We consider the problem of online multi-agent Nash social welfare (NSW) maximization.

Instead, we seek to learn a fair distribution overthearms. Drawing onalong lineofresearch ineconomics and computer science, we use theNash social welfareas our notion of fairness.

We consider the problem of online multi-agent Nash social welfare (NSW) maximization.

Our goal in this section is twofold. To avoid excessive repetition of notation and proof arguments, we purposefully leave this section not self-contained and only outline these adjustments needed. We refer an interested reader to the work of Auer et al. We leave it to future work to optimize the dependence on N and K . In our MA-MAB problem, an algorithm is allowed to "pull" a distribution over the arms In the proof of their Lemma A.1, in the explanation of their Equation (30), they cite the assumption that given the rewards observed in the first Finally, in the proof of their Theorem A.2, they again consider the probability Thus, we have the following lower bound.

Clustering of time series based on their underlying dynamics is keeping attracting researchers due to its impacts on assisting complex system modelling. Most current time series clustering methods handle only scalar time series, treat them as white noise, or rely on domain knowledge for high-quality feature construction, where the autocorrelation pattern/feature is mostly ignored. Instead of relying on heuristic feature/metric construction, the system identification approach allows treating vector time series clustering by explicitly considering their underlying autoregressive dynamics. We first derive a clustering algorithm based on a mixture autoregressive model. Unfortunately it turns out to have significant computational problems. We then derive a `small-noise' limiting version of the algorithm, which we call k-LMVAR (Limiting Mixture Vector AutoRegression), that is computationally manageable. We develop an associated BIC criterion for choosing the number of clusters and model order. The algorithm performs very well in comparative simulations and also scales well computationally.

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.

We propose a multi-agent variant of the classical multi-armed bandit problem, in which there are N agents and K arms, and pulling an arm generates a (possibly different) stochastic reward to each agent. Unlike the classical multi-armed bandit problem, the goal is not to learn the "best arm", as each agent may perceive a different arm as best for her. Instead, we seek to learn a fair distribution over arms. Drawing on a long line of research in economics and computer science, we use the Nash social welfare as our notion of fairness. We design multi-agent variants of three classic multi-armed bandit algorithms, and show that they achieve sublinear regret, now measured in terms of the Nash social welfare.

This paper studies a stylized, yet natural, learning-to-rank problem and points out the critical incorrectness of a widely used nearest neighbor algorithm. We consider a model with $n$ agents (users) $\{x_i\}_{i \in [n]}$ and $m$ alternatives (items) $\{y_j\}_{j \in [m]}$, each of which is associated with a latent feature vector. Agents rank items nondeterministically according to the Plackett-Luce model, where the higher the utility of an item to the agent, the more likely this item will be ranked high by the agent. Our goal is to find neighbors of an arbitrary agent or alternative in the latent space. We first show that the Kendall-tau distance based kNN produces incorrect results in our model. Next, we fix the problem by introducing a new algorithm with features constructed from "global information" of the data matrix. Our approach is in sharp contrast to most existing feature engineering methods. Finally, we design another new algorithm identifying similar alternatives. The construction of alternative features can be done using "local information," highlighting the algorithmic difference between finding similar agents and similar alternatives.