Papers

A fundamental assumption in statistics and machine learning is that “the future looks like the past,” formalized as exchangeability: the joint data distribution is order-invariant. In practice, this assumption is often violated due to distribution shifts over time. Early detection of exchangeability violations is crucial to prevent performance degradation and enable timely interventions like model retraining. Conformal test martingales offer a flexible, distribution-free framework for sequential exchangeability testing with guaranteed false-alarm rate control by betting against the uniformity of conformal p-values. While alternatives such as plug-in martingales and mixture-based strategies exist, computationally efficient baselines like the Simple Jumper are limited to detecting mean location shifts. We propose a family of conformal test martingales based on shifted Legendre polynomials that extend the Simple Jumper to higher-order moments. The Simple Legendre Jumper replaces linear betting functions with polynomials of arbitrary degree, enabling rapid detection of variance, skewness, and other higher-order deviations. The Product Legendre Jumper combines multiple polynomial degrees into a single betting function but suffers from exponential state-space growth, termed the jumping tax. To resolve this, we introduce the Variational Legendre Jumper, which employs a mean-field approximation to reduce complexity to constant time per step with minimal power loss, providing an expressive, scalable framework for real-time distribution shift monitoring.

Emergent misalignment (EM) is a phenomenon in which models generalize with narrow fine-tuning, leading to broad (yet uneven) misalignment across evaluation questions. We study EM and its variability directly through the components of fine-tuning: training dynamics, model priors, and data. (1) We first explored how in-domain training loss relates to out-of-domain alignment scores across datasets and model families. Then, we tried to induce potential alternative local minima through different learning schedules for one narrow fine-tuning, but did not find strong runs with better broad alignment scores conditioned on similar or lower training loss. (2) We found that although the mean and standard deviations of the misaligned model scores are usually statistically different from those of the pre-trained model, there are some potential signals on overall positive correlation. The evaluation prompt-only activations from both the pre-trained and the original instruct models (prior to narrow fine-tuning) could predict fine-grained alignment scores after narrow fine-tuning. (3) Finally, we compared activation deltas before and after narrow fine-tuning and found moderate-to-high subspace overlap and similarity between the resulting activation shifts for training and evaluation prompts. Subspace overlaps between training and evaluation prompt activations correlate with their shifts' similarities when measuring with the last prompt-token activations. The train-evaluation data prompt overlap is controlled against overlap computed from random vectors and evaluation prompts activations.

A model is multicalibrated on a collection of group weights $G$ if it is calibrated – i.e. unbiased even conditional on its prediction – not just overall, but also after reweighting contexts by each $g \in G$. It is a useful property for many downstream applications and is a basic desideratum of trustworthy machine learning. Before this work, all predictors known to attain the minimax-optimal $\widetilde O(\varepsilon^{-3})$ sample complexity rate for $\varepsilon$-multicalibration were randomized, while deterministic predictors were known only with substantially worse sample complexity. Whether randomization is necessary for optimal sample complexity in multicalibration was explicitly asked by [CLNR26] and implicitly in several prior works. We resolve this open problem by giving a minimax-optimal multicalibration algorithm that outputs a deterministic predictor. We then generalize the algorithm to produce optimal deterministic predictors that satisfy outcome indistinguishability (OI) with respect to finite or finitely covered collections of tests. As an application, this also gives deterministic omnipredictors and panpredictors with optimal sample complexity, resolving open problems posed by [OKK25] and [BHHLZ25].

Variable selection methods that control the false discovery rate often lose power when predictors exhibit complex dependence structures. We previously showed that selecting hierarchically clustered groups of predictors can mitigate this issue while maintaining false discovery rate control. When correlations are less structured, however, overlapping predictor sets may be more effective. We introduce a generalized false discovery rate for hypotheses defined on sets of predictors and propose a hypergraph-based selection method. This approach achieves higher power across diverse settings while preserving rigorous false discovery rate control.

In the last years, an anomalously high spreading of West Nile virus (WNV) has been observed in Italy, with particularly high peaks of infections in southern Lazio, Campania and Veneto regions. The main disease vector for WNV is represented by Culex pipiens mosquitoes, which spread human infections through their bites. Here, we investigate WNV fever epidemic diffusion during summer season 2025 in Italy through a computational approach based on a quantum version of the Game of Life (GOL) cellular automaton model. Specifically, human dynamics evolves according to the GOL rules, while stochastic dynamics of disease vectors, i.e., mosquitoes, as well as their interaction with humans, simultaneously occur. We show that this model fits the curves of cumulative infected individuals with high accuracy, either at local and average-regional level, with only optimization of mosquito birth and removal rates parameters. Furthermore, leveraging model flexibility, we show that changes in model parameters values elucidate system response to environmental variations. For instance, we quantify, e.g., the impact of mosquito spreading containment measures or sudden mosquito increasing abundance due to climatic and ecological changes. Overall, we provide a general, quantitative description of WNV infection spreading in Italy which could represent a supportive tool to test different environmental scenarios and could be useful to devise strategies for decision makers to monitor disease vector dynamics and to control consequent virus diffusion.

We consider contraction of Bayesian posterior distributions in nonparametric settings where coefficients of a function over a basis or dictionary are given priors with $p$–exponential tails, including Laplace tails $(p=1)$ and heavier tails $(p<1)$. It is shown that contraction rates improve as $p$ decreases and that full adaptation to smoothness, up to logarithmic factors, is obtained in an appropriate $p\to 0$ regime. As applications, we consider both series priors in white noise regression and shallow ReLU neural networks in random design regression. In particular, we show that overparametrised shallow ReLU networks can adapt to any regularity $0\le \beta\le 2$. Through a simulation study, we show strong empirical agreement with the behavior predicted by our theory.

Competing risks are commonly observed in engineering fields and can bring challenges to time-to-event data modeling when the application scenarios are complicated. Recently, deep neural networks have received great attention for prediction with competing risks, due to their flexibility and high learning capability. However, the complexity of neural network structure brings extra difficulty in hyperparameter tuning based on different data inputs. Additionally, when an engineered system has complex physical structures with multiple hierarchical levels, treating all structural levels as a single group of inputs may fail to capture critical information. To address the issues, we propose a Structured Segmented Hazard Deep Neural Network (SSH-Net) for failure time prediction under cause-specific competing risks framework. Our approach associates neural network structure with data structures, and allows different covariate groups to impact the failure prediction through separate sub-networks. The neural network is constructed based on a cause-specific competing risks model. The SSH-Net outputs cause-specific hazard functions, and utilizes the penalized log-likelihood as the loss function. The prediction accuracy of SSH-Net is validated through simulation studies by evaluating the Brier score, the area under receiver operating characteristic curves (AUC), and the root mean square error (RMSE) of the predicted cause-specific cumulative incident function. We further demonstrate the model's ability to predict failure time distribution functions using the Titan GPU failure time data.

We reconcile privacy protection and rate-double-robust inference. The privacy of individuals is protected by a local privacy mechanism: injecting noise into their sensitive data, revealing only the noisy data for inference. Hence, privacy protection hinders inference. In contrast, the inference of a target parameter is rate-double-robust when the large-sample bias of an estimator of the parameter is characterised by a trade-off between the estimation errors of two other, nuisance, parameters. Hence, rate-double-robustness facilitates inference. Our starting point of reconciliation is a class of rate-double-robust target parameters indexed linearly by an infinite-dimensional and nonlinearly by a low-dimensional regression. Among others, this includes causal parameters. To infer these targets privately, we show how suitable privacy mechanisms transfer the semiparametric properties of the sensitive-data model to the private setting. Rate-double-robustness is transferred, enabling locally-private, unbiased and semiparametrically efficient inference of our target parameters. Finally, we transform general nonparametric nuisance estimators into private ones, which inherit convergence properties of their nonprivate counterparts. For parametric nuisance models, we develop a private method-of-moments estimator and its large-sample inference theory.

The accurate calibration of interest rate models is central to market-consistent valuation and Economic Scenario Generators (ESGs). Traditional calibration methods for multi-factor models such as the G2++ model often rely on point estimates, neglecting the influence of specific market data and the quantification of estimation uncertainty. This paper develops a diagnostic framework embedding the calibration problem into non-linear regression theory. It shows that the common industry practice of minimizing the Root Mean Squared Relative Error (RMSRE) is equivalent to a Weighted Least Squares (WLS) problem. This equivalence yields the corresponding formulations for diagnostic tools, including the Weighted Hat Matrix for leverage analysis, Influence Functions for local sensitivity diagnostics, and the Functional Delta Method for local, boundary-respecting confidence intervals. The implementation uses an efficient Jacobian factorization that exploits the analytical tractability of At-The-Money (ATM) caps. The framework is applied to a dataset of Euro ATM caps covering the period 2016–2025. Our empirical analysis reveals a boundary-dominated leverage profile, repeated losses of effective dimensionality due to active parameter constraints, and a diagnostic regime shift in local parameter stability around the post-2022 market transition. The resulting message for actuarial model governance is that low RMSRE is not sufficient for calibration validation. We conclude by discussing the framework's applicability to general least-squares problems while highlighting the computational challenges for instruments lacking closed-form gradients, such as swaptions.

We propose a mixture of location-scale skewed-$t$ distributions to fit bimodal, skewed and heavy-tailed data. In particular, the mixture is based on the skewed-$t$ distribution by Fernández and Steel (1998), so that the model-building procedure can be easily extended to mixtures of other symmetric distributions. After studying the properties of the mixture, we develop a maximum likelihood estimation approach via the EM algorithm and a likelihood ratio test of the null hypothesis of no skewness in any given component. A simulation-based comparison to a recently proposed mixture of g-and-h distributions suggests that the performance of the proposed model is excellent, in terms of both estimation precision in well-specified setups and modeling capability in mis-specified frameworks. Fitting the model to the Standard & Poor's 500 distortion allows us to confirm the bimodality of its distribution, with the implication that the US stock market has historically been in bearish or bullish conditions, rather than near its fundamental value.

In this article, we present a robust $Q$-learning algorithm for discrete-time mean-field control problems under Wasserstein uncertainty in the common noise law. The algorithm combines a quantization-and-projection scheme with a Wasserstein dual reformulation on the common-noise space. We establish its convergence together with finite-time iteration bounds for both synchronous and asynchronous learning schemes. Numerical experiments on systemic risk and epidemic models compare the asynchronous implementation with an idealized Bellman iteration, illustrate the robustness-performance tradeoff under common-noise misspecification, and report the observed convergence behavior of the asynchronous $Q$-learning algorithm.

Unanchored indirect treatment comparisons using single-arm studies or disconnected evidence are increasingly used in health technology assessment (HTA) when randomized evidence is unavailable. Existing methods, including matching-adjusted indirect comparison (MAIC) and simulated treatment comparison (STC), are generally limited to pairwise settings and typically estimate marginal effects in the comparator study population, which may differ from the decision-relevant population. We propose multilevel unanchored meta-regression (ML-UMR), a Bayesian regression framework for synthesizing individual patient data and aggregate data from fully disconnected evidence. ML-UMR extends multilevel network meta-regression (ML-NMR) to unanchored settings by jointly modeling individual- and aggregate-level data within a unified likelihood, enabling estimation of treatment-specific outcomes and both marginal and conditional effects across multiple treatments, studies, and target populations. ML-UMR distinguishes assumptions required to identify treatment effects from those required to transport results to target populations. As with all unanchored comparisons, valid inference relies on strong and often unverifiable assumptions, including conditional exchangeability, correct specification of the outcome model, and cross-treatment assumptions (e.g., shared prognostic factor assumption (SPFA)). ML-UMR does not lessen these requirements but makes them explicit within a unified framework and facilitates sensitivity analyses. In simulation studies, ML-UMR produced low bias and nominal coverage for comparator-population effects. Transportability to alternative populations depended critically on identifying assumptions: violations of SPFA led to bias under strong effect modification, whereas incorporating subgroup information restored near-unbiased estimation and nominal coverage.

Deep learning has managed to evade numerous intuitions from classical statistics to achieve unprecedented performance on a number of real-world tasks. In this article, we investigate the key features and surprises of deep learning from a physics-informed perspective, taking care to point out and justify where possible the many choices inherent in constructing a deep learning model. In particular, we review the phenomenon of neural scaling laws and discuss their interplay with the constraints and inductive biases which may be present when applying machine learning to problems in physics.

Two-sample IV is a popular estimation method when the outcome and treatment variables are available in different samples, whereas instruments are available in both samples. The standard estimator is two-sample two-stage least squares estimator, which is efficient under homoskedasticity and homogeneity of the samples. We develop a robust two-step procedure for efficient estimation under general heteroskedasticity and heterogeneity of the samples, and propose a related two-sample Hansen overidentification test. A key feature of our approach is that only summary statistics from the linear regressions of the reduced form and first-stage in the two samples are needed. These are the six objects of the estimated coefficient vectors, and the homoskedastic and heteroskedasticity robust estimated variance matrices. We further show that the first-stage F-statistic in the treatment sample can be used as a test for weak instruments in the standard way under homoskedasticity and homogeneity, with the relative bias here a proportional bias. We propose an extension of the effective F-statistic of Montiel-Olea and Pflueger (2013) for the heteroskedastic case, following the generalization in Windmeijer (2025). We illustrate the estimators and tests in an application studying the effect of education on voting behavior from Marshall (2019), with cluster robust inference.

Uncertain data without frequency stability often arises in experimental design. Classical fixed-effects models can only analyze precise experimental data. Based on an uncertain measure, this paper establishes uncertain fixed-effect models for Latin-square designs. First, we propose three methods with uncertainty to estimate the treatment and blocked effects and construct their confidence intervals. Then, uncertain homogeneity and common tests are conducted to assess the significance of treatment effects. In the numerical simulations, the three estimation methods are compared based on bias, mean squared error, mean absolute error, overall standard deviation, coverage probability, and average interval length. Several examples are given to illustrate the process of estimation and hypothesis. Finally, the uncertain fixed-effects model is applied to real education data, demonstrating its practical value.

In offline Reinforcement Learning, immediate rewards in logged batch data are often unobserved due to sparse or irregular record-keeping, or censored beyond certain reward values. This issue arises in practical settings, including health care and marketing. We investigate off-policy evaluation (OPE) in finite-horizon Markov decision processes when rewards are missing not at random (MNAR), which breaks ignorability and induces selection bias even after conditioning on states and actions. To address this, we formalize a reward-dependent propensity model and use future states as shadow variables to identify the full-data conditional mean reward. We further introduce a bridge function that recovers the conditional mean reward without explicitly modeling the MNAR mechanism, and estimate it via a min-max procedure to avoid double sampling. Building upon these identification results, we propose an Fitted-Q-Evaluation-style estimator that propagates the recovered rewards while allowing target policies to depend on past missingness indicators. Finally, we establish consistency and finite-sample error bounds for our OPE estimator, and show through experiments the strong performance of our method compared to existing methods on simulated and MIMIC-III Sepsis data.

We introduce a novel active-learning framework for failure probability estimation in structural reliability analysis that integrates Active Kriging Monte Carlo simulation with conformal prediction. The proposed approach employs an adaptive cross-conformal strategy specifically designed for small-sample settings and kriging surrogate models using the J+GP conformal estimator. Unlike standard AK-MCS methods, the proposed framework provides distribution-free guarantees on prediction errors, leading to more reliable classification of samples near the limit-state surface. This improved uncertainty quantification enhances both the accuracy and robustness of failure probability estimates, especially for rare-event regimes where such efficiency is crucial. Reproducible numerical results illustrate the effectiveness of the method and also compare it to classical approaches on well-established benchmarks.

We propose a Bayesian nonparametric approach using a truncated Enriched Dirichlet Process mixture (EDPM) model to estimate natural direct (NDE) and indirect (NIE) effects in causal mediation analyses in the presence of post-treatment confounders. We introduce an efficient cluster reallocation Metropolis-Hasting algorithm to improve mixing in the blocked Gibbs sampler. We implement a one-step posterior correction based on the efficient influence function for our setting. This post-processing step solves a critical problem in Bayesian nonparametrics: how to obtain reliable estimates and posteriors for a specific causal estimand of interest (the NDE and NIE) with excellent frequentist properties, such as correct coverage, from a model designed for complex joint distributions. We conduct simulation studies to assess our method's performance and apply it to evaluate causal mediation effects in a weight management clinical trial.

Subset selection problems as MIPs (Mixed Integer Programs) are NP-hard. For large scale problems, it is infeasible to find global optimal solutions in a reasonable time and good-quality incumbent solutions are sought after with MIP solvers in practice. This paper proposes DASH (Decreasing Active Set Hierarchy) – a dimensionality reduction method that improves the MIP solver performance for a subclass of best subset selection problems that can be formulated as MIQPs (Mixed Integer Quadratic Programs). We develop and evaluate the performance of DASH in the subset portfolio selection problem with comparison to Gurobi, a commercial MIP solver. In addition to the problem size, the difficulty of a problem is related to the condition number of the covariance matrix and the box constraint on portfolio weights. An extensive set of numerical experiments with varying problem configurations shows that DASH offers consistent and significant improvement of incumbent solutions when the problem is difficult to solve by Gurobi. In particular, the magnitude and duration of improvement by DASH scale with the difficulty of the problem.

The statistical modeling of consensus in Delphi data faces a critical bottleneck: the high dimensionality of questionnaire items relative to the limited sample size of expert panels. This rank deficiency leads traditional latent variable models, such as Principal Component Analysis, to be structurally unstable and prone to overfitting. Addressing this methodological gap, this study proposes a transition from variable-centric covariance models to network-centric connectivity models. By mapping item correlations onto a weighted graph topology, we present a simulation-based benchmark that utilizes community detection algorithms to identify latent thematic structures, effectively addressing the spectral instability and rank deficiency typical of high-dimensional, low-sample-size regimes. The research systematically evaluates the robustness of topological approaches based on structural density, information flow, and spectral partitioning against synthetic datasets designed to replicate the pathological conditions of consensus data, including ordinal scales and systemic noise. The central methodological contribution lies in demonstrating that collinearity among expert judgments - traditionally treated as statistical redundancy to be regularized - can be effectively reinterpreted as a topological signal of cohesion. This framework provides researchers with a structured and automated procedure for dimensionality reduction, ensuring structural stability and psychometric consistency even in small-sample regimes where standard factor analysis breaks down.

The g-formula is a foundational tool for identifying causal effects in observational data. This tool is based on the law of iterated expectation, a key mathematical identity in statistics. However, the notation with which the law of iterated expectation and the g-formula is expressed can be opaque to those with little background in statistics. We provide a primer introducing the law of iterated expectation, the integration notation used to express it, and its role for causal effect identification via the g-formula. Combined with the assumptions of causal consistency, positivity, and conditional exchangeability, the law of iterated expectation yields a causal standardization formula (the g-formula) in two nonparametrically equivalent forms: a non-iterative conditional expectation (NICE) form involving a single weighted average of conditional outcome means, and an iterative conditional expectation (ICE) form involving nested expectations. We illustrate both forms using three progressively complex numerical examples: a time-fixed example with a single binary confounder, a time-fixed example with discrete and continuous confounders, and a time-varying example with two timepoints. We provide clarity on what the law of iterated expectation is, how it is related to the g-formula, and how to gain intuition of its mathematical formulations in actual data examples that can be generalized to a range of settings.

This paper investigates the problem of comparing the location parameters of two shifted exponential models through a novel double sequential sampling framework. The proposed hypothesis testing procedure is developed by controlling the type I error probability at a preassigned level while minimizing a loss function that incorporates both the type II error probability and the associated sampling cost. The corresponding optimal fixed-sample-size expressions are shown to depend on unknown scale parameters, rendering the desired testing accuracies unattainable in practice under fixed-sample designs. To overcome this difficulty, a double sequential sampling procedure is proposed to test the difference between location parameters when the scale parameters are unknown and unequal. The proposed methodology is shown to possess desirable asymptotic properties, including first-order efficiency, second-order efficiency, and second-order risk efficiency. Extensive simulation studies and a real-data application that involves heavy precipitation episodes at meteorological stations demonstrate the practical effectiveness and applicability of the proposed procedure.

Contextual bandit learning has attracted increasing attention in engineering applications due to its ability to incorporate side/context information in time-varying environments. A widely used approach for contextual bandits is optimism in the face of uncertainty (OFU), in which uncertainty sets for unknown parameters are constructed and decisions are made optimistically over these sets. In the bandit literature, uncertainty sets are typically derived using statistical learning tools, such as confidence intervals. In contrast, set membership estimation (SME), a well-established uncertainty quantification approach in control and system identification, has received little attention in the bandit literature. This paper investigates the connection between SME and contextual bandits by focusing on linear contextual bandits as a starting point. We propose an SME-based OFU algorithm and establish a sublinear regret bound under proper assumptions. Simulations further demonstrate the promising performance of SME-OFU compared with existing methods. This supplementary manuscript provides more proof details in Section IV and the Appendix.

In time-to-event analysis, the hazard ratio (HR) derived from the Cox proportional hazards (PH) model is the most commonly used and widely reported measure for assessing treatment effects. However, hazard ratios are non-collapsible due to their inherent conditioning on survival up to each time point. As a result, they are subject to built-in selection bias in the presence of unmeasured heterogeneity arising from omitted important covariates, even when these covariates are independent of the main exposure at baseline, as is the case in randomized controlled trials. This article aims to provide an overview of key findings from the literature on how unobserved heterogeneity, due to omitted covariates that affect the outcome, can bias the estimation of the treatment hazard ratio in standard proportional hazards models, even in randomized trials where treatment is assigned independently of such covariates. Through simulations, we evaluate the extent of bias in the semi-parametric Cox PH model and parametric PH model under various scenarios of unmeasured heterogeneity. We then compare these standard models to alternative approaches that either account for this issue or are considered robust to it. These alternatives include the hazard ratio estimated from frailty models, regression parameters from an Accelerated Failure Time (AFT) model, and survival differences between treatment groups estimated nonparametrically using Kaplan-Meier curves or based on a Cox model with time-dependent effect of the exposure. We illustrate the practical relevance of the explored alternatives through a real data application to a randomized controlled trial from the Radiation Therapy Oncology Group (RTOG 9202).

Despite the theoretical appeal of their non-reversibility, to date, no Piecewise Deterministic Markov Process (PDMP) samplers have been developed that scale better than $\mathcal{O}(\sqrt{d})$ in computational complexity with respect to the target dimension $d$. We prove that this is a fundamental limitation by establishing an $\Omega(\sqrt{d})$ lower bound on the algorithmic complexity of PDMP samplers in a standard setup. By relaxing the assumption that the target density must remain invariant at all continuous times, we then demonstrate how to bypass this barrier. Specifically, we introduce a novel PDMP sampling scheme and show that it achieves an empirical complexity of $\mathcal{O}(d^\alpha)$, where $\alpha \in [0.2, 0.3]$ for Gaussian-tailed targets. In addition, this PDMP scheme is locally adaptive in both trajectory length and distance between velocity updates.

Recurrent events or competing risks regression models are often applied in the bio-medical setting and both can be considered as marginal models. In presence of right-censoring, such models need to be adjusted to give consistent estimators. When censoring is administrative, marginal regression models are particularly easy to estimate. However, when censoring is instead acting randomly, inverse probability of censoring weighting (IPCW) adjustments are typically considered to obtain parameter estimates. This technique relies on a censoring-weights adjustment via a correct censoring model, but for administrative censoring the adjustment is done correctly simply by modifying the risk-set. In practice for large central registries or some clinical trials, the administrative censoring time will be known for all subjects, but there will typically also be a proportion of subjects that are censored at random. In this work, we consider two frequently used regression approaches, the Ghosh-Lin model for recurrent events with terminal events and the Fine-Gray model for competing events. For these two settings, when both administrative and random censoring are present, we demonstrate how to obtain correct estimation by dealing with the combination of the two different types of censoring relying on a minimum of modeling assumptions.

We study a multi-agent multi-armed bandit problem in the competitive setup with two-sided matching markets under a human centric decision making model. To capture human preferences, we use cumulative prospect theory (CPT) that weighs the actions of the agent in a nonlinear fashion using a ($\alpha$-Hölder continuous) weight function. CPT has been widely used in behavioral economics and risk sensitive machine learning to emulate human preferences. We analyze the state-of-the-art learning algorithm with CPT weight distorted rewards and obtain a player optimal regret of $\mathcal{O}(K\log T \left(\frac{1}{\Delta}\right)^{2/\alpha})$, where $K$ denotes the number of arms, $T$ is the learning horizon, and $\Delta$ represents (suitably defined) players' minimum preference gap. Noticing the dependence on $\Delta$ to be sub-optimal, we further improve this regret by judiciously selecting the active set of arms during exploration, which removes the dependence on $K$ in the dominant term and achieves an improved (optimal) regret guarantees in the setting where the number of arms $K$ is significantly larger than the number of players $N$. In addition, we consider adversarial markets where the observed rewards of the agents may be corrupted. We propose and analyze algorithms for robust markets with CPT as risk sensitive measure in both settings where the total corruption budget is known and where it is unknown, and establish logarithmic player-optimal regret guarantees in both cases.

Recent work on first-order optimizers for empirical risk minimization (ERM) has suggested that smoothness of ERM loss functions in the training data, rather than in the optimization parameters, can be leveraged to improve the oracle complexity of gradient descent (GD) methods. In this paper, we propose an inexact gradient method, piecewise polynomial interpolation-based gradient descent (PPI-GD), which approximates the full gradient in each iteration by querying the first-order oracle at equidistant points in the data domain to construct polynomial interpolants of the resulting gradient samples over appropriately sized patches of the data domain. We analyze the oracle complexity of PPI-GD for strongly convex and non-convex loss functions when the data space dimension is bounded by a polylogarithmic function of the number of training samples, and find it to outperform several GD variants in key regimes when the loss function is sufficiently smooth. Furthermore, our analysis extends several techniques from the error analysis of bicubic spline interpolants to the setting of $d$-variate tensor product polynomial interpolants which may be of independent interest in interpolation analysis.

Recent research on Doeblin coefficients has shed light on their usefulness as a multi-way generalization of the Dobrushin contraction coefficient for TV distance, in a separate vein from their classic role in the theory of Markov chain ergodicity. However, strong conditions, such as being bounded away from 0, are typically necessary for Doeblin coefficients to establish the existence of information contraction. Building on recently formulated concepts of nonlinear information contraction, we aim to propose a finer-grained Doeblin-based characterization of multi-way contraction behavior which yields non-vacuous contraction guarantees even for channels whose Doeblin coefficient is 0. To this end, we introduce the notion of a Doeblin curve – a nonlinear function which quantifies the contraction behavior of a Markov kernel on collections of input distributions at specific levels of divergence and power. Through the course of our analysis, we develop a new variational characterization of Doeblin coefficients, present several properties of Doeblin curves, define several versions of power-constrained Doeblin curves, and derive upper and lower bounds using our aforementioned variational characterization. We then utilize these results in diverse areas, including generalization bounds for noisy iterative optimization, error bounds for reliable computation with noisy circuits, and differential privacy guarantees for online iterative algorithms. In particular, we extend results in these areas to broader domains or group settings, leveraging Doeblin curves to reveal finer-grained contraction phenomena than Doeblin coefficients.

We study dynamic resource allocation in a multicore computing system with a fixed number of processing cores and a stream of malleable jobs. Each job may adjust its level of parallelism during execution, allowing adaptive redistribution of resources across concurrently active jobs. Jobs belong to one of two observable classes, each characterized by a distinct speed-up function with unknown parameters. The objective is to learn a core-allocation policy that minimizes the long-run mean number of jobs in the system, equivalently the mean response time in steady state. \noindent To address this uncertainty, we develop an iterative learning-and-control framework. The system alternates between estimating the unknown speed-up parameters from observed job completions and solving the associated Markov decision process (MDP) to update the allocation policy. Within each job class, cores are shared equally among active jobs; the fraction of capacity assigned to each class is obtained from the MDP formulation of \cite{berg2017}, evaluated at the current parameter estimates. We construct a maximum likelihood estimator based on state-dependent inter-departure times and prove its strong consistency under a fixed allocation policy. We further propose two learning algorithms that combine this estimation step with dynamic programming-based policy updates, and illustrate their through numerical experiments.