Showing 1–50 of 442 results for author: Jordan, M

Post-Training at the Edge of Detectability: A Game-Theoretic Approach to Fine-Tuning

Authors: Keegan Harris, Brian W. Lee, Ian Waudby-Smith, Philip Amortila, Nika Haghtalab, Michael I. Jordan

Abstract: Reinforcement learning (RL) fine-tuning is widely used in language model training to improve model performance on a target task while limiting drift from a reference policy. A standard way to balance this trade-off is via a KL-regularized RL objective, although this formulation does not by itself provide a principled way to set the regularization coefficient. In practice, the coefficient is typica… ▽ More Reinforcement learning (RL) fine-tuning is widely used in language model training to improve model performance on a target task while limiting drift from a reference policy. A standard way to balance this trade-off is via a KL-regularized RL objective, although this formulation does not by itself provide a principled way to set the regularization coefficient. In practice, the coefficient is typically chosen heuristically or via hyperparameter search, which can lead to unnecessary overhead in training cost or undesirable reward-retention trade-offs. We instead propose a game-theoretic framework that gives this trade-off an explicit statistical interpretation. Specifically, we study a sequential game in which an agent chooses a policy to maximize cumulative reward while a monitor observes policy outputs over time and tests for deviations from the reference policy. Although not originating from the same perspective, we show that the resulting equilibrium policy can nonetheless be expressed as the solution to a KL-regularized RL problem for an optimal regularization parameter that can be viewed as maximizing reward per unit of statistical distinguishability. Drawing on classical results from concave-convex fractional programming, we provide a principled method for learning this equilibrium coefficient via reduction to the KL-regularized RL objective, thus allowing for flexible integration into standard fine-tuning pipelines. In experiments with Qwen3-8B and Llama-3.2-1B, we demonstrate that our methods result in competitive reward-retention trade-offs in a continual learning setting, and illustrate how our framework may be used to audit API providers serving open-source models. △ Less

Submitted 28 July, 2026; originally announced July 2026.

Physics-Informed Deep Learning Model for Cross-Modality Super-Resolution in Fluorescence Microscopy

Authors: Mohammad Soltaninezhad, Elena Corbetta, Francisco Paez Larios, Paul M. Jordan, Oliver Werz, Christian Eggeling, Thomas Bocklitz

Abstract: Cross-modality image translation offers a route to super-resolution fluorescence microscopy from low-resolution images while reducing phototoxicity and instrumentation demands. However, purely data-driven models can produce visually plausible outputs that are inconsistent with optical image formation. Here, we propose a physics-informed generative adversarial network for confocal-to-STED image tra… ▽ More Cross-modality image translation offers a route to super-resolution fluorescence microscopy from low-resolution images while reducing phototoxicity and instrumentation demands. However, purely data-driven models can produce visually plausible outputs that are inconsistent with optical image formation. Here, we propose a physics-informed generative adversarial network for confocal-to-STED image translation that incorporates microscope-specific point spread function information into the training objective. Simulated and experimentally measured PSFs were evaluated using a limited paired confocal-STED dataset of TOM20-labeled mitochondria in human primary M2 macrophages acquired across different experimental days. Performance was assessed using reference-based and non-reference-based image-quality metrics, together with complementary frequency- and distribution-sensitive analyses. The no-reference metrics probed physics-relevant image properties, including spatial-frequency content, contrast, and signal-to-noise behavior. PSF-guided models improved structural fidelity, reduced local deviations, and achieved closer agreement with STED references than non-PSF baselines, particularly in frequency-domain analyses. These results demonstrate that optical priors can improve the structural fidelity and physical plausibility of generative microscopy models for cross-modality super-resolution imaging. △ Less

Submitted 23 July, 2026; originally announced July 2026.

Provably Optimal Learning Algorithms for Assistance Games

Authors: Nivasini Ananthakrishnan, Mark Bedaywi, Michael I. Jordan, Stuart Russell, Nika Haghtalab

Abstract: This paper studies an online variant of the assistance games framework, where an informed agent and an uninformed agent repeatedly interact over $T$ timesteps to optimize a common reward function. While the informed agent (the human) observes a latent state of the world, the uninformed agent (the assistant) observes only the human's actions. We provide the first provably efficient learning algorit… ▽ More This paper studies an online variant of the assistance games framework, where an informed agent and an uninformed agent repeatedly interact over $T$ timesteps to optimize a common reward function. While the informed agent (the human) observes a latent state of the world, the uninformed agent (the assistant) observes only the human's actions. We provide the first provably efficient learning algorithms for repeated assistance games. We introduce the notion of assistance regret: the gap between the cumulative utility of interactions and that of the optimal joint policies in hindsight, which map latent states to action pairs. We present decentralized algorithms for both the human and the assistant that achieve a $(1-1/e)$-approximate assistance regret rate of $\widetilde{O}(T^{3/4})$, with runtime polynomial in the size of the action and state spaces. These algorithms are general; in particular, they accommodate any no-regret algorithm for the assistant. We prove that achieving a regret approximation factor better than $(1-1/e)$ is computationally intractable. Furthermore, we demonstrate how these generic no-regret algorithms can be tailored to a pseudo-decentralized setting -- using a shared random string -- to achieve a rate of $\widetilde{O}(T^{1/2})$, optimal up to logarithmic factors. △ Less

Submitted 8 July, 2026; originally announced July 2026.

Blackwell Approachability and Gradient Equilibrium are Equivalent

Authors: Brian W. Lee, Nika Haghtalab, Michael I. Jordan, Ryan J. Tibshirani

Abstract: Gradient equilibrium (GEQ) is a recently introduced online optimization framework that generalizes first-order stationarity from offline optimization and abstracts problems like online conformal prediction. While GEQ has curious similarities with known online learning frameworks, namely regret minimization, prior work has shown that GEQ error and regret are incomparable objectives, leaving open a… ▽ More Gradient equilibrium (GEQ) is a recently introduced online optimization framework that generalizes first-order stationarity from offline optimization and abstracts problems like online conformal prediction. While GEQ has curious similarities with known online learning frameworks, namely regret minimization, prior work has shown that GEQ error and regret are incomparable objectives, leaving open a precise understanding of how GEQ fits into the broader online learning landscape. In this work, we show that GEQ is equivalent to Blackwell approachability in the algorithmic sense. That is, a Blackwell approachability problem can always be solved using queries to a black-box GEQ oracle, with no asymptotic loss in the oracle's error rate, and vice versa. Taken together with known equivalences between approachability, regret minimization, and calibration, these results imply that GEQ is equivalent to these frameworks, as well. Our reductions are efficient and can be used to transfer refined guarantees, such as optimism and strong adaptivity, from regret minimization to GEQ. Along the way, we also identify necessary and sufficient conditions for GEQ, and establish reductions between different notions of GEQ with unconstrained and constrained decision sets. △ Less

Submitted 25 June, 2026; originally announced June 2026.

Comments: 30 pages, 1 figure, accepted for presentation at COLT 2026

SPINE: A Fault Injection Profiler for Quantized Neural Networks under Accumulated Faults

Authors: Nathan Guimarães, Ian Kersz, Leonardo R. Gobatto, Fabio Benevenuti, Michael G. Jordan, Antonio Carlos S. Beck, Fernanda L. Kastensmidt, Jose Rodrigo Azambuja

Abstract: Deploying deep neural networks at the edge demands efficient inference under strict cost and power constraints. Quantized neural networks address these demands by replacing floating-point parameters with low-precision integers, yet their weights remain continuously exposed to radiation-induced bit-flips during inference. Fault Injection can be used to simulate those environments, but existing stud… ▽ More Deploying deep neural networks at the edge demands efficient inference under strict cost and power constraints. Quantized neural networks address these demands by replacing floating-point parameters with low-precision integers, yet their weights remain continuously exposed to radiation-induced bit-flips during inference. Fault Injection can be used to simulate those environments, but existing studies fail to characterize how accumulated upsets translate into mispredictions under realistic memory layouts. This paper presents a GDB-driven profiling framework that injects cumulative weight bit-flips directly onto the target binary of edge CPUs, generating per-layer fault profiles without requiring model retraining or code modification. Evaluated across multiple topologies, quantization efforts, and memory layouts, the results indicate how selective hardening strategies should be applied to effectively protect neural networks. △ Less

Submitted 17 June, 2026; originally announced June 2026.

Comments: ACM/IEEE/SBC/SBMICRO Symposium on Integrated Circuits and Systems Design 2026

ACM Class: B.8.1

Strategic Feature Selection

Authors: Jivat Neet Kaur, Pratik Patil, Divya Shanmugam, Emma Pierson, Michael I. Jordan, Nika Haghtalab, Meena Jagadeesan, Ahmed Alaa, Serena Wang

Abstract: When algorithmic predictors inform resource allocation in high-stakes domains such as healthcare, these predictors must account for strategic manipulation of input features. The typical solution is to redesign the predictor itself to explicitly account for strategic interactions. In practice, however, decision makers are often constrained to adjusting coarser levers within existing prediction pipe… ▽ More When algorithmic predictors inform resource allocation in high-stakes domains such as healthcare, these predictors must account for strategic manipulation of input features. The typical solution is to redesign the predictor itself to explicitly account for strategic interactions. In practice, however, decision makers are often constrained to adjusting coarser levers within existing prediction pipelines. For example, healthcare organizations often select which features to exclude based on perceived manipulability, while using standard regularization procedures to shrink the coefficients of retained features. In this work, we initiate a formal study of strategic classification through feature selection and its interaction with ridge regularization. Our main finding is that excluding individual features based on their manipulability alone is generally suboptimal. We provide a fine-grained characterization of the performance of a feature subset under optimal regularization, yielding new insights for policy design. Motivated by this characterization, we develop a practical algorithm for jointly choosing the feature set and the level of ridge regularization. Through a real-world case study on a healthcare payments benchmark, we illustrate how our algorithm can guide the design of coarse policy levers in practice. Our results provide a principled, practical framework for mitigating the effects of strategic behavior in algorithmic decision-making systems. △ Less

Submitted 17 June, 2026; originally announced June 2026.

Towards Accurate Model Selection in Deep Unsupervised Domain Adaptation

Authors: Kaichao You, Ximei Wang, Mingsheng Long, Michael I. Jordan

Abstract: Deep unsupervised domain adaptation (Deep UDA) methods successfully leverage rich labeled data in a source domain to boost the performance on related but unlabeled data in a target domain. However, algorithm comparison is cumbersome in Deep UDA due to the absence of accurate and standardized model selection method, posing an obstacle to further advances in the field. Existing model selection metho… ▽ More Deep unsupervised domain adaptation (Deep UDA) methods successfully leverage rich labeled data in a source domain to boost the performance on related but unlabeled data in a target domain. However, algorithm comparison is cumbersome in Deep UDA due to the absence of accurate and standardized model selection method, posing an obstacle to further advances in the field. Existing model selection methods for Deep UDA are either highly biased, restricted, unstable, or even controversial (requiring labeled target data). To this end, we propose \textit{Deep Embedded Validation} (\textbf{DEV}), which embeds adapted feature representation into the validation procedure to obtain unbiased estimation of the target risk with bounded variance. The variance is further reduced by the technique of control variate. The efficacy of the method has been justified both theoretically and empirically. △ Less

Submitted 3 June, 2026; originally announced June 2026.

Comments: upload to arxiv for record

Diffusion-Robust Optimization over Graphs

Authors: Liviu Aolaritei, Ricky Huang, Michael I. Jordan, Paul Grigas

Abstract: We introduce a diffusion-based uncertainty model for robust optimization on directed graphs, in which perturbations of edge weights propagate along adjacent edges and satisfy conservation constraints at nodes. This topology-aware structure is natural in networked systems where uncertainty is induced by flows and local interactions, including transportation, logistics, communication, and energy net… ▽ More We introduce a diffusion-based uncertainty model for robust optimization on directed graphs, in which perturbations of edge weights propagate along adjacent edges and satisfy conservation constraints at nodes. This topology-aware structure is natural in networked systems where uncertainty is induced by flows and local interactions, including transportation, logistics, communication, and energy networks. We analyze how such diffusive uncertainty reshapes the computational landscape of robust graph optimization. We focus on two canonical combinatorial graph problems, shortest path and the traveling salesman problem (TSP), which provide complementary benchmarks: shortest path is polynomial-time solvable in the nominal setting, whereas TSP is already NP-hard. We show that, for shortest path, propagation depth induces a sharp transition between tractable and intractable robust counterparts. For the traveling salesman problem, robustness often adds no computational complexity beyond ordinary TSP, because the structure of Hamiltonian cycles makes the fixed-tour adversarial problem collapse to explicit formulas. Together, these results show that topology-aware uncertainty can fundamentally change robust combinatorial optimization, with tractability governed by the interaction between propagation, budget geometry, and the structure of feasible solutions. △ Less

Submitted 19 June, 2026; v1 submitted 29 May, 2026; originally announced May 2026.

Comments: 46 pages, 6 figures

CalArena: A Large-Scale Post-Hoc Calibration Benchmark

Authors: Eugène Berta, David Holzmüller, Francis Bach, Michael I. Jordan

Abstract: Reliable probability estimates are critical in many machine learning applications, yet modern classifiers are often poorly calibrated. Post-hoc calibration provides a simple and widely used solution, but the large number of proposed methods, combined with small-scale and inconsistent evaluations, makes it difficult to determine which approaches are truly effective in practice. We introduce a large… ▽ More Reliable probability estimates are critical in many machine learning applications, yet modern classifiers are often poorly calibrated. Post-hoc calibration provides a simple and widely used solution, but the large number of proposed methods, combined with small-scale and inconsistent evaluations, makes it difficult to determine which approaches are truly effective in practice. We introduce a large-scale, standardized benchmark for post-hoc calibration, covering nearly 2000 experiments across tabular and computer vision tasks, including binary, multiclass, and large-scale classification settings. Our benchmark aggregates predictions from a diverse set of classical models, modern deep learning architectures, and foundation models, and provides unified, reproducible implementations of dozens of calibration methods within a common evaluation framework. We argue that Post-Hoc Improvement (PHI) in proper scoring rules offers a principled alternative to traditional calibration error estimators for comparing post-hoc methods, capturing both calibration quality and potential degradation to the model's predictive performance. Using this framework, we conduct the most comprehensive empirical study of post-hoc calibration to date. Our results reveal consistent patterns across domains: smooth calibration functions outperform binning-based approaches, dedicated multiclass methods are essential in high-dimensional settings, and generic machine learning models are not competitive without calibration-specific design. To facilitate future research, we release all data, code, and evaluation tools, providing a plug-and-play benchmark for developing and comparing calibration methods. △ Less

Submitted 1 June, 2026; v1 submitted 28 May, 2026; originally announced May 2026.

Comments: 30 pages, 9 figures

Response Time Enhances Alignment with Heterogeneous Preferences

Authors: Federico Echenique, Alireza Fallah, Baihe Huang, Michael I. Jordan

Abstract: Aligning large language models (LLMs) to human preferences typically relies on aggregating pooled feedback into a single reward model. However, this standard approach assumes that all labelers share the same underlying preferences, ignoring the fact that real-world labelers are highly heterogeneous and usually anonymous. Consequently, relying solely on binary choice data fundamentally distorts the… ▽ More Aligning large language models (LLMs) to human preferences typically relies on aggregating pooled feedback into a single reward model. However, this standard approach assumes that all labelers share the same underlying preferences, ignoring the fact that real-world labelers are highly heterogeneous and usually anonymous. Consequently, relying solely on binary choice data fundamentally distorts the learned policy, making the true population-average preference unidentifiable. To overcome this critical limitation, we demonstrate that augmenting preference datasets with a simple, secondary signal -- the user's response time -- can restore the identifiability of the population's average preference. By modeling each decision as a Drift-Diffusion Model (DDM), we introduce a novel, consistent estimator of heterogeneous preferences that successfully corrects the distortions of standard choice-only labels. We prove that our estimator asymptotically converges to the true average preference even in extreme cases where each anonymous labeler contributes only a single choice. Empirically, across both synthetic and real-world datasets, our method consistently outperforms standard baselines that otherwise fail and plateau at a bias floor. Because response times are essentially free to record and require zero user tracking or identification, our results bring promises and open up new opportunities for future data-collection pipelines to improve the social benefit without requiring user-level identifiers or repeated elicitations. △ Less

Submitted 7 May, 2026; originally announced May 2026.

Super-Level-Set Regression: Conditional Quantiles via Volume Minimization

Authors: Sacha Braun, Michael I. Jordan, Francis Bach

Abstract: Constructing minimum-volume prediction regions that satisfy conditional coverage is a fundamental challenge in multivariate regression. Standard approaches rely on explicitly estimating the full conditional density and subsequently thresholding it. This two-step plug-in process is notoriously difficult, sensitive to estimation errors, and computationally expensive. One would like to instead optimi… ▽ More Constructing minimum-volume prediction regions that satisfy conditional coverage is a fundamental challenge in multivariate regression. Standard approaches rely on explicitly estimating the full conditional density and subsequently thresholding it. This two-step plug-in process is notoriously difficult, sensitive to estimation errors, and computationally expensive. One would like to instead optimize the region directly. Formulating a direct solution is challenging, however, because it requires minimizing a volume objective that is coupled with the conditional quantiles of the model's own estimation error. In this work, we address this challenge. We introduce super-level-set regression (SLS), a novel mathematical framework that successfully resolves this implicit coupling, allowing us to directly parameterize and optimize the geometric boundaries of the target conditional level sets. By bypassing full distribution estimation and leveraging flexible volume-preserving frontier functions, our approach natively captures complex, multimodal, and disjoint conditional structures end-to-end. Ultimately, SLS offers a new perspective on multivariate conditional quantile regression, replacing the restrictive assumptions of density-first methods with a direct geometric optimization strategy. △ Less

Submitted 7 May, 2026; originally announced May 2026.

Explaining and Preventing Alignment Collapse in Iterative RLHF

Authors: Etienne Gauthier, Francis Bach, Michael I. Jordan

Abstract: Reinforcement learning from human feedback (RLHF) typically assumes a static or non-strategic reward model (RM). In iterative deployment, however, the policy generates the data on which the RM is retrained, creating a feedback loop. Building on the Stackelberg game formulation of this interaction, we derive an analytical decomposition of the policy's true optimization gradient into a standard poli… ▽ More Reinforcement learning from human feedback (RLHF) typically assumes a static or non-strategic reward model (RM). In iterative deployment, however, the policy generates the data on which the RM is retrained, creating a feedback loop. Building on the Stackelberg game formulation of this interaction, we derive an analytical decomposition of the policy's true optimization gradient into a standard policy gradient and a parameter-steering term that captures the policy's influence on the RM's future parameters. We show that standard iterative RLHF, which drops this steering term entirely, suffers from alignment collapse: the policy systematically exploits the RM's blind spots, producing low-quality, high-reward outputs whose feedback reinforces the very errors it exploits. To mitigate this, we propose foresighted policy optimization (FPO), a mechanism-design intervention that restores the missing steering term by regularizing the policy's parameter-steering effect on RM updates. We instantiate FPO via a scalable first-order approximation and demonstrate that it prevents alignment collapse on both controlled environments and an LLM alignment pipeline using Llama-3.2-1B. △ Less

Submitted 5 May, 2026; originally announced May 2026.

Comments: Code at: https://github.com/GauthierE/fpo

Calibeating Made Simple

Authors: Yurong Chen, Zhiyi Huang, Michael I. Jordan, Haipeng Luo

Abstract: 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… ▽ More 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. △ Less

Submitted 23 March, 2026; originally announced March 2026.

Multi-Armed Sequential Hypothesis Testing by Betting

Authors: Ricardo J. Sandoval, Ian Waudby-Smith, Michael I. Jordan

Abstract: We consider a variant of sequential testing by betting where, at each time step, the statistician is presented with multiple data sources (arms) and obtains data by choosing one of the arms. We consider the composite global null hypothesis $\mathscr{P}$ that all arms are null in a certain sense (e.g. all dosages of a treatment are ineffective) and we are interested in rejecting $\mathscr{P}$ in fa… ▽ More We consider a variant of sequential testing by betting where, at each time step, the statistician is presented with multiple data sources (arms) and obtains data by choosing one of the arms. We consider the composite global null hypothesis $\mathscr{P}$ that all arms are null in a certain sense (e.g. all dosages of a treatment are ineffective) and we are interested in rejecting $\mathscr{P}$ in favor of a composite alternative $\mathscr{Q}$ where at least one arm is non-null (e.g. there exists an effective treatment dosage). We posit an optimality desideratum that we describe informally as follows: even if several arms are non-null, we seek $e$-processes and sequential tests whose performance are as strong as the ones that have oracle knowledge about which arm generates the most evidence against $\mathscr{P}$. Formally, we generalize notions of log-optimality and expected rejection time optimality to more than one arm, obtaining matching lower and upper bounds for both. A key technical device in this optimality analysis is a modified upper-confidence-bound-like algorithm for unobservable but sufficiently "estimable" rewards. In the design of this algorithm, we derive nonasymptotic concentration inequalities for optimal wealth growth rates in the sense of Kelly [1956]. These may be of independent interest. △ Less

Submitted 4 June, 2026; v1 submitted 18 March, 2026; originally announced March 2026.

Test-then-Punish: A Statistical Approach to Repeated Games

Authors: Aymeric Capitaine, Antoine Scheid, Etienne Boursier, Alain Durmus, Michael I. Jordan

Abstract: We study discounted infinitely repeated games in which players agree on a cooperative mixed action profile but, at each step, observe only the realized pure actions. This form of imperfect monitoring breaks classical trigger strategies, since deviations cannot be identified with certainty. To address this problem, we study how hypothesis testing can be used to sustain cooperation. First, we develo… ▽ More We study discounted infinitely repeated games in which players agree on a cooperative mixed action profile but, at each step, observe only the realized pure actions. This form of imperfect monitoring breaks classical trigger strategies, since deviations cannot be identified with certainty. To address this problem, we study how hypothesis testing can be used to sustain cooperation. First, we develop a framework that embeds statistical inference directly into strategic behavior. We introduce relaxed equilibrium notions that allow players to ignore vanishing probability histories arising from rare but extreme realizations of the monitoring process. Within this framework, we formalize a generic test then punish strategy: players commit ex ante to a cooperative mixed action profile, continuously test whether observed play is consistent with this prescription, and permanently switch to punishment once sufficient statistical evidence of deviation accumulates. Under mild conditions on the testing procedure, this construction sustains any feasible and individually rational payoff for sufficiently patient players, yielding a Folk theorem type result under imperfect monitoring. We then propose two explicit implementations of this strategy. The first relies on anytime valid sequential tests, providing uniform control of Type I error over an infinite horizon and a finite expected detection time for payoff-relevant deviations. However, this strategy only accounts for stationary deviations and yields a Nash equilibrium. The second uses testing over batches with a fixed size, accommodating arbitrary deviations and achieving subgame perfect Nash equilibrium, at the cost of losing global anytime guarantees on false punishments. △ Less

Submitted 5 March, 2026; originally announced March 2026.

A Variational Estimator for $L_p$ Calibration Errors

Authors: Eugène Berta, Sacha Braun, David Holzmüller, Francis Bach, Michael I. Jordan

Abstract: Calibration$\unicode{x2014}$the problem of ensuring that predicted probabilities align with observed class frequencies$\unicode{x2014}$is a basic desideratum for reliable prediction with machine learning systems. Calibration error is traditionally assessed via a divergence function, using the expected divergence between predictions and empirical frequencies. Accurately estimating this quantity is… ▽ More Calibration$\unicode{x2014}$the problem of ensuring that predicted probabilities align with observed class frequencies$\unicode{x2014}$is a basic desideratum for reliable prediction with machine learning systems. Calibration error is traditionally assessed via a divergence function, using the expected divergence between predictions and empirical frequencies. Accurately estimating this quantity is challenging, especially in the multiclass setting. Here, we show how to extend a recent variational framework for estimating calibration errors beyond divergences induced induced by proper losses, to cover a broad class of calibration errors induced by $L_p$ divergences. Our method can separate over- and under-confidence and, unlike non-variational approaches, avoids overestimation. We provide extensive experiments and integrate our code in the open-source package probmetrics (https://github.com/dholzmueller/probmetrics) for evaluating calibration errors. △ Less

Submitted 27 February, 2026; originally announced February 2026.

Towards Anytime-Valid Statistical Watermarking

Authors: Baihe Huang, Eric Xu, Kannan Ramchandran, Jiantao Jiao, Michael I. Jordan

Abstract: The proliferation of Large Language Models (LLMs) necessitates efficient mechanisms to distinguish machine-generated content from human text. While statistical watermarking has emerged as a promising solution, existing methods suffer from two critical limitations: the lack of a principled approach for selecting sampling distributions and the reliance on fixed-horizon hypothesis testing, which prec… ▽ More The proliferation of Large Language Models (LLMs) necessitates efficient mechanisms to distinguish machine-generated content from human text. While statistical watermarking has emerged as a promising solution, existing methods suffer from two critical limitations: the lack of a principled approach for selecting sampling distributions and the reliance on fixed-horizon hypothesis testing, which precludes valid early stopping. In this paper, we bridge this gap by developing the first e-value-based watermarking framework, Anchored E-Watermarking, that unifies optimal sampling with anytime-valid inference. Unlike traditional approaches where optional stopping invalidates Type-I error guarantees, our framework enables valid, anytime-inference by constructing a test supermartingale for the detection process. By leveraging an anchor distribution to approximate the target model, we characterize the optimal e-value with respect to the worst-case log-growth rate and derive the optimal expected stopping time. Our theoretical claims are substantiated by simulations and evaluations on established benchmarks, showing that our framework can significantly enhance sample efficiency, reducing the average token budget required for detection by 13-15% relative to state-of-the-art baselines. △ Less

Submitted 19 February, 2026; originally announced February 2026.

Signaling in Data Markets via Free Samples

Authors: Nivasini Ananthakrishnan, Alireza Fallah, Michael I. Jordan

Abstract: We study a setting in which a data buyer seeks to estimate an unknown parameter by purchasing samples from one of K data sellers. Each seller has privately known data quality (e.g., high vs. low variance) and a private per-sample cost. We consider a multi-stage game in which the first stage is a free-trial stage in which the sellers have the option of signaling data quality by offering a few sampl… ▽ More We study a setting in which a data buyer seeks to estimate an unknown parameter by purchasing samples from one of K data sellers. Each seller has privately known data quality (e.g., high vs. low variance) and a private per-sample cost. We consider a multi-stage game in which the first stage is a free-trial stage in which the sellers have the option of signaling data quality by offering a few samples of data for free. Buyers update their beliefs based on the sample variance of the free data and then run a procurement auction to buy data in a second stage. For the auction stage, we characterize an approximately optimal Bayesian incentive compatible mechanism: the buyer selects a single seller by minimizing a belief-adjusted virtual cost and chooses the purchased sample size as a function of posterior quality and virtual cost. For the free-trial stage, we characterize the equilibrium, taking the above mechanism as the continuation game. Free trials may fail to emerge: for some parameters, all sellers reveal zero samples. However, under sufficiently strong competition (large K), there is an equilibrium in which sellers reveal the maximum allowable number of samples; in fact, it is the unique equilibrium. △ Less

Submitted 18 February, 2026; originally announced February 2026.

Locally Adaptive Multi-Objective Learning

Authors: Jivat Neet Kaur, Isaac Gibbs, Michael I. Jordan

Abstract: We consider the general problem of learning a predictor that satisfies multiple objectives of interest simultaneously, a broad framework that captures a range of specific learning goals including calibration, regret, and multiaccuracy. We work in an online setting where the data distribution can change arbitrarily over time. Existing approaches to this problem aim to minimize the set of objectives… ▽ More We consider the general problem of learning a predictor that satisfies multiple objectives of interest simultaneously, a broad framework that captures a range of specific learning goals including calibration, regret, and multiaccuracy. We work in an online setting where the data distribution can change arbitrarily over time. Existing approaches to this problem aim to minimize the set of objectives over the entire time horizon in a worst-case sense, and in practice they do not necessarily adapt to distribution shifts. Earlier work has aimed to alleviate this problem by incorporating additional objectives that target local guarantees over contiguous subintervals. Empirical evaluation of these proposals is, however, scarce. In this article, we consider an alternative procedure that achieves local adaptivity by replacing one part of the multi-objective learning method with an adaptive online algorithm. Empirical evaluations on datasets from energy forecasting and algorithmic fairness show that our proposed method improves upon existing approaches and achieves unbiased predictions over subgroups, while remaining robust under distribution shift. △ Less

Submitted 16 February, 2026; originally announced February 2026.

Comments: Code is available at https://github.com/jivatneet/adaptive-multiobjective

Olmix: A Framework for Data Mixing Throughout LM Development

Authors: Mayee F. Chen, Tyler Murray, David Heineman, Matt Jordan, Hannaneh Hajishirzi, Christopher Ré, Luca Soldaini, Kyle Lo

Abstract: Data mixing -- determining the ratios of data from different domains -- is a first-order concern for training language models (LMs). While existing mixing methods show promise, they fall short when applied during real-world LM development. We present Olmix, a framework that addresses two such challenges. First, the configuration space for developing a mixing method is not well understood -- design… ▽ More Data mixing -- determining the ratios of data from different domains -- is a first-order concern for training language models (LMs). While existing mixing methods show promise, they fall short when applied during real-world LM development. We present Olmix, a framework that addresses two such challenges. First, the configuration space for developing a mixing method is not well understood -- design choices across existing methods lack justification or consensus and overlook practical issues like data constraints. We conduct a comprehensive empirical study of this space, identifying which design choices lead to a strong mixing method. Second, in practice, the domain set evolves throughout LM development as datasets are added, removed, partitioned, and revised -- a problem setting largely unaddressed by existing works, which assume fixed domains. We study how to efficiently recompute the mixture after the domain set is updated, leveraging information from past mixtures. We introduce mixture reuse, a mechanism that reuses existing ratios and recomputes ratios only for domains affected by the update. Over a sequence of five domain-set updates mirroring real-world LM development, mixture reuse matches the performance of fully recomputing the mix after each update with 74% less compute and improves over training without mixing by 11.6% on downstream tasks. △ Less

Submitted 12 February, 2026; originally announced February 2026.

How Sampling Shapes LLM Alignment: From One-Shot Optima to Iterative Dynamics

Authors: Yurong Chen, Yu He, Michael I. Jordan, Fan Yao

Abstract: Standard methods for aligning large language models with human preferences learn from pairwise comparisons among sampled candidate responses and regularize toward a reference policy. Despite their effectiveness, the effects of sampling and reference choices are poorly understood theoretically. We investigate these effects through Identity Preference Optimization, a widely used preference alignment… ▽ More Standard methods for aligning large language models with human preferences learn from pairwise comparisons among sampled candidate responses and regularize toward a reference policy. Despite their effectiveness, the effects of sampling and reference choices are poorly understood theoretically. We investigate these effects through Identity Preference Optimization, a widely used preference alignment framework, and show that proper instance-dependent sampling can yield stronger ranking guarantees, while skewed on-policy sampling can induce excessive concentration under structured preferences. We then analyze iterative alignment dynamics in which the learned policy feeds back into future sampling and reference policies, reflecting a common practice of model-generated preference data. We prove that these dynamics can exhibit persistent oscillations or entropy collapse for certain parameter choices, and characterize regimes that guarantee stability. Our theoretical insights extend to Direct Preference Optimization, indicating the phenomena we captured are common to a broader class of preference-alignment methods. Experiments on real-world preference data validate our findings. △ Less

Submitted 12 February, 2026; originally announced February 2026.

Dynamic Programming for Epistemic Uncertainty in Markov Decision Processes

Authors: Axel Benyamine, Julien Grand-Clément, Marek Petrik, Michael I. Jordan, Alain Durmus

Abstract: In this paper, we propose a general theory of ambiguity-averse MDPs, which treats the uncertain transition probabilities as random variables and evaluates a policy via a risk measure applied to its random return. This ambiguity-averse MDP framework unifies several models of MDPs with epistemic uncertainty for specific choices of risk measures. We extend the concepts of value functions and Bellman… ▽ More In this paper, we propose a general theory of ambiguity-averse MDPs, which treats the uncertain transition probabilities as random variables and evaluates a policy via a risk measure applied to its random return. This ambiguity-averse MDP framework unifies several models of MDPs with epistemic uncertainty for specific choices of risk measures. We extend the concepts of value functions and Bellman operators to our setting. Based on these objects, we establish the consequences of dynamic programming principles in this framework (existence of stationary policies, value and policy iteration algorithms), and we completely characterize law-invariant risk measures compatible with dynamic programming. Our work draws connections among several variants of MDP models and fully delineates what is possible under the dynamic programming paradigm and which risk measures require leaving it. △ Less

Submitted 7 August, 2026; v1 submitted 3 February, 2026; originally announced February 2026.

Anytime Detection of Strategic Deviations in Multi-Agent Systems

Authors: Etienne Gauthier, Francis Bach, Michael I. Jordan

Abstract: In many multi-agent systems, agents interact repeatedly and are expected to settle into stable, rational behavior over time. Yet in practice, behavior often drifts, and detecting such deviations in real time remains an open challenge. We introduce a sequential testing framework that monitors whether observed play is consistent with a benchmark of strategic behavior, without assuming a fixed sample… ▽ More In many multi-agent systems, agents interact repeatedly and are expected to settle into stable, rational behavior over time. Yet in practice, behavior often drifts, and detecting such deviations in real time remains an open challenge. We introduce a sequential testing framework that monitors whether observed play is consistent with a benchmark of strategic behavior, without assuming a fixed sample size. Our approach builds on the e-value framework for safe anytime-valid inference: by "betting" against the benchmark, we construct a test supermartingale that accumulates evidence whenever observed payoffs systematically violate the expected conditions. For repeated normal-form games, we take equilibrium as the benchmark, yielding a statistically sound, interpretable measure of departure from equilibrium that can be monitored online; our framework unifies the treatment of Nash, correlated, and coarse correlated equilibria, offering finite-time guarantees and a detailed analysis of detection times. We also leverage Benjamini-Hochberg-type procedures to increase detection power in large games while rigorously controlling the false discovery rate. Finally, we extend our method to stochastic games, verifying online whether observed trajectories adhere to a specified target policy, such as a computed equilibrium, broadening the framework's applicability to dynamic, state-dependent settings. △ Less

Submitted 22 May, 2026; v1 submitted 8 January, 2026; originally announced January 2026.

Comments: Code at: https://github.com/GauthierE/anytime-detection-deviation

Olmo 3

Authors: Team Olmo, :, Allyson Ettinger, Amanda Bertsch, Bailey Kuehl, David Graham, David Heineman, Dirk Groeneveld, Faeze Brahman, Finbarr Timbers, Hamish Ivison, Jacob Morrison, Jake Poznanski, Kyle Lo, Luca Soldaini, Matt Jordan, Mayee Chen, Michael Noukhovitch, Nathan Lambert, Pete Walsh, Pradeep Dasigi, Robert Berry, Saumya Malik, Saurabh Shah, Scott Geng , et al. (44 additional authors not shown)

Abstract: We introduce Olmo 3, a family of state-of-the-art, fully-open language models at the 7B and 32B parameter scales. Olmo 3 model construction targets long-context reasoning, function calling, coding, instruction following, general chat, and knowledge recall. This release includes the entire model flow, i.e., the full lifecycle of the family of models, including every stage, checkpoint, data point, a… ▽ More We introduce Olmo 3, a family of state-of-the-art, fully-open language models at the 7B and 32B parameter scales. Olmo 3 model construction targets long-context reasoning, function calling, coding, instruction following, general chat, and knowledge recall. This release includes the entire model flow, i.e., the full lifecycle of the family of models, including every stage, checkpoint, data point, and dependency used to build it. Our flagship model, Olmo 3 Think 32B, is the strongest fully-open thinking model released to-date. △ Less

Submitted 14 April, 2026; v1 submitted 15 December, 2025; originally announced December 2025.

Comments: minor edit updates

Stopping Rules for Stochastic Gradient Descent via Anytime-Valid Confidence Sequences

Authors: Liviu Aolaritei, Michael I. Jordan

Abstract: The problem of stopping stochastic gradient descent (SGD) in an online manner, based solely on the observed trajectory, is a challenging theoretical problem with significant consequences for applications. While SGD is routinely monitored as it runs, the classical theory of SGD provides guarantees only at pre-specified iteration horizons and offers no valid way to decide, based on the observed traj… ▽ More The problem of stopping stochastic gradient descent (SGD) in an online manner, based solely on the observed trajectory, is a challenging theoretical problem with significant consequences for applications. While SGD is routinely monitored as it runs, the classical theory of SGD provides guarantees only at pre-specified iteration horizons and offers no valid way to decide, based on the observed trajectory, when further computation is justified. We address this longstanding gap by developing anytime-valid confidence sequences for stochastic gradient methods, which remain valid under continuous monitoring and directly induce statistically valid, trajectory-dependent stopping rules: stop as soon as the current upper confidence bound on an appropriate performance measure falls below a user-specified tolerance. The confidence sequences are constructed using nonnegative supermartingales, are time-uniform, and depend only on observable quantities along the SGD trajectory, without requiring prior knowledge of the optimization horizon. In convex optimization, this yields anytime-valid certificates for weighted suboptimality of projected SGD under general stepsize schedules, without assuming smoothness or strong convexity. In nonconvex optimization, it yields time-uniform certificates for weighted first-order stationarity under smoothness assumptions. We further characterize the stopping-time complexity of the resulting stopping rules under standard stepsize schedules. To the best of our knowledge, this is the first framework that provides statistically valid, time-uniform stopping rules for SGD across both convex and nonconvex settings based solely on its observed trajectory. △ Less

Submitted 20 February, 2026; v1 submitted 15 December, 2025; originally announced December 2025.

Conditional Coverage Diagnostics for Conformal Prediction

Authors: Sacha Braun, David Holzmüller, Michael I. Jordan, Francis Bach

Abstract: Evaluating conditional coverage remains one of the most persistent challenges in assessing the reliability of predictive systems. Although conformal methods can give guarantees on marginal coverage, no method can guarantee to produce sets with correct conditional coverage, leaving practitioners without a clear way to interpret local deviations. To overcome sample-inefficiency and overfitting issue… ▽ More Evaluating conditional coverage remains one of the most persistent challenges in assessing the reliability of predictive systems. Although conformal methods can give guarantees on marginal coverage, no method can guarantee to produce sets with correct conditional coverage, leaving practitioners without a clear way to interpret local deviations. To overcome sample-inefficiency and overfitting issues of existing metrics, we cast conditional coverage estimation as a classification problem. Conditional coverage is violated if and only if some classifier can achieve lower risk than the target coverage. Through the choice of a (proper) loss function, the resulting risk difference gives a conservative estimate of natural miscoverage measures such as L1 and L2 distance, and can even separate the effects of over- and under-coverage, and non-constant target coverages. We call the resulting family of metrics excess risk of the target coverage (ERT). We show experimentally that the use of modern classifiers provides much higher statistical power than simple classifiers underlying established metrics like CovGap. Additionally, we use our metric to benchmark different conformal prediction methods. Finally, we release an open-source package for ERT as well as previous conditional coverage metrics. Together, these contributions provide a new lens for understanding, diagnosing, and improving the conditional reliability of predictive systems. △ Less

Submitted 29 May, 2026; v1 submitted 12 December, 2025; originally announced December 2025.

Structured Matrix Scaling for Multi-Class Calibration

Authors: Eugène Berta, David Holzmüller, Michael I. Jordan, Francis Bach

Abstract: Post-hoc recalibration methods are widely used to ensure that classifiers provide faithful probability estimates. We argue that parametric recalibration functions based on logistic regression can be motivated from a simple theoretical setting for both binary and multiclass classification. This insight motivates the use of more expressive calibration methods beyond standard temperature scaling. For… ▽ More Post-hoc recalibration methods are widely used to ensure that classifiers provide faithful probability estimates. We argue that parametric recalibration functions based on logistic regression can be motivated from a simple theoretical setting for both binary and multiclass classification. This insight motivates the use of more expressive calibration methods beyond standard temperature scaling. For multi-class calibration however, a key challenge lies in the increasing number of parameters introduced by more complex models, often coupled with limited calibration data, which can lead to overfitting. Through extensive experiments, we demonstrate that the resulting bias-variance tradeoff can be effectively managed by structured regularization, robust preprocessing and efficient optimization. The resulting methods lead to substantial gains over existing logistic-based calibration techniques. We provide efficient and easy-to-use open-source implementations of our methods, making them an attractive alternative to common temperature, vector, and matrix scaling implementations. △ Less

Submitted 10 March, 2026; v1 submitted 5 November, 2025; originally announced November 2025.

Scalable Utility-Aware Multiclass Calibration

Authors: Mahmoud Hegazy, Michael I. Jordan, Aymeric Dieuleveut

Abstract: Ensuring that classifiers are well-calibrated, i.e., their predictions align with observed frequencies, is a minimal and fundamental requirement for classifiers to be viewed as trustworthy. Existing methods for assessing multiclass calibration often focus on specific aspects associated with prediction (e.g., top-class confidence, class-wise calibration) or utilize computationally challenging varia… ▽ More Ensuring that classifiers are well-calibrated, i.e., their predictions align with observed frequencies, is a minimal and fundamental requirement for classifiers to be viewed as trustworthy. Existing methods for assessing multiclass calibration often focus on specific aspects associated with prediction (e.g., top-class confidence, class-wise calibration) or utilize computationally challenging variational formulations. In this work, we study scalable \emph{evaluation} of multiclass calibration. To this end, we propose utility calibration, a general framework that measures the calibration error relative to a specific utility function that encapsulates the goals or decision criteria relevant to the end user. We demonstrate how this framework can unify and re-interpret several existing calibration metrics, particularly allowing for more robust versions of the top-class and class-wise calibration metrics, and, going beyond such binarized approaches, toward assessing calibration for richer classes of downstream utilities. △ Less

Submitted 29 October, 2025; originally announced October 2025.

Adaptive Coverage Policies in Conformal Prediction

Authors: Etienne Gauthier, Francis Bach, Michael I. Jordan

Abstract: Traditional conformal prediction methods construct prediction sets such that the true label falls within the set with a user-specified coverage level. However, poorly chosen coverage levels can result in uninformative predictions, either producing overly conservative sets when the coverage level is too high, or empty sets when it is too low. Moreover, the fixed coverage level cannot adapt to the s… ▽ More Traditional conformal prediction methods construct prediction sets such that the true label falls within the set with a user-specified coverage level. However, poorly chosen coverage levels can result in uninformative predictions, either producing overly conservative sets when the coverage level is too high, or empty sets when it is too low. Moreover, the fixed coverage level cannot adapt to the specific characteristics of each individual example, limiting the flexibility and efficiency of these methods. In this work, we leverage recent advances in e-values and post-hoc conformal inference, which allow the use of data-dependent coverage levels while maintaining valid statistical guarantees. We propose to optimize an adaptive coverage policy by training a neural network using a leave-one-out procedure on the calibration set, allowing the coverage level and the resulting prediction set size to vary with the difficulty of each individual example. We support our approach with theoretical coverage guarantees and demonstrate its practical benefits through a series of experiments. △ Less

Submitted 2 April, 2026; v1 submitted 5 October, 2025; originally announced October 2025.

Comments: Code at: https://github.com/GauthierE/adaptive-coverage-policies

A Compositional Kernel Model for Feature Learning

Authors: Feng Ruan, Keli Liu, Michael Jordan

Abstract: We study a compositional variant of kernel ridge regression in which the predictor is applied to a coordinate-wise reweighting of the inputs. Formulated as a variational problem, this model provides a simple testbed for feature learning in compositional architectures. From the perspective of variable selection, we show how relevant variables are recovered while noise variables are eliminated. We e… ▽ More We study a compositional variant of kernel ridge regression in which the predictor is applied to a coordinate-wise reweighting of the inputs. Formulated as a variational problem, this model provides a simple testbed for feature learning in compositional architectures. From the perspective of variable selection, we show how relevant variables are recovered while noise variables are eliminated. We establish guarantees showing that both global minimizers and stationary points discard noise coordinates when the noise variables are Gaussian distributed. A central finding is that $\ell_1$-type kernels, such as the Laplace kernel, succeed in recovering features contributing to nonlinear effects at stationary points, whereas Gaussian kernels recover only linear ones. △ Less

Submitted 3 November, 2025; v1 submitted 17 September, 2025; originally announced September 2025.

Comments: Fix Typos

OLMoASR: Open Models and Data for Training Robust Speech Recognition Models

Authors: Huong Ngo, Matt Deitke, Martijn Bartelds, Sarah Pratt, Josh Gardner, Matt Jordan, Ludwig Schmidt

Abstract: Improvements in training data scale and quality have led to significant advances, yet its influence in speech recognition remains underexplored. In this paper, we present a large-scale dataset, OLMoASR-Pool, and series of models, OLMoASR, to study and develop robust zero-shot speech recognition models. Beginning from OLMoASR-Pool, a collection of 3M hours of English audio and 17M transcripts, we d… ▽ More Improvements in training data scale and quality have led to significant advances, yet its influence in speech recognition remains underexplored. In this paper, we present a large-scale dataset, OLMoASR-Pool, and series of models, OLMoASR, to study and develop robust zero-shot speech recognition models. Beginning from OLMoASR-Pool, a collection of 3M hours of English audio and 17M transcripts, we design text heuristic filters to remove low-quality or mistranscribed data. Our curation pipeline produces a new dataset containing 1M hours of high-quality audio-transcript pairs, which we call OLMoASR-Mix. We use OLMoASR-Mix to train the OLMoASR-Mix suite of models, ranging from 39M (tiny.en) to 1.5B (large.en) parameters. Across all model scales, OLMoASR achieves comparable average performance to OpenAI's Whisper on short and long-form speech recognition benchmarks. Notably, OLMoASR-medium.en attains a 12.8\% and 11.0\% word error rate (WER) that is on par with Whisper's largest English-only model Whisper-medium.en's 12.4\% and 10.5\% WER for short and long-form recognition respectively (at equivalent parameter count). OLMoASR-Pool, OLMoASR models, and filtering, training and evaluation code will be made publicly available to further research on robust speech processing. △ Less

Submitted 28 August, 2025; originally announced August 2025.

Comments: 17 pages, 7 figures

The Statistical Fairness-Accuracy Frontier

Authors: Alireza Fallah, Michael I. Jordan, Annie Ulichney

Abstract: We study fairness-accuracy tradeoffs when a single predictive model must serve multiple demographic groups. A useful tool for understanding this tradeoff is the fairness-accuracy (FA) Pareto frontier, which characterizes the set of models that cannot be improved in either fairness or accuracy without worsening the other. While characterizing the FA frontier requires full knowledge of the data dist… ▽ More We study fairness-accuracy tradeoffs when a single predictive model must serve multiple demographic groups. A useful tool for understanding this tradeoff is the fairness-accuracy (FA) Pareto frontier, which characterizes the set of models that cannot be improved in either fairness or accuracy without worsening the other. While characterizing the FA frontier requires full knowledge of the data distribution, we focus on the finite-sample regime, quantifying how well a designer can approximate any point on the frontier from limited data and bounding the worst-case gap. In particular, we derive worst-case-optimal estimators that depend on the designer's knowledge of the covariate distribution. For each estimator, we characterize how finite-sample effects asymmetrically impact each group's welfare and identify optimal sample allocation strategies. Finally, we provide uniform finite-sample bounds for the entire FA frontier, yielding confidence bands that quantify the reliability of welfare comparisons across alternative fairness-accuracy tradeoffs. △ Less

Submitted 16 February, 2026; v1 submitted 24 August, 2025; originally announced August 2025.

Multivariate Standardized Residuals for Conformal Prediction

Authors: Sacha Braun, Eugène Berta, Michael I. Jordan, Francis Bach

Abstract: While split conformal prediction guarantees marginal coverage, approaching the stronger property of conditional coverage is essential for reliable uncertainty quantification. Naive conformal scores, however, suffer from poor conditional coverage in heteroskedastic settings. In univariate regression, this is commonly addressed by normalizing non-conformity scores using an estimated local score vari… ▽ More While split conformal prediction guarantees marginal coverage, approaching the stronger property of conditional coverage is essential for reliable uncertainty quantification. Naive conformal scores, however, suffer from poor conditional coverage in heteroskedastic settings. In univariate regression, this is commonly addressed by normalizing non-conformity scores using an estimated local score variance. In this work, we propose a natural extension of this normalization to the multivariate setting, effectively whitening the residuals to decouple output correlations and standardize local variance. Furthermore, we derive a sufficient condition characterizing a broad class of distributions for which standardized residuals yield asymptotic conditional coverage. We demonstrate that using the Mahalanobis distance induced by a learned local covariance as a non-conformity score provides a closed-form, computationally efficient mechanism for capturing inter-output correlations and heteroskedasticity, avoiding the expensive sampling required by previous methods based on cumulative distribution functions. This structure unlocks several practical extensions, including the handling of missing output values, the refinement of conformal sets when partial information is revealed, and the construction of valid conformal sets for transformations of the output. Finally, we provide extensive empirical evidence on both synthetic and real-world datasets showing that our approach yields conformal sets that improve upon the conditional coverage of existing multivariate baselines. △ Less

Submitted 7 May, 2026; v1 submitted 28 July, 2025; originally announced July 2025.

A General Framework for Estimating Preferences Using Response Time Data

Authors: Federico Echenique, Alireza Fallah, Michael I. Jordan

Abstract: We propose a general methodology for recovering preference parameters from data on choices and response times. Our methods yield estimates with fast ($1/n$ for $n$ data points) convergence rates when specialized to the popular Drift Diffusion Model (DDM), but are broadly applicable to generalizations of the DDM as well as to alternative models of decision making that make use of response time data… ▽ More We propose a general methodology for recovering preference parameters from data on choices and response times. Our methods yield estimates with fast ($1/n$ for $n$ data points) convergence rates when specialized to the popular Drift Diffusion Model (DDM), but are broadly applicable to generalizations of the DDM as well as to alternative models of decision making that make use of response time data. The paper develops an empirical application to an experiment on intertemporal choice, showing that the use of response times delivers predictive accuracy and matters for the estimation of economically relevant parameters. △ Less

Submitted 31 July, 2025; v1 submitted 27 July, 2025; originally announced July 2025.

A Collectivist, Economic Perspective on AI

Authors: Michael I. Jordan

Abstract: Information technology is in the midst of a revolution in which omnipresent data collection and machine learning are impacting the human world as never before. The word ``intelligence'' is being used as a North Star for the development of this technology, with human cognition viewed as a baseline. This view neglects the fact that humans are social animals and that much of our intelligence is socia… ▽ More Information technology is in the midst of a revolution in which omnipresent data collection and machine learning are impacting the human world as never before. The word ``intelligence'' is being used as a North Star for the development of this technology, with human cognition viewed as a baseline. This view neglects the fact that humans are social animals and that much of our intelligence is social and cultural in origin. Moreover, failing to properly situate aspects of intelligence at the social level contributes to the treatment of the societal consequences of technology as an afterthought. The path forward is not merely more data and compute, and not merely more attention paid to cognitive or symbolic representations, but a thorough blending of economic and social concepts with computational and inferential concepts at the level of algorithm design. △ Less

Submitted 15 December, 2025; v1 submitted 7 July, 2025; originally announced July 2025.

Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilities

Authors: Gheorghe Comanici, Eric Bieber, Mike Schaekermann, Ice Pasupat, Noveen Sachdeva, Inderjit Dhillon, Marcel Blistein, Ori Ram, Dan Zhang, Evan Rosen, Luke Marris, Sam Petulla, Colin Gaffney, Asaf Aharoni, Nathan Lintz, Tiago Cardal Pais, Henrik Jacobsson, Idan Szpektor, Nan-Jiang Jiang, Krishna Haridasan, Ahmed Omran, Nikunj Saunshi, Dara Bahri, Gaurav Mishra, Eric Chu , et al. (3410 additional authors not shown)

Abstract: In this report, we introduce the Gemini 2.X model family: Gemini 2.5 Pro and Gemini 2.5 Flash, as well as our earlier Gemini 2.0 Flash and Flash-Lite models. Gemini 2.5 Pro is our most capable model yet, achieving SoTA performance on frontier coding and reasoning benchmarks. In addition to its incredible coding and reasoning skills, Gemini 2.5 Pro is a thinking model that excels at multimodal unde… ▽ More In this report, we introduce the Gemini 2.X model family: Gemini 2.5 Pro and Gemini 2.5 Flash, as well as our earlier Gemini 2.0 Flash and Flash-Lite models. Gemini 2.5 Pro is our most capable model yet, achieving SoTA performance on frontier coding and reasoning benchmarks. In addition to its incredible coding and reasoning skills, Gemini 2.5 Pro is a thinking model that excels at multimodal understanding and it is now able to process up to 3 hours of video content. Its unique combination of long context, multimodal and reasoning capabilities can be combined to unlock new agentic workflows. Gemini 2.5 Flash provides excellent reasoning abilities at a fraction of the compute and latency requirements and Gemini 2.0 Flash and Flash-Lite provide high performance at low latency and cost. Taken together, the Gemini 2.X model generation spans the full Pareto frontier of model capability vs cost, allowing users to explore the boundaries of what is possible with complex agentic problem solving. △ Less

Submitted 19 December, 2025; v1 submitted 7 July, 2025; originally announced July 2025.

Comments: 72 pages, 17 figures

Valid Selection among Conformal Sets

Authors: Mahmoud Hegazy, Liviu Aolaritei, Michael I. Jordan, Aymeric Dieuleveut

Abstract: Conformal prediction offers a distribution-free framework for constructing prediction sets with coverage guarantees. In practice, multiple valid conformal prediction sets may be available, arising from different models or methodologies. However, selecting the most desirable set, such as the smallest, can invalidate the coverage guarantees. To address this challenge, we propose a stability-based ap… ▽ More Conformal prediction offers a distribution-free framework for constructing prediction sets with coverage guarantees. In practice, multiple valid conformal prediction sets may be available, arising from different models or methodologies. However, selecting the most desirable set, such as the smallest, can invalidate the coverage guarantees. To address this challenge, we propose a stability-based approach that ensures coverage for the selected prediction set. We extend our results to the online conformal setting, propose several refinements in settings where additional structure is available, and demonstrate its effectiveness through experiments. △ Less

Submitted 25 June, 2025; originally announced June 2025.

Imaging at the quantum limit with convolutional neural networks

Authors: Andrew H. Proppe, Aaron Z. Goldberg, Guillaume Thekkadath, Noah Lupu-Gladstein, Kyle M. Jordan, Philip J. Bustard, Frédéric Bouchard, Duncan England, Khabat Heshami, Jeff S. Lundeen, Benjamin J. Sussman

Abstract: Deep neural networks have been shown to achieve exceptional performance for computer vision tasks like image recognition, segmentation, and reconstruction or denoising. Here, we evaluate the ultimate performance limits of deep convolutional neural network models for image reconstruction, by comparing them against the standard quantum limit set by shot-noise and the Heisenberg limit on precision. W… ▽ More Deep neural networks have been shown to achieve exceptional performance for computer vision tasks like image recognition, segmentation, and reconstruction or denoising. Here, we evaluate the ultimate performance limits of deep convolutional neural network models for image reconstruction, by comparing them against the standard quantum limit set by shot-noise and the Heisenberg limit on precision. We train U-Net models on images of natural objects illuminated with coherent states of light, and find that the average mean-squared error of the reconstructions can surpass the standard quantum limit, and in some cases reaches the Heisenberg limit. Further, we train models on well-parameterized images for which we can calculate the quantum Cramér-Rao bound to determine the minimum possible measurable variance of an estimated parameter for a given probe state. We find the mean-squared error of the model predictions reaches these bounds calculated for the parameters, across a variety of parameterized images. These results suggest that deep convolutional neural networks can learn to become the optimal estimators allowed by the laws of physics, performing parameter estimation and image reconstruction at the ultimate possible limits of precision for the case of classical illumination of the object. △ Less

Submitted 16 June, 2025; originally announced June 2025.

Generalization or Hallucination? Understanding Out-of-Context Reasoning in Transformers

Authors: Yixiao Huang, Hanlin Zhu, Tianyu Guo, Jiantao Jiao, Somayeh Sojoudi, Michael I. Jordan, Stuart Russell, Song Mei

Abstract: Large language models (LLMs) can acquire new knowledge through fine-tuning, but this process exhibits a puzzling duality: models can generalize remarkably from new facts, yet are also prone to hallucinating incorrect information. However, the reasons for this phenomenon remain poorly understood. In this work, we argue that both behaviors stem from a single mechanism known as out-of-context reasoni… ▽ More Large language models (LLMs) can acquire new knowledge through fine-tuning, but this process exhibits a puzzling duality: models can generalize remarkably from new facts, yet are also prone to hallucinating incorrect information. However, the reasons for this phenomenon remain poorly understood. In this work, we argue that both behaviors stem from a single mechanism known as out-of-context reasoning (OCR): the ability to deduce implications by associating concepts, even those without a causal link. Our experiments across five prominent LLMs confirm that OCR indeed drives both generalization and hallucination, depending on whether the associated concepts are causally related. To build a rigorous theoretical understanding of this phenomenon, we then formalize OCR as a synthetic factual recall task. We empirically show that a one-layer single-head attention-only transformer with factorized output and value matrices can learn to solve this task, while a model with combined weights cannot, highlighting the crucial role of matrix factorization. Our theoretical analysis shows that the OCR capability can be attributed to the implicit bias of gradient descent, which favors solutions that minimize the nuclear norm of the combined output-value matrix. This mathematical structure explains why the model learns to associate facts and implications with high sample efficiency, regardless of whether the correlation is causal or merely spurious. Ultimately, our work provides a theoretical foundation for understanding the OCR phenomenon, offering a new lens for analyzing and mitigating undesirable behaviors from knowledge injection. △ Less

Submitted 2 February, 2026; v1 submitted 12 June, 2025; originally announced June 2025.

Comments: NeurIPS 2025, first three authors contributed equally

Revisiting mean estimation over $\ell_p$ balls: Is the MLE optimal?

Authors: Liviu Aolaritei, Michael I. Jordan, Reese Pathak, Annie Ulichney

Abstract: We revisit the problem of mean estimation in the Gaussian sequence model with $\ell_p$ constraints for $p \in [0, \infty]$. We demonstrate two phenomena for the behavior of the maximum likelihood estimator (MLE), which depend on the noise level, the radius of the (quasi)norm constraint, the dimension, and the norm index $p$. First, if $p$ lies between $0$ and $1 + Θ(\tfrac{1}{\log d})$, inclusive,… ▽ More We revisit the problem of mean estimation in the Gaussian sequence model with $\ell_p$ constraints for $p \in [0, \infty]$. We demonstrate two phenomena for the behavior of the maximum likelihood estimator (MLE), which depend on the noise level, the radius of the (quasi)norm constraint, the dimension, and the norm index $p$. First, if $p$ lies between $0$ and $1 + Θ(\tfrac{1}{\log d})$, inclusive, or if it is greater than or equal to $2$, the MLE is minimax rate-optimal for all noise levels and all constraint radii. On the other hand, for the remaining norm indices -- namely, if $p$ lies between $1 + Θ(\tfrac{1}{\log d})$ and $2$ -- here is a more striking behavior: the MLE is minimax rate-suboptimal, despite its nonlinearity in the observations, for essentially all noise levels and constraint radii for which nonlinear estimates are necessary for minimax-optimal estimation. Our results imply that when given $n$ independent and identically distributed Gaussian samples, the MLE can be suboptimal by a polynomial factor in the sample size. Our lower bounds are constructive: whenever the MLE is rate-suboptimal, we provide explicit instances on which the MLE provably incurs suboptimal risk. Finally, in the non-convex case -- namely when $p < 1$ -- we develop sharp local Gaussian width bounds, which may be of independent interest. △ Less

Submitted 1 July, 2025; v1 submitted 12 June, 2025; originally announced June 2025.

Comments: 43 pages, 3 figures

Sample Complexity and Representation Ability of Test-time Scaling Paradigms

Authors: Baihe Huang, Shanda Li, Tianhao Wu, Yiming Yang, Ameet Talwalkar, Kannan Ramchandran, Michael I. Jordan, Jiantao Jiao

Abstract: Test-time scaling paradigms have significantly advanced the capabilities of large language models (LLMs) on complex tasks. Despite their empirical success, theoretical understanding of the sample efficiency of various test-time strategies -- such as self-consistency, best-of-$n$, and self-correction -- remains limited. In this work, we first establish a separation result between two repeated sampl… ▽ More Test-time scaling paradigms have significantly advanced the capabilities of large language models (LLMs) on complex tasks. Despite their empirical success, theoretical understanding of the sample efficiency of various test-time strategies -- such as self-consistency, best-of-$n$, and self-correction -- remains limited. In this work, we first establish a separation result between two repeated sampling strategies: self-consistency requires $Θ(1/Δ^2)$ samples to produce the correct answer, while best-of-$n$ only needs $Θ(1/Δ)$, where $Δ< 1$ denotes the probability gap between the correct and second most likely answers. Next, we present an expressiveness result for the self-correction approach with verifier feedback: it enables Transformers to simulate online learning over a pool of experts at test time. Therefore, a single Transformer architecture can provably solve multiple tasks without prior knowledge of the specific task associated with a user query, extending the representation theory of Transformers from single-task to multi-task settings. Finally, we empirically validate our theoretical results, demonstrating the practical effectiveness of self-correction methods. △ Less

Submitted 12 June, 2025; v1 submitted 5 June, 2025; originally announced June 2025.

IDA-Bench: Evaluating LLMs on Interactive Guided Data Analysis

Authors: Hanyu Li, Haoyu Liu, Tingyu Zhu, Tianyu Guo, Zeyu Zheng, Xiaotie Deng, Michael I. Jordan

Abstract: Large Language Models (LLMs) show promise as data analysis agents, but existing benchmarks overlook the iterative nature of the field, where experts' decisions evolve with deeper insights of the dataset. To address this, we introduce IDA-Bench, a novel benchmark evaluating LLM agents in multi-round interactive scenarios. Derived from complex Kaggle notebooks, tasks are presented as sequential natu… ▽ More Large Language Models (LLMs) show promise as data analysis agents, but existing benchmarks overlook the iterative nature of the field, where experts' decisions evolve with deeper insights of the dataset. To address this, we introduce IDA-Bench, a novel benchmark evaluating LLM agents in multi-round interactive scenarios. Derived from complex Kaggle notebooks, tasks are presented as sequential natural language instructions by an LLM-simulated user. Agent performance is judged by comparing its final numerical output to the human-derived baseline. Initial results show that even state-of-the-art coding agents (like Claude-3.7-thinking) succeed on < 50% of the tasks, highlighting limitations not evident in single-turn tests. This work underscores the need to improve LLMs' multi-round capabilities for building more reliable data analysis agents, highlighting the necessity of achieving a balance between instruction following and reasoning. △ Less

Submitted 6 June, 2025; v1 submitted 23 May, 2025; originally announced May 2025.

Backward Conformal Prediction

Authors: Etienne Gauthier, Francis Bach, Michael I. Jordan

Abstract: We introduce $\textit{Backward Conformal Prediction}$, a method that guarantees conformal coverage while providing flexible control over the size of prediction sets. Unlike standard conformal prediction, which fixes the coverage level and allows the conformal set size to vary, our approach defines a rule that constrains how prediction set sizes behave based on the observed data, and adapts the cov… ▽ More We introduce $\textit{Backward Conformal Prediction}$, a method that guarantees conformal coverage while providing flexible control over the size of prediction sets. Unlike standard conformal prediction, which fixes the coverage level and allows the conformal set size to vary, our approach defines a rule that constrains how prediction set sizes behave based on the observed data, and adapts the coverage level accordingly. Our method builds on two key foundations: (i) recent results by Gauthier et al. [2025] on post-hoc validity using e-values, which ensure marginal coverage of the form $\mathbb{P}(Y_{\rm test} \in \hat C_n^{\tildeα}(X_{\rm test})) \ge 1 - \mathbb{E}[\tildeα]$ up to a first-order Taylor approximation for any data-dependent miscoverage $\tildeα$, and (ii) a novel leave-one-out estimator $\hatα^{\rm LOO}$ of the marginal miscoverage $\mathbb{E}[\tildeα]$ based on the calibration set, ensuring that the theoretical guarantees remain computable in practice. This approach is particularly useful in applications where large prediction sets are impractical such as medical diagnosis. We provide theoretical results and empirical evidence supporting the validity of our method, demonstrating that it maintains computable coverage guarantees while ensuring interpretable, well-controlled prediction set sizes. △ Less

Submitted 12 February, 2026; v1 submitted 19 May, 2025; originally announced May 2025.

Comments: Code available at: https://github.com/GauthierE/backward-cp

Online Decision-Focused Learning

Authors: Aymeric Capitaine, Maxime Haddouche, Eric Moulines, Michael I. Jordan, Etienne Boursier, Alain Durmus

Abstract: Decision-focused learning (DFL) is an increasingly popular paradigm for training predictive models whose outputs are used in decision-making tasks. Instead of merely optimizing for predictive accuracy, DFL trains models to directly minimize the loss associated with downstream decisions. However, existing studies focus solely on scenarios where a fixed batch of data is available and the objective f… ▽ More Decision-focused learning (DFL) is an increasingly popular paradigm for training predictive models whose outputs are used in decision-making tasks. Instead of merely optimizing for predictive accuracy, DFL trains models to directly minimize the loss associated with downstream decisions. However, existing studies focus solely on scenarios where a fixed batch of data is available and the objective function does not change over time. We instead investigate DFL in dynamic environments where the objective function and data distribution evolve over time. This setting is challenging for online learning because the objective function has zero or undefined gradients, which prevents the use of standard first-order optimization methods, and is generally non-convex. To address these difficulties, we (i) regularize the objective to make it differentiable and (ii) use perturbation techniques along with a near-optimal oracle to overcome non-convexity. Combining those techniques yields two original online algorithms tailored for DFL, for which we establish respectively static and dynamic regret bounds. These are the first provable guarantees for the online decision-focused problem. Finally, we showcase the effectiveness of our algorithms on a knapsack experiment, where they outperform two standard benchmarks. △ Less

Submitted 7 March, 2026; v1 submitted 19 May, 2025; originally announced May 2025.

Understanding In-context Learning of Addition via Activation Subspaces

Authors: Xinyan Hu, Kayo Yin, Michael I. Jordan, Jacob Steinhardt, Lijie Chen

Abstract: To perform few-shot learning, language models extract signals from a few input-label pairs, aggregate these into a learned prediction rule, and apply this rule to new inputs. How is this implemented in the forward pass of modern transformer models? To explore this question, we study a structured family of few-shot learning tasks for which the true prediction rule is to add an integer $k$ to the in… ▽ More To perform few-shot learning, language models extract signals from a few input-label pairs, aggregate these into a learned prediction rule, and apply this rule to new inputs. How is this implemented in the forward pass of modern transformer models? To explore this question, we study a structured family of few-shot learning tasks for which the true prediction rule is to add an integer $k$ to the input. We introduce a novel optimization method that localizes the model's few-shot ability to only a few attention heads. We then perform an in-depth analysis of individual heads, via dimensionality reduction and decomposition. As an example, on Llama-3-8B-instruct, we reduce its mechanism on our tasks to just three attention heads with six-dimensional subspaces, where four dimensions track the unit digit with trigonometric functions at periods $2$, $5$, and $10$, and two dimensions track magnitude with low-frequency components. To deepen our understanding of the mechanism, we also derive a mathematical identity relating ``aggregation'' and ``extraction'' subspaces for attention heads, allowing us to track the flow of information from individual examples to a final aggregated concept. Using this, we identify a self-correction mechanism where mistakes learned from earlier demonstrations are suppressed by later demonstrations. Our results demonstrate how tracking low-dimensional subspaces of localized heads across a forward pass can provide insight into fine-grained computational structures in language models. △ Less

Submitted 9 October, 2025; v1 submitted 8 May, 2025; originally announced May 2025.

Stochastic Optimization with Optimal Importance Sampling

Authors: Liviu Aolaritei, Bart P. G. Van Parys, Henry Lam, Michael I. Jordan

Abstract: Importance Sampling (IS) is a widely used variance reduction technique for enhancing the efficiency of Monte Carlo methods, particularly in rare-event simulation and related applications. Despite its effectiveness, the performance of IS is highly sensitive to the choice of the proposal distribution and often requires stochastic calibration. While the design and analysis of IS have been extensively… ▽ More Importance Sampling (IS) is a widely used variance reduction technique for enhancing the efficiency of Monte Carlo methods, particularly in rare-event simulation and related applications. Despite its effectiveness, the performance of IS is highly sensitive to the choice of the proposal distribution and often requires stochastic calibration. While the design and analysis of IS have been extensively studied in estimation settings, applying IS within stochastic optimization introduces a lesser-known fundamental challenge: the decision variable and the importance sampling distribution are mutually dependent, creating a circular optimization structure. This interdependence complicates both convergence analysis and variance control. In this paper, we consider the generic setting of convex stochastic optimization with linear constraints. We propose a single-loop stochastic approximation algorithm, based on a variant of Nesterov's dual averaging, that jointly updates the decision variable and the importance sampling distribution, notably without time-scale separation or nested optimization. The method is globally convergent and achieves the minimal asymptotic variance among stochastic gradient schemes, which moreover matches the performance of an oracle sampler adapted to the optimal solution and thus effectively resolves the circular optimization challenge. △ Less

Submitted 10 February, 2026; v1 submitted 4 April, 2025; originally announced April 2025.

Minimum Volume Conformal Sets for Multivariate Regression

Authors: Sacha Braun, Liviu Aolaritei, Michael I. Jordan, Francis Bach

Abstract: Conformal prediction provides a principled framework for constructing predictive sets with finite-sample validity. While much of the focus has been on univariate response variables, existing multivariate methods either impose rigid geometric assumptions or rely on flexible but computationally expensive approaches that do not explicitly optimize prediction set volume. We propose an optimization-dri… ▽ More Conformal prediction provides a principled framework for constructing predictive sets with finite-sample validity. While much of the focus has been on univariate response variables, existing multivariate methods either impose rigid geometric assumptions or rely on flexible but computationally expensive approaches that do not explicitly optimize prediction set volume. We propose an optimization-driven framework based on a novel loss function that directly learns minimum-volume covering sets while ensuring valid coverage. This formulation naturally induces a new nonconformity score for conformal prediction, which adapts to the residual distribution and covariates. Our approach optimizes over prediction sets defined by arbitrary norm balls, including single and multi-norm formulations. Additionally, by jointly optimizing both the predictive model and predictive uncertainty, we obtain prediction sets that are tight, informative, and computationally efficient, as demonstrated in our experiments on real-world datasets. △ Less

Submitted 18 March, 2026; v1 submitted 24 March, 2025; originally announced March 2025.

E-Values Expand the Scope of Conformal Prediction

Authors: Etienne Gauthier, Francis Bach, Michael I. Jordan

Abstract: Conformal prediction is a powerful framework for distribution-free uncertainty quantification. The standard approach to conformal prediction relies on comparing the ranks of prediction scores: under exchangeability, the rank of a future test point cannot be too extreme relative to a calibration set. This rank-based method can be reformulated in terms of p-values. In this paper, we explore an alter… ▽ More Conformal prediction is a powerful framework for distribution-free uncertainty quantification. The standard approach to conformal prediction relies on comparing the ranks of prediction scores: under exchangeability, the rank of a future test point cannot be too extreme relative to a calibration set. This rank-based method can be reformulated in terms of p-values. In this paper, we explore an alternative approach based on e-values, known as conformal e-prediction. E-values offer key advantages that cannot be achieved with p-values, enabling new theoretical and practical capabilities. In particular, we present three applications that leverage the unique strengths of e-values: batch anytime-valid conformal prediction, fixed-size conformal sets with data-dependent coverage, and conformal prediction under ambiguous ground truth. Overall, these examples demonstrate that e-value-based constructions provide a flexible expansion of the toolbox of conformal prediction. △ Less

Submitted 6 May, 2025; v1 submitted 17 March, 2025; originally announced March 2025.

Comments: Code available at: https://github.com/GauthierE/evalues-expand-cp

Datasets, Documents, and Repetitions: The Practicalities of Unequal Data Quality

Authors: Alex Fang, Hadi Pouransari, Matt Jordan, Alexander Toshev, Vaishaal Shankar, Ludwig Schmidt, Tom Gunter

Abstract: Data filtering has become a powerful tool for improving model performance while reducing computational cost. However, as large language model compute budgets continue to grow, the limited data volume provided by heavily filtered and deduplicated datasets will become a practical constraint. In efforts to better understand how to proceed, we study model performance at various compute budgets and acr… ▽ More Data filtering has become a powerful tool for improving model performance while reducing computational cost. However, as large language model compute budgets continue to grow, the limited data volume provided by heavily filtered and deduplicated datasets will become a practical constraint. In efforts to better understand how to proceed, we study model performance at various compute budgets and across multiple pre-training datasets created through data filtering and deduplication. We find that, given appropriate modifications to the training recipe, repeating existing aggressively filtered datasets for up to ten epochs can outperform training on the ten times larger superset for a single epoch across multiple compute budget orders of magnitude. While this finding relies on repeating the dataset for many epochs, we also investigate repeats within these datasets at the document level. We find that not all documents within a dataset are equal, and we can create better datasets relative to a token budget by explicitly manipulating the counts of individual documents. We conclude by arguing that even as large language models scale, data filtering remains an important direction of research. △ Less

Submitted 6 November, 2025; v1 submitted 10 March, 2025; originally announced March 2025.

Marketplace Operators Can Induce Competitive Pricing

Authors: Tiffany Ding, Dominique Perrault-Joncas, Orit Ronen, Michael I. Jordan, Dirk Bergemann, Dean Foster, Omer Gottesman

Abstract: As e-commerce marketplaces continue to grow in popularity, it has become increasingly important to understand the role and impact of marketplace operators on competition and social welfare. We model a marketplace operator as an entity that not only facilitates third-party sales but can also choose to directly participate in the market as a competing seller. We formalize this market structure as a… ▽ More As e-commerce marketplaces continue to grow in popularity, it has become increasingly important to understand the role and impact of marketplace operators on competition and social welfare. We model a marketplace operator as an entity that not only facilitates third-party sales but can also choose to directly participate in the market as a competing seller. We formalize this market structure as a price-quantity Stackelberg duopoly in which the leader is a marketplace operator and the follower is an independent seller who shares a fraction of their revenue with the marketplace operator for the privilege of selling on the platform. The objective of the marketplace operator is to maximize a weighted sum of profit and a term capturing positive customer experience, whereas the independent seller seeks solely to maximize their own profit. We derive the subgame-perfect Nash equilibrium and find that it is often optimal for the marketplace operator to induce competition by offering the product at a low price to incentivize the independent seller to match their price. △ Less

Submitted 22 October, 2025; v1 submitted 9 March, 2025; originally announced March 2025.