Mitigating Over-Optimization in PRM-Guided Search in Mathematical Reasoning by Optimizing the Guide

Process reward models (PRMs) provide dense step-level guidance for search-based reasoning, enabling inference-time compute to be allocated toward promising partial solutions. However, recent evidence suggests that PRM-guided search can over-optimize imperfect process rewards, pruning viable trajectories while expanding spurious ones. In this work, we theoretically show that directly leveraging PRM score is vulnerable to verifier noise through an extreme-value effect: non-viable prefixes become more likely to receive spuriously high scores as reasoning depth increase. Therefore, we formulate the PRM-guided search as a robust optimization problem over plausible reward perturbations, termed maximin PRM-guided search, leading to a training-free robust process supervision method that preserves promising alternatives when step-level scores are noisy. Maximin PRM-guided search mitigates this failure mode by reducing sensitivity to over-optimized PRM outliers. Without fine-tuning or online adaptation, maximin search consistently improves the PRM-guided search by 17-35\% on average, outperforming outcome- and step-level baselines in 14 out of 16 settings. Our source code is available at https://github.com/tjoo512/maximin-search.

The Value of Human Expertise

We consider optimization applications with unknown parameters where the decision maker believes that the optimal value of the nominal problem-the optimization problem they would have solved if the true parameters were known-is unlikely to be large. This belief derives from information that humans have that is not captured in datasets, obtained from domain knowledge and interacting with the physical world. We propose an approach to evaluating policies that provides tighter performance guarantees if the decision maker's belief happens to be correct. Our main result shows that if computing a policy's worst-case performance is a convex program, then the value of human expertise-the maximum improvement in performance guarantees that can be obtained from the belief about the nominal problem-is equal to the minimax gap of a max-min problem. We illustrate our developments in assortment optimization and shortest path problems.

Robust Data-Collection Policy Learning for Low-Variance Online Policy Evaluation

In reinforcement learning policy evaluation, classic on-policy methods often suffer from high variance when estimating policy performance. To mitigate this issue, behavior policy search has been proposed to learn data-collecting policies tailored to reduce online evaluation variance. However, these approaches do not account for uncertainties in the transition functions. In practice, simulator transitions often differ from the real world due to modeling errors or approximation limitations. As a result, behavior policies trained in simulation may still yield high variance when deployed in real environments, leading to costly reliance on real-world evaluation samples. In this work, we propose a double-loop gradient-based algorithm for learning behavior policies that are both efficient and robust to transition uncertainty. Theoretically, we derive novel transition-variance gradient expressions and establish global convergence guarantees for the algorithm. Numerically, we demonstrate that our method is less sensitive to transition perturbations than existing approaches, providing supportive evidence for its practical utility.

Robust Risk Under Evolving Uncertainty: A Wasserstein Counterpart of the Entropic Value-at-Risk

An agent still learning its environment should be cautious while ignorant and bold once confident. The entropic value-at-risk captures this through a robust-optimization identity---a confidence level fixes the radius of a relative-entropy ball of alternative models---but that ball cannot reach catastrophes the nominal deems impossible, precisely what a safe agent must hedge. We instead use an optimal-transport ball and study the coherent risk measure it induces, the Wasserstein entropic value-at-risk. It has a variational dual mirroring the entropic formula (an inverse temperature becomes a transport price), occupies a definite place in the risk hierarchy, and provably accounts for the reachable catastrophes the entropic measure ignores; we verify both dualities numerically. Driving the transport radius by belief entropy then yields a closed-form robust dynamic-programming operator whose caution contracts as the belief sharpens, with a certified safety sandwich and a sharp safety switch.

Adaptive Policy Portfolios for Robust Markov Decision Processes

Robust Markov decision processes optimize one policy against a set of plausible transition functions. This can be conservative when the unknown dynamics are fixed and become partially identifiable after deployment. We study adaptive policy portfolios: finite sets of memoryless randomized policies synthesized offline and paired with a lightweight online selector. Robust regret is a natural measure of portfolio quality: for each plausible environment, it measures the loss of the best portfolio member relative to the policy that would have been optimal had that environment been known. Related regret objectives were studied by Ghavamzadeh et al. (2016) with an emphasis on approximations and relaxations for safe policy improvement. We give a complexity-theoretic account of portfolio certification and synthesis. Certifying a given portfolio is $\forall\mathbb{R}$-complete already for deterministic portfolios in acyclic (s,a)-rectangular RMDPs. Synthesizing a portfolio of unary-bounded size is $\exists\forall\mathbb{R}$-complete for general rational polytopes, even with fixed discount and acyclic dynamics. The single-policy case is already hard, both combinatorially and algebraically. Finally, we present an offline portfolio construction that is amenable to runtime specialization.

PERO: Efficient Robust Post-Training Foundation Models for Encrypted Traffic Classification

Encrypted traffic classification is vital for network security, yet real-world deployments are inherently sensitive to rare but high-loss errors such as misclassification of malicious traffic. The encrypted traffic foundation model, as a promising general-purpose technique, can achieve impressive overall performance. However, employing standard objectives such as empirical risk minimization often overlooks high-risk tail events, and commonly used performance metrics hardly reflect robustness limitations in risk-sensitive scenarios. Directly applying robust optimization objectives, such as conditional value-at-risk, to post-training is computationally prohibitive for large models, as identifying high-loss samples exhausts substantial computation. To this end, we propose Pre-Evaluation Robust Optimization (PERO), an efficient robust post-training framework for encrypted traffic foundation models. PERO employs a lightweight proxy to estimate sample-wise risk and selects a subset of high-risk samples to update the foundation model, decoupling risk estimation from expensive large-model optimization. Extensive experiments on typical encrypted traffic datasets show that PERO achieves competitive or superior robustness and average performance compared to outstanding robust post-training methods, while significantly reducing computational and memory costs.

Statistical Properties of Robust Learning under Distributional Shifts

Distributional shifts arise when the target deployment environment differs from the source environment that generated the training data. Robust learning frameworks such as Distributionally Robust Optimization (DRO) and Robust Satisficing (RS) aim to address this challenge, yet their finite-sample guarantees under such shifts, and their systematic comparison, remain underexplored: existing analyses typically establish guarantees either in the source environment or for adversarial worst-case performance over an ambiguity set. This paper instead studies generalization error in the target environment---the excess loss under the shifted target distribution. Our contributions are threefold. First, we derive finite-sample generalization error bounds in the shifted target environment for both DRO and RS. These bounds explicitly characterize the trade-off between reduced sensitivity to shift and the regularization penalty induced by each method's robustness hyperparameter, and they avoid the curse of dimensionality associated with Wasserstein empirical concentration. Second, when partial shift information such as shift magnitude or direction is available, we propose information-directed hyperparameter calibrations and compare the two methods given the same information. Under these calibrations, and in the partial-information regimes we study, DRO and RS exhibit complementary theoretical and empirical behavior. Finally, we apply the framework to a network lot-sizing problem, using it to interpret how robust policies respond to positive shifts in the demand distribution. Together, these results fill a gap in understanding the statistical properties of robust learning methods under distributional shifts and provide a principled basis for comparing DRO and RS.

TOFD: Target-Oriented Feature Decoupling against Poisoning Attacks in Split Federated Learning

Split Federated Learning (SFL) facilitates privacy-preserving collaborative training with reduced client-side overhead. However, its split architecture introduces unique attack surfaces, rendering it vulnerable to diverse poisoning attacks. Most existing defenses fail to exploit the split paradigm, limiting their ability to detect and contain malicious behaviors at an early stage. To bridge this gap, we propose Target-Oriented Feature Decoupling (TOFD), a unified framework that jointly enables proactive detection and robust optimization against a wide range of poisoning attacks. TOFD operates in three stages: (1) Target Inference, which identifies potential attack targets by refining class-wise safe zones via class-specific Margin Perturbation (MP); (2) Sample Purification, which adaptively filters poisoned smashed data using thresholds calibrated through cross-class min-max normalization of MP; and (3) Decoupling Optimization, which leverages an adversarial guidance model to capture attack-induced patterns and decouple their influence during optimization, thereby suppressing residual adversarial effects. We provide theoretical guarantees for the convergence of TOFD. Extensive experiments on five datasets demonstrate that TOFD consistently outperforms state-of-the-art defenses under diverse attack scenarios, achieving superior robustness with low computational overhead suitable for practical deployment.

Posture and Sustainment Optimization Under Adversarial Uncertainty

Pre-commitment posture, the assignment of military assets to theater locations before conflict scenarios resolve, is a critical and formally unsolved problem in joint operational planning. Current practice relies on greedy heuristics that maximize value and ignore geographic coverage and are structurally vulnerable to adversaries that target high-strategic value locations. This paper presents a scenario-weighted adversarially robust posture optimization engine for the Posture and sustainability allocation (PSA) problem, modeled as a finite-horizon Markov Decision Process over assets, theater locations, and time steps. We introduce the Composite Expected Value (CEV) optimizer, which places assets by maximizing scenario-weighted expected posture efficiency over a distribution of threat scenarios, and the RobustCEV extension, which iterates against a Bayesian adversary that updates its targeting distribution in response to observed placement. Across three experiments in an Indo-Pacific basing environment with 20 assets and 5 theater locations, we demonstrate that: (1) the greedy baseline incurs a permanent 25.1% posture efficiency penalty due to geographic under-coverage and a 57.3% scenario-weighted readiness collapse under value-correlated adversarial threat; (2) the CEV optimizer recovers up to 19.8% efficiency over greedy when the threat distribution carries a geographic signal, with a curated set of 5 to 20 scenarios sufficient to capture the majority of this gain; and (3) the RobustCEV extension recovers up to 158% efficiency relative to a naive optimizer when an adaptive adversary employs a deceptive threat prior. All findings are validated using paired t-tests with Bonferroni correction and two-level variance decomposition, confirming that the performance gaps reported are structural properties of placement strategies rather than sampling artifacts.

Equitable System-Prompt Selection via Constrained Mixed-Strategy GroupDRO

Large language models are increasingly used for information seeking, yet semantically equivalent questions phrased in different ways can receive answers of considerably different quality. System prompts are widely employed to steer response behavior, but they are typically optimized for average-case quality, so some question phrasings may still receive incomplete or low-quality answers. To address this, we formulate a constrained mixed-strategy GroupDRO framework for system-prompt selection. Instead of optimizing the system-prompt text, the framework assigns weights to system prompts in an existing pool to minimize the worst-case information-quality loss across evaluation metrics and groups, while constraining the mean loss to stay close to that of average-based selection. Because pool generation and selection are decoupled, the method applies to any system-prompt pool and can leverage an ensemble of complementary system prompts rather than a single one. Across five LLMs on two bilingual medical and consumer-finance benchmarks, the constrained method reduces the Overall Mean, Worst 25% Mean, and Worst by 13.1%, 13.2%, and 13.7% on average relative to no mitigation while keeping overall quality close to Average selection. Its multi-prompt weights reveal complementarity across metric-group pairs. Code and data are available at https://github.com/Rainxu09/equitable-system-prompt-selection.

Robust General Utility for Reinforcement Learning

Reinforcement learning (RL) with general utility extends classic RL by optimizing an arbitrary utility functional of the policy-induced occupancy measure, thereby enabling a broader range of applications. However, previous work on general utility RL typically assumes the evaluation utility is fixed and correctly specified. In practice, the utility used at deployment can deviate from the training one, creating a robustness gap that prior work does not address. Motivated by this, we propose robust general-utility RL, a minimax learning framework that trains policies against utility misspecification within a prescribed uncertainty set. Our framework strictly generalizes standard general-utility RL while also providing a unified view of many existing RL frameworks, including reward-robust RL and constrained RL, through appropriate choices of the utility uncertainty set. We further develop provably convergent stochastic algorithms for two regimes. For concave utilities, we develop a projected stochastic gradient descent-ascent method and establish stationarity guarantees. For the more challenging nonconcave regime, we propose a stochastic prox-extragradient algorithm that mitigates ill-posed behavior induced by nonconcavity, with convergence guarantees to approximate first-order stationarity. Experiments on LLM safety alignment and exploration maximization tasks further corroborate the convergence behavior consistent with our theory.

Expected Survival-Time Bounds for Robust Optimization Over Time under Isotropic Gaussian Dynamics

Robust Optimization Over Time (ROOT) is a recent branch of evolutionary dynamic optimization that seeks solutions capable of remaining effective across multiple consecutive environments. Unlike the traditional track-the-moving-optimum (TMO) paradigm, which reoptimizes after every environmental change, ROOT explicitly values persistence. Although the field has grown considerably, most contributions remain algorithmic and empirical, leaving several fundamental properties poorly understood from a theoretical perspective. One such property is survival time, defined as the number of future environments in which a deployed solution continues to satisfy a prescribed quality threshold. While survival time is widely used as a measure of temporal robustness, little is known about how its expected value depends on environmental dynamics, deployment quality, or problem characteristics. This paper studies expected survival time for a fixed deployed solution under isotropic Gaussian environmental dynamics. Modeling survival as a discrete first-exit problem, we derive a rigorous lower bound and a computable multi-step upper bound. The analysis shows that expected survival scales as $Θ(σ^-{2})$ in slowly varying environments and approaches its minimum value of one future change in high dimensions. A comprehensive Monte Carlo study validates the theoretical predictions, examines sensitivity to modeling assumptions and parameter uncertainty, and illustrates how the bounds can support deployment decisions after optimization. The resulting framework provides an analytical characterization of deployment lifetime and identifies when a required deployment horizon can be guaranteed, ruled out, or remains analytically unresolved.

Smart predict-then-robustly-optimize

In this paper, we propose and study a robust variant of the smart predict-then-optimize approach that accounts for prediction shifts due to disturbance in the covariate feature space. While traditional integrated-learning-and-optimization models assume that side information is perfectly revealed, empirical data-driven features are frequently corrupted or noisy at the time of decision-making, leading to fragile operational policies. To bridge this gap, we integrate principles of robust optimization directly into the predictive-prescriptive pipeline via a smart predict-then-robustly optimize loss and establish a computationally tractable convex surrogate, designed to hedge against worst-case feature perturbations. On the theoretical front, we formalize the structural validity of this surrogate by proving its approximation error probability decays exponentially according to a sub-Gaussian concentration profile. Furthermore, we establish that under mild assumptions, the surrogate is Fisher consistent with high probability. We also prove necessary conditions under which our framework outperforms standard smart predict-then-optimize and maintain its superiority even when the standard method is equipped with regularized upstream predictions. Numerical experiments validate that our robust framework consistently yields significant performance improvements over standard methods, both in out-of-sample terms and in training stability.

Robust Peak-cost Constrained Reinforcement Learning

We study robust peak-cost constrained reinforcement learning (RP-CRL), where the objective is to maximize expected reward while controlling the maximum cost encountered along a trajectory. This setting is motivated by safety-critical applications in which a single large violation can be catastrophic and therefore cannot be adequately captured by the standard CMDP framework based on expected cumulative cost. Existing reachability-constrained RL methods adopt Lagrangian-based approaches, yet the underlying duality properties of peak-cost constrained MDPs remain unclear. We show that, unlike standard CMDPs, peak-cost constrained MDPs may not admit zero duality gap. We further consider a robust formulation to address simulator-to-real-world mismatch in the transition dynamics. To solve this problem, we develop a surrogate optimization framework and a robust value estimation method based on integral probability metrics. We prove that, with appropriate hyperparameter choices, the surrogate solution attains the same robust reward value as the original problem while violating the constraint by at most epsilon. Experiments show that the proposed method effectively enforces safety under dynamics perturbations while retaining strong reward performance.

Maximally Robust Satisficing Bayesian Optimization

Many design tasks can be cast as black-box function optimization, enabling use of Bayesian optimization to find an ideal design with minimal number of trials. However, often we do not actually need the optimum but instead a sufficiently good solution is enough, for instance a material that is durable enough for its intended use. In most cases there are multiple satisfactory solutions, forming a superlevel set of the function, raising a key question of which one to prefer. We answer this by explaining why robustness to input perturbations that may occur when the solution is deployed is a good criterion and by introduce a Bayesian optimization method that efficiently finds satisficing solutions that are robust to maximally large perturbations. In contrast to previous works, we assume the inputs can be accurately controlled during optimization, but will be perturbed after the deployment.

SLAM: Structured and Localized Analytic Manifold Adaptation for Lifelong VPR

Visual Place Recognition (VPR) in lifelong deployment requires continuous adaptation to new environments without catastrophic forgetting. In this paper, we propose SLAM, a Structured and Localized Analytic Manifold adaptation framework. Our framework elegantly unifies uncertainty-aware smoothing via Unscented transformation, topological space partitioning through a Gaussian Mixture Model (GMM), and $H_\infty$ robust bound optimization into a singular, unified closed-form analytical recursion. Exhaustive ablation studies demonstrate that while the synergistic combination of uncertainty smoothing and localized mapping (U+G configuration) achieves the state-of-the-art nominal accuracy of 27.5%, the full deployment of the $H_\infty$ bound does not require an architectural split; rather, it introduces a mathematically guaranteed minimax robust bound. This formulation enables the system to seamlessly modulate the intrinsic trade-off between nominal placement precision and worst-case disturbance attenuation through a single regularization parameter.

Robust Metaheuristics under Uncertainty for Berth Allocation and Quay Crane Assignment: A Review

The berth allocation and quay crane assignment problem (BACAP) is a representative port-terminal scheduling problem in maritime transportation and freight logistics, where vessel arrivals, berth positions, service durations, and quay?crane availability are tightly coupled. Under uncertainties such as arrival deviations, handling-time fluctuations, and resource disruptions, schedules optimized under nominal assumptions may become fragile during execution, motivating the study of robust metaheuristic optimization for BACAP in port-terminal operations. Although population-based metaheuristics have been widely used for BACAP and related port-scheduling problems, existing studies remain fragmented in their uncertainty repre?sentations, robustness criteria, search mechanisms, and empir?ical evaluation protocols. To the best of our knowledge, this paper provides the first focused review dedicated to robust population-based metaheuristics for BACAP under uncertainty. We first summarize uncertainty sources and information repre?sentations in BACAP, and then organize existing methods from a mechanism-oriented perspective, covering solution representation and decoding, robust evaluation and selection, robustness-guided search dynamics, and feasibility preservation and recovery. We further present a benchmark suite for uncertain BACAP to support controlled empirical comparison and report illustrative baseline results by combining representative metaheuristics with different robustness strategies. Finally, we identify open chal?lenges related to benchmark extension, robustness-aware search design, time-adaptive robustness, and non-stationary uncertainty.

Distributionally Robust Listwise Preference Optimization

Existing robust preference optimization for language-model alignment mainly studies pairwise supervision and places robustness at the dataset, prompt, or preference-pair level. We instead study listwise preference optimization under ranking-label uncertainty: given a prompt and a candidate list, the observed ranking over that list may be ambiguous due to annotator inconsistency, near-ties, lossy rankwise feedback, or reward-model noise. We propose a pointwise total-variation robust Plackett--Luce objective that directly robustifies the ranking label conditional on the candidate list. The robust loss admits an exact decomposition into the nominal PL loss plus a worst-case PL correction, and the worst-case ranking is obtained by sorting current implicit scores in ascending order, reducing the inner maximization from $K!$ enumeration to $O(K\log K)$. This tractable structure yields strong offline and online optimization guarantees. In the offline fixed-list setting, the robust objective is convex and projected stochastic subgradient reaches global $ε$-suboptimality with $O(ε^{-2})$ sample complexity. In the online policy-induced setting, where candidate lists are generated by the current policy, we establish weak convexity and $\widetilde O(ε^{-2})$ Moreau-envelope stationarity. Experiments in offline LLM alignment show that the proposed robust correction largely preserves performance under clean labels and improves robustness under noise. In online alignment, it makes reward-model-ranked candidate expansion more reliable and improves both reward-model and external GPT-4 judge metrics.

Is Variational Monte Carlo Robust? Sharp Moment Thresholds and Heavy-tailed Stochastic Optimization

Variational Monte Carlo (VMC) is a central algorithm in electronic structure theory and has gained renewed importance through modern neural-network ansätze such as FermiNet. At its core, VMC seeks ground states by minimizing the Rayleigh quotient by stochastic optimization. In this work, we show that the resulting stochastic optimization problem is intrinsically governed by the nodal geometry of the underlying wave function. More precisely, we establish that properties of the nodal set determine the integrability of the local energy and gradient estimators that drive VMC. For broad and practically relevant ansatz classes, including Slater-Jastrow wave functions with variable-exponent Slater-type orbitals, we prove that these estimators are generically heavy-tailed and fail to admit higher moments. At the same time, for general analytic ansätze, we prove weak moment bounds for the relevant estimators and identify precise low-moment regimes, showing how generic and degenerate nodal structures lead to different integrability thresholds. Building on this analysis, we introduce a new robust variant of VMC $\unicode{x2013}$ coined PS-Clip-VMC $\unicode{x2013}$ which is based on clipping both the local energy and the gradient random variable. We prove that PS-Clip-VMC converges both in expectation and with high probability in the weak moment regime of VMC. Preliminary experiments for training FermiNet on Atoms with up to 18 electrons suggest that PS-Clip-VMC is significantly more robust than standard methods.

Generative Robust Optimisation

Classical uncertainty sets for robust optimisation impose fixed geometric shapes that cannot represent the complex dependencies present in real-world data. We propose Generative Robust Optimisation (GRO), a framework in which a deep generative model defines the uncertainty set as the image of a neural network decoder over a calibrated latent set, naturally accommodating nonlinear correlations, asymmetry, and multimodality. A five-point evaluation framework (reconstruction fidelity, distribution matching, latent regularity, robust relevance, and computational tractability) provides systematic, model-agnostic criteria for assessing any neural network-based uncertainty set. We instantiate this framework with a Wasserstein Adversarial Autoencoder employing Gaussian mixture model-guided training for latent regularity and constraint-consistency regularisation for robust relevance. Restricting the decoder to ReLU activations enables exact worst-case verification through mixed-integer programming embedding. Extensive experiments on a production planning problem across six uncertainty distributions and six generative architectures, together with a multi-period facility location study, validate the framework and demonstrate that systematic attention to all five criteria yields uncertainty sets that are simultaneously expressive, well-calibrated, and optimisation-tractable.

Distribution-Aware Robust Bilevel Optimization: Quantile-Guided Huber Updates in Two-Timescale Stochastic Approximation

Bilevel optimization (BLO) is fundamental to hierarchical decision-making but suffers from critical instability under heavy-tailed stochastic noise. Existing variance-reduction techniques typically rely on myopic magnitude checks, which fail to distinguish informative geometric signals from impulsive outliers. To resolve this, we propose \textbf{RQ-TTSA} (Robust Quantile-guided TTSA), a distribution-aware framework that leverages historical gradient buffers to estimate rolling quantiles for adaptive Huber-style clipping, effectively preserving local optimization geometry while strictly bounding effective variance. Theoretically, we provide a convergence analysis for quantile-guided TTSA under nonconvex-strongly convex assumptions with infinite-variance noise ($p \in (1,2]$), deriving a rate of $\mathcal{O}(T^{-\frac{p-1}{3p-2}})$ that recovers optimal dependence on the heavy-tailed parameter. Empirically, across six diverse tasks, spanning heterogeneous vision benchmarks, dynamic games under momentum poisoning, and offline reinforcement learning, RQ-TTSA consistently outperforms state-of-the-art baselines by eliminating divergence spikes and ensuring stable convergence. Our method demonstrates significant robustness to hyperparameter variations and incurs negligible computational overhead ($\approx 2.7\%$ increase), validating distribution-aware gradient control as a practical and necessary component for reliable bilevel learning.

Robust $Q$-learning for mean-field control under Wasserstein uncertainty in common noise

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.

A Human-in-the-Loop Bayesian Optimization Framework for Constraint-Aware Bioprocess Development

This work presents an extension to Pareto Front Guided Sampling (PFGS), a Human-in-the-Loop (HitL) Bayesian Optimization (BO) framework in which Gaussian process (GP) surrogate-derived quantities are reformulated as objectives of a multi-objective optimization problem, and the resulting Pareto front is exposed to a domain expert for interactive candidate selection rather than returning a single automated recommendation. The framework is extended in two directions: constrained optimization is addressed by incorporating the posterior probability of satisfying output specification limits as an explicit Pareto objective, computed analytically from the GP posterior distribution; robust optimization is addressed by a Monte Carlo sampling strategy that estimates expected lower-confidence performance over a user-defined variability of input perturbations, capturing performance degradation under likely implementation deviations. The resulting multi-dimensional Pareto representation renders trade-offs between predicted performance, model uncertainty, probabilistic constraint satisfaction, and input robustness simultaneously visible through pairwise two-dimensional projections on an interactive dashboard, enabling selection criteria to be iteratively refined as the surrogate model improves and development objectives evolve. The framework is showcased on an eight-dimensional fed-batch Chinese Hamster Ovary (CHO) cell culture simulator demonstrating systematic identification of high-performing, feasibility-compliant, and perturbation-resilient operating conditions, and illustrating how expert-defined requirements provide a principled stopping criterion and support informed allocation of experimental resources.

Which Directions Matter? Sparse Design for Affine Robust Optimization

Robust machine learning and optimization rely on the uncertainty model choice. We investigate which uncertainty directions a model must cover when defined by a finite dictionary and a budget constraint. Selecting a subset forms an atomic uncertainty set with a closed form support function, yielding tractable robust programs for affine objectives. We propose a data driven selection rule based on a coverage objective over evaluation directions, including gradients, adversarial perturbations, or shifts observed on held out data. We prove this objective is monotone and submodular, supporting a greedy method with a $(1-1/e)$ approximation guarantee and a matching hardness barrier. We also provide a certificate bounding the loss from the selected subset and a radius calibration rule with out of sample control.

A Robust Optimization Approach to Sparse Principal Component Analysis

While principal component analysis (PCA) is a fundamental tool for dimensionality reduction, its dense representations make it ill-suited for high-dimensional data. Existing methods address this by promoting sparsity through explicit $\ell_1$-penalties, but these are not obvious to tune due to the unsupervised nature of the task. In contrast, we propose Adversarial PCA (AdvPCA), which leverages robust optimization to achieve sparsity by optimizing the reconstruction objective against bounded, worst-case latent space perturbations. We show that this formulation admits a closed-form reduction, leading to a practical iterative algorithm that alternates between adversarial linear regression-style updates for the sparse encoder and orthogonal updates for the decoder. By theoretically characterizing the solution, we derive a data-adaptive parameterization that allows the algorithm to perform effectively out of the box. We validate these claims through numerical experiments on synthetic and real-world genomics data.

Decision-calibrated prediction sets for robust power system operations

Robust optimization offers a tractable approach to balance operating costs and reliability in power systems dominated by weather-dependent renewable uncertainty, but its performance depends critically on the uncertainty set. Standard data-driven approaches often calibrate uncertainty sets to attain predictive coverage, which can produce unnecessarily large sets and costly operating decisions. In contrast, we introduce decision-calibrated prediction sets and embed them as uncertainty sets in robust optimization problems; these are conditional multivariate prediction sets where calibration is defined in terms of the reliability of downstream decisions, rather than in terms of the coverage. First, we learn these conditional prediction sets as sub-level sets of norm-based score functions represented by partially input-convex neural networks, capturing contextual information and multivariate dependence while preserving convexity and tractability in downstream robust formulations. Second, inspired by conformal risk control, we calibrate a score-threshold parameter that sets the volume of the uncertainty set, thereby controlling the expected violations of downstream operational constraints. We apply our approach to 15-minute-ahead reserve scheduling with network-constrained deliverability, which we formulate as a robust DC optimal power flow problem with affine recourse. Numerical experiments show that decision-calibrated sets attain prescribed constraint-satisfaction targets within about three percentage points, whereas standard coverage-based calibration systematically exceeds these targets by more than eleven percentage points, leading to larger sets and higher operating costs.

Robust and Fast Training via Per-Sample Clipping

We propose a robust gradient estimator based on per-sample gradient clipping and analyze its properties both theoretically and empirically. We show that the resulting method, per-sample clipped SGD (PS-Clip-SGD), achieves optimal in-expectation convergence rates for non-convex optimization problems under heavy-tailed gradient noise. Moreover, we establish high-probability convergence guarantees that match the in-expectation rates up to polylogarithmic factors in the failure probability. We complement our theoretical results with multiple numerical experiments. In particular, we demonstrate that PS-Clip-SGD outperforms both vanilla SGD with momentum and standard gradient clipping when training AlexNet on the CIFAR-100 dataset, even after accounting for the additional computational time caused by per-sample clipping. We also empirically show that, in the presence of gradient accumulation, applying clipping at the mini-batch level can improve training performance while incurring virtually no additional computational cost. This finding is particularly interesting, as it contradicts the common practice of applying clipping only after all accumulation steps have been completed.

ANO: A Principled Approach to Robust Policy Optimization

Proximal Policy Optimization (PPO) dominates deep RL but faces a fundamental dilemma. Its "hard clipping" mechanism discards valuable gradient information from outliers, leading to sample inefficiency. Conversely, removing clipping (as in SPO) exposes optimization to unbounded gradients, causing significant instability and hyperparameter sensitivity. To resolve this, we establish a Unified Trust Region Framework that generalizes existing objectives. Within this framework, we derive Anchored Neighborhood Optimization (ANO) based on a set of design principles. We identify that the failure of standard policy gradients stems from a misapplication of gradient influence on outliers. We propose the Redescending Influence Principle, a paradigm shift from monotonic penalties (SPO) and hard-thresholding (PPO) to dynamic outlier suppression, and prove its necessity for stability in high-variance stochastic optimization. Theoretically, we prove ANO possesses the minimal structural complexity required for robust optimization. Empirically, ANO achieves state-of-the-art performance on MuJoCo benchmarks, significantly outperforming PPO and SPO. Notably, ANO demonstrates superior stability, preventing policy collapse even under aggressive hyperparameters (e.g., learning rates 3x larger than standard) where PPO fails completely.

Semi-Markov Reinforcement Learning for City-Scale EV Ride-Hailing with Feasibility-Guaranteed Actions

We study city-scale control of electric-vehicle (EV) ride-hailing fleets where dispatch, repositioning, and charging decisions must respect charger and feeder limits under uncertain, spatially correlated demand and travel times. We formulate the problem as a hex-grid semi-Markov decision process (semi-MDP) with mixed actions -- discrete actions for serving, repositioning, and charging, together with continuous charging power -- and variable action durations. To guarantee physical feasibility during both training and deployment, the policy learns over high-level intentions produced by a masked, temperature-annealed actor. These intentions are projected at every decision step through a time-limited rolling mixed-integer linear program (MILP) that strictly enforces state-of-charge, port, and feeder constraints. To mitigate distributional shifts, we optimize a Soft Actor--Critic (SAC) agent against a Wasserstein-1 ambiguity set with a graph-aligned Mahalanobis ground metric that captures spatial correlations. The robust backup uses the Kantorovich--Rubinstein dual, a projected subgradient inner loop, and a primal--dual risk-budget update. Our architecture combines a two-layer Graph Convolutional Network (GCN) encoder, twin critics, and a value network that drives the adversary. Experiments on a large-scale EV fleet simulator built from NYC taxi data show that PD--RSAC achieves the highest net profit, reaching \$1.22M, compared with \$0.58M--\$0.70M for strong heuristic, single-agent RL, and multi-agent RL baselines, including Greedy, SAC, MAPPO, and MADDPG, while maintaining zero feeder-limit violations.