Corrupted Learning Dynamics in Games

Learning in games is the problem where multiple players interact in a shared environment, each aiming to minimize their own regret, and it is known that an approximate equilibrium can be obtained when all players employ no-regret algorithms. Notably, by adopting optimistic follow-the-regularized-leader (OFTRL), the regret of each player after $T$ rounds is constant in two-player zero-sum games, implying that an equilibrium can be computed at a faster rate of $O(1/T)$. However, this acceleration is limited to the honest regime, where all players fully adhere to the given algorithms. To address this limitation, this paper presents corrupted learning dynamics that adaptively find an equilibrium at a rate dependent on the degree of deviation by each player from the given algorithm's output. First, in two-player zero-sum games, we provide learning dynamics where the external regret of the x-player (and similarly for the y-player) in the corrupted regime is roughly bounded by $O(\log (m_\mathrm{x} m_\mathrm{y}) + \sqrt{C_\mathrm{y}} + C_\mathrm{x})$, which implies a convergence rate of $\tilde{O}((\sqrt{C_\mathrm{y}} + C_\mathrm{x})/T)$ to a Nash equilibrium. Here, $m_\mathrm{x}$ and $m_\mathrm{y}$ are the number of actions of the x- and y-players, respectively, and $C_\mathrm{x}$ and $C_\mathrm{y}$ are the cumulative deviations of the x- and y-players from their given algorithms. Furthermore, we extend our approach to multi-player general-sum games, showing that the swap regret of player $i$ in the corrupted regime is bounded by $O(\log T + \sqrt{\sum_j C_j \log T} + C_i)$, where $C_i$ is the cumulative deviations of player $i$ from the given algorithm. This implies a convergence rate of $O((\log T + \sqrt{\sum_j C_j \log T} + C_i)/T)$ to a correlated equilibrium. Our learning dynamics are agnostic to the corruption levels and are based on OFTRL with new adaptive learning rates.

Arena Learning: Build Data Flywheel for LLMs Post-training via Simulated Chatbot Arena

Assessing the effectiveness of large language models (LLMs) presents substantial challenges. The method of conducting human-annotated battles in an online Chatbot Arena is a highly effective evaluative technique. However, this approach is limited by the costs and time required for human annotation. In this paper, we introduce Arena Learning, an innovative offline strategy designed to simulate these arena battles using AI-driven annotations to evaluate battle outcomes, thus facilitating the continuous improvement of the target model through both supervised fine-tuning and reinforcement learning. Arena Learning comprises two key elements. First, it ensures precise evaluations and maintains consistency between offline simulations and online competitions via WizardArena, a pipeline developed to accurately predict the Elo rankings of various models using a meticulously designed offline test set. Our results demonstrate that WizardArena's predictions closely align with those from the online Arena. Second, it involves the continuous improvement of training data based on the battle results and the refined model. We establish a data flywheel to iteratively update the training data by highlighting the weaknesses of the target model based on its battle results, enabling it to learn from the strengths of multiple different models. We apply Arena Learning to train our target model, WizardLM-$\beta$, and demonstrate significant performance enhancements across various metrics. This fully automated training and evaluation pipeline sets the stage for continuous advancements in various LLMs via post-training. Notably, Arena Learning plays a pivotal role in the success of WizardLM-2, and this paper serves both as an exploration of its efficacy and a foundational study for future discussions related to WizardLM-2 and its derivatives.

Fast Last-Iterate Convergence of Learning in Games Requires Forgetful Algorithms

Self-play via online learning is one of the premier ways to solve large-scale two-player zero-sum games, both in theory and practice. Particularly popular algorithms include optimistic multiplicative weights update (OMWU) and optimistic gradient-descent-ascent (OGDA). While both algorithms enjoy $O(1/T)$ ergodic convergence to Nash equilibrium in two-player zero-sum games, OMWU offers several advantages including logarithmic dependence on the size of the payoff matrix and $\widetilde{O}(1/T)$ convergence to coarse correlated equilibria even in general-sum games. However, in terms of last-iterate convergence in two-player zero-sum games, an increasingly popular topic in this area, OGDA guarantees that the duality gap shrinks at a rate of $O(1/\sqrt{T})$, while the best existing last-iterate convergence for OMWU depends on some game-dependent constant that could be arbitrarily large. This begs the question: is this potentially slow last-iterate convergence an inherent disadvantage of OMWU, or is the current analysis too loose? Somewhat surprisingly, we show that the former is true. More generally, we prove that a broad class of algorithms that do not forget the past quickly all suffer the same issue: for any arbitrarily small $\delta>0$, there exists a $2\times 2$ matrix game such that the algorithm admits a constant duality gap even after $1/\delta$ rounds. This class of algorithms includes OMWU and other standard optimistic follow-the-regularized-leader algorithms.

No-Regret Learning for Fair Multi-Agent Social Welfare Optimization

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.

Provably Efficient Interactive-Grounded Learning with Personalized Reward

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.

Optimal Multiclass U-Calibration Error and Beyond

We consider the problem of online multiclass U-calibration, where a forecaster aims to make sequential distributional predictions over $K$ classes with low U-calibration error, that is, low regret with respect to all bounded proper losses simultaneously. Kleinberg et al. (2023) developed an algorithm with U-calibration error $O(K\sqrt{T})$ after $T$ rounds and raised the open question of what the optimal bound is. We resolve this question by showing that the optimal U-calibration error is $\Theta(\sqrt{KT})$ -- we start with a simple observation that the Follow-the-Perturbed-Leader algorithm of Daskalakis and Syrgkanis (2016) achieves this upper bound, followed by a matching lower bound constructed with a specific proper loss (which, as a side result, also proves the optimality of the algorithm of Daskalakis and Syrgkanis (2016) in the context of online learning against an adversary with finite choices). We also strengthen our results under natural assumptions on the loss functions, including $\Theta(\log T)$ U-calibration error for Lipschitz proper losses, $O(\log T)$ U-calibration error for a certain class of decomposable proper losses, U-calibration error bounds for proper losses with a low covering number, and others.

Near-Optimal Regret in Linear MDPs with Aggregate Bandit Feedback

In many real-world applications, it is hard to provide a reward signal in each step of a Reinforcement Learning (RL) process and more natural to give feedback when an episode ends. To this end, we study the recently proposed model of RL with Aggregate Bandit Feedback (RL-ABF), where the agent only observes the sum of rewards at the end of an episode instead of each reward individually. Prior work studied RL-ABF only in tabular settings, where the number of states is assumed to be small. In this paper, we extend ABF to linear function approximation and develop two efficient algorithms with near-optimal regret guarantees: a value-based optimistic algorithm built on a new randomization technique with a Q-functions ensemble, and a policy optimization algorithm that uses a novel hedging scheme over the ensemble.

Tractable Local Equilibria in Non-Concave Games

While Online Gradient Descent and other no-regret learning procedures are known to efficiently converge to coarse correlated equilibrium in games where each agent's utility is concave in their own strategy, this is not the case when the utilities are non-concave, a situation that is common in machine learning applications where the agents' strategies are parameterized by deep neural networks, or the agents' utilities are computed by a neural network, or both. Indeed, non-concave games present a host of game-theoretic and optimization challenges: (i) Nash equilibria may fail to exist; (ii) local Nash equilibria exist but are intractable; and (iii) mixed Nash, correlated, and coarse correlated equilibria have infinite support in general, and are intractable. To sidestep these challenges we propose a new solution concept, termed $(\varepsilon, \Phi(\delta))$-local equilibrium, which generalizes local Nash equilibrium in non-concave games, as well as (coarse) correlated equilibrium in concave games. Importantly, we show that two instantiations of this solution concept capture the convergence guarantees of Online Gradient Descent and no-regret learning, which we show efficiently converge to this type of equilibrium in non-concave games with smooth utilities.

Contextual Multinomial Logit Bandits with General Value Functions

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.

Efficient Contextual Bandits with Uninformed Feedback Graphs

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.