Pure exploration is one of the fundamental problems in multi-armed bandits (MAB). However, existing works mostly focus on specific pure exploration tasks, without a holistic view of the general pure exploration problem. This work fills this gap by introducing a versatile framework to study pure exploration, with a focus on identifying the pairwise relationships between targeted arm pairs. Moreover, unlike existing works that only optimize the stopping time (i.e., sample complexity), this work considers that arms are associated with potentially different costs and targets at optimizing the cumulative cost that occurred during learning. Under the general framework of pairwise pure exploration with arm-specific costs, a performance lower bound is derived. Then, a novel algorithm, termed CAET (Cost-Aware Pairwise Exploration Task), is proposed. CAET builds on the track-and-stop principle with a novel design to handle the arm-specific costs, which can potentially be zero and thus represent a very challenging case. Theoretical analyses prove that the performance of CAET approaches the lower bound asymptotically. Special cases are further discussed, including an extension to regret minimization, which is another major focus of MAB. The effectiveness and efficiency of CAET are also verified through experimental results under various settings.

The in-context learning (ICL) capability of pre-trained models based on the transformer architecture has received growing interest in recent years. While theoretical understanding has been obtained for ICL in reinforcement learning (RL), the previous results are largely confined to the single-agent setting. This work proposes to further explore the in-context learning capabilities of pre-trained transformer models in competitive multi-agent games, i.e., in-context game-playing (ICGP). Focusing on the classical two-player zero-sum games, theoretical guarantees are provided to demonstrate that pre-trained transformers can provably learn to approximate Nash equilibrium in an in-context manner for both decentralized and centralized learning settings. As a key part of the proof, constructional results are established to demonstrate that the transformer architecture is sufficiently rich to realize celebrated multi-agent game-playing algorithms, in particular, decentralized V-learning and centralized VI-ULCB.

The remarkable instruction-following capability of large language models (LLMs) has sparked a growing interest in automatically learning suitable prompts. However, while many effective methods have been proposed, the cost incurred during the learning process (e.g., accessing LLM and evaluating the responses) has not been considered. To overcome this limitation, this work explicitly incorporates a finite budget constraint into prompt learning. Towards developing principled solutions, a novel connection is established between prompt learning and fixed-budget best arm identification (BAI-FB) in multi-armed bandits (MAB). Based on this connection, a general framework TRIPLE (besT aRm Identification for Prompt LEarning) is proposed to harness the power of BAI-FB in prompt learning systematically. Unique characteristics of prompt learning further lead to two embedding-based enhancements of TRIPLE by exploiting the ideas of clustering and function approximation. Extensive experiments on multiple well-adopted tasks using both GPT 3.5 and Llama2 demonstrate the significant performance improvement of TRIPLE over the previous baselines while satisfying the limited budget constraints.

Most of the existing federated multi-armed bandits (FMAB) designs are based on the presumption that clients will implement the specified design to collaborate with the server. In reality, however, it may not be possible to modify the clients' existing protocols. To address this challenge, this work focuses on clients who always maximize their individual cumulative rewards, and introduces a novel idea of ``reward teaching'', where the server guides the clients towards global optimality through implicit local reward adjustments. Under this framework, the server faces two tightly coupled tasks of bandit learning and target teaching, whose combination is non-trivial and challenging. A phased approach, called Teaching-After-Learning (TAL), is first designed to encourage and discourage clients' explorations separately. General performance analyses of TAL are established when the clients' strategies satisfy certain mild requirements. With novel technical approaches developed to analyze the warm-start behaviors of bandit algorithms, particularized guarantees of TAL with clients running UCB or epsilon-greedy strategies are then obtained. These results demonstrate that TAL achieves logarithmic regrets while only incurring logarithmic adjustment costs, which is order-optimal w.r.t. a natural lower bound. As a further extension, the Teaching-While-Learning (TWL) algorithm is developed with the idea of successive arm elimination to break the non-adaptive phase separation in TAL. Rigorous analyses demonstrate that when facing clients with UCB1, TWL outperforms TAL in terms of the dependencies on sub-optimality gaps thanks to its adaptive design. Experimental results demonstrate the effectiveness and generality of the proposed algorithms.

Existing theoretical studies on offline reinforcement learning (RL) mostly consider a dataset sampled directly from the target task. In practice, however, data often come from several heterogeneous but related sources. Motivated by this gap, this work aims at rigorously understanding offline RL with multiple datasets that are collected from randomly perturbed versions of the target task instead of from itself. An information-theoretic lower bound is derived, which reveals a necessary requirement on the number of involved sources in addition to that on the number of data samples. Then, a novel HetPEVI algorithm is proposed, which simultaneously considers the sample uncertainties from a finite number of data samples per data source and the source uncertainties due to a finite number of available data sources. Theoretical analyses demonstrate that HetPEVI can solve the target task as long as the data sources collectively provide a good data coverage. Moreover, HetPEVI is demonstrated to be optimal up to a polynomial factor of the horizon length. Finally, the study is extended to offline Markov games and offline robust RL, which demonstrates the generality of the proposed designs and theoretical analyses.

Most existing studies on linear bandits focus on the one-dimensional characterization of the overall system. While being representative, this formulation may fail to model applications with high-dimensional but favorable structures, such as the low-rank tensor representation for recommender systems. To address this limitation, this work studies a general tensor bandits model, where actions and system parameters are represented by tensors as opposed to vectors, and we particularly focus on the case that the unknown system tensor is low-rank. A novel bandit algorithm, coined TOFU (Tensor Optimism in the Face of Uncertainty), is developed. TOFU first leverages flexible tensor regression techniques to estimate low-dimensional subspaces associated with the system tensor. These estimates are then utilized to convert the original problem to a new one with norm constraints on its system parameters. Lastly, a norm-constrained bandit subroutine is adopted by TOFU, which utilizes these constraints to avoid exploring the entire high-dimensional parameter space. Theoretical analyses show that TOFU improves the best-known regret upper bound by a multiplicative factor that grows exponentially in the system order. A novel performance lower bound is also established, which further corroborates the efficiency of TOFU.

Offline reinforcement learning (RL) aims at learning an optimal strategy using a pre-collected dataset without further interactions with the environment. While various algorithms have been proposed for offline RL in the previous literature, the minimax optimality has only been (nearly) established for tabular Markov decision processes (MDPs). In this paper, we focus on offline RL with linear function approximation and propose a new pessimism-based algorithm for offline linear MDP. At the core of our algorithm is the uncertainty decomposition via a reference function, which is new in the literature of offline RL under linear function approximation. Theoretical analysis demonstrates that our algorithm can match the performance lower bound up to logarithmic factors. We also extend our techniques to the two-player zero-sum Markov games (MGs), and establish a new performance lower bound for MGs, which tightens the existing result, and verifies the nearly minimax optimality of the proposed algorithm. To the best of our knowledge, these are the first computationally efficient and nearly minimax optimal algorithms for offline single-agent MDPs and MGs with linear function approximation.

Incentivized exploration in multi-armed bandits (MAB) has witnessed increasing interests and many progresses in recent years, where a principal offers bonuses to agents to do explorations on her behalf. However, almost all existing studies are confined to temporary myopic agents. In this work, we break this barrier and study incentivized exploration with multiple and long-term strategic agents, who have more complicated behaviors that often appear in real-world applications. An important observation of this work is that strategic agents' intrinsic needs of learning benefit (instead of harming) the principal's explorations by providing "free pulls". Moreover, it turns out that increasing the population of agents significantly lowers the principal's burden of incentivizing. The key and somewhat surprising insight revealed from our results is that when there are sufficiently many learning agents involved, the exploration process of the principal can be (almost) free. Our main results are built upon three novel components which may be of independent interest: (1) a simple yet provably effective incentive-provision strategy; (2) a carefully crafted best arm identification algorithm for rewards aggregated under unequal confidences; (3) a high-probability finite-time lower bound of UCB algorithms. Experimental results are provided to complement the theoretical analysis.

Despite the significant interests and many progresses in decentralized multi-player multi-armed bandits (MP-MAB) problems in recent years, the regret gap to the natural centralized lower bound in the heterogeneous MP-MAB setting remains open. In this paper, we propose BEACON -- Batched Exploration with Adaptive COmmunicatioN -- that closes this gap. BEACON accomplishes this goal with novel contributions in implicit communication and efficient exploration. For the former, we propose a novel adaptive differential communication (ADC) design that significantly improves the implicit communication efficiency. For the latter, a carefully crafted batched exploration scheme is developed to enable incorporation of the combinatorial upper confidence bound (CUCB) principle. We then generalize the existing linear-reward MP-MAB problems, where the system reward is always the sum of individually collected rewards, to a new MP-MAB problem where the system reward is a general (nonlinear) function of individual rewards. We extend BEACON to solve this problem and prove a logarithmic regret. BEACON bridges the algorithm design and regret analysis of combinatorial MAB (CMAB) and MP-MAB, two largely disjointed areas in MAB, and the results in this paper suggest that this previously ignored connection is worth further investigation. Supplementary Material: pdf

Federated learning (FL) is an emerging distributed machine learning paradigm that has many attractive properties. In particular, FL is motivated by the growing trend that massive amounts of real-world data are exogenously generated at edge devices, which are non-independent and identically distributed (non-IID) and highly imbalanced (Bonawitz et al., 2019). FL focuses on many clients collaboratively training a machine learning model under the coordination of a central server while keeping the local data private at each client (McMahan et al., 2017). Earlier FL approaches focus on training a single global model that can perform well on the aggregated global dataset. However, the performance of the FL-trained global model on an individual client dataset degrades dramatically when significant heterogeneity among the local datasets exists, which raises the concern of using one global model for all individual clients in edge inference. To address this issue, FL with personalization (Smith et al., 2017) has been proposed. Instead of learning a single global model, each device aims at learning a mixture of the global model and its own local model (Hanzely and Richtárik, 2020; Deng et al., 2020), which provides an explicit trade-off between the two potentially competing learning goals. While the main focus of the state of the art FL with personalization is on the supervised learning setting, we propose to extend its core principles to the multi-armed bandits (MAB) problem.