A recent work by Schlisselberg et al. (2024) studies a delay-as-payoff model for stochastic multi-armed bandits, where the payoff (either loss or reward) is delayed for a period that is proportional to the payoff itself. While this captures many real-world applications, the simple multi-armed bandit setting limits the practicality of their results. In this paper, we address this limitation by studying the delay-as-payoff model for contextual linear bandits. Specifically, we start from the case with a fixed action set and propose an efficient algorithm whose regret overhead compared to the standard no-delay case is at most $D\Delta_{\max}\log T$, where $T$ is the total horizon, $D$ is the maximum delay, and $\Delta_{\max}$ is the maximum suboptimality gap. When payoff is loss, we also show further improvement of the bound, demonstrating a separation between reward and loss similar to Schlisselberg et al. (2024). Contrary to standard linear bandit algorithms that construct least squares estimator and confidence ellipsoid, the main novelty of our algorithm is to apply a phased arm elimination procedure by only picking actions in a volumetric spanner of the action set, which addresses challenges arising from both payoff-dependent delays and large action sets. We further extend our results to the case with varying action sets by adopting the reduction from Hanna et al. (2023). Finally, we implement our algorithm and showcase its effectiveness and superior performance in experiments.

Motivated by alternating learning dynamics in two-player games, a recent work by Cevher et al.(2024) shows that $o(\sqrt{T})$ alternating regret is possible for any $T$-round adversarial Online Linear Optimization (OLO) problem, and left as an open question whether the same is true for general Online Convex Optimization (OCO). We answer this question in the affirmative by showing that the continuous Hedge algorithm achieves $\tilde{\mathcal{O}}(d^{\frac{2}{3}}T^{\frac{1}{3}})$ alternating regret for any adversarial $d$-dimensional OCO problems. We show that this implies an alternating learning dynamic that finds a Nash equilibrium for any convex-concave zero-sum games or a coarse correlated equilibrium for any convex two-player general-sum games at a rate of $\tilde{\mathcal{O}}(d^{\frac{2}{3}}/T^{\frac{2}{3}})$. To further improve the time complexity and/or the dimension dependence, we propose another simple algorithm, Follow-the-Regularized-Leader with a regularizer whose convex conjugate is 3rd-order smooth, for OCO with smooth and self-concordant loss functions (such as linear or quadratic losses). We instantiate our algorithm with different regularizers and show that, for example, when the decision set is the $\ell_2$ ball, our algorithm achieves $\tilde{\mathcal{O}}(T^{\frac{2}{5}})$ alternating regret with no dimension dependence (and a better $\tilde{\mathcal{O}}(T^{\frac{1}{3}})$ bound for quadratic losses). We complement our results by showing some algorithm-specific alternating regret lower bounds, including a somewhat surprising $\Omega(\sqrt{T})$ lower bound for a Regret Matching variant that is widely used in alternating learning dynamics.

Machine learning (ML) models have become essential tools in various scenarios. Their effectiveness, however, hinges on a substantial volume of data for satisfactory performance. Model marketplaces have thus emerged as crucial platforms bridging model consumers seeking ML solutions and data owners possessing valuable data. These marketplaces leverage model trading mechanisms to properly incentive data owners to contribute their data, and return a well performing ML model to the model consumers. However, existing model trading mechanisms often assume the data owners are willing to share their data before being paid, which is not reasonable in real world. Given that, we propose a novel mechanism, named Structural Importance based Model Trading (SIMT) mechanism, that assesses the data importance and compensates data owners accordingly without disclosing the data. Specifically, SIMT procures feature and label data from data owners according to their structural importance, and then trains a graph neural network for model consumers. Theoretically, SIMT ensures incentive compatible, individual rational and budget feasible. The experiments on five popular datasets validate that SIMT consistently outperforms vanilla baselines by up to $40\%$ in both MacroF1 and MicroF1.

Interactive-Grounded Learning (IGL) [Xie et al., 2021] is a powerful framework in which a learner aims at maximizing unobservable rewards through interacting with an environment and observing reward-dependent feedback on the taken actions. To deal with personalized rewards that are ubiquitous in applications such as recommendation systems, Maghakian et al. [2022] study a version of IGL with context-dependent feedback, but their algorithm does not come with theoretical guarantees. In this work, we consider the same problem and provide the first provably efficient algorithms with sublinear regret under realizability. Our analysis reveals that the step-function estimator of prior work can deviate uncontrollably due to finite-sample effects. Our solution is a novel Lipschitz reward estimator which underestimates the true reward and enjoys favorable generalization performances. Building on this estimator, we propose two algorithms, one based on explore-then-exploit and the other based on inverse-gap weighting. We apply IGL to learning from image feedback and learning from text feedback, which are reward-free settings that arise in practice. Experimental results showcase the importance of using our Lipschitz reward estimator and the overall effectiveness of our algorithms.

We consider the problem of online multi-agent Nash social welfare (NSW) maximization. While previous works of Hossain et al. [2021], Jones et al. [2023] study similar problems in stochastic multi-agent multi-armed bandits and show that $\sqrt{T}$-regret is possible after $T$ rounds, their fairness measure is the product of all agents' rewards, instead of their NSW (that is, their geometric mean). Given the fundamental role of NSW in the fairness literature, it is more than natural to ask whether no-regret fair learning with NSW as the objective is possible. In this work, we provide a complete answer to this question in various settings. Specifically, in stochastic $N$-agent $K$-armed bandits, we develop an algorithm with $\widetilde{\mathcal{O}}\left(K^{\frac{2}{N}}T^{\frac{N-1}{N}}\right)$ regret and prove that the dependence on $T$ is tight, making it a sharp contrast to the $\sqrt{T}$-regret bounds of Hossain et al. [2021], Jones et al. [2023]. We then consider a more challenging version of the problem with adversarial rewards. Somewhat surprisingly, despite NSW being a concave function, we prove that no algorithm can achieve sublinear regret. To circumvent such negative results, we further consider a setting with full-information feedback and design two algorithms with $\sqrt{T}$-regret: the first one has no dependence on $N$ at all and is applicable to not just NSW but a broad class of welfare functions, while the second one has better dependence on $K$ and is preferable when $N$ is small. Finally, we also show that logarithmic regret is possible whenever there exists one agent who is indifferent about different arms.

Contextual multinomial logit (MNL) bandits capture many real-world assortment recommendation problems such as online retailing/advertising. However, prior work has only considered (generalized) linear value functions, which greatly limits its applicability. Motivated by this fact, in this work, we consider contextual MNL bandits with a general value function class that contains the ground truth, borrowing ideas from a recent trend of studies on contextual bandits. Specifically, we consider both the stochastic and the adversarial settings, and propose a suite of algorithms, each with different computation-regret trade-off. When applied to the linear case, our results not only are the first ones with no dependence on a certain problem-dependent constant that can be exponentially large, but also enjoy other advantages such as computational efficiency, dimension-free regret bounds, or the ability to handle completely adversarial contexts and rewards.

Bandits with feedback graphs are powerful online learning models that interpolate between the full information and classic bandit problems, capturing many real-life applications. A recent work by Zhang et al. (2023) studies the contextual version of this problem and proposes an efficient and optimal algorithm via a reduction to online regression. However, their algorithm crucially relies on seeing the feedback graph before making each decision, while in many applications, the feedback graph is uninformed, meaning that it is either only revealed after the learner makes her decision or even never fully revealed at all. This work develops the first contextual algorithm for such uninformed settings, via an efficient reduction to online regression over both the losses and the graphs. Importantly, we show that it is critical to learn the graphs using log loss instead of squared loss to obtain favorable regret guarantees. We also demonstrate the empirical effectiveness of our algorithm on a bidding application using both synthetic and real-world data.

While contextual bandit has a mature theory, effectively leveraging different feedback patterns to enhance the pace of learning remains unclear. Bandits with feedback graphs, which interpolates between the full information and bandit regimes, provides a promising framework to mitigate the statistical complexity of learning. In this paper, we propose and analyze an approach to contextual bandits with feedback graphs based upon reduction to regression. The resulting algorithms are computationally practical and achieve established minimax rates, thereby reducing the statistical complexity in real-world applications.

Supply chain management (SCM) has been recognized as an important discipline with applications to many industries, where the two-echelon stochastic inventory model, involving one downstream retailer and one upstream supplier, plays a fundamental role for developing firms' SCM strategies. In this work, we aim at designing online learning algorithms for this problem with an unknown demand distribution, which brings distinct features as compared to classic online optimization problems. Specifically, we consider the two-echelon supply chain model introduced in [Cachon and Zipkin, 1999] under two different settings: the centralized setting, where a planner decides both agents' strategy simultaneously, and the decentralized setting, where two agents decide their strategy independently and selfishly. We design algorithms that achieve favorable guarantees for both regret and convergence to the optimal inventory decision in both settings, and additionally for individual regret in the decentralized setting. Our algorithms are based on Online Gradient Descent and Online Newton Step, together with several new ingredients specifically designed for our problem. We also implement our algorithms and show their empirical effectiveness.

We study online learning in contextual pay-per-click auctions where at each of the $T$ rounds, the learner receives some context along with a set of ads and needs to make an estimate on their click-through rate (CTR) in order to run a second-price pay-per-click auction. The learner's goal is to minimize her regret, defined as the gap between her total revenue and that of an oracle strategy that always makes perfect CTR predictions. We first show that $\sqrt{T}$-regret is obtainable via a computationally inefficient algorithm and that it is unavoidable since our algorithm is no easier than the classical multi-armed bandit problem. A by-product of our results is a $\sqrt{T}$-regret bound for the simpler non-contextual setting, improving upon a recent work of [Feng et al., 2023] by removing the inverse CTR dependency that could be arbitrarily large. Then, borrowing ideas from recent advances on efficient contextual bandit algorithms, we develop two practically efficient contextual auction algorithms: the first one uses the exponential weight scheme with optimistic square errors and maintains the same $\sqrt{T}$-regret bound, while the second one reduces the problem to online regression via a simple epsilon-greedy strategy, albeit with a worse regret bound. Finally, we conduct experiments on a synthetic dataset to showcase the effectiveness and superior performance of our algorithms.