Haipeng Luo

Haipeng Luo, Qingfeng Sun, Can Xu et al. · microsoft-research

Large language models (LLMs), such as GPT-4, have shown remarkable performance in natural language processing (NLP) tasks, including challenging mathematical reasoning. However, most existing open-source models are only pre-trained on large-scale internet data and without math-related optimization. In this paper, we present WizardMath, which enhances the mathematical CoT reasoning abilities of LLMs without using external python tools, by applying our proposed Reinforcement Learning from Evol-Instruct Feedback (RLEIF) method to the domain of math. Through extensive experiments on two mathematical reasoning benchmarks, namely GSM8k and MATH, we reveal the extraordinary capabilities of our model. Remarkably, WizardMath-Mistral 7B surpasses top-tier open-source LLMs by a substantial margin with higher data efficiency. Furthermore, WizardMath 70B even outperforms GPT-3.5-Turbo, Claude 2, Gemini Pro and GPT-4-early-version. Additionally, our preliminary exploration highlights the pivotal role of instruction evolution and process supervision in achieving exceptional math performance. For more details refer to https://github.com/nlpxucan/WizardLM

Wenhao Wu, Haipeng Luo, Bo Fang et al. · amazon-science

Most existing text-video retrieval methods focus on cross-modal matching between the visual content of videos and textual query sentences. However, in real-world scenarios, online videos are often accompanied by relevant text information such as titles, tags, and even subtitles, which can be utilized to match textual queries. This insight has motivated us to propose a novel approach to text-video retrieval, where we directly generate associated captions from videos using zero-shot video captioning with knowledge from web-scale pre-trained models (e.g., CLIP and GPT-2). Given the generated captions, a natural question arises: what benefits do they bring to text-video retrieval? To answer this, we introduce Cap4Video, a new framework that leverages captions in three ways: i) Input data: video-caption pairs can augment the training data. ii) Intermediate feature interaction: we perform cross-modal feature interaction between the video and caption to produce enhanced video representations. iii) Output score: the Query-Caption matching branch can complement the original Query-Video matching branch for text-video retrieval. We conduct comprehensive ablation studies to demonstrate the effectiveness of our approach. Without any post-processing, Cap4Video achieves state-of-the-art performance on four standard text-video retrieval benchmarks: MSR-VTT (51.4%), VATEX (66.6%), MSVD (51.8%), and DiDeMo (52.0%). The code is available at https://github.com/whwu95/Cap4Video .

Wenhao Wu, Xiaohan Wang, Haipeng Luo et al. · amazon-science

Vision-language models (VLMs) pre-trained on large-scale image-text pairs have demonstrated impressive transferability on various visual tasks. Transferring knowledge from such powerful VLMs is a promising direction for building effective video recognition models. However, current exploration in this field is still limited. We believe that the greatest value of pre-trained VLMs lies in building a bridge between visual and textual domains. In this paper, we propose a novel framework called BIKE, which utilizes the cross-modal bridge to explore bidirectional knowledge: i) We introduce the Video Attribute Association mechanism, which leverages the Video-to-Text knowledge to generate textual auxiliary attributes for complementing video recognition. ii) We also present a Temporal Concept Spotting mechanism that uses the Text-to-Video expertise to capture temporal saliency in a parameter-free manner, leading to enhanced video representation. Extensive studies on six popular video datasets, including Kinetics-400 & 600, UCF-101, HMDB-51, ActivityNet and Charades, show that our method achieves state-of-the-art performance in various recognition scenarios, such as general, zero-shot, and few-shot video recognition. Our best model achieves a state-of-the-art accuracy of 88.6% on the challenging Kinetics-400 using the released CLIP model. The code is available at https://github.com/whwu95/BIKE .

Haipeng Luo, Qingfeng Sun, Can Xu et al. · microsoft-research

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-$β$, 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.

Yan Dai, Haipeng Luo, Liyu Chen · tsinghua

We consider regret minimization for Adversarial Markov Decision Processes (AMDPs), where the loss functions are changing over time and adversarially chosen, and the learner only observes the losses for the visited state-action pairs (i.e., bandit feedback). While there has been a surge of studies on this problem using Online-Mirror-Descent (OMD) methods, very little is known about the Follow-the-Perturbed-Leader (FTPL) methods, which are usually computationally more efficient and also easier to implement since it only requires solving an offline planning problem. Motivated by this, we take a closer look at FTPL for learning AMDPs, starting from the standard episodic finite-horizon setting. We find some unique and intriguing difficulties in the analysis and propose a workaround to eventually show that FTPL is also able to achieve near-optimal regret bounds in this case. More importantly, we then find two significant applications: First, the analysis of FTPL turns out to be readily generalizable to delayed bandit feedback with order-optimal regret, while OMD methods exhibit extra difficulties (Jin et al., 2022). Second, using FTPL, we also develop the first no-regret algorithm for learning communicating AMDPs in the infinite-horizon setting with bandit feedback and stochastic transitions. Our algorithm is efficient assuming access to an offline planning oracle, while even for the easier full-information setting, the only existing algorithm (Chandrasekaran and Tewari, 2021) is computationally inefficient.

Yan Dai, Haipeng Luo, Chen-Yu Wei et al. · tsinghua

We consider learning in an adversarial Markov Decision Process (MDP) where the loss functions can change arbitrarily over $K$ episodes and the state space can be arbitrarily large. We assume that the Q-function of any policy is linear in some known features, that is, a linear function approximation exists. The best existing regret upper bound for this setting (Luo et al., 2021) is of order $\tilde{\mathcal O}(K^{2/3})$ (omitting all other dependencies), given access to a simulator. This paper provides two algorithms that improve the regret to $\tilde{\mathcal O}(\sqrt K)$ in the same setting. Our first algorithm makes use of a refined analysis of the Follow-the-Regularized-Leader (FTRL) algorithm with the log-barrier regularizer. This analysis allows the loss estimators to be arbitrarily negative and might be of independent interest. Our second algorithm develops a magnitude-reduced loss estimator, further removing the polynomial dependency on the number of actions in the first algorithm and leading to the optimal regret bound (up to logarithmic terms and dependency on the horizon). Moreover, we also extend the first algorithm to simulator-free linear MDPs, which achieves $\tilde{\mathcal O}(K^{8/9})$ regret and greatly improves over the best existing bound $\tilde{\mathcal O}(K^{14/15})$. This algorithm relies on a better alternative to the Matrix Geometric Resampling procedure by Neu & Olkhovskaya (2020), which could again be of independent interest.

Ioannis Anagnostides, Gabriele Farina, Christian Kroer et al.

In this paper we establish efficient and \emph{uncoupled} learning dynamics so that, when employed by all players in a general-sum multiplayer game, the \emph{swap regret} of each player after $T$ repetitions of the game is bounded by $O(\log T)$, improving over the prior best bounds of $O(\log^4 (T))$. At the same time, we guarantee optimal $O(\sqrt{T})$ swap regret in the adversarial regime as well. To obtain these results, our primary contribution is to show that when all players follow our dynamics with a \emph{time-invariant} learning rate, the \emph{second-order path lengths} of the dynamics up to time $T$ are bounded by $O(\log T)$, a fundamental property which could have further implications beyond near-optimally bounding the (swap) regret. Our proposed learning dynamics combine in a novel way \emph{optimistic} regularized learning with the use of \emph{self-concordant barriers}. Further, our analysis is remarkably simple, bypassing the cumbersome framework of higher-order smoothness recently developed by Daskalakis, Fishelson, and Golowich (NeurIPS'21).

Gabriele Farina, Ioannis Anagnostides, Haipeng Luo et al.

A recent line of work has established uncoupled learning dynamics such that, when employed by all players in a game, each player's \emph{regret} after $T$ repetitions grows polylogarithmically in $T$, an exponential improvement over the traditional guarantees within the no-regret framework. However, so far these results have only been limited to certain classes of games with structured strategy spaces -- such as normal-form and extensive-form games. The question as to whether $O(\text{polylog} T)$ regret bounds can be obtained for general convex and compact strategy sets -- which occur in many fundamental models in economics and multiagent systems -- while retaining efficient strategy updates is an important question. In this paper, we answer this in the positive by establishing the first uncoupled learning algorithm with $O(\log T)$ per-player regret in general \emph{convex games}, that is, games with concave utility functions supported on arbitrary convex and compact strategy sets. Our learning dynamics are based on an instantiation of optimistic follow-the-regularized-leader over an appropriately \emph{lifted} space using a \emph{self-concordant regularizer} that is, peculiarly, not a barrier for the feasible region. Further, our learning dynamics are efficiently implementable given access to a proximal oracle for the convex strategy set, leading to $O(\log\log T)$ per-iteration complexity; we also give extensions when access to only a \emph{linear} optimization oracle is assumed. Finally, we adapt our dynamics to guarantee $O(\sqrt{T})$ regret in the adversarial regime. Even in those special cases where prior results apply, our algorithm improves over the state-of-the-art regret bounds either in terms of the dependence on the number of iterations or on the dimension of the strategy sets.

Yue Wu, Weiqiang Zheng, Yang Cai et al.

We revisit the convergence guarantees of the Extragradient (EG) method for unconstrained biaffine min-max optimization. It is known that EG with a fixed stepsize achieves a $Θ(T^{-1/2})$ last-iterate convergence rate, which is slower than the optimal $\mathcal{O}(T^{-1})$ rate attainable by incorporating additional mechanisms such as anchoring. Motivated by recent advances showing that dynamic stepsizes alone can significantly accelerate gradient descent, we ask whether dynamic stepsizes can similarly accelerate the last-iterate convergence of EG. We present the first positive result in this direction. Specifically, we provide a deterministic dynamic stepsize schedule that accelerates the convergence rate of EG to $\mathcal{O}(T^{-2/3+\varepsilon})$ for any $\varepsilon > 0$. We also show that this rate is tight when the extrapolation and update steps of EG use the same stepsize. We then show that allowing different stepsizes for the extrapolation and update steps further improves the convergence rate to the near-optimal $\mathcal{O}(T^{-1+\varepsilon})$. Our analysis reduces stepsize scheduling to an optimization problem, whose solution leads to a stepsize schedule that follows (a discretization of) a power-law distribution. Our proposed stepsize schedules and analysis extend to other methods, such as Optimistic Gradient (OG), and suggest broader applicability to general min-max optimization problems.

Tencent Hunyuan Team, Ao Liu, Botong Zhou et al. · tencent-ai

As Large Language Models (LLMs) rapidly advance, we introduce Hunyuan-TurboS, a novel large hybrid Transformer-Mamba Mixture of Experts (MoE) model. It synergistically combines Mamba's long-sequence processing efficiency with Transformer's superior contextual understanding. Hunyuan-TurboS features an adaptive long-short chain-of-thought (CoT) mechanism, dynamically switching between rapid responses for simple queries and deep "thinking" modes for complex problems, optimizing computational resources. Architecturally, this 56B activated (560B total) parameter model employs 128 layers (Mamba2, Attention, FFN) with an innovative AMF/MF block pattern. Faster Mamba2 ensures linear complexity, Grouped-Query Attention minimizes KV cache, and FFNs use an MoE structure. Pre-trained on 16T high-quality tokens, it supports a 256K context length and is the first industry-deployed large-scale Mamba model. Our comprehensive post-training strategy enhances capabilities via Supervised Fine-Tuning (3M instructions), a novel Adaptive Long-short CoT Fusion method, Multi-round Deliberation Learning for iterative improvement, and a two-stage Large-scale Reinforcement Learning process targeting STEM and general instruction-following. Evaluations show strong performance: overall top 7 rank on LMSYS Chatbot Arena with a score of 1356, outperforming leading models like Gemini-2.0-Flash-001 (1352) and o4-mini-2025-04-16 (1345). TurboS also achieves an average of 77.9% across 23 automated benchmarks. Hunyuan-TurboS balances high performance and efficiency, offering substantial capabilities at lower inference costs than many reasoning models, establishing a new paradigm for efficient large-scale pre-trained models.

Yang Cai, Haipeng Luo, Chen-Yu Wei et al.

We revisit the problem of learning in two-player zero-sum Markov games, focusing on developing an algorithm that is uncoupled, convergent, and rational, with non-asymptotic convergence rates. We start from the case of stateless matrix game with bandit feedback as a warm-up, showing an $O(t^{-\frac{1}{8}})$ last-iterate convergence rate. To the best of our knowledge, this is the first result that obtains finite last-iterate convergence rate given access to only bandit feedback. We extend our result to the case of irreducible Markov games, providing a last-iterate convergence rate of $O(t^{-\frac{1}{9+\varepsilon}})$ for any $\varepsilon>0$. Finally, we study Markov games without any assumptions on the dynamics, and show a path convergence rate, which is a new notion of convergence we defined, of $O(t^{-\frac{1}{10}})$. Our algorithm removes the coordination and prior knowledge requirement of [Wei et al., 2021], which pursued the same goals as us for irreducible Markov games. Our algorithm is related to [Chen et al., 2021, Cen et al., 2021] and also builds on the entropy regularization technique. However, we remove their requirement of communications on the entropy values, making our algorithm entirely uncoupled.

Tiancheng Jin, Junyan Liu, Haipeng Luo

We study the problem of designing adaptive multi-armed bandit algorithms that perform optimally in both the stochastic setting and the adversarial setting simultaneously (often known as a best-of-both-world guarantee). A line of recent works shows that when configured and analyzed properly, the Follow-the-Regularized-Leader (FTRL) algorithm, originally designed for the adversarial setting, can in fact optimally adapt to the stochastic setting as well. Such results, however, critically rely on an assumption that there exists one unique optimal arm. Recently, Ito (2021) took the first step to remove such an undesirable uniqueness assumption for one particular FTRL algorithm with the $\frac{1}{2}$-Tsallis entropy regularizer. In this work, we significantly improve and generalize this result, showing that uniqueness is unnecessary for FTRL with a broad family of regularizers and a new learning rate schedule. For some regularizers, our regret bounds also improve upon prior results even when uniqueness holds. We further provide an application of our results to the decoupled exploration and exploitation problem, demonstrating that our techniques are broadly applicable.

Haipeng Luo, Hanghang Tong, Mengxiao Zhang et al.

We study high-probability regret bounds for adversarial $K$-armed bandits with time-varying feedback graphs over $T$ rounds. For general strongly observable graphs, we develop an algorithm that achieves the optimal regret $\widetilde{\mathcal{O}}((\sum_{t=1}^Tα_t)^{1/2}+\max_{t\in[T]}α_t)$ with high probability, where $α_t$ is the independence number of the feedback graph at round $t$. Compared to the best existing result [Neu, 2015] which only considers graphs with self-loops for all nodes, our result not only holds more generally, but importantly also removes any $\text{poly}(K)$ dependence that can be prohibitively large for applications such as contextual bandits. Furthermore, we also develop the first algorithm that achieves the optimal high-probability regret bound for weakly observable graphs, which even improves the best expected regret bound of [Alon et al., 2015] by removing the $\mathcal{O}(\sqrt{KT})$ term with a refined analysis. Our algorithms are based on the online mirror descent framework, but importantly with an innovative combination of several techniques. Notably, while earlier works use optimistic biased loss estimators for achieving high-probability bounds, we find it important to use a pessimistic one for nodes without self-loop in a strongly observable graph.

Mengxiao Zhang, Yuheng Zhang, Olga Vrousgou et al.

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.

Liyu Chen, Haipeng Luo

We initiate the study of dynamic regret minimization for goal-oriented reinforcement learning modeled by a non-stationary stochastic shortest path problem with changing cost and transition functions. We start by establishing a lower bound $Ω((B_{\star} SAT_{\star}(Δ_c + B_{\star}^2Δ_P))^{1/3}K^{2/3})$, where $B_{\star}$ is the maximum expected cost of the optimal policy of any episode starting from any state, $T_{\star}$ is the maximum hitting time of the optimal policy of any episode starting from the initial state, $SA$ is the number of state-action pairs, $Δ_c$ and $Δ_P$ are the amount of changes of the cost and transition functions respectively, and $K$ is the number of episodes. The different roles of $Δ_c$ and $Δ_P$ in this lower bound inspire us to design algorithms that estimate costs and transitions separately. Specifically, assuming the knowledge of $Δ_c$ and $Δ_P$, we develop a simple but sub-optimal algorithm and another more involved minimax optimal algorithm (up to logarithmic terms). These algorithms combine the ideas of finite-horizon approximation [Chen et al., 2022a], special Bernstein-style bonuses of the MVP algorithm [Zhang et al., 2020], adaptive confidence widening [Wei and Luo, 2021], as well as some new techniques such as properly penalizing long-horizon policies. Finally, when $Δ_c$ and $Δ_P$ are unknown, we develop a variant of the MASTER algorithm [Wei and Luo, 2021] and integrate the aforementioned ideas into it to achieve $\widetilde{O}(\min\{B_{\star} S\sqrt{ALK}, (B_{\star}^2S^2AT_{\star}(Δ_c+B_{\star}Δ_P))^{1/3}K^{2/3}\})$ regret, where $L$ is the unknown number of changes of the environment.

Yang Cai, Gabriele Farina, Julien Grand-Clément et al.

We study last-iterate convergence properties of algorithms for solving two-player zero-sum games based on Regret Matching$^+$ (RM$^+$). Despite their widespread use for solving real games, virtually nothing is known about their last-iterate convergence. A major obstacle to analyzing RM-type dynamics is that their regret operators lack Lipschitzness and (pseudo)monotonicity. We start by showing numerically that several variants used in practice, such as RM$^+$, predictive RM$^+$ and alternating RM$^+$, all lack last-iterate convergence guarantees even on a simple $3\times 3$ matrix game. We then prove that recent variants of these algorithms based on a smoothing technique, extragradient RM$^{+}$ and smooth Predictive RM$^+$, enjoy asymptotic last-iterate convergence (without a rate), $1/\sqrt{t}$ best-iterate convergence, and when combined with restarting, linear-rate last-iterate convergence. Our analysis builds on a new characterization of the geometric structure of the limit points of our algorithms, marking a significant departure from most of the literature on last-iterate convergence. We believe that our analysis may be of independent interest and offers a fresh perspective for studying last-iterate convergence in algorithms based on non-monotone operators.

Akhil Agnihotri, Rahul Jain, Haipeng Luo

Reinforcement Learning (RL) for constrained MDPs (CMDPs) is an increasingly important problem for various applications. Often, the average criterion is more suitable than the discounted criterion. Yet, RL for average-CMDPs (ACMDPs) remains a challenging problem. Algorithms designed for discounted constrained RL problems often do not perform well for the average CMDP setting. In this paper, we introduce a new policy optimization with function approximation algorithm for constrained MDPs with the average criterion. The Average-Constrained Policy Optimization (ACPO) algorithm is inspired by trust region-based policy optimization algorithms. We develop basic sensitivity theory for average CMDPs, and then use the corresponding bounds in the design of the algorithm. We provide theoretical guarantees on its performance, and through extensive experimental work in various challenging OpenAI Gym environments, show its superior empirical performance when compared to other state-of-the-art algorithms adapted for the ACMDPs.

Mengxiao Zhang, Shi Chen, Haipeng Luo et al.

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.

Yang Cai, Constantinos Daskalakis, Haipeng Luo et al.

Learning and computation of equilibria are central problems in game theory, theory of computation, and artificial intelligence. In this work, we introduce proximal regret, a new notion of regret based on proximal operators that lies strictly between external and swap regret. When every player employs a no-proximal-regret algorithm in a general convex game, the empirical distribution of play converges to proximal correlated equilibria (PCE), a refinement of coarse correlated equilibria. Our framework unifies several emerging notions in online learning and game theory-such as gradient equilibrium and semicoarse correlated equilibrium-and introduces new ones. Our main result shows that the classic Online Gradient Descent (GD) algorithm achieves an optimal $O(\sqrt{T})$ bound on proximal regret, revealing that GD, without modification, minimizes a stronger regret notion than external regret. This provides a new explanation for the empirically superior performance of gradient descent in online learning and games. We further extend our analysis to Mirror Descent in the Bregman setting and to Optimistic Gradient Descent, which yields faster convergence in smooth convex games.

Yurong Chen, Zhiyi Huang, Michael I. Jordan et al. · pku

We study calibeating, the problem of post-processing external forecasts online to minimize cumulative losses and match an informativeness-based benchmark. Unlike prior work, which analyzed calibeating for specific losses with specific arguments, we reduce calibeating to existing online learning techniques and obtain results for general proper losses. More concretely, we first show that calibeating is minimax-equivalent to regret minimization. This recovers the $O(\log T)$ calibeating rate of Foster and Hart [FH23] for the Brier and log losses and its optimality, and yields new optimal calibeating rates for mixable losses and general bounded losses. Second, we prove that multi-calibeating is minimax-equivalent to the combination of calibeating and the classical expert problem. This yields new optimal multi-calibeating rates for mixable losses, including Brier and log losses, and general bounded losses. Finally, we obtain new bounds for achieving calibeating and calibration simultaneously for the Brier loss. For binary predictions, our result gives the first calibrated algorithm that at the same time also achieves the optimal $O(\log T)$ calibeating rate.

Mengxiao Zhang, Haipeng Luo

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.

Lunjia Hu, Haipeng Luo, Spandan Senapati et al.

Multicalibration [HJKRR18] is an algorithmic fairness perspective that demands that the predictions of a predictor are correct conditional on themselves and membership in a collection of potentially overlapping subgroups of a population. The work of [NR23] established a surprising connection between multicalibration for an arbitrary property $Γ$ (e.g., mean or median) and property elicitation: a property $Γ$ can be multicalibrated if and only if it is elicitable, where elicitability is the notion that the true property value of a distribution can be obtained by solving a regression problem over the distribution. In the online setting, [NR23] proposed an inefficient algorithm that achieves $\sqrt T$ $\ell_2$-multicalibration error for a hypothesis class of group membership functions and an elicitable property $Γ$, after $T$ rounds of interaction between a forecaster and adversary. In this paper, we generalize multicalibration for an elicitable property $Γ$ from group membership functions to arbitrary bounded hypothesis classes and introduce a stronger notion -- swap multicalibration, following [GKR23]. Subsequently, we propose an oracle-efficient algorithm which, when given access to an online agnostic learner, achieves $T^{1/(r+1)}$ $\ell_r$-swap multicalibration error with high probability (for $r\ge2$) for a hypothesis class with bounded sequential Rademacher complexity and an elicitable property $Γ$. For the special case of $r=2$, this implies an oracle-efficient algorithm that achieves $T^{1/3}$ $\ell_2$-swap multicalibration error, which significantly improves on the previously established bounds for the problem [NR23, GMS25, LSS25a], and completely resolves an open question raised in [GJRR24] on the possibility of an oracle-efficient algorithm that achieves $\sqrt{T}$ $\ell_2$-mean multicalibration error by answering it in a strongly affirmative sense.

Aadirupa Saha, Amith Bhat, Haipeng Luo

We study $K$-armed Multiarmed Bandit (MAB) problem with $M$ heterogeneous data sources, each exhibiting unknown and distinct noise variances $\{σ_j^2\}_{j=1}^M$. The learner's objective is standard MAB regret minimization, with the additional complexity of adaptively selecting which data source to query from at each round. We propose Source-Optimistic Adaptive Regret minimization (SOAR), a novel algorithm that quickly prunes high-variance sources using sharp variance-concentration bounds, followed by a `balanced min-max LCB-UCB approach' that seamlessly integrates the parallel tasks of identifying the best arm and the optimal (minimum-variance) data source. Our analysis shows SOAR achieves an instance-dependent regret bound of $\tilde{O}\left({σ^*}^2\sum_{i=2}^K \frac{\log T}{Δ_i} + \sqrt{K \sum_{j=1}^M σ_j^2}\right)$, up to preprocessing costs depending only on problem parameters, where ${σ^*}^2 := \min_j σ_j^2$ is the minimum source variance and $Δ_i$ denotes the suboptimality gap of the $i$-th arm. This result is both surprising as despite lacking prior knowledge of the minimum-variance source among $M$ alternatives, SOAR attains the optimal instance-dependent regret of standard single-source MAB with variance ${σ^*}^2$, while incurring only an small (and unavoidable) additive cost of $\tilde O(\sqrt{K \sum_{j=1}^M σ_j^2})$ towards the optimal (minimum variance) source identification. Our theoretical bounds represent a significant improvement over some proposed baselines, e.g. Uniform UCB or Explore-then-Commit UCB, which could potentially suffer regret scaling with $σ_{\max}^2$ in place of ${σ^*}^2$-a gap that can be arbitrarily large when $σ_{\max} \gg σ^*$. Experiments on multiple synthetic problem instances and the real-world MovieLens\;25M dataset, demonstrating the superior performance of SOAR over the baselines.

Haipeng Luo, Huawen Feng, Qingfeng Sun et al.

Large Reasoning Models (LRMs) like o3 and DeepSeek-R1 have achieved remarkable progress in natural language reasoning with long chain-of-thought. However, they remain computationally inefficient and struggle with accuracy when solving problems requiring complex mathematical operations. In this work, we present AgentMath, an agent framework that seamlessly integrates language models' reasoning capabilities with code interpreters' computational precision to efficiently tackle complex mathematical problems. Our approach introduces three key innovations: (1) An automated method that converts natural language chain-of-thought into structured tool-augmented trajectories, generating high-quality supervised fine-tuning (SFT) data to alleviate data scarcity; (2) A novel agentic reinforcement learning (RL) paradigm that dynamically interleaves natural language generation with real-time code execution. This enables models to autonomously learn optimal tool-use strategies through multi-round interactive feedback, while fostering emergent capabilities in code refinement and error correction; (3) An efficient training system incorporating innovative techniques, including request-level asynchronous rollout scheduling, agentic partial rollout, and prefix-aware weighted load balancing, achieving 4-5x speedup and making efficient RL training feasible on ultra-long sequences with scenarios with massive tool calls.Extensive evaluations show that AgentMath achieves state-of-the-art performance on challenging mathematical competition benchmarks including AIME24, AIME25, and HMMT25. Specifically, AgentMath-30B-A3B attains 90.6%, 86.4%, and 73.8% accuracy respectively, achieving advanced capabilities.These results validate the effectiveness of our approach and pave the way for building more sophisticated and scalable mathematical reasoning agents.

Mengxiao Zhang, Yuheng Zhang, Haipeng Luo et al.

In this paper, we study Interaction-Grounded Learning (IGL) [Xie et al., 2021], a paradigm designed for realistic scenarios where the learner receives indirect feedback generated by an unknown mechanism, rather than explicit numerical rewards. While prior work on IGL provides efficient algorithms with provable guarantees, those results are confined to single-step settings, restricting their applicability to modern sequential decision-making systems such as multi-turn Large Language Model (LLM) deployments. To bridge this gap, we propose a computationally efficient algorithm that achieves a sublinear regret guarantee for contextual episodic Markov Decision Processes (MDPs) with personalized feedback. Technically, we extend the reward-estimator construction of Zhang et al. [2024a] from the single-step to the multi-step setting, addressing the unique challenges of decoding latent rewards under MDPs. Building on this estimator, we design an Inverse-Gap-Weighting (IGW) algorithm for policy optimization. Finally, we demonstrate the effectiveness of our method in learning personalized objectives from multi-turn interactions through experiments on both a synthetic episodic MDP and a real-world user booking dataset.

Yang Cai, Haipeng Luo, Chen-Yu Wei et al.

We consider bidding in repeated Bayesian first-price auctions. Bidding algorithms that achieve optimal regret have been extensively studied, but their strategic robustness to the seller's manipulation remains relatively underexplored. Bidding algorithms based on no-swap-regret algorithms achieve both desirable properties, but are suboptimal in terms of statistical and computational efficiency. In contrast, online gradient ascent is the only algorithm that achieves $O(\sqrt{TK})$ regret and strategic robustness [KSS24], where $T$ denotes the number of auctions and $K$ the number of bids. In this paper, we explore whether simple online linear optimization (OLO) algorithms suffice for bidding algorithms with both desirable properties. Our main result shows that sublinear linearized regret is sufficient for strategic robustness. Specifically, we construct simple black-box reductions that convert any OLO algorithm into a strategically robust no-regret bidding algorithm, in both known and unknown value distribution settings. For the known value distribution case, our reduction yields a bidding algorithm that achieves $O(\sqrt{T \log K})$ regret and strategic robustness (with exponential improvement on the $K$-dependence compared to [KSS24]). For the unknown value distribution case, our reduction gives a bidding algorithm with high-probability $O(\sqrt{T (\log K+\log(T/δ)})$ regret and strategic robustness, while removing the bounded density assumption made in [KSS24].

Shinji Ito, Haipeng Luo, Arnab Maiti et al.

Learning to play zero-sum games is a fundamental problem in game theory and machine learning. While significant progress has been made in minimizing external regret in the self-play settings or with full-information feedback, real-world applications often force learners to play against unknown, arbitrary opponents and restrict learners to bandit feedback where only the payoff of the realized action is observable. In such challenging settings, it is well-known that $Ω(\sqrt{T})$ external regret is unavoidable (where T is the number of rounds). To overcome this barrier, we investigate adversarial learning in zero-sum games under bandit feedback, aiming to minimize the deficit against the maximin pure strategy -- a metric we term Pure-Strategy Maximin Regret. We analyze this problem under two bandit feedback models: uninformed (only the realized reward is revealed) and informed (both the reward and the opponent's action are revealed). For uninformed bandit learning of normal-form games, we show that the Tsallis-INF algorithm achieves $O(c \log T)$ instance-dependent regret with a game-dependent parameter $c$. Crucially, we prove an information-theoretic lower bound showing that the dependence on c is necessary. To overcome this hardness, we turn to the informed setting and introduce Maximin-UCB, which obtains another regret bound of the form $O(c' \log T)$ for a different game-dependent parameter $c'$ that could potentially be much smaller than $c$. Finally, we generalize both results to bilinear games over an arbitrary, large action set, proposing Tsallis-FTRL-SPM and Maximin-LinUCB for the uninformed and informed setting respectively and establishing similar game-dependent logarithmic regret bounds.

Ioannis Anagnostides, Gabriele Farina, Maxwell Fishelson et al.

We consider the problem of minimizing different notions of swap regret in online optimization. These forms of regret are tightly connected to correlated equilibrium concepts in games, and have been more recently shown to guarantee non-manipulability against strategic adversaries. The only computationally efficient algorithm for minimizing linear swap regret over a general convex set in $\mathbb{R}^d$ was developed recently by Daskalakis, Farina, Fishelson, Pipis, and Schneider (STOC '25). However, it incurs a highly suboptimal regret bound of $Ω(d^4 \sqrt{T})$ and also relies on computationally intensive calls to the ellipsoid algorithm at each iteration. In this paper, we develop a significantly simpler, computationally efficient algorithm that guarantees $O(d^{3/2} \sqrt{T})$ linear swap regret for a general convex set and $O(d \sqrt{T})$ when the set is centrally symmetric. Our approach leverages the powerful response-based approachability framework of Bernstein and Shimkin (JMLR '15) -- previously overlooked in the line of work on swap regret minimization -- combined with geometric preconditioning via the John ellipsoid. Our algorithm simultaneously minimizes profile swap regret, which was recently shown to guarantee non-manipulability. Moreover, we establish a matching information-theoretic lower bound: any learner must incur in expectation $Ω(d \sqrt{T})$ linear swap regret for large enough $T$, even when the set is centrally symmetric. This also shows that the classic algorithm of Gordon, Greenwald, and Marks (ICML '08) is existentially optimal for minimizing linear swap regret, although it is computationally inefficient. Finally, we extend our approach to minimize regret with respect to the set of swap deviations with polynomial dimension, unifying and strengthening recent results in equilibrium computation and online learning.

Taira Tsuchiya, Haipeng Luo, Shinji Ito

Scale-invariance in games has recently emerged as a widely valued desirable property. Yet, almost all fast convergence guarantees in learning in games require prior knowledge of the utility scale. To address this, we develop learning dynamics that achieve fast convergence while being both scale-free, requiring no prior information about utilities, and scale-invariant, remaining unchanged under positive rescaling of utilities. For two-player zero-sum games, we obtain scale-free and scale-invariant dynamics with external regret bounded by $\tilde{O}(A_{\mathrm{diff}})$, where $A_{\mathrm{diff}}$ is the payoff range, which implies an $\tilde{O}(A_{\mathrm{diff}} / T)$ convergence rate to Nash equilibrium after $T$ rounds. For multiplayer general-sum games with $n$ players and $m$ actions, we obtain scale-free and scale-invariant dynamics with swap regret bounded by $O(U_{\mathrm{max}} \log T)$, where $U_{\mathrm{max}}$ is the range of the utilities, ignoring the dependence on the number of players and actions. This yields an $O(U_{\mathrm{max}} \log T / T)$ convergence rate to correlated equilibrium. Our learning dynamics are based on optimistic follow-the-regularized-leader with an adaptive learning rate that incorporates the squared path length of the opponents' gradient vectors, together with a new stopping-time analysis that exploits negative terms in regret bounds without scale-dependent tuning. For general-sum games, scale-free learning is enabled also by a technique called doubling clipping, which clips observed gradients based on past observations.

Junyan Liu, Haipeng Luo, Lillian J. Ratliff

Making calibrated online predictions is a central challenge in modern AI systems. Much of the existing literature focuses on fully adversarial environments where outcomes may be arbitrary, leading to conservative algorithms that can perform suboptimally in more benign settings, such as when outcomes are nearly stationary. This gap raises a natural question: can we design online prediction algorithms whose calibration error automatically adapts to the degree of non-stationarity in the environment, smoothly interpolating between i.i.d. and adversarial regimes? We answer this question in the affirmative and develop a suite of algorithms that achieve adaptive calibration guarantees under multiple calibration measures. Specifically, with $T$ being the number of rounds and $C\in[0,T]$ being an unknown non-stationary measure defined as the minimal $\ell_1$ deviation of the mean outcomes, our algorithms attain $\widetilde{O}(\sqrt{T}+(TC)^{\frac{1}{3}})$ for $\ell_1$ calibration error and $\widetilde{O}((1+C)^{\frac{1}{3}})$ for both $\ell_2$ and pseudo KL calibration error. These bounds match the optimal rates in the stationary case ($C=0$) and recover known guarantees in the fully adversarial regime ($C=T$). Our approach builds on and extends prior work [Hu et al., 2026, Luo et al., 2025], introducing an epoch-based scheduling together with a novel non-uniform partition of the prediction space that allocates finer resolution near the underlying ground truth.

Soumita Hait, Ping Li, Haipeng Luo et al.

Last-iterate convergence of learning dynamics in games has attracted significant recent attention. In two-player zero-sum games with bandit feedback, where only the loss of the selected action pair is observed, Fiegel et al. (2025) show a separation between average-iterate and last-iterate convergence in duality gap: while the optimal t^(-1/2) rate after t rounds is achievable for the former via standard no-regret algorithms, the latter cannot converge faster than t^(-1/3) in expectation or t^(-1/4) with high probability. However, in many practical settings, such as preference learning, the players observe not only their loss but also the opponent's action. This raises a natural question: can such additional information enable faster last-iterate convergence? We answer this question affirmatively, showing that t^(-1/2) last-iterate convergence is achievable with high probability in this setting, via an efficient algorithm that updates its strategy infrequently by solving an estimated log-barrier-regularized game. We identify fundamental obstacles preventing standard analysis for multi-armed bandits, the single-player case, from generalizing to games, and develop a novel analysis to overcome them. Experiments confirm that our algorithm indeed converges faster than naive baselines and prior methods that do not exploit opponent-action feedback. Finally, we note that our results also improve those for dueling bandits, a special case with skew-symmetric game matrices.

Junyan Liu, Haipeng Luo, Zihan Zhang et al.

We study online learning in two-player uninformed Markov games, where the opponent's actions and policies are unobserved. In this setting, Tian et al. (2021) show that achieving no-external-regret is impossible without incurring an exponential dependence on the episode length $H$. They then turn to the weaker notion of Nash-value regret and propose a V-learning algorithm with regret $O(K^{2/3})$ after $K$ episodes. However, their algorithm and guarantee do not adapt to the difficulty of the problem: even in the case where the opponent follows a fixed policy and thus $O(\sqrt{K})$ external regret is well-known to be achievable, their result is still the worse rate $O(K^{2/3})$ on a weaker metric. In this work, we fully address both limitations. First, we introduce empirical Nash-value regret, a new regret notion that is strictly stronger than Nash-value regret and naturally reduces to external regret when the opponent follows a fixed policy. Moreover, under this new metric, we propose a parameter-free algorithm that achieves an $O(\min \{\sqrt{K} + (CK)^{1/3},\sqrt{LK}\})$ regret bound, where $C$ quantifies the variance of the opponent's policies and $L$ denotes the number of policy switches (both at most $O(K)$). Therefore, our results not only recover the two extremes -- $O(\sqrt{K})$ external regret when the opponent is fixed and $O(K^{2/3})$ Nash-value regret in the worst case -- but also smoothly interpolate between these extremes by automatically adapting to the opponent's non-stationarity. We achieve so by first providing a new analysis of the epoch-based V-learning algorithm by Mao et al. (2022), establishing an $O(ηC + \sqrt{K/η})$ regret bound, where $η$ is the epoch incremental factor. Next, we show how to adaptively restart this algorithm with an appropriate $η$ in response to the potential non-stationarity of the opponent, eventually achieving our final results.

Haipeng Luo

This short note describes a simple variant of the Squint algorithm of Koolen and Van Erven [2015] for the classic expert problem. Via an equally simple modification of their proof, we prove that this variant ensures a regret bound that resembles the one shown in a recent work by Freund et al. [2026] for a variant of the NormalHedge algorithm [Chaudhuri et al., 2009].

Haipeng Luo, Spandan Senapati, Vatsal Sharan

Calibration is a fundamental concept that aims at ensuring the reliability of probabilistic predictions by aligning them with real-world outcomes. There is a surge of studies on new calibration measures that are easier to optimize compared to the classical $\ell_1$-Calibration while still having strong implications for downstream applications. One recent such example is the work by Fishelson et al. (2025) who show that it is possible to achieve $O(T^{1/3})$ pseudo $\ell_2$-Calibration error via minimizing pseudo swap regret of the squared loss, which in fact implies the same bound for all bounded proper losses with a smooth univariate form. In this work, we significantly generalize their result in the following ways: (a) in addition to smooth univariate forms, our algorithm also simultaneously achieves $O(T^{1/3})$ swap regret for any proper loss with a twice continuously differentiable univariate form (such as Tsallis entropy); (b) our bounds hold not only for pseudo swap regret that measures losses using the forecaster's distributions on predictions, but also hold for the actual swap regret that measures losses using the forecaster's actual realized predictions. We achieve so by introducing a new stronger notion of calibration called (pseudo) KL-Calibration, which we show is equivalent to the (pseudo) swap regret for log loss. We prove that there exists an algorithm that achieves $O(T^{1/3})$ KL-Calibration error and provide an explicit algorithm that achieves $O(T^{1/3})$ pseudo KL-Calibration error. Moreover, we show that the same algorithm achieves $O(T^{1/3}(\log T)^{-1/3}\log(T/δ))$ swap regret w.p. $\ge 1-δ$ for any proper loss with a smooth univariate form, which implies $O(T^{1/3})$ $\ell_2$-Calibration error. A technical contribution of our work is a new randomized rounding procedure and a non-uniform discretization scheme to minimize the swap regret for log loss.

Shinji Ito, Haipeng Luo, Taira Tsuchiya et al.

No-regret self-play learning dynamics have become one of the premier ways to solve large-scale games in practice. Accelerating their convergence via improving the regret of the players over the naive $O(\sqrt{T})$ bound after $T$ rounds has been extensively studied in recent years, but almost all studies assume access to exact gradient feedback. We address the question of whether acceleration is possible under bandit feedback only and provide an affirmative answer for two-player zero-sum normal-form games. Specifically, we show that if both players apply the Tsallis-INF algorithm of Zimmert and Seldin (2018, arXiv:1807.07623), then their regret is at most $O(c_1 \log T + \sqrt{c_2 T})$, where $c_1$ and $c_2$ are game-dependent constants that characterize the difficulty of learning -- $c_1$ resembles the complexity of learning a stochastic multi-armed bandit instance and depends inversely on some gap measures, while $c_2$ can be much smaller than the number of actions when the Nash equilibria have a small support or are close to the boundary. In particular, for the case when a pure strategy Nash equilibrium exists, $c_2$ becomes zero, leading to an optimal instance-dependent regret bound as we show. We additionally prove that in this case, our algorithm also enjoys last-iterate convergence and can identify the pure strategy Nash equilibrium with near-optimal sample complexity.

Asaf Cassel, Haipeng Luo, Aviv Rosenberg et al.

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.

Mengxiao Zhang, Yuheng Zhang, Haipeng Luo et al.

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.

Yang Cai, Constantinos Daskalakis, Haipeng Luo et al.

While Online Gradient Descent and other no-regret learning procedures are known to efficiently converge to a coarse correlated equilibrium in games where each agent's utility is concave in their own strategy, this is not the case when utilities are non-concave -- a common scenario in machine learning applications involving strategies parameterized by deep neural networks, or when agents' utilities are computed by neural networks, or both. Non-concave games introduce significant game-theoretic and optimization challenges: (i) Nash equilibria may not exist; (ii) local Nash equilibria, though they exist, are intractable; and (iii) mixed Nash, correlated, and coarse correlated equilibria generally have infinite support and are intractable. To sidestep these challenges, we revisit the classical solution concept of $Φ$-equilibria introduced by Greenwald and Jafari [2003], which is guaranteed to exist for an arbitrary set of strategy modifications $Φ$ even in non-concave games [Stolz and Lugosi, 2007]. However, the tractability of $Φ$-equilibria in such games remains elusive. In this paper, we initiate the study of tractable $Φ$-equilibria in non-concave games and examine several natural families of strategy modifications. We show that when $Φ$ is finite, there exists an efficient uncoupled learning algorithm that converges to the corresponding $Φ$-equilibria. Additionally, we explore cases where $Φ$ is infinite but consists of local modifications. We show that approximating local $Φ$-equilibria beyond the first-order stationary regime is computationally intractable. In contrast, within this regime, we show Online Gradient Descent efficiently converges to $Φ$-equilibria for several natural infinite families of modifications, including a new structural family of modifications inspired by the well-studied proximal operator.

Yang Cai, Haipeng Luo, Chen-Yu Wei et al.

The convergence of online learning algorithms in games under self-play is a fundamental question in game theory and machine learning. Among various notions of convergence, last-iterate convergence is particularly desirable, as it reflects the actual decisions made by the learners and captures the day-to-day behavior of the learning dynamics. While many algorithms are known to converge in the average-iterate, achieving last-iterate convergence typically requires considerably more effort in both the design and the analysis of the algorithm. Somewhat surprisingly, we show in this paper that for a large family of games, there exists a simple black-box reduction that transforms the average iterates of an uncoupled learning dynamics into the last iterates of a new uncoupled learning dynamics, thus also providing a reduction from last-iterate convergence to average-iterate convergence. Our reduction applies to games where each player's utility is linear in both their own strategy and the joint strategy of all opponents. This family includes two-player bimatrix games and generalizations such as multi-player polymatrix games. By applying our reduction to the Optimistic Multiplicative Weights Update algorithm, we obtain new state-of-the-art last-iterate convergence rates for uncoupled learning dynamics in multi-player zero-sum polymatrix games: (1) an $O(\frac{\log d}{T})$ last-iterate convergence rate under gradient feedback, representing an exponential improvement in the dependence on the dimension $d$ (i.e., the maximum number of actions available to either player); and (2) an $\widetilde{O}(d^{\frac{1}{5}} T^{-\frac{1}{5}})$ last-iterate convergence rate under bandit feedback, improving upon the previous best rates of $\widetilde{O}(\sqrt{d} T^{-\frac{1}{8}})$ and $\widetilde{O}(\sqrt{d} T^{-\frac{1}{6}})$.

Yang Cai, Gabriele Farina, Julien Grand-Clément et al.

Non-ergodic convergence of learning dynamics in games is widely studied recently because of its importance in both theory and practice. Recent work (Cai et al., 2024) showed that a broad class of learning dynamics, including Optimistic Multiplicative Weights Update (OMWU), can exhibit arbitrarily slow last-iterate convergence even in simple $2 \times 2$ matrix games, despite many of these dynamics being known to converge asymptotically in the last iterate. It remains unclear, however, whether these algorithms achieve fast non-ergodic convergence under weaker criteria, such as best-iterate convergence. We show that for $2\times 2$ matrix games, OMWU achieves an $O(T^{-1/6})$ best-iterate convergence rate, in stark contrast to its slow last-iterate convergence in the same class of games. Furthermore, we establish a lower bound showing that OMWU does not achieve any polynomial random-iterate convergence rate, measured by the expected duality gaps across all iterates. This result challenges the conventional wisdom that random-iterate convergence is essentially equivalent to best-iterate convergence, with the former often used as a proxy for establishing the latter. Our analysis uncovers a new connection to dynamic regret and presents a novel two-phase approach to best-iterate convergence, which could be of independent interest.

Soumita Hait, Ping Li, Haipeng Luo et al.

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 $Ω(\sqrt{T})$ lower bound for a Regret Matching variant that is widely used in alternating learning dynamics.

Mengxiao Zhang, Yingfei Wang, Haipeng Luo

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 $DΔ_{\max}\log T$, where $T$ is the total horizon, $D$ is the maximum delay, and $Δ_{\max}$ 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.

Taira Tsuchiya, Shinji Ito, Haipeng Luo

Learning in games refers to scenarios where multiple players interact in a shared environment, each aiming to minimize their regret. An equilibrium can be computed at a fast rate of $O(1/T)$ when all players follow the optimistic follow-the-regularized-leader (OFTRL). However, this acceleration is limited to the honest regime, in which all players adhere to a prescribed algorithm -- a situation that may not be realistic in practice. To address this issue, we present corrupted learning dynamics that adaptively find an equilibrium at a rate that depends on the extent to which each player deviates from the strategy suggested by the prescribed algorithm. First, in two-player zero-sum corrupted games, we provide learning dynamics for which the external regret of $x$-player (and similarly for $y$-player) is roughly bounded by $O(\log (m_x m_y) + \sqrt{\hat{C}_y} + \hat{C}_x)$, where $m_x$ and $m_y$ denote the number of actions of $x$- and $y$-players, respectively, and $\hat{C}_x$ and $\hat{C}_y$ represent their cumulative deviations. We then extend our approach to multi-player general-sum corrupted games, providing learning dynamics for which the swap regret of player $i$ is bounded by $O(\log T + \sqrt{\sum_{k} \hat{C}_k \log T} + \hat{C}_i)$ ignoring dependence on the number of players and actions, where $\hat{C}_i$ is the cumulative deviation of player $i$ from the prescribed algorithm. Our learning dynamics are agnostic to the levels of corruption. A key technical contribution is a new analysis that ensures the stability of a Markov chain under a new adaptive learning rate, thereby allowing us to achieve the desired bound in the corrupted regime while matching the best existing bound in the honest regime. Notably, our framework can be extended to address not only corruption in strategies but also corruption in the observed expected utilities, and we provide several matching lower bounds.

Mengxiao Zhang, Haipeng Luo

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.

Shinji Ito, Kevin Jamieson, Haipeng Luo et al.

We study online learning in finite-horizon episodic Markov decision processes (MDPs) under the challenging aggregate bandit feedback model, where the learner observes only the cumulative loss incurred in each episode, rather than individual losses at each state-action pair. While prior work in this setting has focused exclusively on worst-case analysis, we initiate the study of best-of-both-worlds (BOBW) algorithms that achieve low regret in both stochastic and adversarial environments. We propose the first BOBW algorithms for episodic tabular MDPs with aggregate bandit feedback. In the case of known transitions, our algorithms achieve $O(\log T)$ regret in stochastic settings and ${O}(\sqrt{T})$ regret in adversarial ones. Importantly, we also establish matching lower bounds, showing the optimality of our algorithms in this setting. We further extend our approach to unknown-transition settings by incorporating confidence-based techniques. Our results rely on a combination of FTRL over occupancy measures, self-bounding techniques, and new loss estimators inspired by recent advances in online shortest path problems. Along the way, we also provide the first individual-gap-dependent lower bounds and demonstrate near-optimal BOBW algorithms for shortest path problems with bandit feedback.

Haipeng Luo, Spandan Senapati, Vatsal Sharan

In this paper, we consider the related problems of multicalibration -- a multigroup fairness notion and omniprediction -- a simultaneous loss minimization paradigm, both in the distributional and online settings. The recent work of Garg et al. (2024) raised the open problem of whether it is possible to efficiently achieve $O(\sqrt{T})$ $\ell_{2}$-multicalibration error against bounded linear functions. In this paper, we answer this question in a strongly affirmative sense. We propose an efficient algorithm that achieves $O(T^{\frac{1}{3}})$ $\ell_{2}$-swap multicalibration error (both in high probability and expectation). On propagating this bound onward, we obtain significantly improved rates for $\ell_{1}$-swap multicalibration and swap omniprediction for a loss class of convex Lipschitz functions. In particular, we show that our algorithm achieves $O(T^{\frac{2}{3}})$ $\ell_{1}$-swap multicalibration and swap omniprediction errors, thereby improving upon the previous best-known bound of $O(T^{\frac{7}{8}})$. As a consequence of our improved online results, we further obtain several improved sample complexity rates in the distributional setting. In particular, we establish a $O(\varepsilon ^ {-3})$ sample complexity of efficiently learning an $\varepsilon$-swap omnipredictor for the class of convex and Lipschitz functions, $O(\varepsilon ^{-2.5})$ sample complexity of efficiently learning an $\varepsilon$-swap agnostic learner for the squared loss, and $O(\varepsilon ^ {-5}), O(\varepsilon ^ {-2.5})$ sample complexities of learning $\ell_{1}, \ell_{2}$-swap multicalibrated predictors against linear functions, all of which significantly improve on the previous best-known bounds.

Soumita Hait, Ping Li, Haipeng Luo et al.

In the classic expert problem, $Φ$-regret measures the gap between the learner's total loss and that achieved by applying the best action transformation $φ\in Φ$. A recent work by Lu et al., [2025] introduces an adaptive algorithm whose regret against a comparator $φ$ depends on a certain sparsity-based complexity measure of $φ$, (almost) recovering and interpolating optimal bounds for standard regret notions such as external, internal, and swap regret. In this work, we propose a general idea to achieve an even better comparator-adaptive $Φ$-regret bound via much simpler algorithms compared to Lu et al., [2025]. Specifically, we discover a prior distribution over all possible binary transformations and show that it suffices to achieve prior-dependent regret against these transformations. Then, we propose two concrete and efficient algorithms to achieve so, where the first one learns over multiple copies of a prior-aware variant of the Kernelized MWU algorithm of Farina et al., [2022], and the second one learns over multiple copies of a prior-aware variant of the BM-reduction [Blum and Mansour, 2007]. To further showcase the power of our methods and the advantages over Lu et al., [2025] besides the simplicity and better regret bounds, we also show that our second approach can be extended to the game setting to achieve accelerated and adaptive convergence rate to $Φ$-equilibria for a class of general-sum games. When specified to the special case of correlated equilibria, our bound improves over the existing ones from Anagnostides et al., [2022a,b]

Taira Tsuchiya, Shinji Ito, Haipeng Luo

We introduce a new framework of episodic tabular Markov decision processes (MDPs) with adversarial preferences, which we refer to as preference-based MDPs (PbMDPs). Unlike standard episodic MDPs with adversarial losses, where the numerical value of the loss is directly observed, in PbMDPs the learner instead observes preferences between two candidate arms, which represent the choices being compared. In this work, we focus specifically on the setting where the reward functions are determined by Borda scores. We begin by establishing a regret lower bound for PbMDPs with Borda scores. As a preliminary step, we present a simple instance to prove a lower bound of $Ω(\sqrt{HSAT})$ for episodic MDPs with adversarial losses, where $H$ is the number of steps per episode, $S$ is the number of states, $A$ is the number of actions, and $T$ is the number of episodes. Leveraging this construction, we then derive a regret lower bound of $Ω( (H^2 S K)^{1/3} T^{2/3} )$ for PbMDPs with Borda scores, where $K$ is the number of arms. Next, we develop algorithms that achieve a regret bound of order $T^{2/3}$. We first propose a global optimization approach based on online linear optimization over the set of all occupancy measures, achieving a regret bound of $\tilde{O}((H^2 S^2 K)^{1/3} T^{2/3} )$ under known transitions. However, this approach suffers from suboptimal dependence on the potentially large number of states $S$ and computational inefficiency. To address this, we propose a policy optimization algorithm whose regret is roughly bounded by $\tilde{O}( (H^6 S K^5)^{1/3} T^{2/3} )$ under known transitions, and further extend the result to the unknown-transition setting.

Junyan Liu, Ziyun Chen, Kun Wang et al.

We study the Pandora's Box problem in an online learning setting with semi-bandit feedback. In each round, the learner sequentially pays to open up to $n$ boxes with unknown reward distributions, observes rewards upon opening, and decides when to stop. The utility of the learner is the maximum observed reward minus the cumulative cost of opened boxes, and the goal is to minimize regret defined as the gap between the cumulative expected utility and that of the optimal policy. We propose a new algorithm that achieves $\widetilde{O}(\sqrt{nT})$ regret after $T$ rounds, which improves the $\widetilde{O}(n\sqrt{T})$ bound of Agarwal et al. [2024] and matches the known lower bound up to logarithmic factors. To better capture real-life applications, we then extend our results to a natural but challenging contextual linear setting, where each box's expected reward is linear in some known but time-varying $d$-dimensional context and the noise distribution is fixed over time. We design an algorithm that learns both the linear function and the noise distributions, achieving $\widetilde{O}(nd\sqrt{T})$ regret. Finally, we show that our techniques also apply to the online Prophet Inequality problem, where the learner must decide immediately whether or not to accept a revealed reward. In both non-contextual and contextual settings, our approach achieves similar improvements and regret bounds.

Haomin Bai, Dingzhi Yu, Shuai Li et al.

Group distributionally robust optimization (GDRO) aims to develop models that perform well across $m$ distributions simultaneously. Existing GDRO algorithms can only process a fixed number of samples per iteration, either 1 or $m$, and therefore can not support scenarios where the sample size varies dynamically. To address this limitation, we investigate GDRO with flexible sample queries and cast it as a two-player game: one player solves an online convex optimization problem, while the other tackles a prediction with limited advice (PLA) problem. Within such a game, we propose a novel PLA algorithm, constructing appropriate loss estimators for cases where the sample size is either 1 or not, and updating the decision using follow-the-regularized-leader. Then, we establish the first high-probability regret bound for non-oblivious PLA. Building upon the above approach, we develop a GDRO algorithm that allows an arbitrary and varying sample size per round, achieving a high-probability optimization error bound of $O\left(\frac{1}{t}\sqrt{\sum_{j=1}^t \frac{m}{r_j}\log m}\right)$, where $r_t$ denotes the sample size at round $t$. This result demonstrates that the optimization error decreases as the number of samples increases and implies a consistent sample complexity of $O(m\log (m)/ε^2)$ for any fixed sample size $r\in[m]$, aligning with existing bounds for cases of $r=1$ or $m$. We validate our approach on synthetic binary and real-world multi-class datasets.