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Deep Learning · Robustness

Fengchun Qiao, Yanlin Chen, Xi Peng

Ensemble learning is a powerful approach for improving generalization under distribution shifts, but its effectiveness heavily depends on how individual models are combined. Existing methods often optimize ensemble weights based on validation data, which may not represent unseen test distributions, leading to suboptimal performance in out-of-distribution (OoD) settings. Inspired by Distributionally Robust Optimization (DRO), we propose Structure-informed Risk Minimization (SRM), a principled framework that learns robust ensemble weights without access to test data. Unlike standard DRO, which defines uncertainty sets based on divergence metrics alone, SRM incorporates structural information of training distributions, ensuring that the uncertainty set aligns with plausible real-world shifts. This approach mitigates the over-pessimism of traditional worst-case optimization while maintaining robustness. We introduce a computationally efficient optimization algorithm with theoretical guarantees and demonstrate that SRM achieves superior OoD generalization compared to existing ensemble combination strategies across diverse benchmarks. Code is available at: https://github.com/deep-real/SRM.

General Machine Learning · Evaluation

Chenyin Gao, Peter Gilbert, Larry Han

Standard conformal prediction ensures marginal coverage but consistently undercovers underrepresented groups, limiting its reliability for fair uncertainty quantification. Group fairness requires prediction sets to achieve a user-specified coverage level within each protected group. While group-wise conformal inference meets this requirement, it often produces excessively wide prediction sets due to limited sample sizes in underrepresented groups, highlighting a fundamental tradeoff between fairness and efficiency. To bridge this gap, we introduce Surrogate-Assisted Group-Clustered Conformal Inference (SAGCCI), a framework that improves efficiency through two key innovations: (1) clustering protected groups with similar conformal score distributions to enhance precision while maintaining fairness, and (2) deriving an efficient influence function that optimally integrates surrogate outcomes to construct tighter prediction sets. Theoretically, SAGCCI guarantees approximate group-conditional coverage in a doubly robust manner under mild convergence conditions, enabling flexible nuisance model estimation. Empirically, through simulations and an analysis of the phase 3 Moderna COVE COVID-19 vaccine trial, we demonstrate that SAGCCI outperforms existing methods, producing narrower prediction sets while maintaining valid group-conditional coverage, effectively balancing fairness and efficiency in uncertainty quantification.

Theory · Reinforcement Learning and Planning

Audrey Huang, Adam Block, Qinghua Liu, Nan Jiang, Akshay Krishnamurthy, Dylan Foster

Recent work on inference-time alignment has established benefits of increasing inference-time computation in language models, but naively scaling compute through techniques like Best-of-N sampling can cause performance to degrade due to reward hacking. Toward a theoretical understanding of how to best leverage additional computation, we formalize inference-time alignment as improving a pre-trained policy’s responses for a prompt of interest, given access to an imperfect reward model. We analyze the performance of inference-time alignment algorithms in terms of (i) response quality, and (ii) compute, and provide new results that highlight the importance of the pre-trained policy’s coverage over high-quality responses for performance and compute scaling: (1) We show that Best-of-N alignment with an ideal N can achieve optimal performance under stringent notions of coverage, but provably suffers from reward hacking when N is large, and fails to achieve tight guarantees under more realistic coverage conditions; (2) We introduce InferenceTimePessimism, a new algorithm which mitigates reward hacking through deliberate use of inference-time compute, implementing pessimism in the face of uncertainty; we prove that its performance is optimal and scaling-monotonic, i.e., does ot degrade as N increases. We complement our theoretical results with experiments that demonstrate the practicality of our algorithm across a variety of tasks and models.

Theory · Reinforcement Learning and Planning

Tong Yang, Bo Dai, Lin Xiao, Yuejie Chi

Multi-agent reinforcement learning (MARL) lies at the heart of a plethora of applications involving the interaction of a group of agents in a shared unknown environment. A prominent framework for studying MARL is Markov games, with the goal of finding various notions of equilibria in a sample-efficient manner, such as the Nash equilibrium (NE) and the coarse correlated equilibrium (CCE). However, existing sample-efficient approaches either require tailored uncertainty estimation under function approximation, or careful coordination of the players. In this paper, we propose a novel model-based algorithm, called VMG, that incentivizes exploration via biasing the empiricalestimate of the model parameters towards those with a higher collective best-response values of all the players when fixing the other players' policies, thus encouraging the policy to deviate from its current equilibrium for more exploration. VMG is oblivious to different forms of function approximation, and permits simultaneous and uncoupled policy updates of all players. Theoretically, we also establish that VMG achieves a near-optimal regret for finding both the NEs of two-player zero-sum Markov games and CCEs of multi-player general-sum Markov games under linear function approximation in an online environment, which nearly match their counterparts with sophisticated uncertainty quantification.

Deep Learning · Large Language Models

Allen Nie, Yi Su, Bo Chang, Jonathan Lee, Ed Chi, Quoc Le, Minmin Chen

Despite their success in many domains, large language models (LLMs) remain under-studied in scenarios requiring optimal decision-making under uncertainty. This is crucial as many real-world applications, ranging from personalized recommendations to healthcare interventions, demand that LLMs not only predict but also actively learn to make optimal decisions through exploration. In this work, we measure LLMs' (in)ability to make optimal decisions in bandits, a state-less reinforcement learning setting relevant to many applications. We develop a comprehensive suite of environments, including both context-free and contextual bandits with varying task difficulties, to benchmark LLMs' performance. Motivated by the existence of optimal exploration algorithms, we propose efficient ways to integrate this algorithmic knowledge into LLMs: by providing explicit algorithm-guided support during inference; and through algorithm distillation via in-context demonstrations and fine-tuning, using synthetic data generated from these algorithms. Impressively, these techniques allow us to achieve superior exploration performance with smaller models, surpassing larger models on various tasks. We conducted an extensive ablation study to shed light on various factors, such as task difficulty and data representation, that influence the efficiency of LLM exploration. Additionally, we conduct a rigorous analysis of the LLM's exploration efficiency using the concept of regret, linking its ability to explore to the model size and underlying algorithm.

Reinforcement Learning · Multi-agent

Laixi Shi, Jingchu Gai, Eric Mazumdar, Yuejie Chi, Adam Wierman

Standard multi-agent reinforcement learning (MARL) algorithms are vulnerable to sim-to-real gaps. To address this, distributionally robust Markov games (RMGs) have been proposed to enhance robustness in MARL by optimizing the worst-case performance when game dynamics shift within a prescribed uncertainty set. RMGs remains under-explored, from reasonable problem formulation to the development of sample-efficient algorithms. Two notorious and open challenges are the formulation of the uncertainty set and whether the corresponding RMGs can overcome the curse of multiagency, where the sample complexity scales exponentially with the number of agents. In this work, we propose a natural class of RMGs inspired by behavioral economics, where each agent's uncertainty set is shaped by both the environment and the integrated behavior of other agents. We first establish the well-posedness of this class of RMGs by proving the existence of game-theoretic solutions such as robust Nash equilibria and coarse correlated equilibria (CCE). Assuming access to a generative model, we then introduce a sample-efficient algorithm for learning the CCE whose sample complexity scales polynomially with all relevant parameters. To the best of our knowledge, this is the first algorithm to break the curse of multiagency for RMGs, regardless of the uncertainty set formulation.

Applications · Chemistry, Physics, and Earth Sciences

Louis Serrano, Armand Kassaï Koupaï, Thomas Wang, Pierre ERBACHER, patrick gallinari

Solving time-dependent parametric partial differential equations (PDEs) is challenging for data-driven methods, as these models must adapt to variations in parameters such as coefficients, forcing terms, and initial conditions. State-of-the-art neural surrogates perform adaptation through gradient-based optimization and meta-learning to implicitly encode the variety of dynamics from observations. This often comes with increased inference complexity. Inspired by the in-context learning capabilities of large language models (LLMs), we introduce Zebra, a novel generative auto-regressive transformer designed to solve parametric PDEs without requiring gradient adaptation at inference. By leveraging in-context information during both pre-training and inference, Zebra dynamically adapts to new tasks by conditioning on input sequences that incorporate context example trajectories. As a generative model, Zebra can be used to generate new trajectories and allows quantifying the uncertainty of the predictions. We evaluate Zebra across a variety of challenging PDE scenarios, demonstrating its adaptability, robustness, and superior performance compared to existing approaches.

Applications · Time Series

Thomas Y.L. Lin, Jerry Yao-Chieh Hu, Wan-Jiun Paul Chiou, Peter Lin

We revisit the Bayesian Black–Litterman (BL) portfolio model and remove its reliance on subjective investor views. Classical BL requires an investor “view”: a forecast vector $q$ and its uncertainty matrix $\Omega$ that describe how much a chosen portfolio should outperform the market.Our key idea is to treat $(q,\Omega)$ as latent variables and learn them from market data within a single Bayesian network.Consequently, the resulting posterior estimation admits closed-form expression, enabling fast inference and stable portfolio weights.Building on these, we propose two mechanisms to capture how features interact with returns: shared-latent parametrization and feature-influenced views; both recover classical BL and Markowitz portfolios as special cases.Empirically, on 30-year Dow-Jones and 20-year sector-ETF data, we improve Sharpe ratios by 50\% and cut turnover by 55\% relative to Markowitz and the index baselines.This work turns BL into a fully data-driven, view-free, and coherent Bayesian framework for portfolio optimization.

Probabilistic Methods · Bayesian Models and Methods

Antoine Wehenkel, Juan L. Gamella, Ozan Sener, Jens Behrmann, Guillermo Sapiro, Jörn Jacobsen, Marco Cuturi

Driven by steady progress in deep generative modeling, simulation-based inference (SBI) has emerged as the workhorse for inferring the parameters of stochastic simulators. However, recent work has demonstrated that model misspecification can harm SBI's reliability, preventing its adoption in important applications where only misspecified simulators are available.This work introduces robust posterior estimation (RoPE), a framework that overcomes model misspecification with a small real-world calibration set of ground truth parameter measurements.We formalize the misspecification gap as the solution of an optimal transport (OT) problem between learned representations of real-world and simulated observations, allowing RoPE to learn a model of the misspecification without placing additional assumptions on its nature. RoPE shows how the calibration set and OT together offer a controllable balance between calibrated uncertainty and informative inference even under severely misspecified simulators. Results on four synthetic tasks and two real-world problems with ground-truth labels demonstrate that RoPE outperforms baselines and consistently returns informative and calibrated credible intervals.

General Machine Learning · Supervised Learning

Tian Bai, Yue Zhao, Xiang Yu, Archer Yang

Selecting high-quality candidates from large datasets is critical in applications such as drug discovery, precision medicine, and alignment of large language models (LLMs). While Conformal Selection (CS) provides rigorous uncertainty quantification, it is limited to univariate responses and scalar criteria. To address this, we propose Multivariate Conformal Selection (mCS), a generalization of CS designed for multivariate response settings. Our method introduces regional monotonicity and employs multivariate nonconformity scores to construct conformal $p$-values, enabling finite-sample False Discovery Rate (FDR) control. We present two variants: $\texttt{mCS-dist}$, using distance-based scores, and $\texttt{mCS-learn}$, which learns optimal scores via differentiable optimization. Experiments on simulated and real-world datasets demonstrate that mCS significantly improves selection power while maintaining FDR control, establishing it as a robust framework for multivariate selection tasks.

Reinforcement Learning · Online

Yiting He, Zhishuai Liu, Weixin Wang, Pan Xu

Off-dynamics reinforcement learning (RL), where training and deployment transition dynamics are different, can be formulated as learning in a robust Markov decision process (RMDP) where uncertainties in transition dynamics are imposed. Existing literature mostly assumes access to generative models allowing arbitrary state-action queries or pre-collected datasets with a good state coverage of the deployment environment, bypassing the challenge of exploration. In this work, we study a more realistic and challenging setting where the agent is limited to online interaction with the training environment. To capture the intrinsic difficulty of exploration in online RMDPs, we introduce the supremal visitation ratio, a novel quantity that measures the mismatch between the training dynamics and the deployment dynamics. We show that if this ratio is unbounded, online learning becomes exponentially hard. We propose the first computationally efficient algorithm that achieves sublinear regret in online RMDPs with $f$-divergence based transition uncertainties. We also establish matching regret lower bounds, demonstrating that our algorithm achieves optimal dependence on both the supremal visitation ratio and the number of interaction episodes. Finally, we validate our theoretical results through comprehensive numerical experiments.

Reinforcement Learning

Zachary Roch, George Atia, Yue Wang

Robust reinforcement learning (RL) under the average reward criterion, which seeks to optimize long-term system performance in uncertain environments, remains a largely unexplored area. To address this challenge, we propose a reduction-based framework that transforms robust average reward optimization into the more extensively studied robust discounted reward optimization by employing a specific discount factor. Our framework provides two key advantages. **Data Efficiency**: We design a model-based reduction algorithm that achieves near-optimal sample complexity, enabling efficient identification of optimal robust policies; **Scalability**: By bypassing the inherent challenges of scaling up average reward optimization, our framework facilitates the design of scalable, convergent algorithms for robust average reward optimization leveraging function approximation. Our algorithmic design, supported by theoretical and empirical analyses, provides a concrete solution to robust average reward RL with the first data efficiency and scalability guarantees, highlighting the framework’s potential to optimize long-term performance under model uncertainty in practical problems.

General Machine Learning · Transfer, Multitask and Meta-learning

Zitao Wang, Ziyuan Wang, Molei Liu, Nian Si

Wasserstein Distributionally Robust Optimization (WDRO) is a principled framework for robust estimation under distributional uncertainty. However, its standard formulation can be overly conservative, particularly in small-sample regimes. We propose a novel knowledge-guided WDRO (KG-WDRO) framework for transfer learning, which adaptively incorporates multiple sources of external knowledge to improve generalization accuracy. Our method constructs smaller Wasserstein ambiguity sets by controlling the transportation along directions informed by the source knowledge. This strategy can alleviate perturbations on the predictive projection of the covariates and protect against information loss. Theoretically, we establish the equivalence between our WDRO formulation and the knowledge-guided shrinkage estimation based on collinear similarity, ensuring tractability and geometrizing the feasible set. This also reveals a novel and general interpretation for recent shrinkage-based transfer learning approaches from the perspective of distributional robustness. In addition, our framework can adjust for scaling differences in the regression models between the source and target and accommodates general types of regularization such as lasso and ridge. Extensive simulations demonstrate the superior performance and adaptivity of KG-WDRO in enhancing small-sample transfer learning.

General Machine Learning

Gauthier Thurin, Kimia Nadjahi, Claire Boyer

Conformal Prediction (CP) is a principled framework for quantifying uncertainty in black-box learning models, by constructing prediction sets with finite-sample coverage guarantees. Traditional approaches rely on scalar nonconformity scores, which fail to fully exploit the geometric structure of multivariate outputs, such as in multi-output regression or multiclass classification. Recent methods addressing this limitation impose predefined convex shapes for the prediction sets, potentially misaligning with the intrinsic data geometry. We introduce a novel CP procedure handling multivariate score functions through the lens of optimal transport. Specifically, we leverage Monge-Kantorovich vector ranks and quantiles to construct prediction region with flexible, potentially non-convex shapes, better suited to the complex uncertainty patterns encountered in multivariate learning tasks. We prove that our approach ensures finite-sample, distribution-free coverage properties, similar to typical CP methods. We then adapt our method for multi-output regression and multiclass classification, and also propose simple adjustments to generate adaptive prediction regions with asymptotic conditional coverage guarantees. Finally, we evaluate our method on practical regression and classification problems, illustrating its advantages in terms of (conditional) coverage and efficiency.

Deep Learning · Large Language Models

Zeyu Tang, Zhenhao Chen, Xiangchen Song, Loka Li, Yunlong Deng, Yifan Shen, Guangyi Chen, Peter Spirtes, Kun Zhang

The autoregressive decoding for text generation in large language models (LLMs), while widely used, is inherently suboptimal due to the lack of a built-in mechanism to perform refinement and/or correction of the generated content. In this paper, we consider optimality in terms of the joint probability over the generated response, when jointly considering all tokens at the same time. We theoretically characterize the potential deviation of the autoregressively generated response from its globally optimal counterpart that is of the same length. Our analysis suggests that we need to be cautious when noticeable uncertainty arises during text generation, which may signal the sub-optimality of the generation history. To address the pitfall of autoregressive decoding for text generation, we propose an approach that incorporates a sliding reflection window and a pausing criterion, such that refinement and generation can be carried out interchangeably as the decoding proceeds. Our selective refinement framework strikes a balance between efficiency and optimality, and our extensive experimental results demonstrate the effectiveness of our approach.

Reinforcement Learning · Planning

Tuan Dam, Pascal Stenger, Lukas Schneider, Joni Pajarinen, Carlo D'Eramo, Odalric-Ambrym Maillard

This paper introduces a novel backup strategy for Monte-Carlo Tree Search (MCTS) tailored for highly stochastic and partially observable Markov decision processes. We adopt a probabilistic approach, modeling both value and action-value nodes as Gaussian distributions, to introduce a novel backup operator that computes value nodes as the Wasserstein barycenter of their action-value children nodes; thus, propagating the uncertainty of the estimate across the tree to the root node. We study our novel backup operator when using a novel combination of $L^1$-Wasserstein barycenter with $\alpha$-divergence, by drawing a crucial connection to the generalized mean backup operator. We complement our probabilistic backup operator with two sampling strategies, based on optimistic selection and Thompson sampling, obtaining our Wasserstein MCTS algorithm. We provide theoretical guarantees of asymptotic convergence of $\mathcal{O}(n^{-1/2})$, with $n$ as the number of visited trajectories, to the optimal policy and an empirical evaluation on several stochastic and partially observable environments, where our approach outperforms well-known related baselines.

Probabilistic Methods · Bayesian Models and Methods

Pankaj Bhagwat, Linglong Kong, Bei Jiang

Conformal prediction has emerged as a popular technique for facilitating valid predictive inference across a spectrum of machine learning models, under minimal assumption of exchangeability. Recently, Hoff (2023) showed that full conformal Bayes provides the most efficient prediction sets (smallest by expected volume) among all prediction sets that are valid at the $(1 - \alpha)$ level if the model is correctly specified. However, a critical issue arises when the Bayesian model itself may be mis-specified, resulting in prediction interval that might be suboptimal, even though it still enjoys the frequentist coverage guarantee. To address this limitation, we propose an innovative solution that combines Bayesian model averaging (BMA) with conformal prediction. This hybrid not only leverages the strengths of Bayesian conformal prediction but also introduces a layer of robustness through model averaging. Theoretically, we prove that the resulting prediction interval will converge to the optimal level of efficiency, if the true model is included among the candidate models. This assurance of optimality, even under potential model uncertainty, provides a significant improvement over existing methods, ensuring more reliable and precise uncertainty quantification.

Deep Learning · Robustness

Shuang Liu, Yihan Wang, Yifan Zhu, Yibo Miao, XIAOSHAN GAO

Wasserstein distributionally robust optimization (WDRO) optimizes against worst-case distributional shifts within a specified uncertainty set, leading to enhanced generalization on unseen adversarial examples, compared to standard adversarial training which focuses on pointwise adversarial perturbations. However, WDRO still suffers fundamentally from the robust overfitting problem, as it does not consider statistical error. We address this gap by proposing a novel robust optimization framework under a new uncertainty set for adversarial noise via Wasserstein distance and statistical error via Kullback-Leibler divergence, called the Statistically Robust WDRO. We establish a robust generalization bound for the new optimization framework, implying that out-of-distribution adversarial performance is at least as good as the statistically robust training loss with high probability. Furthermore, we derive conditions under which Stackelberg and Nash equilibria exist between the learner and the adversary, giving an optimal robust model in certain sense.Finally, through extensive experiments, we demonstrate that our method significantly mitigates robust overfitting and enhances robustness within the framework of WDRO.

Reinforcement Learning · Everything Else

Zexu Sun, Yiju Guo, Yankai Lin, Xu Chen, Qi Qi, Xing Tang, xiuqiang He, Ji-Rong Wen

Reinforcement Learning from Human Feedback (RLHF) has emerged as a standard and effective approach for training large language models (LLMs) with human preferences. In this framework, a learned reward model approximates human preferences and guides policy optimization, making it crucial to develop an accurate reward model. However, without the ``true'' reward function, challenges arise when the reward model is an imperfect proxy for human preference. Since the policy optimization continuously shifts the human preference training dataset's distribution. The fixed reward model suffers from this problem of off-distribution, especially the on policy methods. While collecting new preference data can mitigate this issue, it is costly and challenging to optimize. Thus, reusing the policy interaction samples becomes a possible way to further refine the reward model. To tackle these challenges, we introduce a novel method \textbf{U}ncertainty-\textbf{G}radient based \textbf{D}ata \textbf{A}ugmentation (\textbf{UGDA} for short) to enhance reward modeling by leveraging policy samples to maintain on-distribution performance. Specifically, UGDA selects interaction samples based on the uncertainty of the reward ensembles and the gradient based influence of policy optimization. After the reward relabeling of selected samples, we use supervised learning to refine the reward ensembles, then get the retrained policy. Extensive experiments demonstrate that by leveraging UGDA to select a few samples without the costly human preference data collection, we can improve the ability of the policy and surpass the state-of-the-art methods.

Social Aspects · Trustworthy Machine Learning

Santiago Cortes-Gomez, Carlos Patiño, Yewon Byun, Steven Wu, Eric Horvitz, Bryan Wilder

There is increasing interest in ``decision-focused" machine learning methods which train models to account for how their predictions are used in downstream optimization problems. Doing so can often improve performance on subsequent decision problems. However, current methods for uncertainty quantification do not incorporate any information at all about downstream decisions. We develop a framework based on conformal prediction to produce prediction sets that account for a downstream decision loss function, making them more appropriate to inform high-stakes decision-making. Our approach harnesses the strengths of conformal methods—modularity, model-agnosticism, and statistical coverage guarantees—while incorporating downstream decisions and user-specified utility functions. We prove that our methods retain standard coverage guarantees. Empirical evaluation across a range of datasets and utility metrics demonstrates that our methods achieve significantly lower decision loss compared to standard conformal methods. Additionally, we present a real-world use case in healthcare diagnosis, where our method effectively incorporates the hierarchical structure of dermatological diseases. It successfully generates sets with coherent diagnostic meaning, aiding the triage process during dermatology diagnosis and illustrating how our method can ground high-stakes decision-making on external domain knowledge.