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.

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.

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.

Multi-Armed Sequential Hypothesis Testing by Betting

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.

A Variational Estimator for $L_p$ Calibration Errors

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.

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.

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.

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.

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.

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

We study stopping rules for stochastic gradient descent (SGD) for convex optimization from the perspective of anytime-valid confidence sequences. Classical analyses of SGD provide convergence guarantees in expectation or at a fixed horizon, but offer no statistically valid way to assess, at an arbitrary time, how close the current iterate is to the optimum. We develop an anytime-valid, data-dependent upper confidence sequence for the weighted average suboptimality of projected SGD, constructed via nonnegative supermartingales and requiring no smoothness or strong convexity. This confidence sequence yields a simple stopping rule that is provably $\varepsilon$-optimal with probability at least $1-α$, with explicit bounds on the stopping time under standard stochastic approximation stepsizes. To the best of our knowledge, these are the first rigorous, time-uniform performance guarantees and finite-time $\varepsilon$-optimality certificates for projected SGD with general convex objectives, based solely on observable trajectory quantities.