Zeroth-order (ZO) methods are widely used when gradients are unavailable or prohibitively expensive, including black-box learning and memory-efficient fine-tuning of large models, yet their optimization dynamics in deep learning remain underexplored. In this work, we provide an explicit step size condition that exactly captures the (mean-square) linear stability of a family of ZO methods based on the standard two-point estimator. Our characterization reveals a sharp contrast with first-order (FO) methods: whereas FO stability is governed solely by the largest Hessian eigenvalue, mean-square stability of ZO methods depends on the entire Hessian spectrum. Since computing the full Hessian spectrum is infeasible in practical neural network training, we further derive tractable stability bounds that depend only on the largest eigenvalue and the Hessian trace. Empirically, we find that full-batch ZO methods operate at the edge of stability: ZO-GD, ZO-GDM, and ZO-Adam consistently stabilize near the predicted stability boundary across a range of deep learning training problems. Our results highlight an implicit regularization effect specific to ZO methods, where large step sizes primarily regularize the Hessian trace, whereas in FO methods they regularize the top eigenvalue.

In online clustering problems, there is often a large amount of uncertainty over possible cluster assignments that cannot be resolved until more data are observed. This difficulty is compounded when clusters follow complex distributions, as is the case with text data. Sequential Monte Carlo (SMC) methods give a natural way of representing and updating this uncertainty over time, but have prohibitive memory requirements for large-scale problems. We propose a novel SMC algorithm that decomposes clustering problems into approximately independent subproblems, allowing a more compact representation of the algorithm state. Our approach is motivated by the knowledge base construction problem, and we show that our method is able to accurately and efficiently solve clustering problems in this setting and others where traditional SMC struggles.

Data-driven operations management often relies on parameters estimated from costly human-generated labels. Recent advances in large language models (LLMs) and other AI systems offer inexpensive auxiliary data, but introduce a new challenge: AI outputs are not direct observations of the target outcomes, but could involve high-dimensional representations with complex and unknown relationships to human labels. Conventional methods leverage AI predictions as direct proxies for true labels, which can be inefficient or unreliable when this relationship is weak or misspecified. We propose Generative Augmented Inference (GAI), a general framework that incorporates AI-generated outputs as informative features for estimating models of human-labeled outcomes. GAI uses an orthogonal moment construction that enables consistent estimation and valid inference with flexible, nonparametric relationship between LLM-generated outputs and human labels. We establish asymptotic normality and show a "safe default" property: relative to human-data-only estimators, GAI weakly improves estimation efficiency under arbitrary auxiliary signals and yields strict gains whenever the auxiliary information is predictive. Empirically, GAI outperforms benchmarks across diverse settings. In conjoint analysis with weak auxiliary signals, GAI reduces estimation error by about 50% and lowers human labeling requirements by over 75%. In retail pricing, where all methods access the same auxiliary inputs, GAI consistently outperforms alternative estimators, highlighting the value of its construction rather than differences in information. In health insurance choice, it cuts labeling requirements by over 90% while maintaining decision accuracy. Across applications, GAI improves confidence interval coverage without inflating width. Overall, GAI provides a principled and scalable approach to integrating AI-generated information.

We study bandit best-arm identification with arbitrary and potentially adversarial rewards. A simple random uniform learner obtains the optimal rate of error in the adversarial scenario. However, this type of strategy is suboptimal when the rewards are sampled stochastically. Therefore, we ask: Can we design a learner that performs optimally in both the stochastic and adversarial problems while not being aware of the nature of the rewards? First, we show that designing such a learner is impossible in general. In particular, to be robust to adversarial rewards, we can only guarantee optimal rates of error on a subset of the stochastic problems. We give a lower bound that characterizes the optimal rate in stochastic problems if the strategy is constrained to be robust to adversarial rewards. Finally, we design a simple parameter-free algorithm and show that its probability of error matches (up to log factors) the lower bound in stochastic problems, and it is also robust to adversarial ones.

When considering a model selection or, more generally, an aggregation approach for adaptive statistical inference, it is often necessary to compute estimators over a wide range of model complexities including unnecessarily large models even when the true data-generating process is relatively simple, due to the lack of prior knowledge. This requirement can lead to substantial computational inefficiency. In this work, we propose a novel framework for efficient model aggregation called the early-stopped aggregation (ESA): instead of computing and aggregating estimators for all candidate models, we compute only a small number of simpler ones using an early-stopping criterion and aggregate only these for final inference. Our framework is versatile and applies to both Bayesian model selection, in particular, within the variational Bayes framework, and frequentist estimation, including a general penalized estimation setting. We investigate adaptive optimal property of the ESA approach across three learning paradigms. We first show that ESA achieves optimal adaptive contraction rates in the variational Bayes setting under mild conditions. We extend this result to variational empirical Bayes, where prior hyperparameters are chosen in a data-dependent manner. In addition, we apply the ESA approach to frequentist aggregation including both penalization-based and sample-splitting implementations, and establish corresponding theory. As we demonstrate, there is a clear unification between early-stopped Bayes and frequentist penalized aggregation, with a common "energy" functional comprising a data-fitting term and a complexity-control term that drives both procedures. We further present several applications and numerical studies that highlight the efficiency and strong performance of the proposed approach.

We study downlink beam and rate adaptation in a multi-user mmWave MISO system where multiple base stations (BSs), each using analog beamforming from finite codebooks, serve multiple single-antenna user equipments (UEs) with a unique beam per UE and discrete data transmission rates. BSs learn about transmission success based on ACK/NACK feedback. To encode service goals, we introduce a satisficing throughput threshold $τ_r$ and cast joint beam and rate adaptation as a combinatorial semi-bandit over beam-rate tuples. Within this framework, we propose SAT-CTS, a lightweight, threshold-aware policy that blends conservative confidence estimates with posterior sampling, steering learning toward meeting $τ_r$ rather than merely maximizing. Our main theoretical contribution provides the first finite-time regret bounds for combinatorial semi-bandits with satisficing objective: when $τ_r$ is realizable, we upper bound the cumulative satisficing regret to the target with a time-independent constant, and when $τ_r$ is non-realizable, we show that SAT-CTS incurs only a finite expected transient outside committed CTS rounds, after which its regret is governed by the sum of the regret contributions of restarted CTS rounds, yielding an $O((\log T)^2)$ standard regret bound. On the practical side, we evaluate the performance via cumulative satisficing regret to $τ_r$ alongside standard regret and fairness. Experiments with time-varying sparse multipath channels show that SAT-CTS consistently reduces satisficing regret and maintains competitive standard regret, while achieving favorable average throughput and fairness across users, indicating that feedback-efficient learning can equitably allocate beams and rates to meet QoS targets without channel state knowledge.

We study post-training interpretability for Support Vector Machines (SVMs) built from truncated orthogonal polynomial kernels. Since the associated reproducing kernel Hilbert space is finite-dimensional and admits an explicit tensor-product orthonormal basis, the fitted decision function can be expanded exactly in intrinsic RKHS coordinates. This leads to Orthogonal Representation Contribution Analysis (ORCA), a diagnostic framework based on normalized Orthogonal Kernel Contribution (OKC) indices. These indices quantify how the squared RKHS norm of the classifier is distributed across interaction orders, total polynomial degrees, marginal coordinate effects, and pairwise contributions. The methodology is fully post-training and requires neither surrogate models nor retraining. We illustrate its diagnostic value on a synthetic double-spiral problem and on a real five-dimensional echocardiogram dataset. The results show that the proposed indices reveal structural aspects of model complexity that are not captured by predictive accuracy alone.

As deep neural networks are deployed in safety-critical domains such as autonomous driving and medical diagnosis, stakeholders need explanations that are interpretable but also trustworthy with formal guarantees. Existing XAI methods fall short: heuristic attribution techniques (e.g., LIME, Integrated Gradients) highlight influential features but offer no mathematical guarantees about decision boundaries, while formal methods verify robustness yet remain untargeted, analyzing the nearest boundary regardless of whether it represents a critical risk. In safety-critical systems, not all misclassifications carry equal consequences; confusing a "Stop" sign for a "60 kph" sign is far more dangerous than confusing it with a "No Passing" sign. We introduce ViTaX (Verified and Targeted Explanations), a formal XAI framework that generates targeted semifactual explanations with mathematical guarantees. For a given input (class y) and a user-specified critical alternative (class t), ViTaX: (1) identifies the minimal feature subset most sensitive to the y->t transition, and (2) applies formal reachability analysis to guarantee that perturbing these features by epsilon cannot flip the classification to t. We formalize this through Targeted epsilon-Robustness, certifying whether a feature subset remains robust under perturbation toward a specific target class. ViTaX is the first method to provide formally guaranteed explanations of a model's resilience against user-identified alternatives. Evaluations on MNIST, GTSRB, EMNIST, and TaxiNet demonstrate over 30% fidelity improvement with minimal explanation cardinality.

Applying kernel methods to matchings is challenging due to their discrete, non-Euclidean nature. In this paper, we develop a principled framework for constructing geometric kernels that respect the natural geometry of the space of matchings. To this end, we first provide a complete characterization of stationary kernels, i.e. kernels that respect the inherent symmetries of this space. Because the class of stationary kernels is too broad, we specifically focus on the heat and Matérn kernel families, adding an appropriate inductive bias of smoothness to stationarity. While these families successfully extend widely popular Euclidean kernels to matchings, evaluating them naively incurs a prohibitive super-exponential computational cost. To overcome this difficulty, we introduce and analyze a novel, sub-exponential algorithm leveraging zonal polynomials for efficient kernel evaluation. Finally, motivated by the known bijective correspondence between matchings and phylogenetic trees-a crucial data modality in biology-we explore whether our framework can be seamlessly transferred to the space of trees, establishing novel negative results and identifying a significant open problem.

We derive a robust update rule for the online infinite hidden Markov model (iHMM) for when the streaming data contains outliers and the model is misspecified. Leveraging recent advances in generalised Bayesian inference, we define robustness via the posterior influence function (PIF), and provide conditions under which the online iHMM has bounded PIF. Imposing robustness inevitably induces an adaptation lag for regime switching. Our method, which is called Batched Robust iHMM (BR-iHMM), balances adaptivity and robustness with two additional tunable parameters. Across limit order book data, hourly electricity demand, and a synthetic high-dimensional linear system, BR-iHMM reduces one-step-ahead forecasting error by up to 67% relative to competing online Bayesian methods. Together with theoretical guarantees of bounded PIF, our results highlight the practicality of our approach for both forecasting and interpretable online learning.