Aleksandrs Slivkins

Aleksandrs Slivkins, Xingyu Zhou, Karthik Abinav Sankararaman et al. · mit

We consider contextual bandits with linear constraints (CBwLC), a variant of contextual bandits in which the algorithm consumes multiple resources subject to linear constraints on total consumption. This problem generalizes contextual bandits with knapsacks (CBwK), allowing for packing and covering constraints, as well as positive and negative resource consumption. We provide the first algorithm for CBwLC (or CBwK) that is based on regression oracles. The algorithm is simple, computationally efficient, and statistically optimal under mild assumptions. Further, we provide the first vanishing-regret guarantees for CBwLC (or CBwK) that extend beyond the stochastic environment. We side-step strong impossibility results from prior work by identifying a weaker (and, arguably, fairer) benchmark to compare against. Our algorithm builds on LagrangeBwK (Immorlica et al., FOCS 2019), a Lagrangian-based technique for CBwK, and SquareCB (Foster and Rakhlin, ICML 2020), a regression-based technique for contextual bandits. Our analysis leverages the inherent modularity of both techniques.

Xinyan Hu, Dung Daniel Ngo, Aleksandrs Slivkins et al. · berkeley

Consider a bandit algorithm that recommends actions to self-interested users in a recommendation system. The users are free to choose other actions and need to be incentivized to follow the algorithm's recommendations. While the users prefer to exploit, the algorithm can incentivize them to explore by leveraging the information collected from the previous users. All published work on this problem, known as incentivized exploration, focuses on small, unstructured action sets and mainly targets the case when the users' beliefs are independent across actions. However, realistic exploration problems often feature large, structured action sets and highly correlated beliefs. We focus on a paradigmatic exploration problem with structure: combinatorial semi-bandits. We prove that Thompson Sampling, when applied to combinatorial semi-bandits, is incentive-compatible when initialized with a sufficient number of samples of each arm (where this number is determined in advance by the Bayesian prior). Moreover, we design incentive-compatible algorithms for collecting the initial samples.

Brendan Lucier, Sarath Pattathil, Aleksandrs Slivkins et al.

We study a game between autobidding algorithms that compete in an online advertising platform. Each autobidder is tasked with maximizing its advertiser's total value over multiple rounds of a repeated auction, subject to budget and return-on-investment constraints. We propose a gradient-based learning algorithm that is guaranteed to satisfy all constraints and achieves vanishing individual regret. Our algorithm uses only bandit feedback and can be used with the first- or second-price auction, as well as with any "intermediate" auction format. Our main result is that when these autobidders play against each other, the resulting expected liquid welfare over all rounds is at least half of the expected optimal liquid welfare achieved by any allocation. This holds whether or not the bidding dynamics converges to an equilibrium.

Kiarash Banihashem, MohammadTaghi Hajiaghayi, Suho Shin et al.

We study social learning dynamics motivated by reviews on online platforms. The agents collectively follow a simple multi-armed bandit protocol, but each agent acts myopically, without regards to exploration. We allow the greedy (exploitation-only) algorithm, as well as a wide range of behavioral biases. Specifically, we allow myopic behaviors that are consistent with (parameterized) confidence intervals for the arms' expected rewards. We derive stark learning failures for any such behavior, and provide matching positive results. The learning-failure results extend to Bayesian agents and Bayesian bandit environments. In particular, we obtain general, quantitatively strong results on failure of the greedy bandit algorithm, both for ``frequentist" and ``Bayesian" versions. Failure results known previously are quantitatively weak, and either trivial or very specialized. Thus, we provide a theoretical foundation for designing non-trivial bandit algorithms, \ie algorithms that intentionally explore, which has been missing from the literature. Our general behavioral model can be interpreted as agents' optimism or pessimism. The matching positive results entail a maximal allowed amount of optimism. Moreover, we find that no amount of pessimism helps against the learning failures, whereas even a small-but-constant fraction of extreme optimists avoids the failures and leads to near-optimal regret rates.

Lequn Wang, Akshay Krishnamurthy, Aleksandrs Slivkins

We consider offline policy optimization (OPO) in contextual bandits, where one is given a fixed dataset of logged interactions. While pessimistic regularizers are typically used to mitigate distribution shift, prior implementations thereof are either specialized or computationally inefficient. We present the first general oracle-efficient algorithm for pessimistic OPO: it reduces to supervised learning, leading to broad applicability. We obtain statistical guarantees analogous to those for prior pessimistic approaches. We instantiate our approach for both discrete and continuous actions and perform experiments in both settings, showing advantage over unregularized OPO across a wide range of configurations.

Keegan Harris, Nicole Immorlica, Brendan Lucier et al.

We study a Bayesian persuasion game where a sender wants to persuade a receiver to take a binary action, such as purchasing a product. The sender is informed about the (real-valued) state of the world, such as the quality of the product, but only has limited information about the receiver's beliefs and utilities. Motivated by customer surveys, user studies, and recent advances in AI, we allow the sender to learn more about the receiver by querying an oracle that simulates the receiver's behavior. After a fixed number of queries, the sender commits to a messaging policy and the receiver takes the action that maximizes her expected utility given the message she receives. We characterize the sender's optimal messaging policy given any distribution over receiver types. We then design a polynomial-time querying algorithm that optimizes the sender's expected utility in this game. We also consider approximate oracles, more general query structures, and costly queries.

Brendan Lucier, Nicole Immorlica, Markus Mobius et al.

Motivated by agentic markets -- two-sided markets in which consumers and businesses are assisted by AI tools that facilitate consumers' search -- we study the impact of improved search technology on learning and welfare in markets. We put forth a model where consumers engage in costly search to acquire signals of product fit prior to purchase. The market tracks indications of fit for searched products and indications of quality for chosen products, thereby guiding searches. We characterize the long-run steady-state of the resulting dynamics as well as the impact of improving search technology. We find cheaper search improves learning and consumer surplus, whereas more informative search can degrade both unless the market learns as much as consumers about the products by, for example, ``reading the transcripts'' of agentic conversations. Finally, we consider the impact of search improvements on how businesses set prices. At equilibrium prices in symmetric markets, consumer surplus is improved by cheaper search but may be decreased by more informative search, due to weakened inter-business competition.

Akshay Krishnamurthy, Keegan Harris, Dylan J. Foster et al.

We investigate the extent to which contemporary Large Language Models (LLMs) can engage in exploration, a core capability in reinforcement learning and decision making. We focus on native performance of existing LLMs, without training interventions. We deploy LLMs as agents in simple multi-armed bandit environments, specifying the environment description and interaction history entirely in-context, i.e., within the LLM prompt. We experiment with GPT-3.5, GPT-4, and Llama2, using a variety of prompt designs, and find that the models do not robustly engage in exploration without substantial interventions: i) Across all of our experiments, only one configuration resulted in satisfactory exploratory behavior: GPT-4 with chain-of-thought reasoning and an externally summarized interaction history, presented as sufficient statistics; ii) All other configurations did not result in robust exploratory behavior, including those with chain-of-thought reasoning but unsummarized history. Although these findings can be interpreted positively, they suggest that external summarization -- which may not be possible in more complex settings -- is important for obtaining desirable behavior from LLM agents. We conclude that non-trivial algorithmic interventions, such as fine-tuning or dataset curation, may be required to empower LLM-based decision making agents in complex settings.

Nicole Immorlica, Jieming Mao, Aleksandrs Slivkins et al.

We propose and design recommendation systems that incentivize efficient exploration. Agents arrive sequentially, choose actions and receive rewards, drawn from fixed but unknown action-specific distributions. The recommendation system presents each agent with actions and rewards from a subsequence of past agents, chosen ex ante. Thus, the agents engage in sequential social learning, moderated by these subsequences. We asymptotically attain optimal regret rate for exploration, using a flexible frequentist behavioral model and mitigating rationality and commitment assumptions inherent in prior work. We suggest three components of effective recommendation systems: independent focus groups, group aggregators, and interlaced information structures.

Aleksandrs Slivkins

How to incentivize self-interested agents to explore when they prefer to exploit? Consider a population of self-interested agents that make decisions under uncertainty. They "explore" to acquire new information and "exploit" this information to make good decisions. Collectively they need to balance these two objectives, but their incentives are skewed toward exploitation. This is because exploration is costly, but its benefits are spread over many agents in the future. "Incentivized Exploration" addresses this issue via strategic communication. Consider a benign ``principal" which can communicate with the agents and make recommendations, but cannot force the agents to comply. Moreover, suppose the principal can observe the agents' decisions and the outcomes of these decisions. The goal is to design a communication and recommendation policy which (i) achieves a desirable balance between exploration and exploitation, and (ii) incentivizes the agents to follow recommendations. What makes it feasible is "information asymmetry": the principal knows more than any one agent, as it collects information from many. It is essential that the principal does not fully reveal all its knowledge to the agents. Incentivized exploration combines two important problems in, resp., machine learning and theoretical economics. First, if agents always follow recommendations, the principal faces a multi-armed bandit problem: essentially, design an algorithm that balances exploration and exploitation. Second, interaction with a single agent corresponds to "Bayesian persuasion", where a principal leverages information asymmetry to convince an agent to take a particular action. We provide a brief but self-contained introduction to each problem through the lens of incentivized exploration, solving a key special case of the former as a sub-problem of the latter.

Kate Donahue, Nicole Immorlica, Meena Jagadeesan et al.

When deployed in the world, a learning agent such as a recommender system or a chatbot often repeatedly interacts with another learning agent (such as a user) over time. In many such two-agent systems, each agent learns separately and the rewards of the two agents are not perfectly aligned. To better understand such cases, we examine the learning dynamics of the two-agent system and the implications for each agent's objective. We model these systems as Stackelberg games with decentralized learning and show that standard regret benchmarks (such as Stackelberg equilibrium payoffs) result in worst-case linear regret for at least one player. To better capture these systems, we construct a relaxed regret benchmark that is tolerant to small learning errors by agents. We show that standard learning algorithms fail to provide sublinear regret, and we develop algorithms to achieve near-optimal $O(T^{2/3})$ regret for both players with respect to these benchmarks. We further design relaxed environments under which faster learning ($O(\sqrt{T})$) is possible. Altogether, our results take a step towards assessing how two-agent interactions in sequential and decentralized learning environments affect the utility of both agents.

Keegan Harris, Aleksandrs Slivkins

We evaluate the ability of the current generation of large language models (LLMs) to help a decision-making agent facing an exploration-exploitation tradeoff. We use LLMs to explore and exploit in silos in various (contextual) bandit tasks. We find that while the current LLMs often struggle to exploit, in-context mitigations may be used to substantially improve performance for small-scale tasks. However even then, LLMs perform worse than a simple linear regression. On the other hand, we find that LLMs do help at exploring large action spaces with inherent semantics, by suggesting suitable candidates to explore.

Seyed A. Esmaeili, Suho Shin, Aleksandrs Slivkins

Motivated by applications such as online labor markets we consider a variant of the stochastic multi-armed bandit problem where we have a collection of arms representing strategic agents with different performance characteristics. The platform (principal) chooses an agent in each round to complete a task. Unlike the standard setting, when an arm is pulled it can modify its reward by absorbing it or improving it at the expense of a higher cost. The principle has to solve a mechanism design problem to incentivize the arms to give their best performance. However, since even with an effective mechanism agents may still deviate from rational behavior, the principal wants a robust algorithm that also gives a non-vacuous guarantee on the total accumulated rewards under non-equilibrium behavior. In this paper, we introduce a class of bandit algorithms that meet the two objectives of performance incentivization and robustness simultaneously. We do this by identifying a collection of intuitive properties that a bandit algorithm has to satisfy to achieve these objectives. Finally, we show that settings where the principal has no information about the arms' performance characteristics can be handled by combining ideas from second price auctions with our algorithms.

Anand Kalvit, Aleksandrs Slivkins, Yonatan Gur

We study "incentivized exploration" (IE) in social learning problems where the principal (a recommendation algorithm) can leverage information asymmetry to incentivize sequentially-arriving agents to take exploratory actions. We identify posterior sampling, an algorithmic approach that is well known in the multi-armed bandits literature, as a general-purpose solution for IE. In particular, we expand the existing scope of IE in several practically-relevant dimensions, from private agent types to informative recommendations to correlated Bayesian priors. We obtain a general analysis of posterior sampling in IE which allows us to subsume these extended settings as corollaries, while also recovering existing results as special cases.

Aleksandrs Slivkins, Yunzong Xu, Shiliang Zuo

We study the greedy (exploitation-only) algorithm in bandit problems with a known reward structure. We allow arbitrary finite reward structures, while prior work focused on a few specific ones. We fully characterize when the greedy algorithm asymptotically succeeds or fails, in the sense of sublinear vs. linear regret as a function of time. Our characterization identifies a partial identifiability property of the problem instance as the necessary and sufficient condition for the asymptotic success. Notably, once this property holds, the problem becomes easy -- any algorithm will succeed (in the same sense as above), provided it satisfies a mild non-degeneracy condition. Our characterization extends to contextual bandits and interactive decision-making with arbitrary feedback. Examples demonstrating broad applicability and extensions to infinite reward structures are provided.

Gagan Bansal, Wenyue Hua, Zezhou Huang et al. · microsoft-research

As LLM agents advance, they are increasingly mediating economic decisions, ranging from product discovery to transactions, on behalf of users. Such applications promise benefits but also raise many questions about agent accountability and value for users. Addressing these questions requires understanding how agents behave in realistic market conditions. However, previous research has largely evaluated agents in constrained settings, such as single-task marketplaces (e.g., negotiation) or structured two-agent interactions. Real-world markets are fundamentally different: they require agents to handle diverse economic activities and coordinate within large, dynamic ecosystems where multiple agents with opaque behaviors may engage in open-ended dialogues. To bridge this gap, we investigate two-sided agentic marketplaces where Assistant agents represent consumers and Service agents represent competing businesses. To study these interactions safely, we develop Magentic-Marketplace -- a simulated environment where Assistants and Services can operate. This environment enables us to study key market dynamics: the utility agents achieve, behavioral biases, vulnerability to manipulation, and how search mechanisms shape market outcomes. Our experiments show that frontier models can approach optimal welfare -- but only under ideal search conditions. Performance degrades sharply with scale, and all models exhibit severe first-proposal bias, creating 10-30x advantages for response speed over quality. These findings reveal how behaviors emerge across market conditions, informing the design of fair and efficient agentic marketplaces.

Mathias Lécuyer, Sang Hoon Kim, Mihir Nanavati et al.

We observe that many system policies that make threshold decisions involving a resource (e.g., time, memory, cores) naturally reveal additional, or implicit feedback. For example, if a system waits X min for an event to occur, then it automatically learns what would have happened if it waited <X min, because time has a cumulative property. This feedback tells us about alternative decisions, and can be used to improve the system policy. However, leveraging implicit feedback is difficult because it tends to be one-sided or incomplete, and may depend on the outcome of the event. As a result, existing practices for using feedback, such as simply incorporating it into a data-driven model, suffer from bias. We develop a methodology, called Sayer, that leverages implicit feedback to evaluate and train new system policies. Sayer builds on two ideas from reinforcement learning -- randomized exploration and unbiased counterfactual estimators -- to leverage data collected by an existing policy to estimate the performance of new candidate policies, without actually deploying those policies. Sayer uses implicit exploration and implicit data augmentation to generate implicit feedback in an unbiased form, which is then used by an implicit counterfactual estimator to evaluate and train new policies. The key idea underlying these techniques is to assign implicit probabilities to decisions that are not actually taken but whose feedback can be inferred; these probabilities are carefully calculated to ensure statistical unbiasedness. We apply Sayer to two production scenarios in Azure, and show that it can evaluate arbitrary policies accurately, and train new policies that outperform the production policies.

Max Simchowitz, Aleksandrs Slivkins

How do you incentivize self-interested agents to $\textit{explore}$ when they prefer to $\textit{exploit}$? We consider complex exploration problems, where each agent faces the same (but unknown) MDP. In contrast with traditional formulations of reinforcement learning, agents control the choice of policies, whereas an algorithm can only issue recommendations. However, the algorithm controls the flow of information, and can incentivize the agents to explore via information asymmetry. We design an algorithm which explores all reachable states in the MDP. We achieve provable guarantees similar to those for incentivizing exploration in static, stateless exploration problems studied previously. To the best of our knowledge, this is the first work to consider mechanism design in a stateful, reinforcement learning setting.

Guy Aridor, Yishay Mansour, Aleksandrs Slivkins et al.

Most online platforms strive to learn from interactions with users, and many engage in exploration: making potentially suboptimal choices for the sake of acquiring new information. We study the interplay between exploration and competition: how such platforms balance the exploration for learning and the competition for users. Here users play three distinct roles: they are customers that generate revenue, they are sources of data for learning, and they are self-interested agents which choose among the competing platforms. We consider a stylized duopoly model in which two firms face the same multi-armed bandit problem. Users arrive one by one and choose between the two firms, so that each firm makes progress on its bandit problem only if it is chosen. Through a mix of theoretical results and numerical simulations, we study whether and to what extent competition incentivizes the adoption of better bandit algorithms, and whether it leads to welfare increases for users. We find that stark competition induces firms to commit to a "greedy" bandit algorithm that leads to low welfare. However, weakening competition by providing firms with some "free" users incentivizes better exploration strategies and increases welfare. We investigate two channels for weakening the competition: relaxing the rationality of users and giving one firm a first-mover advantage. Our findings are closely related to the "competition vs. innovation" relationship, and elucidate the first-mover advantage in the digital economy.

Chara Podimata, Aleksandrs Slivkins

Lipschitz bandits is a prominent version of multi-armed bandits that studies large, structured action spaces such as the $[0,1]$ interval, where similar actions are guaranteed to have similar rewards. A central theme here is the adaptive discretization of the action space, which gradually ``zooms in'' on the more promising regions thereof. The goal is to take advantage of ``nicer'' problem instances, while retaining near-optimal worst-case performance. While the stochastic version of the problem is well-understood, the general version with adversarial rewards is not. We provide the first algorithm (\emph{Adversarial Zooming}) for adaptive discretization in the adversarial version, and derive instance-dependent regret bounds. In particular, we recover the worst-case optimal regret bound for the adversarial version, and the instance-dependent regret bound for the stochastic version. We apply our algorithm to several fundamental applications -- including dynamic pricing and auction reserve tuning -- all under adversarial reward models. While these domains often violate Lipschitzness, our analysis only requires a weaker version thereof, allowing for meaningful regret bounds without additional smoothness assumptions. Notably, we extend our results to multi-product dynamic pricing with non-smooth reward structures, a setting which does not even satisfy one-sided Lipschitzness.

Maryam Majzoubi, Chicheng Zhang, Rajan Chari et al.

We create a computationally tractable algorithm for contextual bandits with continuous actions having unknown structure. Our reduction-style algorithm composes with most supervised learning representations. We prove that it works in a general sense and verify the new functionality with large-scale experiments.

Kianté Brantley, Miroslav Dudik, Thodoris Lykouris et al.

We propose an algorithm for tabular episodic reinforcement learning with constraints. We provide a modular analysis with strong theoretical guarantees for settings with concave rewards and convex constraints, and for settings with hard constraints (knapsacks). Most of the previous work in constrained reinforcement learning is limited to linear constraints, and the remaining work focuses on either the feasibility question or settings with a single episode. Our experiments demonstrate that the proposed algorithm significantly outperforms these approaches in existing constrained episodic environments.

Manish Raghavan, Aleksandrs Slivkins, Jennifer Wortman Vaughan et al.

Online learning algorithms, widely used to power search and content optimization on the web, must balance exploration and exploitation, potentially sacrificing the experience of current users in order to gain information that will lead to better decisions in the future. While necessary in the worst case, explicit exploration has a number of disadvantages compared to the greedy algorithm that always "exploits" by choosing an action that currently looks optimal. We ask under what conditions inherent diversity in the data makes explicit exploration unnecessary. We build on a recent line of work on the smoothed analysis of the greedy algorithm in the linear contextual bandits model. We improve on prior results to show that a greedy approach almost matches the best possible Bayesian regret rate of any other algorithm on the same problem instance whenever the diversity conditions hold, and that this regret is at most $\tilde O(T^{1/3})$.

Mark Sellke, Aleksandrs Slivkins

We consider incentivized exploration: a version of multi-armed bandits where the choice of arms is controlled by self-interested agents, and the algorithm can only issue recommendations. The algorithm controls the flow of information, and the information asymmetry can incentivize the agents to explore. Prior work achieves optimal regret rates up to multiplicative factors that become arbitrarily large depending on the Bayesian priors, and scale exponentially in the number of arms. A more basic problem of sampling each arm once runs into similar factors. We focus on the price of incentives: the loss in performance, broadly construed, incurred for the sake of incentive-compatibility. We prove that Thompson Sampling, a standard bandit algorithm, is incentive-compatible if initialized with sufficiently many data points. The performance loss due to incentives is therefore limited to the initial rounds when these data points are collected. The problem is largely reduced to that of sample complexity: how many rounds are needed? We address this question, providing matching upper and lower bounds and instantiating them in various corollaries. Typically, the optimal sample complexity is polynomial in the number of arms and exponential in the "strength of beliefs".

Karthik Abinav Sankararaman, Aleksandrs Slivkins

Bandits with Knapsacks (BwK) is a general model for multi-armed bandits under supply/budget constraints. While worst-case regret bounds for BwK are well-understood, we present three results that go beyond the worst-case perspective. First, we provide upper and lower bounds which amount to a full characterization for logarithmic, instance-dependent regret rates. Second, we consider "simple regret" in BwK, which tracks algorithm's performance in a given round, and prove that it is small in all but a few rounds. Third, we provide a general "reduction" from BwK to bandits which takes advantage of some known helpful structure, and apply this reduction to combinatorial semi-bandits, linear contextual bandits, and multinomial-logit bandits. Our results build on the BwK algorithm from \citet{AgrawalDevanur-ec14}, providing new analyses thereof.

Thodoris Lykouris, Max Simchowitz, Aleksandrs Slivkins et al.

We initiate the study of multi-stage episodic reinforcement learning under adversarial corruptions in both the rewards and the transition probabilities of the underlying system extending recent results for the special case of stochastic bandits. We provide a framework which modifies the aggressive exploration enjoyed by existing reinforcement learning approaches based on "optimism in the face of uncertainty", by complementing them with principles from "action elimination". Importantly, our framework circumvents the major challenges posed by naively applying action elimination in the RL setting, as formalized by a lower bound we demonstrate. Our framework yields efficient algorithms which (a) attain near-optimal regret in the absence of corruptions and (b) adapt to unknown levels corruption, enjoying regret guarantees which degrade gracefully in the total corruption encountered. To showcase the generality of our approach, we derive results for both tabular settings (where states and actions are finite) as well as linear-function-approximation settings (where the dynamics and rewards admit a linear underlying representation). Notably, our work provides the first sublinear regret guarantee which accommodates any deviation from purely i.i.d. transitions in the bandit-feedback model for episodic reinforcement learning.

Aleksandrs Slivkins

Multi-armed bandits a simple but very powerful framework for algorithms that make decisions over time under uncertainty. An enormous body of work has accumulated over the years, covered in several books and surveys. This book provides a more introductory, textbook-like treatment of the subject. Each chapter tackles a particular line of work, providing a self-contained, teachable technical introduction and a brief review of the further developments; many of the chapters conclude with exercises. The book is structured as follows. The first four chapters are on IID rewards, from the basic model to impossibility results to Bayesian priors to Lipschitz rewards. The next three chapters cover adversarial rewards, from the full-feedback version to adversarial bandits to extensions with linear rewards and combinatorially structured actions. Chapter 8 is on contextual bandits, a middle ground between IID and adversarial bandits in which the change in reward distributions is completely explained by observable contexts. The last three chapters cover connections to economics, from learning in repeated games to bandits with supply/budget constraints to exploration in the presence of incentives. The appendix provides sufficient background on concentration and KL-divergence. The chapters on "bandits with similarity information", "bandits with knapsacks" and "bandits and agents" can also be consumed as standalone surveys on the respective topics.

Nicole Immorlica, Jieming Mao, Aleksandrs Slivkins et al.

It is common in recommendation systems that users both consume and produce information as they make strategic choices under uncertainty. While a social planner would balance "exploration" and "exploitation" using a multi-armed bandit algorithm, users' incentives may tilt this balance in favor of exploitation. We consider Bayesian Exploration: a simple model in which the recommendation system (the "principal") controls the information flow to the users (the "agents") and strives to incentivize exploration via information asymmetry. A single round of this model is a version of a well-known "Bayesian Persuasion game" from [Kamenica and Gentzkow]. We allow heterogeneous users, relaxing a major assumption from prior work that users have the same preferences from one time step to another. The goal is now to learn the best personalized recommendations. One particular challenge is that it may be impossible to incentivize some of the user types to take some of the actions, no matter what the principal does or how much time she has. We consider several versions of the model, depending on whether and when the user types are reported to the principal, and design a near-optimal "recommendation policy" for each version. We also investigate how the model choice and the diversity of user types impact the set of actions that can possibly be "explored" by each type.

Guy Aridor, Kevin Liu, Aleksandrs Slivkins et al.

We empirically study the interplay between exploration and competition. Systems that learn from interactions with users often engage in exploration: making potentially suboptimal decisions in order to acquire new information for future decisions. However, when multiple systems are competing for the same market of users, exploration may hurt a system's reputation in the near term, with adverse competitive effects. In particular, a system may enter a "death spiral", when the short-term reputation cost decreases the number of users for the system to learn from, which degrades its performance relative to competition and further decreases its market share. We ask whether better exploration algorithms are incentivized under competition. We run extensive numerical experiments in a stylized duopoly model in which two firms deploy multi-armed bandit algorithms and compete for myopic users. We find that duopoly and monopoly tend to favor a primitive "greedy algorithm" that does not explore and leads to low consumer welfare, whereas a temporary monopoly (a duopoly with an early entrant) may incentivize better bandit algorithms and lead to higher consumer welfare. Our findings shed light on the first-mover advantage in the digital economy by exploring the role that data can play as a barrier to entry in online markets.

Akshay Krishnamurthy, John Langford, Aleksandrs Slivkins et al.

We study contextual bandit learning with an abstract policy class and continuous action space. We obtain two qualitatively different regret bounds: one competes with a smoothed version of the policy class under no continuity assumptions, while the other requires standard Lipschitz assumptions. Both bounds exhibit data-dependent "zooming" behavior and, with no tuning, yield improved guarantees for benign problems. We also study adapting to unknown smoothness parameters, establishing a price-of-adaptivity and deriving optimal adaptive algorithms that require no additional information.

Nicole Immorlica, Karthik Abinav Sankararaman, Robert Schapire et al.

We consider Bandits with Knapsacks (henceforth, BwK), a general model for multi-armed bandits under supply/budget constraints. In particular, a bandit algorithm needs to solve a well-known knapsack problem: find an optimal packing of items into a limited-size knapsack. The BwK problem is a common generalization of numerous motivating examples, which range from dynamic pricing to repeated auctions to dynamic ad allocation to network routing and scheduling. While the prior work on BwK focused on the stochastic version, we pioneer the other extreme in which the outcomes can be chosen adversarially. This is a considerably harder problem, compared to both the stochastic version and the "classic" adversarial bandits, in that regret minimization is no longer feasible. Instead, the objective is to minimize the competitive ratio: the ratio of the benchmark reward to the algorithm's reward. We design an algorithm with competitive ratio O(log T) relative to the best fixed distribution over actions, where T is the time horizon; we also prove a matching lower bound. The key conceptual contribution is a new perspective on the stochastic version of the problem. We suggest a new algorithm for the stochastic version, which builds on the framework of regret minimization in repeated games and admits a substantially simpler analysis compared to prior work. We then analyze this algorithm for the adversarial version and use it as a subroutine to solve the latter.

Nicole Immorlica, Jieming Mao, Aleksandrs Slivkins et al.

We propose and design recommendation systems that incentivize efficient exploration. Agents arrive sequentially, choose actions and receive rewards, drawn from fixed but unknown action-specific distributions. The recommendation system presents each agent with actions and rewards from a subsequence of past agents, chosen ex ante. Thus, the agents engage in sequential social learning, moderated by these subsequences. We asymptotically attain optimal regret rate for exploration, using a flexible frequentist behavioral model and mitigating rationality and commitment assumptions inherent in prior work. We suggest three components of effective recommendation systems: independent focus groups, group aggregators, and interlaced information structures.

Manish Raghavan, Aleksandrs Slivkins, Jennifer Wortman Vaughan et al.

Online learning algorithms, widely used to power search and content optimization on the web, must balance exploration and exploitation, potentially sacrificing the experience of current users for information that will lead to better decisions in the future. Recently, concerns have been raised about whether the process of exploration could be viewed as unfair, placing too much burden on certain individuals or groups. Motivated by these concerns, we initiate the study of the externalities of exploration - the undesirable side effects that the presence of one party may impose on another - under the linear contextual bandits model. We introduce the notion of a group externality, measuring the extent to which the presence of one population of users impacts the rewards of another. We show that this impact can in some cases be negative, and that, in a certain sense, no algorithm can avoid it. We then study externalities at the individual level, interpreting the act of exploration as an externality imposed on the current user of a system by future users. This drives us to ask under what conditions inherent diversity in the data makes explicit exploration unnecessary. We build on a recent line of work on the smoothed analysis of the greedy algorithm that always chooses the action that currently looks optimal, improving on prior results to show that a greedy approach almost matches the best possible Bayesian regret rate of any other algorithm on the same problem instance whenever the diversity conditions hold, and that this regret is at most $\tilde{O}(T^{1/3})$. Returning to group-level effects, we show that under the same conditions, negative group externalities essentially vanish under the greedy algorithm. Together, our results uncover a sharp contrast between the high externalities that exist in the worst case, and the ability to remove all externalities if the data is sufficiently diverse.

Karthik Abinav Sankararaman, Aleksandrs Slivkins

We unify two prominent lines of work on multi-armed bandits: bandits with knapsacks (BwK) and combinatorial semi-bandits. The former concerns limited "resources" consumed by the algorithm, e.g., limited supply in dynamic pricing. The latter allows a huge number of actions but assumes combinatorial structure and additional feedback to make the problem tractable. We define a common generalization, support it with several motivating examples, and design an algorithm for it. Our regret bounds are comparable with those for BwK and combinatorial semi- bandits.

Yishay Mansour, Aleksandrs Slivkins, Zhiwei Steven Wu

Most modern systems strive to learn from interactions with users, and many engage in exploration: making potentially suboptimal choices for the sake of acquiring new information. We initiate a study of the interplay between exploration and competition--how such systems balance the exploration for learning and the competition for users. Here the users play three distinct roles: they are customers that generate revenue, they are sources of data for learning, and they are self-interested agents which choose among the competing systems. In our model, we consider competition between two multi-armed bandit algorithms faced with the same bandit instance. Users arrive one by one and choose among the two algorithms, so that each algorithm makes progress if and only if it is chosen. We ask whether and to what extent competition incentivizes the adoption of better bandit algorithms. We investigate this issue for several models of user response, as we vary the degree of rationality and competitiveness in the model. Our findings are closely related to the "competition vs. innovation" relationship, a well-studied theme in economics.

Aaron Roth, Aleksandrs Slivkins, Jonathan Ullman et al.

We study the problem of a seller dynamically pricing $d$ distinct types of indivisible goods, when faced with the online arrival of unit-demand buyers drawn independently from an unknown distribution. The goods are not in limited supply, but can only be produced at a limited rate and are costly to produce. The seller observes only the bundle of goods purchased at each day, but nothing else about the buyer's valuation function. Our main result is a dynamic pricing algorithm for optimizing welfare (including the seller's cost of production) that runs in time and a number of rounds that are polynomial in $d$ and the approximation parameter. We are able to do this despite the fact that (i) the price-response function is not continuous, and even its fractional relaxation is a non-concave function of the prices, and (ii) the welfare is not observable to the seller. We derive this result as an application of a general technique for optimizing welfare over \emph{divisible} goods, which is of independent interest. When buyers have strongly concave, Hölder continuous valuation functions over $d$ divisible goods, we give a general polynomial time dynamic pricing technique. We are able to apply this technique to the setting of unit demand buyers despite the fact that in that setting the goods are not divisible, and the natural fractional relaxation of a unit demand valuation is not strongly concave. In order to apply our general technique, we introduce a novel price randomization procedure which has the effect of implicitly inducing buyers to "regularize" their valuations with a strongly concave function. Finally, we also extend our results to a limited-supply setting in which the number of copies of each good cannot be replenished.

Yishay Mansour, Aleksandrs Slivkins, Vasilis Syrgkanis et al.

We consider a ubiquitous scenario in the Internet economy when individual decision-makers (henceforth, agents) both produce and consume information as they make strategic choices in an uncertain environment. This creates a three-way tradeoff between exploration (trying out insufficiently explored alternatives to help others in the future), exploitation (making optimal decisions given the information discovered by other agents), and incentives of the agents (who are myopically interested in exploitation, while preferring the others to explore). We posit a principal who controls the flow of information from agents that came before, and strives to coordinate the agents towards a socially optimal balance between exploration and exploitation, not using any monetary transfers. The goal is to design a recommendation policy for the principal which respects agents' incentives and minimizes a suitable notion of regret. We extend prior work in this direction to allow the agents to interact with one another in a shared environment: at each time step, multiple agents arrive to play a Bayesian game, receive recommendations, choose their actions, receive their payoffs, and then leave the game forever. The agents now face two sources of uncertainty: the actions of the other agents and the parameters of the uncertain game environment. Our main contribution is to show that the principal can achieve constant regret when the utilities are deterministic (where the constant depends on the prior distribution, but not on the time horizon), and logarithmic regret when the utilities are stochastic. As a key technical tool, we introduce the concept of explorable actions, the actions which some incentive-compatible policy can recommend with non-zero probability. We show how the principal can identify (and explore) all explorable actions, and use the revealed information to perform optimally.

Miroslav Dudík, Katja Hofmann, Robert E. Schapire et al.

We consider the problem of learning to choose actions using contextual information when provided with limited feedback in the form of relative pairwise comparisons. We study this problem in the dueling-bandits framework of Yue et al. (2009), which we extend to incorporate context. Roughly, the learner's goal is to find the best policy, or way of behaving, in some space of policies, although "best" is not always so clearly defined. Here, we propose a new and natural solution concept, rooted in game theory, called a von Neumann winner, a randomized policy that beats or ties every other policy. We show that this notion overcomes important limitations of existing solutions, particularly the Condorcet winner which has typically been used in the past, but which requires strong and often unrealistic assumptions. We then present three efficient algorithms for online learning in our setting, and for approximating a von Neumann winner from batch-like data. The first of these algorithms achieves particularly low regret, even when data is adversarial, although its time and space requirements are linear in the size of the policy space. The other two algorithms require time and space only logarithmic in the size of the policy space when provided access to an oracle for solving classification problems on the space.

Ittai Abraham, Omar Alonso, Vasilis Kandylas et al.

Crowdsourcing has been part of the IR toolbox as a cheap and fast mechanism to obtain labels for system development and evaluation. Successful deployment of crowdsourcing at scale involves adjusting many variables, a very important one being the number of workers needed per human intelligence task (HIT). We consider the crowdsourcing task of learning the answer to simple multiple-choice HITs, which are representative of many relevance experiments. In order to provide statistically significant results, one often needs to ask multiple workers to answer the same HIT. A stopping rule is an algorithm that, given a HIT, decides for any given set of worker answers if the system should stop and output an answer or iterate and ask one more worker. Knowing the historic performance of a worker in the form of a quality score can be beneficial in such a scenario. In this paper we investigate how to devise better stopping rules given such quality scores. We also suggest adaptive exploration as a promising approach for scalable and automatic creation of ground truth. We conduct a data analysis on an industrial crowdsourcing platform, and use the observations from this analysis to design new stopping rules that use the workers' quality scores in a non-trivial manner. We then perform a simulation based on a real-world workload, showing that our algorithm performs better than the more naive approaches.

Chien-Ju Ho, Aleksandrs Slivkins, Jennifer Wortman Vaughan

Crowdsourcing markets have emerged as a popular platform for matching available workers with tasks to complete. The payment for a particular task is typically set by the task's requester, and may be adjusted based on the quality of the completed work, for example, through the use of "bonus" payments. In this paper, we study the requester's problem of dynamically adjusting quality-contingent payments for tasks. We consider a multi-round version of the well-known principal-agent model, whereby in each round a worker makes a strategic choice of the effort level which is not directly observable by the requester. In particular, our formulation significantly generalizes the budget-free online task pricing problems studied in prior work. We treat this problem as a multi-armed bandit problem, with each "arm" representing a potential contract. To cope with the large (and in fact, infinite) number of arms, we propose a new algorithm, AgnosticZooming, which discretizes the contract space into a finite number of regions, effectively treating each region as a single arm. This discretization is adaptively refined, so that more promising regions of the contract space are eventually discretized more finely. We analyze this algorithm, showing that it achieves regret sublinear in the time horizon and substantially improves over non-adaptive discretization (which is the only competing approach in the literature). Our results advance the state of art on several different topics: the theory of crowdsourcing markets, principal-agent problems, multi-armed bandits, and dynamic pricing.

Ashwinkumar Badanidiyuru, John Langford, Aleksandrs Slivkins

We study contextual bandits with ancillary constraints on resources, which are common in real-world applications such as choosing ads or dynamic pricing of items. We design the first algorithm for solving these problems that handles constrained resources other than time, and improves over a trivial reduction to the non-contextual case. We consider very general settings for both contextual bandits (arbitrary policy sets, e.g. Dudik et al. (UAI'11)) and bandits with resource constraints (bandits with knapsacks, Badanidiyuru et al. (FOCS'13)), and prove a regret guarantee with near-optimal statistical properties.

Robert Kleinberg, Aleksandrs Slivkins, Eli Upfal

In a multi-armed bandit problem, an online algorithm chooses from a set of strategies in a sequence of trials so as to maximize the total payoff of the chosen strategies. While the performance of bandit algorithms with a small finite strategy set is quite well understood, bandit problems with large strategy sets are still a topic of very active investigation, motivated by practical applications such as online auctions and web advertisement. The goal of such research is to identify broad and natural classes of strategy sets and payoff functions which enable the design of efficient solutions. In this work we study a very general setting for the multi-armed bandit problem in which the strategies form a metric space, and the payoff function satisfies a Lipschitz condition with respect to the metric. We refer to this problem as the "Lipschitz MAB problem". We present a solution for the multi-armed bandit problem in this setting. That is, for every metric space we define an isometry invariant which bounds from below the performance of Lipschitz MAB algorithms for this metric space, and we present an algorithm which comes arbitrarily close to meeting this bound. Furthermore, our technique gives even better results for benign payoff functions. We also address the full-feedback ("best expert") version of the problem, where after every round the payoffs from all arms are revealed.

Aleksandrs Slivkins, Jennifer Wortman Vaughan

Over the past decade, crowdsourcing has emerged as a cheap and efficient method of obtaining solutions to simple tasks that are difficult for computers to solve but possible for humans. The popularity and promise of crowdsourcing markets has led to both empirical and theoretical research on the design of algorithms to optimize various aspects of these markets, such as the pricing and assignment of tasks. Much of the existing theoretical work on crowdsourcing markets has focused on problems that fall into the broad category of online decision making; task requesters or the crowdsourcing platform itself make repeated decisions about prices to set, workers to filter out, problems to assign to specific workers, or other things. Often these decisions are complex, requiring algorithms that learn about the distribution of available tasks or workers over time and take into account the strategic (or sometimes irrational) behavior of workers. As human computation grows into its own field, the time is ripe to address these challenges in a principled way. However, it appears very difficult to capture all pertinent aspects of crowdsourcing markets in a single coherent model. In this paper, we reflect on the modeling issues that inhibit theoretical research on online decision making for crowdsourcing, and identify some steps forward. This paper grew out of the authors' own frustration with these issues, and we hope it will encourage the community to attempt to understand, debate, and ultimately address them. The authors welcome feedback for future revisions of this paper.

Aleksandrs Slivkins

We consider an application of multi-armed bandits to internet advertising (specifically, to dynamic ad allocation in the pay-per-click model, with uncertainty on the click probabilities). We focus on an important practical issue that advertisers are constrained in how much money they can spend on their ad campaigns. This issue has not been considered in the prior work on bandit-based approaches for ad allocation, to the best of our knowledge. We define a simple, stylized model where an algorithm picks one ad to display in each round, and each ad has a \emph{budget}: the maximal amount of money that can be spent on this ad. This model admits a natural variant of UCB1, a well-known algorithm for multi-armed bandits with stochastic rewards. We derive strong provable guarantees for this algorithm.

Ashwinkumar Badanidiyuru, Robert Kleinberg, Aleksandrs Slivkins

Multi-armed bandit problems are the predominant theoretical model of exploration-exploitation tradeoffs in learning, and they have countless applications ranging from medical trials, to communication networks, to Web search and advertising. In many of these application domains the learner may be constrained by one or more supply (or budget) limits, in addition to the customary limitation on the time horizon. The literature lacks a general model encompassing these sorts of problems. We introduce such a model, called "bandits with knapsacks", that combines aspects of stochastic integer programming with online learning. A distinctive feature of our problem, in comparison to the existing regret-minimization literature, is that the optimal policy for a given latent distribution may significantly outperform the policy that plays the optimal fixed arm. Consequently, achieving sublinear regret in the bandits-with-knapsacks problem is significantly more challenging than in conventional bandit problems. We present two algorithms whose reward is close to the information-theoretic optimum: one is based on a novel "balanced exploration" paradigm, while the other is a primal-dual algorithm that uses multiplicative updates. Further, we prove that the regret achieved by both algorithms is optimal up to polylogarithmic factors. We illustrate the generality of the problem by presenting applications in a number of different domains including electronic commerce, routing, and scheduling. As one example of a concrete application, we consider the problem of dynamic posted pricing with limited supply and obtain the first algorithm whose regret, with respect to the optimal dynamic policy, is sublinear in the supply.

Ittai Abraham, Omar Alonso, Vasilis Kandylas et al.

Very recently crowdsourcing has become the de facto platform for distributing and collecting human computation for a wide range of tasks and applications such as information retrieval, natural language processing and machine learning. Current crowdsourcing platforms have some limitations in the area of quality control. Most of the effort to ensure good quality has to be done by the experimenter who has to manage the number of workers needed to reach good results. We propose a simple model for adaptive quality control in crowdsourced multiple-choice tasks which we call the \emph{bandit survey problem}. This model is related to, but technically different from the well-known multi-armed bandit problem. We present several algorithms for this problem, and support them with analysis and simulations. Our approach is based in our experience conducting relevance evaluation for a large commercial search engine.

Sebastien Bubeck, Aleksandrs Slivkins

We present a new bandit algorithm, SAO (Stochastic and Adversarial Optimal), whose regret is, essentially, optimal both for adversarial rewards and for stochastic rewards. Specifically, SAO combines the square-root worst-case regret of Exp3 (Auer et al., SIAM J. on Computing 2002) and the (poly)logarithmic regret of UCB1 (Auer et al., Machine Learning 2002) for stochastic rewards. Adversarial rewards and stochastic rewards are the two main settings in the literature on (non-Bayesian) multi-armed bandits. Prior work on multi-armed bandits treats them separately, and does not attempt to jointly optimize for both. Our result falls into a general theme of achieving good worst-case performance while also taking advantage of "nice" problem instances, an important issue in the design of algorithms with partially known inputs.