Papers

Time series causal discovery — the process of detecting causation between time series — has its roots in signal processing research from over a half century ago. It has since received active and widespread interest, evolving into different forms across domains such as statistics, dynamical systems and machine learning. Entering this fragmented and constantly evolving research landscape can be daunting. This tutorial provides a unified overview of time series causal discovery across research communities. We introduce the key principles invoked by modern discovery methods, and we discuss their limitations and assumptions. We also discuss how discovery methods are evaluated, benchmarked and fine-tuned. We further discuss how causal discovery for time series is becoming crucial for downstream tasks in agriculture and Earth science, in healthcare, and in artificial intelligence systems. We believe that this tutorial will serve as a valuable resource in navigating the deluge of recent causality research, as well as an invitation to signal processing audiences to contribute to causal problems. To further help readers, we host a companion repository to link to useful code packages and resources at https://github.com/CHAI-UK/Awesome-Time-Series-Causal-Discovery.

We develop a model-free, noise-aware Bayesian clustering framework that constructs a martingale posterior over clusterings from a predictive Bayesian perspective. This avoids explicit likelihood-prior model specification, expensive MCMC sampling, and mode-seeking variational approximations, while accommodating irregular cluster shapes and enabling parallel generation of posterior draws. To interpret the posterior over high-dimensional, noise-aware clusterings, we introduce a Wasserstein-type summarization method. Using a novel noise-aware loss, this approach compresses the large collection of posterior draws into a small set of weighted representatives, capturing distinct cluster structures and noise assignments. Further, we marginalise the uncertainty in the density level, avoiding reliance on a specific choice. We establish theoretical properties of the proposed loss and guarantees for the level-marginal posterior approximation. Results demonstrate the framework's effectiveness in capturing clustering and noise uncertainty.

Kernel Ridge Regression (KRR) is a theoretically sound algorithm for nonparametric learning, which is also often used to investigate properties of deep learning. However, the exact KRR formulation incurs time and memory costs preventing applicability to large datasets, motivating the development of approximate solvers. These algorithms rely on (i) iterative methods, to avoid the computational burden of direct solutions; (ii) sketching or batching, to reduce the problem size; and (iii) preconditioning, to accelerate convergence. While crucial for fast convergence, preconditioning incurs memory costs that can be managed by lowering the number of free parameters of the model, potentially at the expense of expressivity and hence accuracy. To address this challenge, we propose BLOKS, a Conjugate Gradient-based solver for Nyström-sketched KRR built around a novel preconditioner that combines a block Jacobi approximation with sketching. The BLOKS preconditioner has memory cost independent of both the dataset size and the number of parameters, allowing scaling to large models. We provide a memory-efficient, GPU-accelerated implementation and show that BLOKS achieves excellent accuracy while retaining favorable time/memory complexity. Indeed, on standard large-scale datasets used to benchmark kernel solvers, BLOKS achieves comparable or better performances than state-of-the-art methods.

Dynamic Time Warping (DTW) is a common choice for measuring the discrepancy between time series. It proceeds by finding an optimal warping between timestamps through dynamic programming, which can be slow and is not differentiable. Consequently, several variants have been proposed. Among them, Soft-DTW proposes a smooth formulation of DTW, amenable to gradient computation, while Continuous-DTW (CDTW) introduces a continuous formulation through parametrizable warping functions. Yet, computational costs remain high for both methods and impractical for large datasets in modern machine learning pipelines. This paper revisits CDTW in several ways. After establishing its differentiability under mild conditions, we introduce FastCDTW, which is an efficient approximate solver with two different computational strategies: an exact closed-form scheme for piecewise-linear warping functions, and an unbiased Monte Carlo estimator for smooth ones, such as implicit neural networks. Both strategies provide gradients with respect to the warping function and the input time series, and the resulting algorithm is GPU-friendly. We demonstrate its benefits through its integration into more complex learning problems, including time series averaging, classification and deep domain adaptation.

A solid understanding of the predictive success of deep neural networks (DNNs) remains elusive. For example, while many data-dependent metrics exist that seek to predict model quality of DNNs, these metrics may be expensive to compute for large datasets and/or may be impossible to compute when training datasets are not released. A practical and effective predictor of model quality involves analyzing DNN weight matrices, as prior work has established that heavy-tailed spectral distributions in trained weight matrices can be strongly predictive of model quality. However, such metrics lack a rigorous statistical justification. In this work, we aim to understand why these metrics perform well, from a statistical perspective. To do so, we employ a probabilistic framework to derive the marginal likelihood for a DNN, evaluated using only the final layer weights. Across large-scale convolutional DNNs and LLMs, we find that this marginal likelihood predicts model quality. To understand the role heavy-tailed spectra play in predictive performance, we derive the limiting log-marginal likelihood for spectra that follow the (heavy-tailed) High-Temperature Marchenko-Pastur (HTMP) distribution. We show that this limiting quantity is convex in a heaviness parameter, and we derive an optimal heaviness that empirically predicts over-fitting for large-scale DNNs.

Trajectory extrapolation from a partial target trajectory is underdetermined, but completed related trajectories provide information about the unresolved continuation. We propose target-constrained pooling (TCP), which separates the finite-horizon endpoint from continuation shape and estimates that endpoint from two complementary source signals. Uniform pooling captures final response levels, while matched-increment pooling captures the remaining change after aligning source and target states. TCP selects representative sources, projects both endpoint estimates onto a target-consistent region, converts them into corrections of the free forecast, and controls their influence through strength calibration and degradation-aware weighting. Across 25,245 targets from five domains and both forecasting directions, TCP reduces mean RMSE by 35.8% to 42.1% for all ten target-only forecasters and improves all 100 forecaster, dataset, and direction comparisons. The two endpoint signals consistently outperform either alone, while representative selection remains within 0.64% to 1.02% of the full donor pool using 12.7x fewer donor inputs and 138.8x fewer matching distance computations.

If a second text corpus were collected from the same population, how much would a predictor's outputs change? We study this question for masked-text reconstruction using a smoothed predictor built from nearby token-pair counts and evaluated over the full 50,257-token GPT-2 vocabulary. Treating each source document as one observation, we derive a first-order formula for the expected disagreement between predictors fitted to two independent corpora. The formula accounts for variable document length, random context counts, and correlations across token offsets, and it can be evaluated from a separate reference corpus without access to either target corpus. On TinyStories, WikiText-103, and CNN/DailyMail, growing each corpus from 512 to 2,048 chunks of 64 tokens lowers disagreement by only 5–40%, far short of the 75% implied by $1/n$ scaling. The forecast tracks this slow decline, which we trace to smoothing of rarely seen contexts. Forecast accuracy depends less on how these correlations are modeled than on the size of the reference corpus: small references systematically underpredict disagreement. A source-split jackknife that uses only reference data reduces this calibration error by 57–59%. After this correction, a simpler smoothed-multinomial approximation is most accurate when the reference corpus is smaller than the target corpus, and a nonlinear source bootstrap is most accurate when it is larger. Finally, small masked diffusion language models trained on the same corpora show that the count-based forecasts partly identify which inputs are most sensitive to the corpus, but miss the neural models' much faster decline in disagreement.

Continuous-time latent state-space models are widely used for irregularly observed time series, typically representing uncertainty through a random initial state followed by smooth ordinary differential equation (ODE) dynamics, or Brownian perturbations accumulated by a stochastic differential equation (SDE). While these constructions are natural for smooth or diffusive systems, they do not directly encode a finite set of unobserved shocks whose timing, magnitude, and effect on memory are themselves uncertain. We instead introduce latent jump models (LJMs): the latent state follows closed-form deterministic dynamics between Poisson-driven event times and undergoes a learned instantaneous jump at each event, with state-dependent intensity and mark distributions. This defines a time-homogeneous Markov process whose jumps are intrinsic state transitions rather than numerical integration steps. For training, we derive a tractable event-augmented path-space evidence lower bound (ELBO) and introduce Scan-Parallel Variational Inference (SPVI): an observation-conditioned variational posterior samples the complete latent event configuration before the state recurrence is evaluated using triangular affine scans. This enables fast and scalable training for LJMs in logarithmic temporal depth. Across synthetic stochastic systems and real irregular time series, LJMs remain competitive with continuous-time latent baselines in forecast accuracy, often with fewer latent updates than numerical solver evaluations, while training one to two orders of magnitude faster than the sequential baselines.

Spatial prediction is a critically important task in data analyses that involve spatially or spatio-temporally-referenced data. It is commonly carried out using hierarchical models, which comprise a process model describing the latent process of interest, and a data model describing how the process is observed. The target quantity is the predictive distribution, which is the conditional distribution of the latent spatial process given the observed data. Unfortunately, the predictive distribution is rarely available in closed form and can be difficult to sample from. Here, we show how recent developments in guided diffusion modelling from machine learning can address a wide class of problems in spatial prediction when the likelihood function derived from the data model is known and differentiable, possibly after relaxation. To show the versatility of the approach, we apply it to both Gaussian and non-Gaussian process models and a variety of data models that describe direct partial observations, linear and areal aggregates, counts, right censoring, and area-exceedance functionals. Where possible, we validate the framework against exact and Markov-chain-Monte-Carlo (MCMC)-derived predictive distributions. We demonstrate our approach on several simulation studies and in a controlled experiment of methane-flux inversion over the UK and Ireland. We show that guided diffusion can sample from the correct predictive distribution in a variety of settings, that it can be applied to models where MCMC is intractable, and that it scales well with the dimensionality of the latent process.

Comparing distributions under severe sample imbalance requires effectively using abundant majority observations while retaining sensitivity to differences involving the smaller sample. This challenge is particularly relevant for complex, non-Euclidean data, such as graphs, whose structural characteristics call for flexible testing procedures. Classifier two-sample tests (C2STs) provide such flexibility, but existing procedures are primarily designed for balanced or comparably sized samples. In this paper, we study C2STs in a highly imbalanced regime, and show that severe sample imbalance may substantially degrade testing performance. Different from conventional imbalanced learning approaches that treat sample imbalance merely as a nuisance for classifier training, we turn it into a source of repeated testing evidence. Building on this insight, we propose an Enhanced Imbalance-Aware Deep Classifier Two-Sample Test (EIAD-C2ST), a classifier-agnostic framework designed for sample imbalance that accommodates diverse data modalities by tailoring classifiers to data structure. For computationally expensive deep classifiers, we further develop a scalable representation-amortized implementation strategy. Theoretically, we establish rigorous guarantees on asymptotic size and characterize conditions for power consistency, allowing the imbalance ratio to diverge. We demonstrate the empirical gains using simulations and an application to blockchain forensics. The results reveal that the majority-sample surplus, when properly leveraged, can be organized into useful statistical evidence, rather than an obstacle, for distributional comparisons with complex data.

We study dictionary learning for probability measure-valued data by embedding measures into Hilbert spaces using features derived from optimal transport geometry. We learn data-adapted dictionaries through sparse autoencoder parameterizations, producing fast, scalable representations of probability measures as sparse finite-dimensional codes. The resulting framework combines the rich, geometric structure of optimal transport with the interpretability and efficiency of sparse representation learning. We demonstrate its utility across diverse data modalities, including hyperspectral images, geometric point clouds, and document embeddings in large language model latent spaces. We also prove generalization bounds for sparse dictionary learning objectives on broad classes of $L^2$-valued measure embeddings, expressed in terms of the metric complexity of the atom class. For optimal transport and entropic transport map dictionaries, we derive new covering number bounds that yield explicit rates. Together, our computational and theoretical results support a new, scalable model for representation learning in the space of probability measures via embedding into Hilbert spaces.

We consider online binary prediction with paid stochastic features, where the learner can spend resources to obtain better observations of the underlying features. This setting is motivated by applications such as medical testing, where additional cost can provide more precise measurements of patient's features for predicting a binary outcome. At each round, a latent feature is drawn i.i.d., the response is generated through a logistic model, and the learner chooses how much to pay before observing a noisy version of the feature. The goal of learner is to jointly control prediction loss and acquisition cost. In contrast to the former result on the online linear regression with paid features, covariance information alone is no longer sufficient: two noise distributions with the same covariance may induce different optimal logistic losses, and lower variance can even lead to higher loss. Under a structural ordering in which larger payments yield better observations by removing independent noise, we obtain $\widetilde O(\sqrt{T})$ regret when the noise structure is known and $\widetilde O(T^{2/3})$ regret when it is unknown, with matching lower bounds up to logarithmic factors.

Determinantal point processes (DPPs) model subset selection probability with repulsion, but many applications also exhibit attraction arising from pairwise relations or shared latent structures. Nonsymmetric DPPs (NDPPs) capture both effects, but their kernels are difficult to interpret and specify. We propose the determinant-ratio point process (DRPP), which models repulsion and attraction via a ratio of the determinants of two symmetric positive-definite kernels. This formulation preserves the feature-based interpretation of standard DPPs while allowing the two effects to be specified separately. Furthermore, we introduce an auxiliary-variable representation that enables efficient MCMC sampling and yields tractable expectation formulas for key statistics including the normalizing constant, which together facilitate maximum-likelihood learning. We theoretically characterize the interaction structures well suited to DRPPs: for clustered attraction with $C$ latent groups, DRPPs exactly realize the target pairwise interactions with an attraction rank $C$, whereas NDPPs can require a rank that grows with the number of items given the same repulsion component. Conversely, nontransitive pair-specific attraction may require compensation through the DRPP repulsion kernel. Experiments support these predictions, showing that DRPPs fit transitive attraction with substantially fewer parameters than NDPPs.

This work studies an entropy-based estimator of Kullback-Leibler (KL) divergence constructed from normalized importance weights. We prove a universal finite-sample bound showing that its expectation can exceed the target-proposal KL divergence by at most $\log 2$ for any sample size. We then characterize the KL divergence between the distribution induced by importance-weighted resampling and the proposal, showing that it is upper bounded by both the expectation of the entropy-based estimator and the target-proposal KL divergence. Importantly, these bounds require neither bounded importance weights nor finite higher moments. We apply these results to test-time alignment of generative models via Soft Best-of-$N$ sampling, an importance-sampling-based scaling technique proposed by Verdun et al. (2025), and to relative entropy coding, obtaining finite-sample policy-divergence certificates and moment-free bounds on coding cost.

Bayesian simulation-based inference (SBI) can struggle to recover increasingly concentrated posteriors as observation size grows. We consider learning sufficient summary statistics jointly with the posterior as a general principle for scalable inference in various SBI methods. A common measure taken in SBI is dimension reduction through predetermined summaries, which may partially discard posterior information. Our approach instead imposes structure on how the posterior learner depends on a learned summary (e.g., affinity), and learns the summary simultaneously with the posterior. We identify an important mechanism named two-way guidance: under a structural restriction on the summary, the posterior sufficiency of the learned summary simplifies posterior learning, whereas a suitable posterior objective guides the learned summary towards posterior sufficiency. This mechanism explains a counterintuitive finding that the structural restriction on the summary statistics improves posterior learning, rather than limit its performance. We establish theoretical guarantees on both posterior estimation error and the learned summary's departure from sufficiency, thus identifying conditions on the simulation budget for recovering increasingly concentrated posteriors as observation size grows.

Determinantal point processes (DPPs) on a finite set are probability models that exhibit repulsion where inter-item similarities are encoded by a positive definite kernel matrix. Scaling DPP learning to large datasets typically requires imposing structural constraints on the kernel matrix, such as a low-rank-plus-diagonal form. However, structural assumptions introduce a severe trade-off between expressive power and computational scalability. Our study addresses this challenge by (a) modeling the full-rank interactions via a nonlinear kernel function and latent features, (b) designing a surrogate function for the maximum likelihood estimator, and (c) adopting stochastic optimization using random features of the nonlinear kernel. Numerical experiments on real-world datasets demonstrate that our method significantly improves this trade-off, achieving better generalization performance with remarkable parameter efficiency.

Estimating (heterogeneous) treatment effects is still challenging when the outcome is semicontinuous (i.e., exhibiting both a discrete point mass at zero and a skewed continuous distribution for positive values), despite the strong presence of this data type in real world settings. Existing methods either ignore the zero mass, impose fragile parametric assumptions, or suffer from extreme computational costs and data augmentation noise. This paper proposes Zero-Inflated Continuous Bayesian Causal Forests with Duan's Smearing (ZIC-BCF-Smear), a decoupled two-part nonparametric estimator that fits a Probit BCF on the full sample to model participation and a Gaussian BCF strictly on the active subpopulation to model positive intensity. Subpopulation Propensity Adjustment (SPA) mitigates regularization-induced confounding within the active subpopulation, and Duan's non-parametric smearing estimator re-transforms predictions to the response scale, bypassing the fragile scale multipliers of parametric models. A comprehensive simulation benchmark shows that ZIC-BCF-Smear matches well-specified parametric models without their assumptions and remains stable and calibrated where competitors either diverge or pathologically over-cover for nonlinear data. The method is then applied to individual-participant data from the SPECTRUM randomized phase III head-and-neck cancer trial, estimating the causal effect of assignment to panitumumab plus chemotherapy on the number of days patients spend with severe toxicity, an outcome for which a quarter of patients contribute an exact zero. Panitumumab assignment raises that burden by about ten days per patient under the trial's own differential treatment duration, and the decoupled structure localizes the effect mainly in the positive-intensity part rather than in whether the reconstructed severe-AE burden is positive.

We study the two-sample estimation of the powered Sliced Wasserstein functional under projected $r$-th moment assumptions. Let $N=n\wedge m$, where $n$ and $m$ are two sample sizes. We establish the minimax rates in the noncritical regimes. When $p<r<2p$, the minimax risk is of order $N^{-(r-p)/r}$, while for $r>2p$ it is of order $N^{-1/2}$. In both cases, the natural plug-in estimator is minimax rate-optimal. At the endpoint $r=p$, the minimax risk is bounded away from zero, so uniformly consistent estimation is impossible. At the critical order $r=2p$, we obtain the exact worst-case rate of the plug-in estimator and a lower bound for the minimax risk, leaving a logarithmic gap between the lower and upper bounds. The exact minimax rate in this critical case remains open.

Gradient-based attribution provides input-specific information about model parameters, but computing it via backpropagation for many samples is expensive, while static importance scores discard input dependence. We derive KPSA (Kernelized Parameter Sensitivity Atlas) from a block decomposition of the empirical neural tangent kernel. It measures normalized parameter-group sensitivity, embeds those sources in an explicit query space, and uses a lightweight kernel to estimate sensitivities for unseen inputs. We derive bounds that separate finite-source from irreducible representation error and give margin-based top-$k$ recovery. This paper studies two interpretability applications. Firstly, KPSA identifies parameter groups with high susceptibility to perturbations across vision, forecasting, and protein-modeling setups. Across a fivefold range of parameter budgets ($0.02$-$0.1\%$), its selected groups attain $39$-$85\%$ of the perturbation susceptibility of a first-order backpropagation oracle. In vision and forecasting, these parameter groups form sparse sensitivity circuits that extend across depth and vary systematically with the input. Secondly, we use KPSA on an activation-attribution problem for MLP features in transformers. It retains $81\%$ of the gap between a backpropagation oracle and a constant-sensitivity baseline on vision while increasing batched query-time throughput from 551 to 2,041 images/s and reducing peak GPU memory from 4.55 to 0.44GB. On the protein-modeling task, it recovers $68\%$ of this gap. An ablation of KPSA's kernel aggregation shows that nearest-neighbor retrieval in KPSA space can remain competitive. The empirics show that compact kernel atlases can store useful input-dependent parameter sensitivity while amortizing repeated gradient attribution, a step toward scalable conditional interpretability. Code can be found at https://github.com/Axym-Labs/kpsa .

Large language models are stochastic by nature. Repeated executions under the same experimental conditions may produce different responses. We propose a framework that can be applied across the model lifecycle, from task design to residual behavior and continuous monitoring. The method consists of building an empirical response distribution estimated through Monte Carlo simulation. Semantic interventions are introduced to study how these distributions change. We illustrate the framework with two different tasks, borrower classification and music album recommendation, and quantify the resulting distributional changes. The framework is linked to model risk management and governance by measuring robustness and control effectiveness, and characterizing residual behavior. Since it relies only on observed responses, it can also be applied to black-box models accessed through commercial APIs.

Efficient sampling from Bayesian posterior distributions in high dimensions remains a hard problem, despite advances in theory and algorithms. Multiple steps are usually required to draw unbiased samples, for example, iterations of an MCMC algorithm or integration steps in a diffusion-based amortised sampler. However, scalable one-step amortised inference remains a challenging open problem. In this work, we build upon recent progress in data-driven generative modelling to offer a possible solution for one-step amortised sampling. Our one-step implicit sampler (OSIS) uses short-time MCMC dynamics to define a local target for the evolution of an implicitly modelled distribution. In the asymptotic limit, the modelled distribution is a fixed point of the MCMC kernel and thus samples from the target. We analyse the convergence properties of OSIS and show connections with other generative modelling and sampling algorithms. We study various design choices and show that OSIS performs well compared to multi-step amortised samplers and non-amortised MCMC baselines while being far more efficient at sampling time.

Masked diffusion models (MDMs) generate discrete data by iteratively unmasking multiple tokens in parallel using a learned mask predictor. Their output carries two errors: parallelization error due to parallel sampling, and estimation error from the mask predictor learned with finite samples. Existing theory treats these errors separately, and the available statistical bounds typically scale with the number $V^L$ of length-$L$ sequences over a vocabulary of size $V$, suffering from the curse of dimensionality. In this work, we establish end-to-end statistical guarantees showing that transformer-based MDMs learn structured distributions with a sample complexity governed by their latent structure. Specifically, we model this structure through mixtures of hidden Markov trees, which capture latent dependencies within components and heterogeneity across components. When the data distribution is well approximated by such a mixture, a transformer-based MDM with parallel unmasking schedules achieves $\varepsilon$-accurate sampling in KL divergence with $\widetilde{O}(LK_{\mathrm{par}}/\varepsilon)$ training samples, where $K_{\mathrm{par}}$ is polynomial in $L$, $V$, the number of hidden states, and the number of mixture components. The transformer network and sampling procedure require no knowledge of the tree topologies or component parameters, thereby adapting to the unknown latent structure automatically. Overall, our results provide theoretical justification for the empirical effectiveness of MDMs on real-world data.

METR's 50% time horizon measures the human completion time of software tasks that an AI solves with 50% probability, allowing AI capabilities to be expressed in interpretable units. On 228 tasks and 26 AIs, we recompute the time horizons using splines and item-response theory to relax the assumption that the AI difficulty of a task depends linearly on the log of human time. Our fitted spline can be interpreted as a function that converts human time to AI difficulty; it is nearly flat in a region from 2–30 min but close to linear elsewhere. Hence, a time-horizon jump from 3 min to 30 min is much easier than one from 30 min to 5 hours despite the same multiplier of $10 \times$. Overall, we contribute time-horizon point estimates that perform better under a cross-validated suite of proper scoring rules, as well as diagnostic plots for assessing time horizons' construct validity. We suggest that time horizons be interpreted together with the diagnostic plots, especially as new time-horizon-based benchmarks are proposed or existing ones grow to include longer tasks.

Density ratios quantify distribution shift from a probability-mass point of view, whereas displacement fields describe, from a dynamical point of view, how one distribution is transported onto another. Although both offer complementary insights, they are usually estimated separately, and converting one into the other requires post-processing. In this paper, we estimate the density ratio between a target and a base distribution by parametrizing it through a displacement field acting on the base: the log-ratio is modeled as minus the Stein operator of the base applied to the field, up to a normalizing constant. This gives both statistical and dynamical descriptions of the distribution shift through a single convex optimization problem. Iterating this estimate-and-move step gives two inference algorithms: push-forward moves the model and corrects a pretrained sampler without retraining it, whereas pull-back moves the data closer to the base and fits a transformation model one layer at a time. Applications to distribution shift in simulation-based inference and to nonlinear independent component analysis illustrate the benefits and limitations of the approach.

Koopman operator theory provides a linear framework for analyzing nonlinear dynamical systems and has become a major tool for data-driven modeling. A central challenge, however, is that finite-dimensional approximations computed by methods such as extended dynamic mode decomposition (EDMD) require the dictionary to be specified a priori. Recent machine-learning approaches address this limitation by learning the dictionary from data, predominantly using artificial neural network (ANN) autoencoder architectures. Although kernel methods offer an alternative with greater interpretability and tractability for theoretical analysis, they have received little attention in this setting. We introduce extended dynamic mode decomposition with kernel-based dictionary learning (EDMD-kDL), a kernel-based method for learning finite-dimensional Koopman embeddings directly from data. The method combines ideas from collocation methods and bilevel optimization to simultaneously learn a kernel dictionary and the corresponding Koopman approximation. We evaluate EDMD-kDL against state-of-the-art ANN-based approaches on a range of numerical experiments, including global sea-surface-temperature forecasting and learning directly from video data. Across all tested settings, EDMD-kDL achieves performance comparable to or better than the ANN-based methods. Moreover, in contrast to standard kernel methods, the proposed approach is scalable to large datasets by design since the size of the required kernel matrices depends on the number of collocation points rather than the size of the training dataset.

We consider policy optimization for online episodic tabular Markov decision processes (MDPs) with adversarial losses and bandit feedback. Policy optimization updates the policy locally at each state and avoids optimization over the occupancy-measure polytope, but its existing regret bounds are larger by a factor of the horizon $H$ than those of occupancy-measure-based algorithms. We close this gap by using regularized $Q$-functions, which allow us to control the stability of the local updates jointly over all state-action pairs rather than separately at each state. The resulting algorithm attains high-probability regret bounds of $\widetilde O(\sqrt{HS(H+A)T})$ for known transitions and $\widetilde O(HS\sqrt{AT})$ for unknown transitions, where $S$ is the number of states, $A$ the number of actions, and $T$ the number of episodes. Both bounds improve the horizon dependence of existing policy optimization bounds, and the latter matches the best-known bound. We further extend the algorithm to adversarial linear-mixture MDPs and obtain the same improvement in the horizon dependence.

Randomized controlled trials (RCTs) identify treatment effects without confounding but are often small, whereas observational studies (OBS) are large but may be confounded. Many estimators combining a small RCT with a large OBS have been developed for the average treatment effect (ATE) and the conditional ATE (CATE). However, existing ATE estimators either make assumptions on the OBS or do not borrow enough power from them. The CATE has been studied less than the ATE. Existing CATE methods either assume the OBS are unconfounded, rely on a model of the confounding function, or accept bias in exchange for lower variance. We therefore propose a framework that, without special assumptions on the OBS, fuses the OBS and the RCT by preserving the unbiasedness of RCT-based estimation while borrowing power from the large OBS to boost precision. Applying this principle, we build an ATE estimator, AIPW-Fusion, with closed-form weights and confidence intervals, and two CATE learners, DR-Fusion and R-Fusion. Experiments corroborate our findings.

We consider stochastic nonsmooth nonconvex composite optimization, which includes several important problems such as constrained optimization and the regularized training of neural networks. The objective is the sum of a possibly nonsmooth nonconvex Lipschitz function and a convex regularizer, and the function is accessed through stochastic gradients or function values. The goal is to find a point that satisfies a Goldstein-type stationarity condition designed for composite objectives. To our knowledge, no oracle complexity bound for this setting is known under first-order access, and existing complexities under zeroth-order access are suboptimal. To handle this issue, we employ the framework of online-to-nonconvex conversion, which chooses update directions by an online learner and is known to achieve optimal rates for noncomposite problems. We extend the framework to our composite scenario by introducing new losses for the learner, which contain the regularizer itself rather than its linearization and for which a variant of online mirror descent achieves low regret. We show that the resulting algorithm finds such a point with $O(\delta^{-1}\varepsilon^{-3})$ stochastic gradient queries or $O(d\delta^{-1}\varepsilon^{-3})$ function-value queries, where $\delta$ is the Goldstein radius, $\varepsilon$ is the stationarity tolerance, and $d$ is the dimension. These rates match the optimal ones for noncomposite nonsmooth nonconvex optimization, demonstrating that the additional convex regularizer does not worsen the oracle complexity. We also give rates for the smooth case and present numerical experiments.

Given independent and identically distributed samples samples from a probability distribution in a potentially high-dimensional real space, we study the problem of testing whether the distribution is concentrated near a real algebraic complete intersection of prescribed dimension, bounded degree, and bounded condition number. We design an explicit and effective learning procedure which either certifies the nonexistence of such a manifold, up to a controlled relaxation of the approximation threshold, or returns a candidate regression manifold with controlled geometric complexity. Equivalently, the procedure tests the manifold hypothesis within this hypothesis class. The proposed procedure relies on quantitative geometric estimates for regular polynomial systems, which lead to a tractable auxiliary optimization problem. We then develop a data-driven algorithm to solve this auxiliary optimization problem, establishing explicit bounds on its sample and arithmetic complexity.

Bayesian Optimization (BO) is a popular method for efficiently optimizing expensive black-box objectives. However, BO utilizing standard Gaussian Processes is ill-suited for doubly stochastic Cox Processes that are often used in spatio-temporal problem spaces. We introduce INLA-SPDE Spatio-Temporal Bayesian Optimization (ISBO): the first scalable BO framework for spatio-temporal data, that models the log-intensity with a Log-Gaussian Cox Process(LGCP) and performs inference via Integrated Nested Laplace Approximation and Stochastic Partial Differential Equations (INLA-SPDE) approach. Using a Matern field on meshes yields a sparse Gaussian Markov Random Field, where INLA provides fast and accurate posterior inference throughout sequential optimization. ISBO stably locates high-intensity regions and the peak of the latent intensity with minimal evaluations. A time-varying Upper Confidence Bound acquisition with masking avoids revisits, while penalized-complexity priors regularize early rounds. Experiments on synthetic and real-world spatio-temporal datasets show accurate peak discovery, intensity recovery, and substantial speedups over an RKHS-based baseline, positioning ISBO as a practical choice for BO with point-process data.