Large-scale hypothesis testing is central to modern science, where controlling the False Discovery Rate (FDR) has become the standard approach to managing false positives across many simultaneous tests. Hypotheses rarely exist in isolation; they often exhibit structure through proximity, connectivity, or hierarchy. This structure represents both a challenge and an opportunity: while classical methods treat these dependencies as obstacles requiring conservative correction, leveraging them can substantially increase discovery power. Here, we reframe structured FDR control as a regularized learning problem. By optimizing within a suitable Reproducing Kernel Hilbert Space (RKHS), we introduce a framework that unifies continuous domains, graphs, and hierarchies under a single algorithm through kernel choice alone. This formulation enables smooth solutions in place of the piecewise-constant fits of prior methods, principled likelihood-based hyperparameter selection rather than heuristic tuning, and inference at unobserved locations which in turn supports sample-efficient experimental design. Building on this estimator, we provide two decision rules which we prove to control the FDR. We validate our method on two sources: spatial locations derived from high-dimensional real-world datasets, and a differential gene expression task utilizing protein-protein interaction graphs.

Many real-world combinatorial problems involve uncertain parameters, which can be predicted given contextual features and historical data. These `predict-then-optimize' or `contextual optimization' problems have gained significant attention: end-to-end training methods can now minimize the downstream task cost rather than the predictive error. However, despite their effectiveness, these decision-focused learning (DFL) approaches often rely on repeated solving of the underlying combinatorial optimization problem during training, making them computationally expensive and difficult to scale. We reframe the learning problem as a cost-sensitive multi-output regression problem: multi-output due to the combinatorial problem having multiple uncertain parameters, and cost-sensitive due to the downstream task cost being the real target. Our technical contribution is the formalization of multiple loss function components that follow from this reframing: cost-insensitive normalization, decision-aware asymmetric penalization of over- and underpredictions, and instance-based costs that mimic the true downstream task-based loss locally. These components require zero or one solve per training data instance, while requiring no further solves during training. Experiments show that the combination of loss components achieves comparable downstream task quality to the state of the art, while being significantly more efficient, enabling scaling to problem sizes that have not been tackled before with DFL.

Muon has recently emerged as a strong alternative to AdamW for training neural networks, with encouraging large-scale pretraining results and growing evidence that matrix-structured updates can be faster in practice. Yet Muon, and more generally Linear Minimization Oracle (LMO) based methods, are typically used synchronously. This is problematic in heterogeneous distributed systems, where workers complete gradient computations at different speeds and synchronous training must repeatedly wait for slower workers. In this work, we introduce Ringmaster LMO, an asynchronous LMO-based momentum method for unconstrained stochastic nonconvex optimization. Our method builds on the delay-thresholding idea of Ringmaster ASGD. For SGD-type methods, Ringmaster ASGD achieves optimal time complexity by discarding overly stale gradients. Ringmaster LMO extends this mechanism to general LMO-based updates. We establish convergence guarantees under generalized $(L_0, L_1)$-smoothness and further develop a parameter-agnostic variant with decreasing stepsizes and adaptive delay thresholds. Finally, we translate our iteration guarantees into time complexity bounds under heterogeneous worker computation times. In the classical Euclidean smooth setting, these bounds recover the optimal time complexity of Ringmaster ASGD. Experiments on stochastic quadratic problems and NanoChat language-model pretraining show that the advantages of Ringmaster LMO grow with system heterogeneity and that the method outperforms strong synchronous and asynchronous baselines.

Learning dynamical systems through operator-theoretic representations provides a powerful framework for analyzing complex dynamics, as spectral quantities such as eigenvalues and invariant structures encode characteristic time scales and long-term behavior. However, dynamical operators are typically estimated independently for each system, preventing the discovery of shared structure across related dynamics. To address this limitation, we posit that related dynamical systems lie near a low-dimensional manifold in spectral operator space. Based on this hypothesis, we introduce DOODL (Dynamical OperatOr Dictionary Learning), a framework that learns a dictionary of characteristic spectral dynamics whose combinations approximate this manifold and yield compact, interpretable embeddings of individual systems. Beyond representation learning, DOODL enables fast and interpretable operator estimation from short and partially observed trajectories by constraining the estimation to the learned operator manifold. Experiments on metastable Langevin dynamics and turbulent plasma simulations demonstrate that DOODL scales to highly complex multiscale regimes while capturing characteristic spectral structure governing the dynamics rather than merely fitting trajectories, achieving errors one to two orders of magnitude lower than independent operator estimation methods in challenging low-data regimes.

Directed Acyclic Graphs (DAGs) are central to uncovering causal structure in complex systems, yet learning a single DAG from data is often challenging: model uncertainty, finite samples, and a combinatorially large search space frequently yield unstable estimates. We propose DAGgr, a model averaging framework that aggregates multiple candidate DAGs into a single stable representation. Candidate graphs are weighted by their out-of-sample predictive likelihood across repeated data splits, and a thresholding rule on the resulting edge-importance scores guarantees that the aggregated graph is itself acyclic. We establish a finite-sample risk bound, prove that the procedure preserves acyclicity, and show that edge selection is consistent under mild conditions on the weights. Simulations across random, hub, and chain structures, together with an analysis of the Sachs et al. (2005) protein-signaling network, show that DAGgr matches or exceeds the best individual candidate while consistently outperforming bootstrap-aggregation baselines across structural recovery metrics.

Conformal prediction provides a principled framework for uncertainty quantification with finite-sample coverage guarantees. While recent work has extended conformal prediction to online and sequential settings, existing methods typically focus on a single coverage level and do not ensure consistency across multiple confidence levels. In many real-world applications, such as weather forecasting, macroeconomic prediction, and risk management, different users operate under heterogeneous risk tolerances and require calibrated uncertainty estimates across a range of coverage levels. In such settings, it is desirable to produce prediction sets corresponding to different coverage levels that are nested and valid simultaneously. In this paper, we propose two novel online conformal prediction methods that output \emph{nested prediction sets} across a range of coverage levels, enabling simultaneous uncertainty quantification across the entire risk spectrum. Beyond interpretability, jointly estimating multiple coverage levels is known to improve statistical efficiency in classical quantile regression by enforcing non-crossing constraints and sharing information across quantiles. Our approaches leverage an online optimization perspective with small regret that translates to quantile estimation error control while enforcing nestedness of prediction sets. Empirical results on synthetic and real-world datasets, including applications in forecasting tasks with heterogeneous risk requirements, demonstrate that our method achieves stable coverage across all levels, strictly nested prediction sets, and improved efficiency compared to existing online conformal baselines.

Experimental design has emerged as a powerful approach for improving the sample efficiency of A/B testing, yet existing designs rely critically on correctly specified models. We study robust sequential experimental design under model misspecification and develop a unified framework that covers both contextual bandit and dynamic settings. Theoretically, we prove that our design bounds the worst-case mean squared error of the estimated treatment effect. Empirically, we demonstrate the effectiveness of the proposed approach using synthetic and real-world datasets from a leading technology company.

Diffusion models typically generate image batches from independent Gaussian initial noises. We argue that this independence assumption is only one choice within a broader class of valid joint noise designs. Instead, one can specify a coupling of the initial noises: each noise remains marginally standard Gaussian, so the pretrained diffusion model receives the same single-sample input distribution, while the dependence across samples is chosen by design. This reframes initial-noise control from selecting or optimizing individual seeds to designing the dependence structure of a multi-sample gallery. This view gives a general framework for initial-noise design, covering several existing methods as special cases and leading naturally to new coupled-noise constructions. Coupled noise can improve generation on its own without adding sampling cost, and it is flexible enough to serve as a structured initialization for optimization-based pipelines when additional computation is available. Empirically, repulsive Gaussian coupling improves gallery diversity on SD1.5, SDXL, and SD3 while largely preserving prompt alignment and image quality. It matches or outperforms recent test-time noise-optimization baselines on several diversity metrics at the same sampling cost as independent generation. Subspace couplings also support fixed-object background generation, producing diverse, natural backgrounds compared with specialized inpainting baselines, with a tunable trade-off in foreground fidelity.

Tree ensembles such as random forests (RFs) and gradient boosting machines (GBMs) are among the most widely used supervised learners, yet their theoretical properties remain incompletely understood. We adopt a spectral perspective on these algorithms, with two main contributions. First, we derive minimax-optimal convergence for RF regression, showing that, under mild regularity conditions on tree growth, the eigenvalue decay of the induced kernel operator governs the statistical rate. Second, we exploit this spectral viewpoint to develop compression schemes for tree ensembles. For RFs, leading eigenfunctions of the kernel operator capture the dominant predictive directions; for GBMs, leading singular vectors of the smoother matrix play an analogous role. Learning nonlinear maps for these spectral representations yields distilled models that are orders of magnitude smaller than the originals while maintaining competitive predictive performance. Our methods compare favorably to state of the art algorithms for forest pruning and rule extraction, with applications to resource constrained computing.

We study optimal policy learning under combined budget and minimum coverage constraints. We show that the problem admits a knapsack-type structure and that the optimal policy can be characterized by an affine threshold rule involving both budget and coverage shadow prices. We establish that the linear programming relaxation of the combinatorial solution has an O(1) integrality gap, implying asymptotic equivalence with the optimal discrete allocation. Building on this result, we analyze two implementable approaches: a Greedy-Lagrangian (GLC) and a rank-and-cut (RC) algorithm. We show that the GLC closely approximates the optimal solution and achieves near-optimal performance in finite samples. By contrast, RC is approximately optimal whenever the coverage constraint is slack or costs are homogeneous, while misallocation arises only when cost heterogeneity interacts with a binding coverage constraint. Monte Carlo evidence supports these findings.