arXiv subjects

Michael I. Jordan

Publications and source records attributed to Michael I. Jordan.

At least 55 records · Page 3Linked to original sources

arXiv2026

The Sample Complexity of Online Reinforcement Learning: A Multi-model Perspective

We study the sample complexity of online reinforcement learning in the general \hzyrev{non-episodic} setting of nonlinear dynamical systems with continuous state and action spaces. Our analysis accommodates a large class of dynamical systems ranging from a finite set of nonlinear candidate models to models with bounded and Lipschitz continuous dynamics, to systems that are parametrized by a compact and real-valued set of parameters. In the most general setting, our algorithm achieves a policy regret of $\mathcal{O}(N ε^2 + d_\mathrm{u}\mathrm{ln}(m(ε))/ε^2)$, where $N$ is the time horizon, $ε$ is a user-specified discretization width, $d_\mathrm{u}$ the input dimension, and $m(ε)$ measures the complexity of the function class under consideration via its packing number. In the special case where the dynamics are parametrized by a compact and real-valued set of parameters (such as neural networks, transformers, etc.), we prove a policy regret of $\mathcal{O}(\sqrt{d_\mathrm{u}N p})$, where $p$ denotes the number of parameters, recovering earlier sample-complexity results that were derived for linear time-invariant dynamical systems. While this article focuses on characterizing sample complexity, the proposed algorithms are likely to be useful in practice, due to their simplicity, their ability to incorporate prior knowledge, and their benign transient behaviors.

cs.LG↗

arXiv2026

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

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.

math.OC↗

arXiv2026

Towards Anytime-Valid Statistical Watermarking

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.

cs.LG↗

arXiv2026

Signaling in Data Markets via Free Samples

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.

cs.GT↗

arXiv2026

The Statistical Fairness-Accuracy Frontier

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.

stat.ML↗

arXiv2026

Locally Adaptive Multi-Objective Learning

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.

cs.LG↗

arXiv2026

Backward Conformal Prediction

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.

stat.ML↗

arXiv2026

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

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.

cs.LG↗

arXiv2026

Stochastic Optimization with Optimal Importance Sampling

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.

math.OC↗

arXiv2026

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

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.

cs.CL↗

arXiv2026

On Nonasymptotic Confidence Intervals for Treatment Effects in Randomized Experiments

We study nonasymptotic (finite-sample) confidence intervals for treatment effects in randomized experiments. In the existing literature, the effective sample sizes of nonasymptotic confidence intervals tend to be looser than the corresponding central-limit-theorem-based confidence intervals by a factor depending on the square root of the propensity score. We show that this performance gap can be closed, designing nonasymptotic confidence intervals that have the same effective sample size as their asymptotic counterparts. Our approach involves systematic exploitation of negative dependence or variance adaptivity (or both). We also show that the nonasymptotic rates that we achieve are unimprovable in an information-theoretic sense.

stat.ME↗

arXiv2025

A Collectivist, Economic Perspective on AI

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.

cs.CY↗

arXiv2025

On Three-Layer Data Markets

We study a three-layer data market comprising users (data owners), platforms, and a data buyer. Each user benefits from platform services in exchange for data, incurring privacy loss when their data, albeit noisily, is shared with the buyer. The user chooses platforms to share data with, while platforms decide on data noise levels and pricing before selling to the buyer. The buyer selects platforms to purchase data from. We model these interactions via a multi-stage game, focusing on the subgame Nash equilibrium. We find that when the buyer places a high value on user data (and platforms can command high prices), all platforms offer services to the user who joins and shares data with every platform. Conversely, when the buyer's valuation of user data is low, only large platforms with low service costs can afford to serve users. In this scenario, users exclusively join and share data with these low-cost platforms. Interestingly, increased competition benefits the buyer, not the user: as the number of platforms increases, the user utility does not necessarily improve while the buyer utility improves. However, increasing the competition improves the overall utilitarian welfare. Building on our analysis, we then study regulations to improve the user utility. We discover that banning data sharing maximizes user utility only when all platforms are low-cost. In mixed markets of high- and low-cost platforms, users prefer a minimum noise mandate over a sharing ban. Imposing this mandate on high-cost platforms and banning data sharing for low-cost ones further enhances user utility.

econ.TH↗

arXiv2025

Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computation

Equilibrium computation on Riemannian manifolds provides a unifying framework for numerous problems in machine learning and data analytics. One of the simplest yet most fundamental methods is Riemannian gradient descent (RGD). While its Euclidean counterpart has been extensively studied, it remains unclear how the manifold curvature affects RGD in game-theoretic settings. This paper addresses this gap by establishing new convergence results for \textit{geodesic strongly monotone} games. Our key result shows that RGD attains last-iterate linear convergence in a \textit{geometry-agnostic} fashion, a key property for applications in machine learning. We extend this guarantee to stochastic and adaptive variants -- SRGD and FARGD -- and establish that: (i) the sample complexity of SRGD is geometry-agnostic and optimal with respect to noise; (ii) FARGD matches the convergence rate of its non-adaptive counterpart up to constant factors, while avoiding reliance on the condition number. Overall, this paper presents the first geometry-agnostic last-iterate convergence analysis for games beyond the Euclidean settings, underscoring the surprising power of RGD -- despite its simplicity -- in solving a wide spectrum of machine learning problems.

math.OC↗

arXiv2025

Cross-Validated Causal Inference: a Modern Method to Combine Experimental and Observational Data

We develop new methods to integrate experimental and observational data in causal inference. While randomized controlled trials offer strong internal validity, they are often costly and therefore limited in sample size. Observational data, though cheaper and often with larger sample sizes, are prone to biases due to unmeasured confounders. To harness their complementary strengths, we propose a systematic framework that formulates causal estimation as an empirical risk minimization (ERM) problem. A full model containing the causal parameter is obtained by minimizing a weighted combination of experimental and observational losses--capturing the causal parameter's validity and the full model's fit, respectively. The weight is chosen through cross-validation on the causal parameter across experimental folds. Our experiments on real and synthetic data show the efficacy and reliability of our method. We also provide theoretical non-asymptotic error bounds.

econ.EM↗

arXiv2025

Scalable Utility-Aware Multiclass Calibration

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.

cs.LG↗

arXiv2025

Online Decision-Making in Tree-Like Multi-Agent Games with Transfers

The widespread deployment of Machine Learning systems everywhere raises challenges, such as dealing with interactions or competition between multiple learners. In that goal, we study multi-agent sequential decision-making by considering principal-agent interactions in a tree structure. In this problem, the reward of a player is influenced by the actions of her children, who are all self-interested and non-cooperative, hence the complexity of making good decisions. Our main finding is that it is possible to steer all the players towards the globally optimal set of actions by simply allowing single-step transfers between them. A transfer is established between a principal and one of her agents: the principal actually offers the proposed payment if the agent picks the recommended action. The analysis poses specific challenges due to the intricate interactions between the nodes of the tree and the propagation of the regret within this tree. Considering a bandit setup, we propose algorithmic solutions for the players to end up being no-regret with respect to the optimal pair of actions and incentives. In the long run, allowing transfers between players makes them act as if they were collaborating together, although they remain self-interested non-cooperative: transfers restore efficiency.

cs.GT↗

arXiv2025

Marketplace Operators Can Induce Competitive Pricing

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.

econ.TH↗