game theory, uncertainties are even more important and robust games - or game with non-unique prior - have gained a lot of interest recently, notably in auctions. The existence of robust equilibria in those games is guaranteed using standard fixed point theorems as in classical finite games, so we focus on the problem of finding and characterizing them. Under some linear assumption on the structure of the uncertainties, we provide a polynomial reduction of the robust Nash problem to a standard Nash problem (on some auxiliary different game). This is possible by proving the existence of a lifting transforming robust linear programs into standard linear programs. In the general case, the above direct reduction is not always possible. However, we prove how to adapt the Lemke-Howson algorithm to find robust Nash equilibria in non-degenerate games.

Motivated by cognitive radios, stochastic multi-player multi-armed bandits gained a lot of interest recently. In this class of problems, several players simultaneously pull arms and encounter a collision – with 0 reward – if some of them pull the same arm at the same time. While the cooperative case where players maximize the collective reward (obediently following some fixed protocol) has been mostly considered, robustness to malicious players is a crucial and challenging concern. Existing approaches consider only the case of adversarial jammers whose objective is to blindly minimize the collective reward. We shall consider instead the more natural class of selfish players whose incentives are to maximize their individual rewards, potentially at the expense of the social welfare. We provide the first algorithm robust to selfish players (a.k.a. Nash equilibrium) with a logarithmic regret, when the arm performance is observed. When collisions are also observed, Grim Trigger type of strategies enable some implicit communication-based algorithms and we construct robust algorithms in two different settings: the homogeneous (with a regret comparable to the centralized optimal one) and heterogeneous cases (for an adapted and relevant notion of regret). We also provide impossibility results when only the reward is observed or when arm means vary arbitrarily among players.

We investigate \emph{stochastic combinatorial semi-bandits}, where the entire joint distribution of outcomes impacts the complexity of the problem instance (unlike in the standard bandits). Typical distributions considered depend on specific parameter values, whose prior knowledge is required in theory but quite difficult to estimate in practice; an example is the commonly assumed \emph{sub-Gaussian} family. We alleviate this issue by instead considering a new general family of \emph{sub-exponential} distributions, which contains bounded and Gaussian ones. We prove a new lower bound on the regret on this family, that is parameterized by the \emph{unknown} covariance matrix, a tighter quantity than the sub-Gaussian matrix. We then construct an algorithm that uses covariance estimates, and provide a tight asymptotic analysis of the regret. Finally, we apply and extend our results to the family of sparse outcomes, which has applications in many recommender systems.

Expanding Non Markovian Reward Decision Processes (NMRDP) into Markov Decision Processes (MDP) enables the use of state of the art Reinforcement Learning (RL) techniques to identify optimal policies. In this paper an approach to exploring NMRDPs and expanding them into MDPs, without the prior knowledge of the reward structure, is proposed. The non Markovianity of the reward function is disentangled under the assumption that sets of similar and dissimilar trajectory batches can be sampled. More precisely, within the same batch, measuring the similarity between any couple of trajectories is permitted, although comparing trajectories from different batches is not possible. A modified version of the triplet loss is optimised to construct a representation of the trajectories under which rewards become Markovian.

We consider the stochastic block model where connection between vertices is perturbed by some latent (and unobserved) random geometric graph. The objective is to prove that spectral methods are robust to this type of noise, even if they are agnostic to the presence (or not) of the random graph. We provide explicit regimes where the second eigenvector of the adjacency matrix is highly correlated to the true community vector (and therefore when weak/exact recovery is possible). This is possible thanks to a detailed analysis of the spectrum of the latent random graph, of its own interest.

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.

State of the art online learning procedures focus either on selecting the best alternative (“best arm identification”) or on minimizing the cost (the “regret”). We merge these two objectives by providing the theoretical analysis of cost minimizing algorithms that are also $\delta$-PAC (with a proven guaranteed bound on the decision time), hence fulfilling at the same time regret minimization and best arm identification. This analysis sheds light on the common observation that ill-callibrated UCB-algorithms minimize regret while still identifying quickly the best arm. We also extend these results to the non-iid case faced by many practitioners. This provides a technique to make cost versus decision time compromise when doing adaptive tests with applications ranging from website A/B testing to clinical trials.

We consider the stochastic contextual bandit problem with additional regularization. The motivation comes from problems where the policy of the agent must be close to some baseline policy which is known to perform well on the task. To tackle this problem we use a nonparametric model and propose an algorithm splitting the context space into bins, and solving simultaneously — and independently — regularized multi-armed bandit instances on each bin. We derive slow and fast rates of convergence, depending on the unknown complexity of the problem. We also consider a new relevant margin condition to get problem-independent convergence rates, ending up in intermediate convergence rates interpolating between the aforementioned slow and fast rates.

We consider the problem where an agent wants to find a hidden object that is randomly located in some vertex of a directed acyclic graph (DAG) according to a fixed but possibly unknown distribution. The agent can only examine vertices whose in-neighbors have already been examined. In this paper, we address a learning setting where we allow the agent to stop before having found the object and restart searching on a new independent instance of the same problem. Our goal is to maximize the total number of hidden objects found given a time budget. The agent can thus skip an instance after realizing that it would spend too much time on it. Our contributions are both to the search theory and multi-armed bandits. If the distribution is known, we provide a quasi-optimal and efficient stationary strategy. If the distribution is unknown, we additionally show how to sequentially approximate it and, at the same time, act near-optimally in order to collect as many hidden objects as possible.

Motivated by posted price auctions where buyers are grouped in an unknown number of latent types characterized by their private values for the good on sale, we investigate regret minimization in stochastic dynamic pricing when the distribution of buyers’ private values is supported on an unknown set of points in $[0,1]$ of unknown cardinality $K$.

We consider the problem of the optimization of bidding strategies in prior-dependent revenue-maximizing auctions, when the seller fixes the reserve prices based on the bid distributions. Our study is done in the setting where one bidder is strategic. Using a variational approach, we study the complexity of the original objective and we introduce a relaxation of the objective functional in order to use gradient descent methods. Our approach is simple, general and can be applied to various value distributions and revenue-maximizing mechanisms. The new strategies we derive yield massive uplifts compared to the traditional truthfully bidding strategy.

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.

Motivated by cognitive radio networks, we consider the stochastic multiplayer multi-armed bandit problem, where several players pull arms simultaneously and collisions occur if one of them is pulled by several players at the same stage. We present a decentralized algorithm that achieves the same performance as a centralized one, contradicting the existing lower bounds for that problem. This is possible by ``hacking'' the standard model by constructing a communication protocol between players that deliberately enforces collisions, allowing them to share their information at a negligible cost. This motivates the introduction of a more appropriate dynamic setting without sensing, where similar communication protocols are no longer possible. However, we show that the logarithmic growth of the regret is still achievable for this model with a new algorithm.

We introduce a new stochastic multi-armed bandit setting where arms are grouped inside ``ordered'' categories. The motivating example comes from e-commerce, where a customer typically has a greater appetence for items of a specific well-identified but unknown category than any other one. We introduce three concepts of ordering between categories, inspired by stochastic dominance between random variables, which are gradually weaker so that more and more bandit scenarios satisfy at least one of them. We first prove instance-dependent lower bounds on the cumulative regret for each of these models, indicating how the complexity of the bandit problems increases with the generality of the ordering concept considered. We also provide algorithms that fully leverage the structure of the model with their associated theoretical guarantees. Finally, we have conducted an analysis on real data to highlight that those ordered categories actually exist in practice.

Blackwell approachability is an online learning setup generalizing the classical problem of regret minimization by allowing for instance multi-criteria optimization, global (online) optimization of a convex loss, or online linear optimization under some cumulative constraint. We consider partial monitoring where the decision maker does not necessarily observe the outcomes of his decision (unlike the traditional regret/bandit literature). Instead, he receives a random signal correlated to the decision–outcome pair, or only to the outcome. We construct, for the first time, approachability algorithms with convergence rate of order $O(T^-1/2)$ when the signal is independent of the decision and of order $O(T^-1/3)$ in the case of general signals. Those rates are optimal in the sense that they cannot be improved without further assumption on the structure of the objectives and/or the signals.

In the classical multi-armed bandit problem, $d$ arms are available to the decision maker who pulls them sequentially in order to maximize his cumulative reward. Guarantees can be obtained on a relative quantity called regret, which scales linearly with $d$ (or with $\sqrt{d}$ in the minimax sense). We here consider the \emphsparse case of this classical problem in the sense that only a small number of arms, namely $s Cite this Paper BibTeX @InProceedings{pmlr-v65-kwon17a, title = {Sparse Stochastic Bandits}, author = {Kwon, Joon and Perchet, Vianney and Vernade, Claire}, booktitle = {Proceedings of the 2017 Conference on Learning Theory}, pages = {1269--1270}, year = {2017}, editor = {Kale, Satyen and Shamir, Ohad}, volume = {65}, series = {Proceedings of Machine Learning Research}, month = {07--10 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v65/kwon17a/kwon17a.pdf}, url = {https://proceedings.mlr.press/v65/kwon17a.html}, abstract = {In the classical multi-armed bandit problem, $d$ arms are available to the decision maker who pulls them sequentially in order to maximize his cumulative reward. Guarantees can be obtained on a relative quantity called regret, which scales linearly with $d$ (or with $\sqrt{d}$ in the minimax sense). We here consider the \emphsparse case of this classical problem in the sense that only a small number of arms, namely $s Copy to Clipboard Download Endnote %0 Conference Paper %T Sparse Stochastic Bandits %A Joon Kwon %A Vianney Perchet %A Claire Vernade %B Proceedings of the 2017 Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2017 %E Satyen Kale %E Ohad Shamir %F pmlr-v65-kwon17a %I PMLR %P 1269--1270 %U https://proceedings.mlr.press/v65/kwon17a.html %V 65 %X In the classical multi-armed bandit problem, $d$ arms are available to the decision maker who pulls them sequentially in order to maximize his cumulative reward. Guarantees can be obtained on a relative quantity called regret, which scales linearly with $d$ (or with $\sqrt{d}$ in the minimax sense). We here consider the \emphsparse case of this classical problem in the sense that only a small number of arms, namely $s Copy to Clipboard Download APA Kwon, J., Perchet, V. & Vernade, C.. (2017). Sparse Stochastic Bandits. Proceedings of the 2017 Conference on Learning Theory, in Proceedings of Machine Learning Research 65:1269-1270 Available from https://proceedings.mlr.press/v65/kwon17a.html. Copy to Clipboard Download Related Material Download PDF This site last compiled Sun, 05 Jul 2026 15:10:29 +0000 Github Account Copyright © The authors and PMLR 2026. MLResearchPress

We consider the problem of bandit optimization, inspired by stochastic optimization and online learning problems with bandit feedback. In this problem, the objective is to minimize a global loss function of all the actions, not necessarily a cumulative loss. This framework allows us to study a very general class of problems, with applications in statistics, machine learning, and other fields. To solve this problem, we analyze the Upper-Confidence Frank-Wolfe algorithm, inspired by techniques for bandits and convex optimization. We give theoretical guarantees for the performance of this algorithm over various classes of functions, and discuss the optimality of these results.

The minimization of convex functions which are only available through partial and noisy information is a key methodological problem in many disciplines. In this paper we consider convex optimization with noisy zero-th order information, that is noisy function evaluations at any desired point. We focus on problems with high degrees of smoothness, such as logistic regression. We show that as opposed to gradient-based algorithms, high-order smoothness may be used to improve estimation rates, with a precise dependence of our upper-bounds on the degree of smoothness. In particular, we show that for infinitely differentiable functions, we recover the same dependence on sample size as gradient-based algorithms, with an extra dimension-dependent factor. This is done for both convex and strongly-convex functions, with finite horizon and anytime algorithms. Finally, we also recover similar results in the online optimization setting.

We consider the sequential decision problem known as regret minimization, or more precisely its generalization to the vectorial or multi-criteria setup called Blackwell approachability. We assume that Nature, the decision maker, or both, might have some quitting (or terminating) actions so that the stream of payoffs is constant whenever they are chosen. We call those environments “quitting games”. We characterize convex target sets \cC that are Blackwell approachable, in the sense that the decision maker has a policy ensuring that the expected average vector payoff converges to \cC at some given horizon known in advance. Moreover, we also compare these results to the cases where the horizon is not known and show that, unlike in standard online learning literature, the necessary or sufficient conditions for the anytime version of this problem are drastically different than those for the fixed horizon.

Motivated by online advertising auctions, we consider repeated Vickrey auctions where goods of unknown value are sold sequentially and bidders only learn (potentially noisy) information about a good’s value once it is purchased. We adopt an online learning approach with bandit feedback to model this problem and derive bidding strategies for two models: stochastic and adversarial. In the stochastic model, the observed values of the goods are random variables centered around the true value of the good. In this case, logarithmic regret is achievable when competing against well behaved adversaries. In the adversarial model, the goods need not be identical. Comparing our performance against that of the best fixed bid in hindsight, we show that sublinear regret is also achievable in this case. For both the stochastic and adversarial models, we prove matching minimax lower bounds showing our strategies to be optimal up to lower-order terms. To our knowledge, this is the first complete set of strategies for bidders participating in auctions of this type.

We introduce an anytime algorithm for stochastic multi-armed bandit with optimal distribution free and distribution dependent bounds (for a specific family of parameters). The performances of this algorithm (as well as another one motivated by the conjectured optimal bound) are evaluated empirically. A similar analysis is provided with full information, to serve as a benchmark.

The combinatorial stochastic semi-bandit problem is an extension of the classical multi-armed bandit problem in which an algorithm pulls more than one arm at each stage and the rewards of all pulled arms are revealed. One difference with the single arm variant is that the dependency structure of the arms is crucial. Previous works on this setting either used a worst-case approach or imposed independence of the arms. We introduce a way to quantify the dependency structure of the problem and design an algorithm that adapts to it. The algorithm is based on linear regression and the analysis uses techniques from the linear bandit literature. By comparing its performance to a new lower bound, we prove that it is optimal, up to a poly-logarithmic factor in the number of arms pulled.

We demonstrate that, in the classical non-stochastic regret minimization problem with $d$ decisions, gains and losses to be respectively maximized or minimized are fundamentally different. Indeed, by considering the additional sparsity assumption (at each stage, at most $s$ decisions incur a nonzero outcome), we derive optimal regret bounds of different orders. Specifically, with gains, we obtain an optimal regret guarantee after $T$ stages of order $\sqrt{T\log s}$, so the classical dependency in the dimension is replaced by the sparsity size. With losses, we provide matching upper and lower bounds of order $\sqrt{Ts\log(d)/d}$, which is decreasing in $d$. Eventually, we also study the bandit setting, and obtain an upper bound of order $\sqrt{Ts\log (d/s)}$ when outcomes are losses. This bound is proven to be optimal up to the logarithmic factor $\sqrt{\log(d/s)}$.

Motivated by practical applications, chiefly clinical trials, we study the regret achievable for stochastic multi-armed bandits under the constraint that the employed policy must split trials into a small number of batches. Our results show that a very small number of batches gives already close to minimax optimal regret bounds and we also evaluate the number of trials in each batch. As a byproduct, we derive optimal policies with low switching cost for stochastic bandits.

In the standard setting of approachability there are two players and a target set. The players play a repeated vector-valued game where one of them wants to have the average vector-valued payoff converge to the target set which the other player tries to exclude. We revisit the classical setting and consider the setting where the player has a preference relation between target sets: she wishes to approach the smallest (“best”) set possible given the observed average payoffs in hindsight. Moreover, as opposed to previous works on approachability, and in the spirit of online learning, we do not assume that there is a known game structure with actions for two players. Rather, the player receives an arbitrary vector-valued reward vector at every round. We show that it is impossible, in general, to approach the best target set in hindsight. We further propose a concrete strategy that approaches a non-trivial relaxation of the best-in-hindsight given the actual rewards. Our approach does not require projection onto a target set and amounts to switching between scalar regret minimization algorithms that are performed in episodes.

In this paper, we analyze a generic algorithm scheme for sequential global optimization using Gaussian processes. The upper bounds we derive on the cumulative regret for this generic algorithm improve by an exponential factor the previously known bounds for algorithms like GP-UCB. We also introduce the novel Gaussian Process Mutual Information algorithm (GP-MI), which significantly improves further these upper bounds for the cumulative regret. We confirm the efficiency of this algorithm on synthetic and real tasks against the natural competitor, GP-UCB, and also the Expected Improvement heuristic.

We study the stochastic multi-armed bandit problem when one knows the value μ^(⋆) of an optimal arm, as a well as a positive lower bound on the smallest positive gap ∆. We propose a new randomized policy that attains a regret uniformly bounded over time in this setting. We also prove several lower bounds, which show in particular that bounded regret is not possible if one only knows ∆, and bounded regret of order 1/∆is not possible if one only knows μ^(⋆).

Approachability has become a central tool in the analysis of repeated games and online learning. A player plays a repeated vector-valued game against Nature and her objective is to have her long-term average reward inside some target set. The celebrated results of Blackwell provide a 1/\sqrtn convergence rate of the expected point-to-set distance if this is achievable, i.e., if the set is approachable. In this paper we provide a characterization for the convergence rates of approachability and show that in some cases a set can be approached with a 1/n rate. Our characterization is solely based on a combination of geometric properties of the set with properties of the repeated game, and not on additional restrictive assumptions on Nature’s behavior.