In applications with significant class imbalance or asymmetric costs, metrics such as the $F_β$-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.

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.

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.

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.

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 $ρ$-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.

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.

Modern machine learning embeddings provide powerful compression of high-dimensional data, yet they typically destroy the geometric structure required for classical likelihood-based statistical inference. This paper develops a rigorous theory of likelihood-preserving embeddings: learned representations that can replace raw data in likelihood-based workflows -- hypothesis testing, confidence interval construction, model selection -- without altering inferential conclusions. We introduce the Likelihood-Ratio Distortion metric $Δ_n$, which measures the maximum error in log-likelihood ratios induced by an embedding. Our main theoretical contribution is the Hinge Theorem, which establishes that controlling $Δ_n$ is necessary and sufficient for preserving inference. Specifically, if the distortion satisfies $Δ_n = o_p(1)$, then (i) all likelihood-ratio based tests and Bayes factors are asymptotically preserved, and (ii) surrogate maximum likelihood estimators are asymptotically equivalent to full-data MLEs. We prove an impossibility result showing that universal likelihood preservation requires essentially invertible embeddings, motivating the need for model-class-specific guarantees. We then provide a constructive framework using neural networks as approximate sufficient statistics, deriving explicit bounds connecting training loss to inferential guarantees. Experiments on Gaussian and Cauchy distributions validate the sharp phase transition predicted by exponential family theory, and applications to distributed clinical inference demonstrate practical utility.

Rank-based input normalization is a workhorse of modern machine learning, prized for its robustness to scale, monotone transformations, and batch-to-batch variation. In many real systems, the ordering of feature values matters far more than their raw magnitudes - yet the structural conditions that a rank-based normalization operator must satisfy to remain stable under these invariances have never been formally pinned down. We show that widely used differentiable sorting and ranking operators fundamentally fail these criteria. Because they rely on value gaps and batch-level pairwise interactions, they are intrinsically unstable under strictly monotone transformations, shifts in mini-batch composition, and even tiny input perturbations. Crucially, these failures stem from the operators' structural design, not from incidental implementation choices. To address this, we propose three axioms that formalize the minimal invariance and stability properties required of rank-based input normalization. We prove that any operator satisfying these axioms must factor into (i) a feature-wise rank representation and (ii) a scalarization map that is both monotone and Lipschitz-continuous. We then construct a minimal operator that meets these criteria and empirically show that the resulting constraints are non-trivial in realistic setups. Together, our results sharply delineate the design space of valid rank-based normalization operators and formally separate them from existing continuous-relaxation-based sorting methods.

Ensemble learning is traditionally justified as a variance-reduction strategy, explaining its strong performance for unstable predictors such as decision trees. This explanation, however, does not account for ensembles constructed from intrinsically stable estimators-including smoothing splines, kernel ridge regression, Gaussian process regression, and other regularized reproducing kernel Hilbert space (RKHS) methods whose variance is already tightly controlled by regularization and spectral shrinkage. This paper develops a general weighting theory for ensemble learning that moves beyond classical variance-reduction arguments. We formalize ensembles as linear operators acting on a hypothesis space and endow the space of weighting sequences with geometric and spectral constraints. Within this framework, we derive a refined bias-variance approximation decomposition showing how non-uniform, structured weights can outperform uniform averaging by reshaping approximation geometry and redistributing spectral complexity, even when variance reduction is negligible. Our main results provide conditions under which structured weighting provably dominates uniform ensembles, and show that optimal weights arise as solutions to constrained quadratic programs. Classical averaging, stacking, and recently proposed Fibonacci-based ensembles appear as special cases of this unified theory, which further accommodates geometric, sub-exponential, and heavy-tailed weighting laws. Overall, the work establishes a principled foundation for structure-driven ensemble learning, explaining why ensembles remain effective for smooth, low-variance base learners and setting the stage for distribution-adaptive and dynamically evolving weighting schemes developed in subsequent work.