Papers

Blind inverse problems arise in many experimental settings where both the signal of interest and the forward operator are (partially) unknown. In this context, methods developed for the non-blind case cannot be adapted in a straightforward manner due to identifiability issues and symmetric solutions inherent to the blind setting. Recently, data-driven approaches have been proposed to address such problems, demonstrating strong empirical performance and adaptability. However, these methods often lack interpretability and are not supported by theoretical guarantees, limiting their reliability in domains such as applied imaging where a blind approach often relates to a calibration of the acquisition device. In this work, we shed light on learning in blind inverse problems within the insightful framework of Linear Minimum Mean Square Estimators (LMMSEs). We provide a theoretical analysis, deriving closed-form expressions for optimal estimators and extending classical recovery results to the blind setting. In particular, we establish equivalences with tailored Tikhonov-regularized formulations, where the regularization structure depends explicitly on the distributions of the unknown signal, of the noise, and of the random forward operator. We also show how the reconstruction error converges as the noise and the randomness of the operator diminish when we use a source condition assumption. Furthermore, we derive finite-sample error bounds that characterize the performance of the learned estimators as a function of the noise level, problem conditioning, and number of available samples. These bounds explicitly quantify the impact of operator randomness and show explicitly the dependence of the associated convergence rates to this randomness factors. Finally, we validate our theoretical findings through illustrative exemplar numerical experiments that confirm the predicted convergence behavior.

Intrinsic Gaussian fields are used in many areas of statistics as models for spatial or spatio-temporal dependence, or as priors for latent variables. However, there are two major gaps in the literature: first, the number and flexibility of existing intrinsic models are very limited; second, theory, fast inference, and software are currently underdeveloped for intrinsic fields. We tackle these challenges by introducing the new flexible class of intrinsic Whittle–Matérn Gaussian random fields obtained as the solution to a stochastic partial differential equation (SPDE). Exploiting sparsity resulting from finite-element approximations, we develop fast estimation and simulation methods for these models. We demonstrate the benefits of this intrinsic SPDE approach for the important task of kriging under extrapolation settings. Leveraging the connection of intrinsic fields to spatial extreme value processes, we translate our theory to an SPDE approach for Brown–Resnick processes for sparse modeling of spatial extreme events. This new paradigm paves the way for efficient inference in unprecedented dimensions. To demonstrate the wide applicability of our new methodology, we apply it in two very different areas: a longitudinal study of renal function data, and the modeling of marine heat waves using high-resolution sea surface temperature data.

Link prediction, a foundational task in complex network analysis, has extensive applications in critical scenarios such as social recommendation, drug target discovery, and knowledge graph completion. However, existing evaluations of algorithmic often rely on experiments conducted on a limited number of networks, assuming consistent performance rankings across domains. Despite the significant disparities in generative mechanisms and semantic contexts, previous studies often improperly highlight “universally optimal" algorithms based solely on naive average over networks across domains. This paper systematically evaluates 12 mainstream link prediction algorithms across 740 real-world networks spanning seven domains. We present substantial empirical evidence elucidating the performance of algorithms in specific domains. This findings reveal a notably low degree of consistency in inter-domain algorithm rankings, a phenomenon that stands in stark contrast to the high degree of consistency observed within individual domains. Principal Component Analysis shows that response vectors formed by the rankings of the 12 algorithms cluster distinctly by domain in low-dimensional space, thus confirming domain attributes as a pivotal factor affecting algorithm performance. We propose a metric called Winner Score that could identify the superior algorithm in each domain: Non-Negative Matrix Factorization for social networks, Neighborhood Overlap-aware Graph Neural Networks for economics, Graph Convolutional Networks for chemistry, and L3-based Resource Allocation for biology. However, these domain-specific top-performing algorithms tend to exhibit suboptimal performance in other domains. This finding underscores the importance of aligning an algorithm's mechanism with the network structure.

Conformal prediction (CP) is widely presented as distribution-free predictive inference with finite-sample marginal coverage under exchangeability. We argue that CP is best understood as a rank-calibrated descendant of the Fisher-Dempster-Hill fiducial/direct-probability tradition rather than as Bayesian conditioning in disguise. We establish four separations from coherent countably additive predictive semantics. First, canonical conformal constructions violate conditional extensionality: prediction sets can depend on the marginal design P(X) even when P(Y|X) is fixed. Second, any finitely additive sequential extension preserving rank calibration is nonconglomerable, implying countable Dutch-book vulnerabilities. Third, rank-calibrated updates cannot be realized as regular conditionals of any countably additive exchangeable law on Y^infty. Fourth, formalizing both paradigms as families of one-step predictive kernels, conformal and Bayesian kernels coincide only on a Baire-meagre subset of the space of predictive laws. We further show that rank- and proxy-based reductions are generically Blackwell-deficient relative to full-data experiments, yielding positive Le Cam deficiency for suitable losses. Extending the analysis to prediction-powered inference (PPI) yields an analogous message: bias-corrected, proxy-rectified estimators can be valid as confidence devices while failing to define transportable belief states across stages, shifts, or adaptive selection. Together, the results sharpen a general limitation of wrappers: finite-sample calibration guarantees do not by themselves supply composable semantics for sequential updating or downstream decision-making.

This paper presents a test for wide-sense stationarity (WSS) based on the geometry of the covariance function. We estimate local patches of the covariance surface and then check whether the directional derivative in the $(1,1,0)$ direction is zero on each patch. The method only requires the covariance function to be locally smooth and does not assume stationarity in advance. It can be applied to general stochastic dynamical systems and provides a time-resolved view. We apply the test method to an SDOF system and to a stochastic Duffing oscillator. These examples show that the method is numerically stable and can detect departures from WSS in practice.

Estimating covariance matrices with high-dimensional complex data presents significant challenges, particularly concerning positive definiteness, sparsity, and numerical stability. Existing robust sparse estimators often fail to guarantee positive definiteness in finite samples, while subsequent positive-definite correction can degrade sparsity and lack explicit control over the condition number. To address these limitations, we propose a novel robust and well-conditioned sparse covariance matrix estimator. Our key innovation is the direct incorporation of a condition number constraint within a robust adaptive thresholding framework. This constraint simultaneously ensures positive definiteness, enforces a controllable level of numerical stability, and preserves the desired sparse structure without resorting to post-hoc modifications that compromise sparsity. We formulate the estimation as a convex optimization problem and develop an efficient alternating direction algorithm with guaranteed convergence. Theoretically, we establish that the proposed estimator achieves the minimax optimal convergence rate under the Frobenius norm. Comprehensive simulations and real-data applications demonstrate that our method consistently produces positive definite, well-conditioned, and sparse estimates, and achieves comparable or superior numerical stability to eigenvalue-bound methods while requiring less tuning parameters.

We study identification of differentiated product demand from market-level data when product characteristics can be endogenous. Past work suggests nonparametric identification may be impossible: that is, in addition to standard price instruments, exogenous characteristic-based instruments are essentially necessary to identify sufficiently flexible demand models with standard index restrictions. We show, however, that price counterfactuals are nonparametrically identified using recentered instruments – which combine exogenous price instruments with possibly endogenous product characteristics – under a weaker index restriction and a new condition we term faithfulness. We argue that faithfulness, like the usual completeness condition for nonparametric instrumental variable identification, is best viewed as a technical requirement on the strength of identifying variation rather than a substantive economic or statistical restriction. We show the two conditions are closely related, though generally distinct. We conclude with several practical implications for the parametric estimation of demand counterfactuals.

Online eXp-concave Optimization (OXO) is a fundamental problem in online learning, where the goal is to minimize regret when loss functions are exponentially concave. The standard algorithm, Online Newton Step (ONS), guarantees an optimal $O(d \log T)$ regret, where $d$ is the dimension and $T$ is the time horizon. Despite its simplicity, ONS may face a computational bottleneck due to the Mahalanobis projection at each round. This step costs $\tilde{O}(d^\omega)$ arithmetic operations for bounded domains, even for simple domains such as the unit ball, where $\omega \in (2,3]$ is the matrix-multiplication exponent. As a result, the total runtime can reach $\tilde{O}(d^\omega T)$, particularly when iterates frequently oscillate near the domain boundary. This paper proposes a simple variant of ONS, called LightONS, which reduces the total runtime to $O(d^2 T + d^\omega \sqrt{T \log T})$ while preserving the optimal regret. Deploying LightONS with the online-to-batch conversion implies a method for stochastic exp-concave optimization with runtime $\tilde{O}(d^3/\varepsilon)$, thereby answering an open problem posed by Koren [2013]. The design leverages domain-conversion techniques from parameter-free online learning and defers expensive Mahalanobis projections until necessary, thereby preserving the elegant structure of ONS and enabling LightONS to act as an efficient plug-in replacement in broader scenarios, including gradient-norm adaptivity, parametric stochastic bandits, and memory-efficient OXO.

Optimization under heavy-tailed noise has become popular recently, since it better fits many modern machine learning tasks, as captured by empirical observations. Concretely, instead of a finite second moment on gradient noise, a bounded ${\frak p}$-th moment where ${\frak p}\in(1,2]$ has been recognized to be more realistic (say being upper bounded by $\sigma_{\frak l}^{\frak p}$ for some $\sigma_{\frak l}\ge0$). A simple yet effective operation, gradient clipping, is known to handle this new challenge successfully. Specifically, Clipped Stochastic Gradient Descent (Clipped SGD) guarantees a high-probability rate ${\cal O}(\sigma_{\frak l}\ln(1/\delta)T^{1/{\frak p}-1})$ (resp. ${\cal O}(\sigma_{\frak l}^2\ln^2(1/\delta)T^{2/{\frak p}-2})$) for nonsmooth convex (resp. strongly convex) problems, where $\delta\in(0,1]$ is the failure probability and $T\in\mathbb{N}$ is the time horizon. In this work, we provide a refined analysis for Clipped SGD and offer two rates, ${\cal O}(\sigma_{\frak l}d_{\rm eff}^{-1/2{\frak p}}\ln^{1-1/{\frak p}}(1/\delta)T^{1/{\frak p}-1})$ and ${\cal O}(\sigma_{\frak l}^2d_{\rm eff}^{-1/{\frak p}}\ln^{2-2/{\frak p}}(1/\delta)T^{2/{\frak p}-2})$, faster than the aforementioned best results, where $d_{\rm eff}\ge1$ is a quantity we call the $\textit{generalized effective dimension}$. Our analysis improves upon the existing approach on two sides: better utilization of Freedman's inequality and finer bounds for clipping error under heavy-tailed noise. In addition, we extend the refined analysis to convergence in expectation and obtain new rates that break the known lower bounds. Lastly, to complement the study, we establish new lower bounds for both high-probability and in-expectation convergence. Notably, the in-expectation lower bounds match our new upper bounds, indicating the optimality of our refined analysis for convergence in expectation.

Attenuation bias – the systematic underestimation of regression coefficients due to measurement errors in input variables – affects astronomical data-driven models. For linear regression, this problem was solved by treating the true input values as latent variables to be estimated alongside model parameters. In this paper, we show that neural networks suffer from the same attenuation bias and that the latent variable solution generalizes directly to neural networks. We introduce LatentNN, a method that jointly optimizes network parameters and latent input values by maximizing the joint likelihood of observing both inputs and outputs. We demonstrate the correction on one-dimensional regression, multivariate inputs with correlated features, and stellar spectroscopy applications. LatentNN reduces attenuation bias across a range of signal-to-noise ratios where standard neural networks show large bias. This provides a framework for improved neural network inference in the low signal-to-noise regime characteristic of astronomical data. This bias correction is most effective when measurement errors are less than roughly half the intrinsic data range; in the regime of very low signal-to-noise and few informative features. Code is available at https://github.com/tingyuansen/LatentNN.

In applications with significant class imbalance or asymmetric costs, metrics such as the $F_\beta$-measure, AM measure, Jaccard similarity coefficient, and weighted accuracy offer more suitable evaluation criteria than standard binary classification loss. However, optimizing these metrics present significant computational and statistical challenges. Existing approaches often rely on the characterization of the Bayes-optimal classifier, and use threshold-based methods that first estimate class probabilities and then seek an optimal threshold. This leads to algorithms that are not tailored to restricted hypothesis sets and lack finite-sample performance guarantees. In this work, we introduce principled algorithms for optimizing generalized metrics, supported by $H$-consistency and finite-sample generalization bounds. Our approach reformulates metric optimization as a generalized cost-sensitive learning problem, enabling the design of novel surrogate loss functions with provable $H$-consistency guarantees. Leveraging this framework, we develop new algorithms, METRO (Metric Optimization), with strong theoretical performance guarantees. We report the results of experiments demonstrating the effectiveness of our methods compared to prior baselines.

The use of neural networks trained on individual claims data has become increasingly popular in the actuarial reserving literature. We consider how to best input historical payment data in neural network models. Additionally, case estimates are also available in the format of a time series, and we extend our analysis to assessing their predictive power. In this paper, we compare a feed-forward neural network trained on summarised transactions to a recurrent neural network equipped to analyse a claim's entire payment history and/or case estimate development history. We draw conclusions from training and comparing the performance of the models on multiple, comparable highly complex datasets simulated from SPLICE (Avanzi, Taylor and Wang, 2023). We find evidence that case estimates will improve predictions significantly, but that equipping the neural network with memory only leads to meagre improvements. Although the case estimation process and quality will vary significantly between insurers, we provide a standardised methodology for assessing their value.

We introduce Neural Optimal Design of Experiments, a learning-based framework for optimal experimental design in inverse problems that avoids classical bilevel optimization and indirect sparsity regularization. NODE jointly trains a neural reconstruction model and a fixed-budget set of continuous design variables representing sensor locations, sampling times, or measurement angles, within a single optimization loop. By optimizing measurement locations directly rather than weighting a dense grid of candidates, the proposed approach enforces sparsity by design, eliminates the need for l1 tuning, and substantially reduces computational complexity. We validate NODE on an analytically tractable exponential growth benchmark, on MNIST image sampling, and illustrate its effectiveness on a real world sparse view X ray CT example. In all cases, NODE outperforms baseline approaches, demonstrating improved reconstruction accuracy and task-specific performance.

Advances in data acquisition and computational methods have accelerated the use of differential equation based modelling for complex systems. Such systems are often described by coupled (or more) variables, yet governing equation is typically available for one variable, while the remaining variable can be accessed only through data. This mismatch between known physics and observed data poses a fundamental challenge for existing physics-informed machine learning approaches, which generally assume either complete knowledge of the governing equations or full data availability across all variables. In this paper, we introduce MUSIC (Multitask Learning Under Sparse and Incomplete Constraints), a sparsity induced multitask neural network framework that integrates partial physical constraints with data-driven learning to recover full-dimensional solutions of coupled systems when physics-constrained and data-informed variables are mutually exclusive. MUSIC employs mesh-free (random) sampling of training data and sparsity regularization, yielding highly compressed models with improved training and evaluation efficiency. We demonstrate that MUSIC accurately learns solutions (shock wave solutions, discontinuous solutions, pattern formation solutions) to complex coupled systems under data-scarce and noisy conditions, consistently outperforming non-sparse formulations. These results highlight MUSIC as a flexible and effective approach for modeling partially observed systems with incomplete physical knowledge.

This research analyses the axiomatic link among health expenditures, inflation rate, and gross national income (GNI) per capita concerning the child mortality (CMU5) rate in D-8 nations, employing panel data analysis from 1995 to 2014. Utilising conventional panel unit root tests and linear regression models, we establish that education expenditures, in conjunction with health expenditures, inflation rate, and GNI per capita, display stationarity at level. Additionally, we examine fixed effects and random effects estimators for the pertinent variables, utilising metrics such as the Hausman Test (HT) and comparisons with CCMR correlations. Our data demonstrate that the CMU5 rate in D-8 nations has steadily decreased, according to a somewhat negative linear regression model, therefore slightly undermining the fourth Millennium Development Goal (MDG4) of the World Health Organisation (WHO).

Modern generative and vision-language models (VLMs) are increasingly used in scientific and medical decision support, where predicted probabilities must be both accurate and well calibrated. Despite strong empirical results with moderate data, it remains unclear when such predictions generalize uniformly across inputs, classes, or subpopulations, rather than only on average-a critical issue in biomedicine, where rare conditions and specific groups can exhibit large errors even when overall loss is low. We study this question from a finite-sample perspective and ask: under what structural assumptions can generative and VLM-based predictors achieve uniformly accurate and calibrated behavior with practical sample sizes? Rather than analyzing arbitrary parameterizations, we focus on induced families of classifiers obtained by varying prompts or semantic embeddings within a restricted representation space. When model outputs depend smoothly on a low-dimensional semantic representation-an assumption supported by spectral structure in text and joint image-text embeddings-classical uniform convergence tools yield meaningful non-asymptotic guarantees. Our main results give finite-sample uniform convergence bounds for accuracy and calibration functionals of VLM-induced classifiers under Lipschitz stability with respect to prompt embeddings. The implied sample complexity depends on intrinsic/effective dimension, not ambient embedding dimension, and we further derive spectrum-dependent bounds that make explicit how eigenvalue decay governs data requirements. We conclude with implications for data-limited biomedical modeling, including when current dataset sizes can support uniformly reliable predictions and why average calibration metrics may miss worst-case miscalibration.

Reinforcement Learning (RL) for Large Language Models (LLMs) faces a fundamental tension: the numerical divergence between high-throughput inference engines and numerically precise training engines. Although these systems share the same parameters, they produce slightly different probability distributions, creating a training-inference mismatch. We prove that the bound on the log-probability divergence arising from this mismatch scales as $(1-p)$, where $p$ is the token probability. This scaling induces a highly asymmetric effect: the bound vanishes for high-probability tokens but remains significant for low-probability tokens in the distribution tail. When sampled, these tail tokens introduce systematically biased errors that accumulate over sequences, thereby destabilizing gradient estimation. Instead of applying post-hoc corrections, we propose Dynamic Vocabulary Pruning (DVP), which constrains the RL objective to a dynamically determined ”safe” vocabulary that excludes the extreme tail. This strategy trades large, destabilizing numerical errors for a small, bounded optimization bias. We validate DVP empirically by demonstrating stable training, and theoretically by deriving strict bounds on the induced bias.

Policy gradient methods for Large Language Models optimize a policy $\pi_\theta$ via a surrogate objective computed from samples of a rollout policy $\pi_{\text{roll}}$. However, modern LLM-RL pipelines suffer from unavoidable implementation divergences – backend discrepancies, Mixture-of-Experts routing discontinuities, and distributed training staleness – causing off-policy mismatch ($\pi_{\text{roll}} \neq \pi_\theta$) and approximation errors between the surrogate and the true objective. We demonstrate that classical trust region bounds on this error scale as $O(T^2)$ with sequence length $T$, rendering them vacuous for long-horizon tasks. To address this, we derive a family of bounds – both KL-based and TV-based – including a Pinsker-Marginal bound ($O(T^{3/2})$), a Mixed bound ($O(T)$), and an Adaptive bound that strictly generalizes the Pinsker-Marginal bound via per-position importance-ratio decomposition. Taking the minimum over all bounds yields the tightest known guarantee across all divergence regimes. Crucially, all bounds depend on the maximum token-level divergence $D_{\mathrm{KL}}^{\mathrm{tok,max}}$ (or $D_{\mathrm{TV}}^{\mathrm{tok,max}}$), a sequence-level quantity that cannot be controlled by token-independent methods like PPO clipping. We propose Trust Region Masking (TRM), which masks entire sequences violating the trust region, enabling the first non-vacuous monotonic improvement guarantees for long-horizon LLM-RL.

Federated Learning (FL) is a distributed machine learning setting that requires multiple clients to collaborate on training a model while maintaining data privacy. The unaddressed inherent sparsity in data and models often results in overly dense models and poor generalizability under data and client participation heterogeneity. We propose FL with an L0 constraint on the density of non-zero parameters, achieved through a reparameterization using probabilistic gates and their continuous relaxation: originally proposed for sparsity in centralized machine learning. We show that the objective for L0 constrained stochastic minimization naturally arises from an entropy maximization problem of the stochastic gates and propose an algorithm based on federated stochastic gradient descent for distributed learning. We demonstrate that the target density (rho) of parameters can be achieved in FL, under data and client participation heterogeneity, with minimal loss in statistical performance for linear and non-linear models: Linear regression (LR), Logistic regression (LG), Softmax multi-class classification (MC), Multi-label classification with logistic units (MLC), Convolution Neural Network (CNN) for multi-class classification (MC). We compare the results with a magnitude pruning-based thresholding algorithm for sparsity in FL. Experiments on synthetic data with target density down to rho = 0.05 and publicly available RCV1, MNIST, and EMNIST datasets with target density down to rho = 0.005 demonstrate that our approach is communication-efficient and consistently better in statistical performance.

For learned models to be trustworthy, it is essential to verify their robustness to perturbations in the training data. Classical approaches involve uncertainty quantification via confidence intervals and bootstrap methods. In contrast, recent work proposes a more stringent form of robustness: stability to the removal of any subset of $k$ samples from the training set. In this paper, we present a theoretical study of this criterion for ordinary least squares (OLS). Our contributions are as follows: (1) Given $n$ i.i.d. training samples from a general misspecified model, we prove that with high probability, OLS is robust to the removal of any $k \ll n $ samples. (2) For data of dimension $p$, OLS can withstand up to ${k\ll \sqrt{np}/\log n}$ sample removals while remaining robust and achieving the same error rate as OLS applied to the full dataset. Conversely, if $k$ is proportional to $n$, OLS is provably non-robust. (3) We revisit prior analyses that found several econometric datasets to be highly non-robust to sample removals. While this appears to contradict our results in (1), we demonstrate that the sensitivity is due to either heavy-tailed responses or correlated samples. Empirically, this sensitivity is considerably attenuated by classical robust methods, such as linear regression with a Huber loss.

As generative AI becomes increasingly embedded in everyday life, the thoughtful and intentional integration of AI-based tools into statistics education has become essential. We address this need with a focus on homework assignments. We propose the use of LLMs as an opportunity for instructors to integrate more pedagogical approaches into homework design, by developing an open-source tool named LLteacher. This LLM-based tool preserves learning processes and it guides students to engage with AI in ways that support their learning, while ensuring alignment with course content and equitable access. We illustrate LLteacher's design and functionality with examples suitable to an undergraduate Statistical Computing course in R, showing how it supports two distinct pedagogical goals: recalling prior knowledge and discovering new concepts. While this is an initial version, LLteacher is an example of one possible pathway for integrating generative AI into statistics courses, with strong potential for adaptation to other types of classes and assignments.

Nominal dimension can greatly overstate how much a Bayesian model actually learns from data. We quantify this gap through a prior-dependent effective dimension based on parameter–data mutual information, normalized so that regular parametric information growth provides the benchmark scale. The main theoretical result shows that, in infinite-dimensional Gaussian experiments, information growth is determined by the spectral counting function: regularly varying spectra yield a sharp asymptotic law with an explicit constant, while exponential spectral decay produces a distinct polylogarithmic regime. In Gaussian regression the same information scale has a spectral representation and is linked exactly to ridge degrees of freedom. For scalar Gaussian scale-mixture priors, we separate information about latent scales from conditional Gaussian information and show that the full marginal information remains finite under polynomially heavy-tailed mixing.

Cold standby 1-out-of-n redundant systems are well-established models in system reliability engineering. To date, reliability analyses of such systems have predominantly assumed exponential, Erlang, or Weibull failure distributions for their components. The Lindley distribution and its generalizations represent a significant class of statistical distributions in reliability engineering. Certain generalized Lindley distributions, due to the appealing characteristics of their hazard functions, can serve as suitable alternatives to other well-known lifetime distributions like the Weibull. This study investigates the reliability of a 1-out-of-n cold standby redundant system with perfect and imperfect switching, assuming that the active component failure times follow the Generalized Lindley distribution. We derive a closed-form expression for the system reliability. To achieve this, the distribution of the sum of n independent and identically distributed random variables following the Generalized Lindley distribution is first determined using the moment-generating function approach.

We consider problems of parameter estimation where design variables can be actively optimized to maximize information gain. To this end, we introduce JADAI, a framework that jointly amortizes Bayesian adaptive design and inference by training a policy, a history network, and an inference network end-to-end. The networks minimize a generic loss that aggregates incremental reductions in posterior error along experimental sequences. Inference networks are instantiated with diffusion-based posterior estimators that can approximate high-dimensional and multimodal posteriors at every experimental step. Across standard adaptive design benchmarks, JADAI achieves superior or competitive performance.

In this paper we study a nonconvex-strongly-concave constrained minimax problem. Specifically, we propose a first-order augmented Lagrangian method for solving it, whose subproblems are nonconvex-strongly-concave unconstrained minimax problems and suitably solved by a first-order method developed in this paper that leverages the strong concavity structure. Under suitable assumptions, the proposed method achieves an operation complexity of $O(\varepsilon^{-3.5}\log\varepsilon^{-1})$, measured in terms of its fundamental operations, for finding an $\varepsilon$-KKT solution of the constrained minimax problem, which improves the previous best-known operation complexity by a factor of $\varepsilon^{-0.5}$.

As AI systems are increasingly used to guide decisions, it is essential that they follow ethical principles. A core principle in medicine is non-maleficence, often equated with “do no harm”. A formal definition of harm based on counterfactual reasoning has been proposed and popularized. This notion of harm has been promoted in simple settings with binary treatments and outcomes. Here, we highlight a problem with this definition in settings involving multiple treatment options. Illustrated by an example with three tuberculosis treatments (say, A, B, and C), we demonstrate that the counterfactual definition of harm can produce intransitive results: B is less harmful than A, C is less harmful than B, yet C is more harmful than A when compared pairwise. This intransitivity poses a challenge as it may lead to practical (clinical) decisions that are difficult to justify or defend. In contrast, an interventionist definition of harm based on expected utility forgoes counterfactual comparisons and ensures transitive treatment rankings.

Large language models (LLMs) have achieved remarkable performance but face critical challenges: hallucinations and high inference costs. Leveraging multiple experts offers a solution: deferring uncertain inputs to more capable experts improves reliability, while routing simpler queries to smaller, distilled models enhances efficiency. This motivates the problem of learning with multiple-expert deferral. This thesis presents a comprehensive study of this problem and the related problem of learning with abstention, supported by strong consistency guarantees. First, for learning with abstention (a special case of deferral), we analyze score-based and predictor-rejector formulations in multi-class classification. We introduce new families of surrogate losses and prove strong non-asymptotic, hypothesis set-specific consistency guarantees, resolving two existing open questions. We analyze both single-stage and practical two-stage settings, with experiments on CIFAR-10, CIFAR-100, and SVHN demonstrating the superior performance of our algorithms. Second, we address general multi-expert deferral in classification. We design new surrogate losses for both single-stage and two-stage scenarios and prove they benefit from strong $H$-consistency bounds. For the two-stage scenario, we show that our surrogate losses are realizable $H$-consistent for constant cost functions, leading to effective new algorithms. Finally, we introduce a novel framework for regression with deferral to address continuous label spaces. Our versatile framework accommodates multiple experts and various cost structures, supporting both single-stage and two-stage methods. It subsumes recent work on regression with abstention. We propose new surrogate losses with proven $H$-consistency and demonstrate the empirical effectiveness of the resulting algorithms.

In machine learning, the loss functions optimized during training often differ from the target loss that defines task performance due to computational intractability or lack of differentiability. We present an in-depth study of the target loss estimation error relative to the surrogate loss estimation error. Our analysis leads to $H$-consistency bounds, which are guarantees accounting for the hypothesis set $H$. These bounds offer stronger guarantees than Bayes-consistency or $H$-calibration and are more informative than excess error bounds. We begin with binary classification, establishing tight distribution-dependent and -independent bounds. We provide explicit bounds for convex surrogates (including linear models and neural networks) and analyze the adversarial setting for surrogates like $\rho$-margin and sigmoid loss. Extending to multi-class classification, we present the first $H$-consistency bounds for max, sum, and constrained losses, covering both non-adversarial and adversarial scenarios. We demonstrate that in some cases, non-trivial $H$-consistency bounds are unattainable. We also investigate comp-sum losses (e.g., cross-entropy, MAE), deriving their first $H$-consistency bounds and introducing smooth adversarial variants that yield robust learning algorithms. We develop a comprehensive framework for deriving these bounds across various surrogates, introducing new characterizations for constrained and comp-sum losses. Finally, we examine the growth rates of $H$-consistency bounds, establishing a universal square-root growth rate for smooth surrogates in binary and multi-class tasks, and analyze minimizability gaps to guide surrogate selection.

The Lindley distribution was first introduced by Lindley in 1958 for Bayesian computations. Over the past years, various generalizations of this distribution have been proposed by different authors. The generalized Lindley distributions sometimes have many parameters, and although they show good flexibility, their statistical form becomes complicated. In this article, we propose a new and simple distribution determined by the recursive relation of the Lindley distribution and the Gamma distribution with specific weights. Subsequently, some statistical properties of this distribution are examined, and with real numerical examples, its superiority over the Lindley generalizations is demonstrated.

This study proposes an end-to-end algorithm for policy learning in causal inference. We observe data consisting of covariates, treatment assignments, and outcomes, where only the outcome corresponding to the assigned treatment is observed. The goal of policy learning is to train a policy from the observed data, where a policy is a function that recommends an optimal treatment for each individual, to maximize the policy value. In this study, we first show that maximizing the policy value is equivalent to minimizing the mean squared error for the conditional average treatment effect (CATE) under $\{-1, 1\}$ restricted regression models. Based on this finding, we modify the causal forest, an end-to-end CATE estimation algorithm, for policy learning. We refer to our algorithm as the causal-policy forest. Our algorithm has three advantages. First, it is a simple modification of an existing, widely used CATE estimation method, therefore, it helps bridge the gap between policy learning and CATE estimation in practice. Second, while existing studies typically estimate nuisance parameters for policy learning as a separate task, our algorithm trains the policy in a more end-to-end manner. Third, as in standard decision trees and random forests, we train the models efficiently, avoiding computational intractability.