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Michael I. Jordan

Publications and source records attributed to Michael I. Jordan.

At least 37 records · Page 2Linked to original sources
arXiv2026

Explicit Second-Order Min-Max Optimization: Practical Algorithms and Complexity Analysis

We propose and analyze several inexact regularized Newton-type methods for finding a global saddle point of convex-concave unconstrained min-max optimization problems. Compared to first-order methods, our understanding of second-order methods for min-max optimization is relatively limited, as obtaining global rates of convergence with second-order information can be much more involved. In this paper, we examine how second-order information is used to speed up extra-gradient methods, even under inexactness. In particular, we show that the proposed methods generate iterates that remain within a bounded set and that the averaged iterates converge to an $ε$-saddle point within $O(ε^{-2/3})$ iterations in terms of a restricted gap function. We also provide a simple routine for solving the subproblem at each iteration, requiring a single Schur decomposition and $O(\log\log(1/ε))$ calls to a linear system solver in a quasi-upper-triangular system. Thus, our method improves the existing line-search-based second-order min-max optimization methods by shaving off an $O(\log\log(1/ε))$ factor in the required number of Schur decompositions. Finally, we evaluate our method on both synthetic benchmarks and a real-world application arising from AUC maximization on standard LIBSVM datasets, and find that the proposed second-order approach delivers stronger practical efficiency than representative first-order methods on these problems.

math.OC↗
arXiv2026

AutoEval Done Right: Using Synthetic Data for Model Evaluation

The evaluation of machine learning models using human-labeled validation data can be expensive and time-consuming. AI-labeled synthetic data can be used to decrease the number of human annotations required for this purpose in a process called autoevaluation. We suggest efficient and statistically principled algorithms for this purpose that improve sample efficiency while remaining unbiased. These algorithms increase the effective human-labeled sample size by up to 50% on experiments with GPT-4.

cs.LG↗
arXiv2026

CalArena: A Large-Scale Post-Hoc Calibration Benchmark

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.

cs.LG↗
arXiv2026

Conditional Coverage Diagnostics for Conformal Prediction

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.

stat.ML↗
arXiv2026

Anytime Detection of Strategic Deviations in Multi-Agent Systems

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.

cs.GT↗
arXiv2026

Multivariate Standardized Residuals for Conformal Prediction

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.

stat.ML↗
arXiv2026

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

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.

stat.ML↗
arXiv2026

Response Time Enhances Alignment with Heterogeneous Preferences

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.

cs.LG↗
arXiv2026

Explaining and Preventing Alignment Collapse in Iterative RLHF

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.

cs.LG↗
arXiv2026

Adaptive Coverage Policies in Conformal Prediction

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.

stat.ML↗
arXiv2026

Calibeating Made Simple

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.

cs.LG↗
arXiv2026

Minimum Volume Conformal Sets for Multivariate Regression

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.

stat.ML↗
arXiv2026

Data-Driven Influence Functions for Optimization-Based Causal Inference

We study a constructive algorithm that approximates Gateaux derivatives for statistical functionals by finite differencing, with a focus on functionals that arise in causal inference. We study the case where probability distributions are not known a priori but need to be estimated from data. These estimated distributions lead to empirical Gateaux derivatives, and we study the relationships between empirical, numerical, and analytical Gateaux derivatives. Starting with a case study of the interventional mean (average potential outcome), we delineate the relationship between finite differences and the analytical Gateaux derivative. We then derive requirements on the rates of numerical approximation in perturbation and smoothing that preserve the statistical benefits of one-step adjustments, such as rate double robustness. We then study more complicated functionals such as dynamic treatment regimes, the linear-programming formulation for policy optimization in infinite-horizon Markov decision processes, and sensitivity analysis in causal inference. More broadly, we study optimization-based estimators, since this begets a class of estimands where identification via regression adjustment is straightforward but obtaining influence functions under minor variations thereof is not. The ability to approximate bias adjustments in the presence of arbitrary constraints illustrates the usefulness of constructive approaches for Gateaux derivatives. We also find that the statistical structure of the functional (rate double robustness) can permit less conservative rates for finite-difference approximation. This property, however, can be specific to particular functionals; e.g., it occurs for the average potential outcome (hence average treatment effect) but not the infinite-horizon MDP policy value.

stat.ME↗
arXiv2026

Structured Matrix Scaling for Multi-Class Calibration

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.

cs.LG↗
arXiv2026

Post-Hoc Large-Sample Statistical Inference

We derive inferential procedures for large sample sizes that remain valid under data-dependent significance levels (so-called "post-hoc valid inference"). Classical statistical tools require that the significance level -- the "type-I error" -- is selected prior to seeing or analyzing any data. This restriction leads to some drawbacks. For instance, if an analyst generates an inconclusive confidence interval, repeating the process with a larger significance level is not an option -- the result is final. Recently, e-values have emerged as the solution to this problem, being both necessary and sufficient tools for performing various forms of post-hoc inference. All such results, however, have thus far been nonasymptotic. As a result, they inherit familiar limitations of nonasymptotic inferential procedures such as requiring strong moment assumptions and being conservative in general. This paper develops a theory of post-hoc inference in the asymptotic setting, yielding asymptotic post-hoc confidence sets and asymptotic post-hoc p-values that make weaker assumptions and are sharper than their nonasymptotic counterparts.

math.ST↗
arXiv2026

Online Decision-Focused Learning

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.

cs.LG↗
arXiv2026

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

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.

stat.AP↗
arXiv2026

A Reinforcement Learning Approach in Multi-Phase Second-Price Auction Design

We study reserve price optimization in multi-phase second price auctions, where the seller's prior actions affect the bidders' later valuations through a Markov Decision Process (MDP). Compared to the bandit setting in existing works, the setting in ours involves three challenges. First, from the seller's perspective, we need to efficiently explore the environment in the presence of potentially untruthful bidders who aim to manipulate the seller's policy. Second, we want to minimize the seller's revenue regret when the market noise distribution is unknown. Third, the seller's per-step revenue is an unknown, nonlinear random variable, and cannot even be directly observed from the environment but realized values. We propose a mechanism addressing all three challenges. To address the first challenge, we use a combination of a new technique named "buffer periods" and inspirations from Reinforcement Learning (RL) with low switching cost to limit bidders' surplus from untruthful bidding, thereby incentivizing approximately truthful bidding. The second one is tackled by a novel algorithm that removes the need for pure exploration when the market noise distribution is unknown. The third challenge is resolved by an extension of LSVI-UCB, where we use the auction's underlying structure to control the uncertainty of the revenue function. The three techniques culminate in the Contextual-LSVI-UCB-Buffer (CLUB) algorithm which achieves $\tilde{O}(H^{5/2}\sqrt{K})$ revenue regret, where $K$ is the number of episodes and $H$ is the length of each episode, when the market noise is known and $\tilde{O}(H^{3}\sqrt{K})$ revenue regret when the noise is unknown with no assumptions on bidders' truthfulness.

cs.LG↗