Jon Schneider

Yishay Mansour, Mehryar Mohri, Jon Schneider et al.

We study repeated two-player games where one of the players, the learner, employs a no-regret learning strategy, while the other, the optimizer, is a rational utility maximizer. We consider general Bayesian games, where the payoffs of both the optimizer and the learner could depend on the type, which is drawn from a publicly known distribution, but revealed privately to the learner. We address the following questions: (a) what is the bare minimum that the optimizer can guarantee to obtain regardless of the no-regret learning algorithm employed by the learner? (b) are there learning algorithms that cap the optimizer payoff at this minimum? (c) can these algorithms be implemented efficiently? While building this theory of optimizer-learner interactions, we define a new combinatorial notion of regret called polytope swap regret, that could be of independent interest in other settings.

Robert Kleinberg, Renato Paes Leme, Jon Schneider et al.

We consider the problem of evaluating forecasts of binary events whose predictions are consumed by rational agents who take an action in response to a prediction, but whose utility is unknown to the forecaster. We show that optimizing forecasts for a single scoring rule (e.g., the Brier score) cannot guarantee low regret for all possible agents. In contrast, forecasts that are well-calibrated guarantee that all agents incur sublinear regret. However, calibration is not a necessary criterion here (it is possible for miscalibrated forecasts to provide good regret guarantees for all possible agents), and calibrated forecasting procedures have provably worse convergence rates than forecasting procedures targeting a single scoring rule. Motivated by this, we present a new metric for evaluating forecasts that we call U-calibration, equal to the maximal regret of the sequence of forecasts when evaluated under any bounded scoring rule. We show that sublinear U-calibration error is a necessary and sufficient condition for all agents to achieve sublinear regret guarantees. We additionally demonstrate how to compute the U-calibration error efficiently and provide an online algorithm that achieves $O(\sqrt{T})$ U-calibration error (on par with optimal rates for optimizing for a single scoring rule, and bypassing lower bounds for the traditionally calibrated learning procedures). Finally, we discuss generalizations to the multiclass prediction setting.

Christoph Dann, Yishay Mansour, Mehryar Mohri et al.

Blackwell's celebrated approachability theory provides a general framework for a variety of learning problems, including regret minimization. However, Blackwell's proof and implicit algorithm measure approachability using the $\ell_2$ (Euclidean) distance. We argue that in many applications such as regret minimization, it is more useful to study approachability under other distance metrics, most commonly the $\ell_\infty$-metric. But, the time and space complexity of the algorithms designed for $\ell_\infty$-approachability depend on the dimension of the space of the vectorial payoffs, which is often prohibitively large. Thus, we present a framework for converting high-dimensional $\ell_\infty$-approachability problems to low-dimensional pseudonorm approachability problems, thereby resolving such issues. We first show that the $\ell_\infty$-distance between the average payoff and the approachability set can be equivalently defined as a pseudodistance between a lower-dimensional average vector payoff and a new convex set we define. Next, we develop an algorithmic theory of pseudonorm approachability, analogous to previous work on approachability for $\ell_2$ and other norms, showing that it can be achieved via online linear optimization (OLO) over a convex set given by the Fenchel dual of the unit pseudonorm ball. We then use that to show, modulo mild normalization assumptions, that there exists an $\ell_\infty$-approachability algorithm whose convergence is independent of the dimension of the original vectorial payoff. We further show that that algorithm admits a polynomial-time complexity, assuming that the original $\ell_\infty$-distance can be computed efficiently. We also give an $\ell_\infty$-approachability algorithm whose convergence is logarithmic in that dimension using an FTRL algorithm with a maximum-entropy regularizer.

Renato Paes Leme, Chara Podimata, Jon Schneider · harvard

We study the problem of contextual search, a generalization of binary search in higher dimensions, in the adversarial noise model. Let $d$ be the dimension of the problem, $T$ be the time horizon and $C$ be the total amount of adversarial noise in the system. We focus on the $ε$-ball and the symmetric loss. For the $ε$-ball loss, we give a tight regret bound of $O(C + d \log(1/ε))$ improving over the $O(d^3 \log(1/ε) \log^2(T) + C \log(T) \log(1/ε))$ bound of Krishnamurthy et al (Operations Research '23). For the symmetric loss, we give an efficient algorithm with regret $O(C+d \log T)$. To tackle the symmetric loss case, we study the more general setting of Corruption-Robust Convex Optimization with Subgradient feedback, which is of independent interest. Our techniques are a significant departure from prior approaches. Specifically, we keep track of density functions over the candidate target vectors instead of a knowledge set consisting of the candidate target vectors consistent with the feedback obtained.

Hossein Esfandiari, Vahab Mirrokni, Jon Schneider

In this work, we present and study a new framework for online learning in systems with multiple users that provide user anonymity. Specifically, we extend the notion of bandits to obey the standard $k$-anonymity constraint by requiring each observation to be an aggregation of rewards for at least $k$ users. This provides a simple yet effective framework where one can learn a clustering of users in an online fashion without observing any user's individual decision. We initiate the study of anonymous bandits and provide the first sublinear regret algorithms and lower bounds for this setting.

Renato Paes Leme, Jon Schneider, Hanrui Zhang

We study mechanism design when agents may have hidden secondary goals which will play a role when the primary utility of the outcomes is the same. We show that in such cases, a mechanism is immune to strategic manipulation if and only if it is incentive compatible with regard to primary utility -- a property we term "primary incentive compatibility" -- and nonbossy -- a well-studied property in the context of matching and allocation mechanisms. We give complete characterizations of primarily incentive-compatible and nonbossy mechanisms in various settings, including auctions with single-parameter agents and public decision settings where all agents share a common outcome. In particular, we show that in the single-item setting, a mechanism is primarily incentive compatible, individually rational, and nonbossy if and only if it is a sequential posted-price mechanism.

Jon Schneider, Kiran Vodrahalli

We study the problem of full-information online learning in the "bounded recall" setting popular in the study of repeated games. An online learning algorithm $\mathcal{A}$ is $M$-$\textit{bounded-recall}$ if its output at time $t$ can be written as a function of the $M$ previous rewards (and not e.g. any other internal state of $\mathcal{A}$). We first demonstrate that a natural approach to constructing bounded-recall algorithms from mean-based no-regret learning algorithms (e.g., running Hedge over the last $M$ rounds) fails, and that any such algorithm incurs constant regret per round. We then construct a stationary bounded-recall algorithm that achieves a per-round regret of $Θ(1/\sqrt{M})$, which we complement with a tight lower bound. Finally, we show that unlike the perfect recall setting, any low regret bound bounded-recall algorithm must be aware of the ordering of the past $M$ losses -- any bounded-recall algorithm which plays a symmetric function of the past $M$ losses must incur constant regret per round.

Guru Guruganesh, Yoav Kolumbus, Jon Schneider et al.

Many real-life contractual relations differ completely from the clean, static model at the heart of principal-agent theory. Typically, they involve repeated strategic interactions of the principal and agent, taking place under uncertainty and over time. While appealing in theory, players seldom use complex dynamic strategies in practice, often preferring to circumvent complexity and approach uncertainty through learning. We initiate the study of repeated contracts with a learning agent, focusing on agents who achieve no-regret outcomes. Optimizing against a no-regret agent is a known open problem in general games; we achieve an optimal solution to this problem for a canonical contract setting, in which the agent's choice among multiple actions leads to success/failure. The solution has a surprisingly simple structure: for some $α> 0$, initially offer the agent a linear contract with scalar $α$, then switch to offering a linear contract with scalar $0$. This switch causes the agent to ``free-fall'' through their action space and during this time provides the principal with non-zero reward at zero cost. Despite apparent exploitation of the agent, this dynamic contract can leave \emph{both} players better off compared to the best static contract. Our results generalize beyond success/failure, to arbitrary non-linear contracts which the principal rescales dynamically. Finally, we quantify the dependence of our results on knowledge of the time horizon, and are the first to address this consideration in the study of strategizing against learning agents.

Maxwell Fishelson, Robert Kleinberg, Princewill Okoroafor et al.

We study the problem of minimizing swap regret in structured normal-form games. Players have a very large (potentially infinite) number of pure actions, but each action has an embedding into $d$-dimensional space and payoffs are given by bilinear functions of these embeddings. We provide an efficient learning algorithm for this setting that incurs at most $\tilde{O}(T^{(d+1)/(d+3)})$ swap regret after $T$ rounds. To achieve this, we introduce a new online learning problem we call \emph{full swap regret minimization}. In this problem, a learner repeatedly takes a (randomized) action in a bounded convex $d$-dimensional action set $\mathcal{K}$ and then receives a loss from the adversary, with the goal of minimizing their regret with respect to the \emph{worst-case} swap function mapping $\mathcal{K}$ to $\mathcal{K}$. For varied assumptions about the convexity and smoothness of the loss functions, we design algorithms with full swap regret bounds ranging from $O(T^{d/(d+2)})$ to $O(T^{(d+1)/(d+2)})$. Finally, we apply these tools to the problem of online forecasting to minimize calibration error, showing that several notions of calibration can be viewed as specific instances of full swap regret. In particular, we design efficient algorithms for online forecasting that guarantee at most $O(T^{1/3})$ $\ell_2$-calibration error and $O(\max(\sqrt{εT}, T^{1/3}))$ \emph{discretized-calibration} error (when the forecaster is restricted to predicting multiples of $ε$).

Rachitesh Kumar, Jon Schneider, Balasubramanian Sivan

Learning to bid in repeated first-price auctions is a fundamental problem at the interface of game theory and machine learning, which has seen a recent surge in interest due to the transition of display advertising to first-price auctions. In this work, we propose a novel concave formulation for pure-strategy bidding in first-price auctions, and use it to analyze natural Gradient-Ascent-based algorithms for this problem. Importantly, our analysis goes beyond regret, which was the typical focus of past work, and also accounts for the strategic backdrop of online-advertising markets where bidding algorithms are deployed -- we provide the first guarantees of strategic-robustness and incentive-compatibility for Gradient Ascent. Concretely, we show that our algorithms achieve $O(\sqrt{T})$ regret when the highest competing bids are generated adversarially, and show that no online algorithm can do better. We further prove that the regret reduces to $O(\log T)$ when the competition is stationary and stochastic, which drastically improves upon the previous best of $O(\sqrt{T})$. Moving beyond regret, we show that a strategic seller cannot exploit our algorithms to extract more revenue on average than is possible under the optimal mechanism. Finally, we prove that our algorithm is also incentive compatible -- it is a (nearly) dominant strategy for the buyer to report her values truthfully to the algorithm as a whole. Altogether, these guarantees make our algorithms the first to simultaneously achieve both optimal regret and strategic-robustness.

Jon Schneider, Julian Zimmert

We consider the problem of designing contextual bandit algorithms in the ``cross-learning'' setting of Balseiro et al., where the learner observes the loss for the action they play in all possible contexts, not just the context of the current round. We specifically consider the setting where losses are chosen adversarially and contexts are sampled i.i.d. from an unknown distribution. In this setting, we resolve an open problem of Balseiro et al. by providing an efficient algorithm with a nearly tight (up to logarithmic factors) regret bound of $\widetilde{O}(\sqrt{TK})$, independent of the number of contexts. As a consequence, we obtain the first nearly tight regret bounds for the problems of learning to bid in first-price auctions (under unknown value distributions) and sleeping bandits with a stochastic action set. At the core of our algorithm is a novel technique for coordinating the execution of a learning algorithm over multiple epochs in such a way to remove correlations between estimation of the unknown distribution and the actions played by the algorithm. This technique may be of independent interest for other learning problems involving estimation of an unknown context distribution.

Eshwar Ram Arunachaleswaran, Natalie Collina, Jon Schneider

We consider the problem of a learning agent who has to repeatedly play a general sum game against a strategic opponent who acts to maximize their own payoff by optimally responding against the learner's algorithm. The learning agent knows their own payoff function, but is uncertain about the payoff of their opponent (knowing only that it is drawn from some distribution $\mathcal{D}$). What learning algorithm should the agent run in order to maximize their own total utility, either in expectation or in the worst-case over $\mathcal{D}$? When the learning algorithm is constrained to be a no-regret algorithm, we demonstrate how to efficiently construct an optimal learning algorithm (asymptotically achieving the optimal utility) in polynomial time for both the in-expectation and worst-case problems, independent of any other assumptions. When the learning algorithm is not constrained to no-regret, we show how to construct an $\varepsilon$-optimal learning algorithm (obtaining average utility within $\varepsilon$ of the optimal utility) for both the in-expectation and worst-case problems in time polynomial in the size of the input and $1/\varepsilon$, when either the size of the game or the support of $\mathcal{D}$ is constant. Finally, for the special case of the maximin objective, where the learner wishes to maximize their minimum payoff over all possible optimizer types, we construct a learner algorithm that runs in polynomial time in each step and guarantees convergence to the optimal learner payoff. All of these results make use of recently developed machinery that converts the analysis of learning algorithms to the study of the class of corresponding geometric objects known as menus.

Maxwell Fishelson, Noah Golowich, Mehryar Mohri et al.

We study the online calibration of multi-dimensional forecasts over an arbitrary convex set $\mathcal{P} \subset \mathbb{R}^d$ relative to an arbitrary norm $\Vert\cdot\Vert$. We connect this with the problem of external regret minimization for online linear optimization, showing that if it is possible to guarantee $O(\sqrt{ρT})$ worst-case regret after $T$ rounds when actions are drawn from $\mathcal{P}$ and losses are drawn from the dual $\Vert \cdot \Vert_*$ unit norm ball, then it is also possible to obtain $ε$-calibrated forecasts after $T = \exp(O(ρ/ε^2))$ rounds. When $\mathcal{P}$ is the $d$-dimensional simplex and $\Vert \cdot \Vert$ is the $\ell_1$-norm, the existence of $O(\sqrt{T\log d})$-regret algorithms for learning with experts implies that it is possible to obtain $ε$-calibrated forecasts after $T = \exp(O(\log{d}/ε^2)) = d^{O(1/ε^2)}$ rounds, recovering a recent result of Peng (2025). Interestingly, our algorithm obtains this guarantee without requiring access to any online linear optimization subroutine or knowledge of the optimal rate $ρ$ -- in fact, our algorithm is identical for every setting of $\mathcal{P}$ and $\Vert \cdot \Vert$. Instead, we show that the optimal regularizer for the above OLO problem can be used to upper bound the above calibration error by a swap regret, which we then minimize by running the recent TreeSwap algorithm with Follow-The-Leader as a subroutine. Finally, we prove that any online calibration algorithm that guarantees $εT$ $\ell_1$-calibration error over the $d$-dimensional simplex requires $T \geq \exp(\mathrm{poly}(1/ε))$ (assuming $d \geq \mathrm{poly}(1/ε)$). This strengthens the corresponding $d^{Ω(\log{1/ε})}$ lower bound of Peng, and shows that an exponential dependence on $1/ε$ is necessary.

Eshwar Ram Arunachaleswaran, Natalie Collina, Yishay Mansour et al.

Swap regret is a notion that has proven itself to be central to the study of general-sum normal-form games, with swap-regret minimization leading to convergence to the set of correlated equilibria and guaranteeing non-manipulability against a self-interested opponent. However, the situation for more general classes of games -- such as Bayesian games and extensive-form games -- is less clear-cut, with multiple candidate definitions for swap-regret but no known efficiently minimizable variant of swap regret that implies analogous non-manipulability guarantees. In this paper, we present a new variant of swap regret for polytope games that we call ``profile swap regret'', with the property that obtaining sublinear profile swap regret is both necessary and sufficient for any learning algorithm to be non-manipulable by an opponent (resolving an open problem of Mansour et al., 2022). Although we show profile swap regret is NP-hard to compute given a transcript of play, we show it is nonetheless possible to design efficient learning algorithms that guarantee at most $O(\sqrt{T})$ profile swap regret. Finally, we explore the correlated equilibrium notion induced by low-profile-swap-regret play, and demonstrate a gap between the set of outcomes that can be implemented by this learning process and the set of outcomes that can be implemented by a third-party mediator (in contrast to the situation in normal-form games).

Mark Braverman, Jingyi Liu, Jieming Mao et al.

The adversarial Bandit with Knapsack problem is a multi-armed bandits problem with budget constraints and adversarial rewards and costs. In each round, a learner selects an action to take and observes the reward and cost of the selected action. The goal is to maximize the sum of rewards while satisfying the budget constraint. The classical benchmark to compare against is the best fixed distribution over actions that satisfies the budget constraint in expectation. Unlike its stochastic counterpart, where rewards and costs are drawn from some fixed distribution (Badanidiyuru et al., 2018), the adversarial BwK problem does not admit a no-regret algorithm for every problem instance due to the "spend-or-save" dilemma (Immorlica et al., 2022). A key problem left open by existing works is whether there exists a weaker but still meaningful benchmark to compare against such that no-regret learning is still possible. In this work, we present a new benchmark to compare against, motivated both by real-world applications such as autobidding and by its underlying mathematical structure. The benchmark is based on the Earth Mover's Distance (EMD), and we show that sublinear regret is attainable against any strategy whose spending pattern is within EMD $o(T^2)$ of any sub-pacing spending pattern. As a special case, we obtain results against the "pacing over windows" benchmark, where we partition time into disjoint windows of size $w$ and allow the benchmark strategies to choose a different distribution over actions for each window while satisfying a pacing budget constraint. Against this benchmark, our algorithm obtains a regret bound of $\tilde{O}(T/\sqrt{w}+\sqrt{wT})$. We also show a matching lower bound, proving the optimality of our algorithm in this important special case. In addition, we provide further evidence of the necessity of the EMD condition for obtaining a sublinear regret.

Christoph Dann, Yishay Mansour, Mehryar Mohri et al.

Abernethy et al. (2011) showed that Blackwell approachability and no-regret learning are equivalent, in the sense that any algorithm that solves a specific Blackwell approachability instance can be converted to a sublinear regret algorithm for a specific no-regret learning instance, and vice versa. In this paper, we study a more fine-grained form of such reductions, and ask when this translation between problems preserves not only a sublinear rate of convergence, but also preserves the optimal rate of convergence. That is, in which cases does it suffice to find the optimal regret bound for a no-regret learning instance in order to find the optimal rate of convergence for a corresponding approachability instance? We show that the reduction of Abernethy et al. (2011) does not preserve rates: their reduction may reduce a $d$-dimensional approachability instance $I_1$ with optimal convergence rate $R_1$ to a no-regret learning instance $I_2$ with optimal regret-per-round of $R_2$, with $R_{2}/R_{1}$ arbitrarily large (in particular, it is possible that $R_1 = 0$ and $R_{2} > 0$). On the other hand, we show that it is possible to tightly reduce any approachability instance to an instance of a generalized form of regret minimization we call improper $φ$-regret minimization (a variant of the $φ$-regret minimization of Gordon et al. (2008) where the transformation functions may map actions outside of the action set). Finally, we characterize when linear transformations suffice to reduce improper $φ$-regret minimization problems to standard classes of regret minimization problems in a rate preserving manner. We prove that some improper $φ$-regret minimization instances cannot be reduced to either subclass of instance in this way, suggesting that approachability can capture some problems that cannot be phrased in the language of online learning.

William Brown, Jon Schneider, Kiran Vodrahalli

We consider a number of questions related to tradeoffs between reward and regret in repeated gameplay between two agents. To facilitate this, we introduce a notion of $\textit{generalized equilibrium}$ which allows for asymmetric regret constraints, and yields polytopes of feasible values for each agent and pair of regret constraints, where we show that any such equilibrium is reachable by a pair of algorithms which maintain their regret guarantees against arbitrary opponents. As a central example, we highlight the case one agent is no-swap and the other's regret is unconstrained. We show that this captures an extension of $\textit{Stackelberg}$ equilibria with a matching optimal value, and that there exists a wide class of games where a player can significantly increase their utility by deviating from a no-swap-regret algorithm against a no-swap learner (in fact, almost any game without pure Nash equilibria is of this form). Additionally, we make use of generalized equilibria to consider tradeoffs in terms of the opponent's algorithm choice. We give a tight characterization for the maximal reward obtainable against $\textit{some}$ no-regret learner, yet we also show a class of games in which this is bounded away from the value obtainable against the class of common "mean-based" no-regret algorithms. Finally, we consider the question of learning reward-optimal strategies via repeated play with a no-regret agent when the game is initially unknown. Again we show tradeoffs depending on the opponent's learning algorithm: the Stackelberg strategy is learnable in exponential time with any no-regret agent (and in polynomial time with any no-$\textit{adaptive}$-regret agent) for any game where it is learnable via queries, and there are games where it is learnable in polynomial time against any no-swap-regret agent but requires exponential time against a mean-based no-regret agent.

Guru Guruganesh, Allen Liu, Jon Schneider et al.

We consider the problem of multi-class classification, where a stream of adversarially chosen queries arrive and must be assigned a label online. Unlike traditional bounds which seek to minimize the misclassification rate, we minimize the total distance from each query to the region corresponding to its correct label. When the true labels are determined via a nearest neighbor partition -- i.e. the label of a point is given by which of $k$ centers it is closest to in Euclidean distance -- we show that one can achieve a loss that is independent of the total number of queries. We complement this result by showing that learning general convex sets requires an almost linear loss per query. Our results build off of regret guarantees for the geometric problem of contextual search. In addition, we develop a novel reduction technique from multiclass classification to binary classification which may be of independent interest.

Sreenivas Gollapudi, Guru Guruganesh, Kostas Kollias et al.

We consider the following variant of contextual linear bandits motivated by routing applications in navigational engines and recommendation systems. We wish to learn a hidden $d$-dimensional value $w^*$. Every round, we are presented with a subset $\mathcal{X}_t \subseteq \mathbb{R}^d$ of possible actions. If we choose (i.e. recommend to the user) action $x_t$, we obtain utility $\langle x_t, w^* \rangle$ but only learn the identity of the best action $\arg\max_{x \in \mathcal{X}_t} \langle x, w^* \rangle$. We design algorithms for this problem which achieve regret $O(d\log T)$ and $\exp(O(d \log d))$. To accomplish this, we design novel cutting-plane algorithms with low "regret" -- the total distance between the true point $w^*$ and the hyperplanes the separation oracle returns. We also consider the variant where we are allowed to provide a list of several recommendations. In this variant, we give an algorithm with $O(d^2 \log d)$ regret and list size $\mathrm{poly}(d)$. Finally, we construct nearly tight algorithms for a weaker variant of this problem where the learner only learns the identity of an action that is better than the recommendation. Our results rely on new algorithmic techniques in convex geometry (including a variant of Steiner's formula for the centroid of a convex set) which may be of independent interest.

Zhe Feng, Sébastien Lahaie, Jon Schneider et al.

The display advertising industry has recently transitioned from second- to first-price auctions as its primary mechanism for ad allocation and pricing. In light of this, publishers need to re-evaluate and optimize their auction parameters, notably reserve prices. In this paper, we propose a gradient-based algorithm to adaptively update and optimize reserve prices based on estimates of bidders' responsiveness to experimental shocks in reserves. Our key innovation is to draw on the inherent structure of the revenue objective in order to reduce the variance of gradient estimates and improve convergence rates in both theory and practice. We show that revenue in a first-price auction can be usefully decomposed into a \emph{demand} component and a \emph{bidding} component, and introduce techniques to reduce the variance of each component. We characterize the bias-variance trade-offs of these techniques and validate the performance of our proposed algorithm through experiments on synthetic data and real display ad auctions data from Google ad exchange.

Allen Liu, Renato Paes Leme, Jon Schneider

In the contextual pricing problem a seller repeatedly obtains products described by an adversarially chosen feature vector in $\mathbb{R}^d$ and only observes the purchasing decisions of a buyer with a fixed but unknown linear valuation over the products. The regret measures the difference between the revenue the seller could have obtained knowing the buyer valuation and what can be obtained by the learning algorithm. We give a poly-time algorithm for contextual pricing with $O(d \log \log T + d \log d)$ regret which matches the $Ω(d \log \log T)$ lower bound up to the $d \log d$ additive factor. If we replace pricing loss by the symmetric loss, we obtain an algorithm with nearly optimal regret of $O(d \log d)$ matching the $Ω(d)$ lower bound up to $\log d$. These algorithms are based on a novel technique of bounding the value of the Steiner polynomial of a convex region at various scales. The Steiner polynomial is a degree $d$ polynomial with intrinsic volumes as the coefficients. We also study a generalized version of contextual search where the hidden linear function over the Euclidean space is replaced by a hidden function $f : \mathcal{X} \rightarrow \mathcal{Y}$ in a certain hypothesis class $\mathcal{H}$. We provide a generic algorithm with $O(d^2)$ regret where $d$ is the covering dimension of this class. This leads in particular to a $\tilde{O}(s^2)$ regret algorithm for linear contextual search if the linear function is guaranteed to be $s$-sparse. Finally we also extend our results to the noisy feedback model, where each round our feedback is flipped with a fixed probability $p < 1/2$.

Yuan Deng, Jon Schneider, Balusubramanian Sivan

How should a player who repeatedly plays a game against a no-regret learner strategize to maximize his utility? We study this question and show that under some mild assumptions, the player can always guarantee himself a utility of at least what he would get in a Stackelberg equilibrium of the game. When the no-regret learner has only two actions, we show that the player cannot get any higher utility than the Stackelberg equilibrium utility. But when the no-regret learner has more than two actions and plays a mean-based no-regret strategy, we show that the player can get strictly higher than the Stackelberg equilibrium utility. We provide a characterization of the optimal game-play for the player against a mean-based no-regret learner as a solution to a control problem. When the no-regret learner's strategy also guarantees him a no-swap regret, we show that the player cannot get anything higher than a Stackelberg equilibrium utility.

Santiago Balseiro, Negin Golrezaei, Mohammad Mahdian et al.

In the classical contextual bandits problem, in each round $t$, a learner observes some context $c$, chooses some action $i$ to perform, and receives some reward $r_{i,t}(c)$. We consider the variant of this problem where in addition to receiving the reward $r_{i,t}(c)$, the learner also learns the values of $r_{i,t}(c')$ for some other contexts $c'$ in set $\mathcal{O}_i(c)$; i.e., the rewards that would have been achieved by performing that action under different contexts $c'\in \mathcal{O}_i(c)$. This variant arises in several strategic settings, such as learning how to bid in non-truthful repeated auctions, which has gained a lot of attention lately as many platforms have switched to running first-price auctions. We call this problem the contextual bandits problem with cross-learning. The best algorithms for the classical contextual bandits problem achieve $\tilde{O}(\sqrt{CKT})$ regret against all stationary policies, where $C$ is the number of contexts, $K$ the number of actions, and $T$ the number of rounds. We design and analyze new algorithms for the contextual bandits problem with cross-learning and show that their regret has better dependence on the number of contexts. Under complete cross-learning where the rewards for all contexts are learned when choosing an action, i.e., set $\mathcal{O}_i(c)$ contains all contexts, we show that our algorithms achieve regret $\tilde{O}(\sqrt{KT})$, removing the dependence on $C$. For any other cases, i.e., under partial cross-learning where $|\mathcal{O}_i(c)|< C$ for some context-action pair of $(i,c)$, the regret bounds depend on how the sets $\mathcal O_i(c)$ impact the degree to which cross-learning between contexts is possible. We simulate our algorithms on real auction data from an ad exchange running first-price auctions and show that they outperform traditional contextual bandit algorithms.

Renato Paes Leme, Jon Schneider

We study the problem of contextual search, a multidimensional generalization of binary search that captures many problems in contextual decision-making. In contextual search, a learner is trying to learn the value of a hidden vector $v \in [0,1]^d$. Every round the learner is provided an adversarially-chosen context $u_t \in \mathbb{R}^d$, submits a guess $p_t$ for the value of $\langle u_t, v\rangle$, learns whether $p_t < \langle u_t, v\rangle$, and incurs loss $\ell(\langle u_t, v\rangle, p_t)$ (for some loss function $\ell$). The learner's goal is to minimize their total loss over the course of $T$ rounds. We present an algorithm for the contextual search problem for the symmetric loss function $\ell(θ, p) = |θ- p|$ that achieves $O_{d}(1)$ total loss. We present a new algorithm for the dynamic pricing problem (which can be realized as a special case of the contextual search problem) that achieves $O_{d}(\log \log T)$ total loss, improving on the previous best known upper bounds of $O_{d}(\log T)$ and matching the known lower bounds (up to a polynomial dependence on $d$). Both algorithms make significant use of ideas from the field of integral geometry, most notably the notion of intrinsic volumes of a convex set. To the best of our knowledge this is the first application of intrinsic volumes to algorithm design.

Mark Braverman, Jieming Mao, Jon Schneider et al.

We consider the problem of a single seller repeatedly selling a single item to a single buyer (specifically, the buyer has a value drawn fresh from known distribution $D$ in every round). Prior work assumes that the buyer is fully rational and will perfectly reason about how their bids today affect the seller's decisions tomorrow. In this work we initiate a different direction: the buyer simply runs a no-regret learning algorithm over possible bids. We provide a fairly complete characterization of optimal auctions for the seller in this domain. Specifically: - If the buyer bids according to EXP3 (or any "mean-based" learning algorithm), then the seller can extract expected revenue arbitrarily close to the expected welfare. This auction is independent of the buyer's valuation $D$, but somewhat unnatural as it is sometimes in the buyer's interest to overbid. - There exists a learning algorithm $\mathcal{A}$ such that if the buyer bids according to $\mathcal{A}$ then the optimal strategy for the seller is simply to post the Myerson reserve for $D$ every round. - If the buyer bids according to EXP3 (or any "mean-based" learning algorithm), but the seller is restricted to "natural" auction formats where overbidding is dominated (e.g. Generalized First-Price or Generalized Second-Price), then the optimal strategy for the seller is a pay-your-bid format with decreasing reserves over time. Moreover, the seller's optimal achievable revenue is characterized by a linear program, and can be unboundedly better than the best truthful auction yet simultaneously unboundedly worse than the expected welfare.

Mark Braverman, Jieming Mao, Jon Schneider et al.

We study a strategic version of the multi-armed bandit problem, where each arm is an individual strategic agent and we, the principal, pull one arm each round. When pulled, the arm receives some private reward $v_a$ and can choose an amount $x_a$ to pass on to the principal (keeping $v_a-x_a$ for itself). All non-pulled arms get reward $0$. Each strategic arm tries to maximize its own utility over the course of $T$ rounds. Our goal is to design an algorithm for the principal incentivizing these arms to pass on as much of their private rewards as possible. When private rewards are stochastically drawn each round ($v_a^t \leftarrow D_a$), we show that: - Algorithms that perform well in the classic adversarial multi-armed bandit setting necessarily perform poorly: For all algorithms that guarantee low regret in an adversarial setting, there exist distributions $D_1,\ldots,D_k$ and an approximate Nash equilibrium for the arms where the principal receives reward $o(T)$. - Still, there exists an algorithm for the principal that induces a game among the arms where each arm has a dominant strategy. When each arm plays its dominant strategy, the principal sees expected reward $μ'T - o(T)$, where $μ'$ is the second-largest of the means $\mathbb{E}[D_{a}]$. This algorithm maintains its guarantee if the arms are non-strategic ($x_a = v_a$), and also if there is a mix of strategic and non-strategic arms.

Xi Chen, Sivakanth Gopi, Jieming Mao et al.

Motivated by applications in recommender systems, web search, social choice and crowdsourcing, we consider the problem of identifying the set of top $K$ items from noisy pairwise comparisons. In our setting, we are non-actively given $r$ pairwise comparisons between each pair of $n$ items, where each comparison has noise constrained by a very general noise model called the strong stochastic transitivity (SST) model. We analyze the competitive ratio of algorithms for the top-$K$ problem. In particular, we present a linear time algorithm for the top-$K$ problem which has a competitive ratio of $\tilde{O}(\sqrt{n})$; i.e. to solve any instance of top-$K$, our algorithm needs at most $\tilde{O}(\sqrt{n})$ times as many samples needed as the best possible algorithm for that instance (in contrast, all previous known algorithms for the top-$K$ problem have competitive ratios of $\tildeΩ(n)$ or worse). We further show that this is tight: any algorithm for the top-$K$ problem has competitive ratio at least $\tildeΩ(\sqrt{n})$.