Assaf Zeevi

Alexander Goldenshluger, Yaakov Malinovsky, Assaf Zeevi

This paper considers a finite horizon optimal stopping problem for a sequence of independent and identically distributed random variables, where the objective is to design stopping rules that attempt to select the random variable with the highest value in the sequence. The performance of any stopping rule may be benchmarked relative to the selection of a ``prophet" that has perfect foreknowledge of the largest value. Such comparisons are typically stated in the form of ``prophet inequalities." In this paper we develop a game-theoretic characterization that supports a principled approach for deriving sharp non-asymptotic prophet inequalities for single threshold stopping rules. We demonstrate that sharp constants in the ratio- and difference-type prophet inequalities are determined by the optimal values of infinite two-person zero-sum game on the unit square with particular payoff kernels, while the the solutions to the game provide optimal stopping rules and least favorable distributions. Among other things, this formulation also allows a systematic way to tackle restricted classes of distributions. The proposed framework leads to a numerically efficient algorithmic paradigm that allows computing sharp constants in prophet inequalities with any prescribed level of accuracy.

Yuhang Wu, Assaf Zeevi

On a platform with many sellers, should a pricing algorithm explicitly model competitors' prices when learning demand? Classical learning arguments suggest an affirmative answer: ignoring competitors induces model misspecification and inefficiency. In contrast, recent work on algorithmic collusion suggests that strategic obliviousness -- deliberately ignoring competitor prices -- may facilitate collusive outcomes and improve profits. We study this modeling choice in a stylized competitive market with unknown noisy demand, in which multiple sellers repeatedly set prices and estimate demand via iterated least squares, and either incorporate competitors' prices into their demand models (informed) or ignore them (oblivious). We first show that, relative to a monopolist, an oblivious seller in a competitive market must explore more aggressively to compensate for the loss of dynamic competitor information. Building on this insight, we characterize market dynamics when all sellers are oblivious and show that prices converge to the competitive outcome under sufficient exploration, while a continuum of pseudo-equilibria arises when exploration decays. Analyzing the resulting price trajectories, we uncover an excursion phenomenon that gives rise to transient collusive patterns that dissipate as learning progresses. In markets with both oblivious and informed sellers, the informed strictly out-earn the oblivious. Read as a strategy game, the modeling choice has a unique Nash equilibrium: the all-informed market, in which prices converge to the competitive outcome efficiently. Overall, our results indicate that collusive patterns are not robust and are not sustained by oblivious modeling; therefore, incorporating competitor information, together with sufficient price exploration, remains a reliable strategy for sellers in competitive markets.

Yunbei Xu, Assaf Zeevi

We develop a general theory to optimize the frequentist regret for sequential learning problems, where efficient bandit and reinforcement learning algorithms can be derived from unified Bayesian principles. We propose a novel optimization approach to generate "algorithmic beliefs" at each round, and use Bayesian posteriors to make decisions. The optimization objective to create "algorithmic beliefs," which we term "Algorithmic Information Ratio," represents an intrinsic complexity measure that effectively characterizes the frequentist regret of any algorithm. To the best of our knowledge, this is the first systematical approach to make Bayesian-type algorithms prior-free and applicable to adversarial settings, in a generic and optimal manner. Moreover, the algorithms are simple and often efficient to implement. As a major application, we present a novel algorithm for multi-armed bandits that achieves the "best-of-all-worlds" empirical performance in the stochastic, adversarial, and non-stationary environments. And we illustrate how these principles can be used in linear bandits, bandit convex optimization, and reinforcement learning.

Wonyoung Kim, Garud Iyengar, Assaf Zeevi

We propose a new regret minimization algorithm for episodic sparse linear Markov decision process (SMDP) where the state-transition distribution is a linear function of observed features. The only previously known algorithm for SMDP requires the knowledge of the sparsity parameter and oracle access to an unknown policy. We overcome these limitations by combining the doubly robust method that allows one to use feature vectors of \emph{all} actions with a novel analysis technique that enables the algorithm to use data from all periods in all episodes. The regret of the proposed algorithm is $\tilde{O}(σ^{-1}_{\min} s_{\star} H \sqrt{N})$, where $σ_{\min}$ denotes the restrictive the minimum eigenvalue of the average Gram matrix of feature vectors, $s_\star$ is the sparsity parameter, $H$ is the length of an episode, and $N$ is the number of rounds. We provide a lower regret bound that matches the upper bound up to logarithmic factors on a newly identified subclass of SMDPs. Our numerical experiments support our theoretical results and demonstrate the superior performance of our algorithm.

Ayoub Foussoul, Vineet Goyal, Orestis Papadigenopoulos et al.

In a recent work, Laforgue et al. introduce the model of last switch dependent (LSD) bandits, in an attempt to capture nonstationary phenomena induced by the interaction between the player and the environment. Examples include satiation, where consecutive plays of the same action lead to decreased performance, or deprivation, where the payoff of an action increases after an interval of inactivity. In this work, we take a step towards understanding the approximability of planning LSD bandits, namely, the (NP-hard) problem of computing an optimal arm-pulling strategy under complete knowledge of the model. In particular, we design the first efficient constant approximation algorithm for the problem and show that, under a natural monotonicity assumption on the payoffs, its approximation guarantee (almost) matches the state-of-the-art for the special and well-studied class of recharging bandits (also known as delay-dependent). In this attempt, we develop new tools and insights for this class of problems, including a novel higher-dimensional relaxation and the technique of mirroring the evolution of virtual states. We believe that these novel elements could potentially be used for approaching richer classes of action-induced nonstationary bandits (e.g., special instances of restless bandits). In the case where the model parameters are initially unknown, we develop an online learning adaptation of our algorithm for which we provide sublinear regret guarantees against its full-information counterpart.

Wonyoung Kim, Garud Iyengar, Assaf Zeevi

We consider the linear contextual multi-class multi-period packing problem (LMMP) where the goal is to pack items such that the total vector of consumption is below a given budget vector and the total value is as large as possible. We consider the setting where the reward and the consumption vector associated with each action is a class-dependent linear function of the context, and the decision-maker receives bandit feedback. LMMP includes linear contextual bandits with knapsacks and online revenue management as special cases. We establish a new estimator which guarantees a faster convergence rate, and consequently, a lower regret in such problems. We propose a bandit policy that is a closed-form function of said estimated parameters. When the contexts are non-degenerate, the regret of the proposed policy is sublinear in the context dimension, the number of classes, and the time horizon $T$ when the budget grows at least as $\sqrt{T}$. We also resolve an open problem posed by Agrawal & Devanur (2016) and extend the result to a multi-class setting. Our numerical experiments clearly demonstrate that the performance of our policy is superior to other benchmarks in the literature.

Anand Kalvit, Assaf Zeevi

We consider a stochastic multi-armed bandit (MAB) problem motivated by ``large'' action spaces, and endowed with a population of arms containing exactly $K$ arm-types, each characterized by a distinct mean reward. The decision maker is oblivious to the statistical properties of reward distributions as well as the population-level distribution of different arm-types, and is precluded also from observing the type of an arm after play. We study the classical problem of minimizing the expected cumulative regret over a horizon of play $n$, and propose algorithms that achieve a rate-optimal finite-time instance-dependent regret of $\mathcal{O}\left( \log n \right)$. We also show that the instance-independent (minimax) regret is $\tilde{\mathcal{O}}\left( \sqrt{n} \right)$ when $K=2$. While the order of regret and complexity of the problem suggests a great degree of similarity to the classical MAB problem, properties of the performance bounds and salient aspects of algorithm design are quite distinct from the latter, as are the key primitives that determine complexity along with the analysis tools needed to study them.

Steven Yin, Shipra Agrawal, Assaf Zeevi

We study the problem of allocating $T$ sequentially arriving items among $n$ homogeneous agents under the constraint that each agent must receive a pre-specified fraction of all items, with the objective of maximizing the agents' total valuation of items allocated to them. The agents' valuations for the item in each round are assumed to be i.i.d. but their distribution is a priori unknown to the central planner. Therefore, the central planner needs to implicitly learn these distributions from the observed values in order to pick a good allocation policy. However, an added challenge here is that the agents are strategic with incentives to misreport their valuations in order to receive better allocations. This sets our work apart both from the online auction design settings which typically assume known valuation distributions and/or involve payments, and from the online learning settings that do not consider strategic agents. To that end, our main contribution is an online learning based allocation mechanism that is approximately Bayesian incentive compatible, and when all agents are truthful, guarantees a sublinear regret for individual agents' utility compared to that under the optimal offline allocation policy.

Wonyoung Kim, Min-Hwan Oh, Garud Iyengar et al.

Reinforcement learning with multinomial logistic (MNL) function approximation has become an important framework due to its flexibility and broad applicability. While existing studies have established regret guarantees under worst-case analysis, they do not capture how performance depends on the variability of the interaction between the learner and the environment. In this paper, we develop a new theoretical analysis for MNL-based Markov decision processes that yields explicit variance-adaptive regret bounds. Our algorithm is computationally efficient and achieves the instance-wise optimal rate of regret, narrowing the gap between upper and lower bounds. Our numerical experiments validate that our method learns optimal policies more efficiently than conventional approaches.

Akshit Kumar, Tianyi Peng, Yuhang Wu et al.

Large language models (LLMs) have exhibited expert-level capabilities across various domains. However, their abilities to solve problems in Operations Research (OR) -- the analysis and optimization of mathematical models derived from real-world problems or their verbal descriptions -- remain underexplored. In this work, we take a first step toward evaluating LLMs' abilities to solve stochastic modeling problems, a core class of OR problems characterized by uncertainty and typically involving tools from probability, statistics, and stochastic processes. We manually procure a representative set of graduate-level homework and doctoral qualification-exam problems and test LLMs' abilities to solve them. We further leverage SimOpt, an open-source library of simulation-optimization problems and solvers, to investigate LLMs' abilities to make real-world decisions under uncertainty. Our results show that, though a nontrivial amount of work is still needed to reliably automate the stochastic modeling pipeline in reality, state-of-the-art LLMs demonstrate proficiency on par with human experts in both classroom and practical settings. These findings highlight the potential of building AI agents that assist OR researchers and amplify the real-world impact of OR through automation.

Kaizheng Wang, Yuhang Wu, Assaf Zeevi

We study adaptive querying for learning user-dependent quantities of interest, such as responses to held-out items and psychometric indicators, within tight question budgets. Classical Bayesian design and computerized adaptive testing typically rely on restrictive parametric assumptions or expensive posterior approximations, limiting their use in heterogeneous, high-dimensional, and cold-start settings. We introduce a persona-induced latent variable model that represents a user's state through membership in a finite dictionary of AI personas, each offering response distributions produced by a large language model. This yields expressive priors with closed-form posterior updates and efficient finite-mixture predictions, enabling scalable Bayesian design for sequential item selection. Experiments on synthetic data and WorldValuesBench demonstrate that persona-based posteriors deliver accurate probabilistic predictions and an interpretable adaptive elicitation pipeline.

Wonyoung Kim, Sungwoo Park, Garud Iyengar et al.

We study the linear bandit problem that accounts for partially observable features. Without proper handling, unobserved features can lead to linear regret in the decision horizon $T$, as their influence on rewards is unknown. To tackle this challenge, we propose a novel theoretical framework and an algorithm with sublinear regret guarantees. The core of our algorithm consists of (i) feature augmentation, by appending basis vectors that are orthogonal to the row space of the observed features; and (ii) the introduction of a doubly robust estimator. Our approach achieves a regret bound of $\tilde{O}(\sqrt{(d + d_h)T})$, where $d$ is the dimension of the observed features and $d_h$ depends on the extent to which the unobserved feature space is contained in the observed one, thereby capturing the intrinsic difficulty of the problem. Notably, our algorithm requires no prior knowledge of the unobserved feature space, which may expand as more features become hidden. Numerical experiments confirm that our algorithm outperforms both non-contextual multi-armed bandits and linear bandit algorithms depending solely on observed features.

Tomer Gafni, Garud Iyengar, Assaf Zeevi

We consider an online learning problem in environments with multiple change points. In contrast to the single change point problem that is widely studied using classical "high confidence" detection schemes, the multiple change point environment presents new learning-theoretic and algorithmic challenges. Specifically, we show that classical methods may exhibit catastrophic failure (high regret) due to a phenomenon we refer to as endogenous confounding. To overcome this, we propose a new class of learning algorithms dubbed Anytime Tracking CUSUM (ATC). These are horizon-free online algorithms that implement a selective detection principle, balancing the need to ignore "small" (hard-to-detect) shifts, while reacting "quickly" to significant ones. We prove that the performance of a properly tuned ATC algorithm is nearly minimax-optimal; its regret is guaranteed to closely match a novel information-theoretic lower bound on the achievable performance of any learning algorithm in the multiple change point problem. Experiments on synthetic as well as real-world data validate the aforementioned theoretical findings.

Yanlin Qu, Hongseok Namkoong, Assaf Zeevi

Thompson Sampling is one of the most widely used and studied bandit algorithms, known for its simple structure, low regret performance, and solid theoretical guarantees. Yet, in stark contrast to most other families of bandit algorithms, the exact mechanism through which posterior sampling (as introduced by Thompson) is able to "properly" balance exploration and exploitation, remains a mystery. In this paper we show that the core insight to address this question stems from recasting Thompson Sampling as an online optimization algorithm. To distill this, a key conceptual tool is introduced, which we refer to as "faithful" stationarization of the regret formulation. Essentially, the finite horizon dynamic optimization problem is converted into a stationary counterpart which "closely resembles" the original objective (in contrast, the classical infinite horizon discounted formulation, that leads to the Gittins index, alters the problem and objective in too significant a manner). The newly crafted time invariant objective can be studied using Bellman's principle which leads to a time invariant optimal policy. When viewed through this lens, Thompson Sampling admits a simple online optimization form that mimics the structure of the Bellman-optimal policy, and where greediness is regularized by a measure of residual uncertainty based on point-biserial correlation. This answers the question of how Thompson Sampling balances exploration-exploitation, and moreover, provides a principled framework to study and further improve Thompson's original idea.

Wonyoung Kim, Garud Iyengar, Assaf Zeevi

We consider Pareto front identification (PFI) for linear bandits (PFILin), i.e., the goal is to identify a set of arms with undominated mean reward vectors when the mean reward vector is a linear function of the context. PFILin includes the best arm identification problem and multi-objective active learning as special cases. The sample complexity of our proposed algorithm is optimal up to a logarithmic factor. In addition, the regret incurred by our algorithm during the estimation is within a logarithmic factor of the optimal regret among all algorithms that identify the Pareto front. Our key contribution is a new estimator that in every round updates the estimate for the unknown parameter along multiple context directions -- in contrast to the conventional estimator that only updates the parameter estimate along the chosen context. This allows us to use low-regret arms to collect information about Pareto optimal arms. Our key innovation is to reuse the exploration samples multiple times; in contrast to conventional estimators that use each sample only once. Numerical experiments demonstrate that the proposed algorithm successfully identifies the Pareto front while controlling the regret.

Anand Kalvit, Assaf Zeevi

We consider a bandit problem where at any time, the decision maker can add new arms to her consideration set. A new arm is queried at a cost from an "arm-reservoir" containing finitely many "arm-types," each characterized by a distinct mean reward. The cost of query reflects in a diminishing probability of the returned arm being optimal, unbeknown to the decision maker; this feature encapsulates defining characteristics of a broad class of operations-inspired online learning problems, e.g., those arising in markets with churn, or those involving allocations subject to costly resource acquisition. The decision maker's goal is to maximize her cumulative expected payoffs over a sequence of n pulls, oblivious to the statistical properties as well as types of the queried arms. We study two natural modes of endogeneity in the reservoir distribution, and characterize a necessary condition for achievability of sub-linear regret in the problem. We also discuss a UCB-inspired adaptive algorithm that is long-run-average optimal whenever said condition is satisfied, thereby establishing its tightness.

Anand Kalvit, Assaf Zeevi

One of the key drivers of complexity in the classical (stochastic) multi-armed bandit (MAB) problem is the difference between mean rewards in the top two arms, also known as the instance gap. The celebrated Upper Confidence Bound (UCB) policy is among the simplest optimism-based MAB algorithms that naturally adapts to this gap: for a horizon of play n, it achieves optimal O(log n) regret in instances with "large" gaps, and a near-optimal O(\sqrt{n log n}) minimax regret when the gap can be arbitrarily "small." This paper provides new results on the arm-sampling behavior of UCB, leading to several important insights. Among these, it is shown that arm-sampling rates under UCB are asymptotically deterministic, regardless of the problem complexity. This discovery facilitates new sharp asymptotics and a novel alternative proof for the O(\sqrt{n log n}) minimax regret of UCB. Furthermore, the paper also provides the first complete process-level characterization of the MAB problem under UCB in the conventional diffusion scaling. Among other things, the "small" gap worst-case lens adopted in this paper also reveals profound distinctions between the behavior of UCB and Thompson Sampling, such as an "incomplete learning" phenomenon characteristic of the latter.

Anand Kalvit, Assaf Zeevi

We consider a stochastic bandit problem with countably many arms that belong to a finite set of types, each characterized by a unique mean reward. In addition, there is a fixed distribution over types which sets the proportion of each type in the population of arms. The decision maker is oblivious to the type of any arm and to the aforementioned distribution over types, but perfectly knows the total number of types occurring in the population of arms. We propose a fully adaptive online learning algorithm that achieves O(log n) distribution-dependent expected cumulative regret after any number of plays n, and show that this order of regret is best possible. The analysis of our algorithm relies on newly discovered concentration and convergence properties of optimism-based policies like UCB in finite-armed bandit problems with "zero gap," which may be of independent interest.

Shipra Agrawal, Steven Yin, Assaf Zeevi

We consider a novel formulation of the dynamic pricing and demand learning problem, where the evolution of demand in response to posted prices is governed by a stochastic variant of the popular Bass model with parameters $α, β$ that are linked to the so-called "innovation" and "imitation" effects. Unlike the more commonly used i.i.d. and contextual demand models, in this model the posted price not only affects the demand and the revenue in the current round but also the future evolution of demand, and hence the fraction of potential market size $m$ that can be ultimately captured. In this paper, we consider the more challenging incomplete information problem where dynamic pricing is applied in conjunction with learning the unknown parameters, with the objective of optimizing the cumulative revenues over a given selling horizon of length $T$. Equivalently, the goal is to minimize the regret which measures the revenue loss of the algorithm relative to the optimal expected revenue achievable under the stochastic Bass model with market size $m$ and time horizon $T$. Our main contribution is the development of an algorithm that satisfies a high probability regret guarantee of order $\tilde O(m^{2/3})$; where the market size $m$ is known a priori. Moreover, we show that no algorithm can incur smaller order of loss by deriving a matching lower bound. Unlike most regret analysis results, in the present problem the market size $m$ is the fundamental driver of the complexity; our lower bound in fact, indicates that for any fixed $α, β$, most non-trivial instances of the problem have constant $T$ and large $m$. We believe that this insight sets the problem of dynamic pricing under the Bass model apart from the typical i.i.d. setting and multi-armed bandit based models for dynamic pricing, which typically focus only on the asymptotics with respect to time horizon $T$.

Daniel Russo, Assaf Zeevi, Tianyi Zhang

We consider a discounted infinite horizon optimal stopping problem. If the underlying distribution is known a priori, the solution of this problem is obtained via dynamic programming (DP) and is given by a well known threshold rule. When information on this distribution is lacking, a natural (though naive) approach is "explore-then-exploit," whereby the unknown distribution or its parameters are estimated over an initial exploration phase, and this estimate is then used in the DP to determine actions over the residual exploitation phase. We show: (i) with proper tuning, this approach leads to performance comparable to the full information DP solution; and (ii) despite common wisdom on the sensitivity of such "plug in" approaches in DP due to propagation of estimation errors, a surprisingly "short" (logarithmic in the horizon) exploration horizon suffices to obtain said performance. In cases where the underlying distribution is heavy-tailed, these observations are even more pronounced: a ${\it single \, sample}$ exploration phase suffices.

Yunbei Xu, Assaf Zeevi

We study problem-dependent rates, i.e., generalization errors that scale near-optimally with the variance, the effective loss, or the gradient norms evaluated at the "best hypothesis." We introduce a principled framework dubbed "uniform localized convergence," and characterize sharp problem-dependent rates for central statistical learning problems. From a methodological viewpoint, our framework resolves several fundamental limitations of existing uniform convergence and localization analysis approaches. It also provides improvements and some level of unification in the study of localized complexities, one-sided uniform inequalities, and sample-based iterative algorithms. In the so-called "slow rate" regime, we provides the first (moment-penalized) estimator that achieves the optimal variance-dependent rate for general "rich" classes; we also establish improved loss-dependent rate for standard empirical risk minimization. In the "fast rate" regime, we establish finite-sample problem-dependent bounds that are comparable to precise asymptotics. In addition, we show that iterative algorithms like gradient descent and first-order Expectation-Maximization can achieve optimal generalization error in several representative problems across the areas of non-convex learning, stochastic optimization, and learning with missing data.

Min-hwan Oh, Garud Iyengar, Assaf Zeevi

We consider a stochastic contextual bandit problem where the dimension $d$ of the feature vectors is potentially large, however, only a sparse subset of features of cardinality $s_0 \ll d$ affect the reward function. Essentially all existing algorithms for sparse bandits require a priori knowledge of the value of the sparsity index $s_0$. This knowledge is almost never available in practice, and misspecification of this parameter can lead to severe deterioration in the performance of existing methods. The main contribution of this paper is to propose an algorithm that does not require prior knowledge of the sparsity index $s_0$ and establish tight regret bounds on its performance under mild conditions. We also comprehensively evaluate our proposed algorithm numerically and show that it consistently outperforms existing methods, even when the correct sparsity index is revealed to them but is kept hidden from our algorithm.

Yunbei Xu, Assaf Zeevi

The principle of optimism in the face of uncertainty is one of the most widely used and successful ideas in multi-armed bandits and reinforcement learning. However, existing optimistic algorithms (primarily UCB and its variants) often struggle to deal with general function classes and large context spaces. In this paper, we study general contextual bandits with an offline regression oracle and propose a simple, generic principle to design optimistic algorithms, dubbed "Upper Counterfactual Confidence Bounds" (UCCB). The key innovation of UCCB is building confidence bounds in policy space, rather than in action space as is done in UCB. We demonstrate that these algorithms are provably optimal and computationally efficient in handling general function classes and large context spaces. Furthermore, we illustrate that the UCCB principle can be seamlessly extended to infinite-action general contextual bandits, provide the first solutions to these settings when employing an offline regression oracle.

Achal Bassamboo, Vikas Deep, Sandeep Juneja et al.

We consider the problem of designing an adaptive sequence of questions that optimally classify a candidate's ability into one of several categories or discriminative grades. A candidate's ability is modeled as an unknown parameter, which, together with the difficulty of the question asked, determines the likelihood with which s/he is able to answer a question correctly. The learning algorithm is only able to observe these noisy responses to its queries. We consider this problem from a fixed confidence-based $δ$-correct framework, that in our setting seeks to arrive at the correct ability discrimination at the fastest possible rate while guaranteeing that the probability of error is less than a pre-specified and small $δ$. In this setting we develop lower bounds on any sequential questioning strategy and develop geometrical insights into the problem structure both from primal and dual formulation. In addition, we arrive at algorithms that essentially match these lower bounds. Our key conclusions are that, asymptotically, any candidate needs to be asked questions at most at two (candidate ability-specific) levels, although, in a reasonably general framework, questions need to be asked only at a single level. Further, and interestingly, the problem structure facilitates endogenous exploration, so there is no need for a separately designed exploration stage in the algorithm.

Asaf Cassel, Shie Mannor, Assaf Zeevi

The stochastic multi-armed bandit (MAB) problem is a common model for sequential decision problems. In the standard setup, a decision maker has to choose at every instant between several competing arms, each of them provides a scalar random variable, referred to as a "reward." Nearly all research on this topic considers the total cumulative reward as the criterion of interest. This work focuses on other natural objectives that cannot be cast as a sum over rewards, but rather more involved functions of the reward stream. Unlike the case of cumulative criteria, in the problems we study here the oracle policy, that knows the problem parameters a priori and is used to "center" the regret, is not trivial. We provide a systematic approach to such problems, and derive general conditions under which the oracle policy is sufficiently tractable to facilitate the design of optimism-based (upper confidence bound) learning policies. These conditions elucidate an interesting interplay between the arm reward distributions and the performance metric. Our main findings are illustrated for several commonly used objectives such as conditional value-at-risk, mean-variance trade-offs, Sharpe-ratio, and more.

Shipra Agrawal, Vashist Avadhanula, Vineet Goyal et al.

We consider a dynamic assortment selection problem, where in every round the retailer offers a subset (assortment) of $N$ substitutable products to a consumer, who selects one of these products according to a multinomial logit (MNL) choice model. The retailer observes this choice and the objective is to dynamically learn the model parameters, while optimizing cumulative revenues over a selling horizon of length $T$. We refer to this exploration-exploitation formulation as the MNL-Bandit problem. Existing methods for this problem follow an "explore-then-exploit" approach, which estimate parameters to a desired accuracy and then, treating these estimates as if they are the correct parameter values, offers the optimal assortment based on these estimates. These approaches require certain a priori knowledge of "separability", determined by the true parameters of the underlying MNL model, and this in turn is critical in determining the length of the exploration period. (Separability refers to the distinguishability of the true optimal assortment from the other sub-optimal alternatives.) In this paper, we give an efficient algorithm that simultaneously explores and exploits, achieving performance independent of the underlying parameters. The algorithm can be implemented in a fully online manner, without knowledge of the horizon length $T$. Furthermore, the algorithm is adaptive in the sense that its performance is near-optimal in both the "well separated" case, as well as the general parameter setting where this separation need not hold.

Shipra Agrawal, Vashist Avadhanula, Vineet Goyal et al.

We consider a sequential subset selection problem under parameter uncertainty, where at each time step, the decision maker selects a subset of cardinality $K$ from $N$ possible items (arms), and observes a (bandit) feedback in the form of the index of one of the items in said subset, or none. Each item in the index set is ascribed a certain value (reward), and the feedback is governed by a Multinomial Logit (MNL) choice model whose parameters are a priori unknown. The objective of the decision maker is to maximize the expected cumulative rewards over a finite horizon $T$, or alternatively, minimize the regret relative to an oracle that knows the MNL parameters. We refer to this as the MNL-Bandit problem. This problem is representative of a larger family of exploration-exploitation problems that involve a combinatorial objective, and arise in several important application domains. We present an approach to adapt Thompson Sampling to this problem and show that it achieves near-optimal regret as well as attractive numerical performance.

Omar Besbes, Yonatan Gur, Assaf Zeevi

In a multi-armed bandit (MAB) problem a gambler needs to choose at each round of play one of K arms, each characterized by an unknown reward distribution. Reward realizations are only observed when an arm is selected, and the gambler's objective is to maximize his cumulative expected earnings over some given horizon of play T. To do this, the gambler needs to acquire information about arms (exploration) while simultaneously optimizing immediate rewards (exploitation); the price paid due to this trade off is often referred to as the regret, and the main question is how small can this price be as a function of the horizon length T. This problem has been studied extensively when the reward distributions do not change over time; an assumption that supports a sharp characterization of the regret, yet is often violated in practical settings. In this paper, we focus on a MAB formulation which allows for a broad range of temporal uncertainties in the rewards, while still maintaining mathematical tractability. We fully characterize the (regret) complexity of this class of MAB problems by establishing a direct link between the extent of allowable reward "variation" and the minimal achievable regret. Our analysis draws some connections between two rather disparate strands of literature: the adversarial and the stochastic MAB frameworks.