Papers

Distribution steering seeks feedback laws that drive the state law of a dynamical system between prescribed initial and terminal distributions. Optimal transport provides a natural geometric approach, but its implementation generally requires a transport map or coupling in the full state space. Sliced optimal transport avoids this full-dimensional construction through one-dimensional projections. Yet, the resulting projected maps specify only directional displacements and do not by themselves prescribe a realizable feedback law. To this end, we develop a finite-horizon control framework based on sliced optimal transport. At each sampling instant, a projected optimal transport map defines a directional terminal condition, whose minimum-energy realization yields a randomized single-direction controller. Averaging over projection directions gives a deterministic sliced feedback. For the single-integrator dynamics, the averaged feedback makes the sliced Wasserstein distance to the target non-increasing. For Gaussian endpoint laws, it is affine, preserves Gaussianity, and steers the mean and covariance to their prescribed terminal values. We further identify a law-dependent gain that yields linear decay of the sliced Wasserstein distance together with an explicit characterization of the control energy. We also prove that the randomized controller converges to the averaged sliced flow as the sampling period vanishes. Finally, we extend the construction to linear dynamical systems. Reachability-normalized coordinates allow instantaneous realization of the sliced velocity for uniformly fully actuated systems, while local controllability Gramians provide exact finite-step realization for general controllable systems. Numerical examples illustrate the resulting distributional flows.

An “Attention Residual U-Net” method is described for probabilistic quantitative precipitation forecasting (PQPF) that predicts the hourly probability of no precipitation plus the distribution of positive precipitation from a weighted mixture of two Gamma distributions. The neural network is trained on patches of numerical weather prediction (NWP) hourly precipitation from The Weather Company's convection-permitting GRAF (Global high-Resolution Atmospheric Forecasting) model along with terrain information and column-average relative humidity from the National Oceanic and Atmospheric Administration's (NOAA's) Global Forecast System (GFS). The target data are NOAA's Multi-Radar, Multi-Sensor (MRMS) gauge-corrected, quality controlled radar data sampled to the same grid as the GRAF data. The network outputs distributional parameters for each model grid point. Training uses negative log-likelihood as a proper scoring rule, with climatological initialization for stable convergence. Inference is performed as a single forward pass over the contiguous United States (CONUS) domain, with edge-replication padding to satisfy the network's spatial-divisibility requirement. The subsequent forecasts are spatially detailed, highly reliable, and skillful with respect to climatology and a simpler reference forecast method. The method is particularly useful for estimating probabilities in regions with large terrain variation.

For solving nonconvex equality-constrained optimization problems, a recent Gradient-Eigenstep Algorithm by Goyens et al.~is an iteration-efficient approach, based on minimizing Fletcher's augmented Lagrangian function, for finding an approximate second-order stationary point from an arbitrary starting point. In this paper, the analysis of this algorithm is extended, offering a two-fold contribution. First, it is shown that a local-linear rate of convergence can be obtained by this method if it is initiated sufficiently close to a strong second-order stationary point and employs a sufficiently small step-size parameter and sufficiently large penalty parameter. In this case, the algorithm reduces to a gradient descent algorithm applied to minimize Fletcher's augmented Lagrangian. Second, as a particularly useful application of the first result, it is shown that the Gradient-Eigenstep algorithm can be used as an iteration-efficient subproblem solver in the context of a progressive sampling strategy for solving equality-constrained optimization problems when the objective and constraint functions are defined by large sample averages, ultimately offering an algorithm with an improved worst-case sample complexity when compared to an approach that solves a full-sample problem directly.

Wildfire susceptibility mapping typically relies on physical variables assembled from multiple remote-sensing, climate, and geospatial products. AlphaEarth Foundations (AEF) provides analysis-ready geospatial embeddings that may reduce this dependence on heavy harmonisation and task-specific feature engineering, but their value for wildfire susceptibility mapping has not been systematically evaluated. Using Victoria, Australia (2017-2025), as a case study, we show that AEF embeddings can reconstruct commonly used variables in wildfire susceptibility analysis with high accuracy. In downstream susceptibility models trained on satellite-derived fire occurrence data, embedding-based susceptibility models achieve ROC-AUC values above 0.92 and consistently identify high wildfire susceptibility across eastern Victoria, particularly Gippsland and the north-eastern uplands, with additional localized hotspots in central and northwestern Victoria. A key feature of AEF embeddings is their strong near-region transferability within climatically similar regions. When embedding-based models trained in Victoria are applied to Canberra and Western Sydney-Blue Mountains, ROC-AUC improves by around 4% at Canberra and declines by around 2% at Western Sydney-Blue Mountains, compared with a mean decrease of approximately 25% for physical-variable models. These findings provide practical guidance for using AEF embeddings and lay a foundation for scalable wildfire susceptibility mapping workflows for downstream users such as government agencies and (re)insurers.

Probabilities of causation (PoCs) characterize individual causal responses that cannot be directly observed and therefore generally require partial identification. Tian and Pearl first derived theoretically sharp bounds for binary PoCs, including the probability of necessity (PN), the probability of sufficiency (PS), and the probability of necessity and sufficiency (PNS). Mueller et al. subsequently tightened the bounds for binary PNS by incorporating causal information encoded in covariates and mediators. More recently, Li and Pearl, as well as Shu et al., extended PoCs to multivalued settings and derived corresponding theoretical bounds. These developments naturally raise the question of whether additional causal knowledge can further tighten the bounds in multivalued settings. This paper addresses this question by deriving tighter bounds for multivalued PoCs through the incorporation of causal information encoded in covariates and mediators. We illustrate the theoretical results with toy examples, while simulation studies further demonstrate that the proposed bounds are tighter than existing nonbinary bounds.

A flat training curve does not reveal whether a neural network has reached a global optimum, is locally trapped, is representation-limited, or is mismatched to its trainer. We introduce Training Under Challenge, an executable-certificate framework in which predeclared, architecture-valid procedures construct complete alternatives in the same certified class and reevaluate the same objective. Any lower-valued candidate is a replayable witness that lower-bounds the checkpoint's empirical global-optimality gap. Passing a finite suite is only suite-relative; global-gap conclusions require a separately justified coverage mechanism. We define a resource-indexed challenge-power modulus that characterizes the largest gap compatible with passage. For squared loss, current block-decrease operators make coverage checkable and yield uniform and realized-residual bounds. We prove the converse frontier: without coverage, a first-order ReLU trainer can reach infinitely many exact conditional head optima while converging to a non-global point. On a channel-gated ResNet-18 distillation problem with known optimum, eight internal challenges cover all 240 audited output directions, and realized-residual bounds lie within factors of 1.74–3.02 of the true gap. Paired predictive certificates separate decoder under-use from representation insufficiency, while quantized-denoising studies demonstrate diagnosis, repair, and current-state recertification.

Causal discovery aims to uncover the underlying causal relationships given data generated from a system. The goal, however, is not merely to predict causal edges given data, but also to be able to interpret and explain either observed or hypothesized phenomena, such as a particularly large causal effect. We consider this task of conditional causal discovery and cast it as a Bayesian inference problem, in which we target the posterior over causal graphs and parameters conditional on an event such as a causal-effect constraint. Unfortunately, this poses a computational challenge: existing approaches to Bayesian causal discovery struggle when the event has small posterior mass. To address this, we adapt rare-event estimation techniques to perform inference the joint graph-parameter space. Our method gradually drives a particle population toward the constrained region while maintaining samples that approximate the conditional posterior. Empirical evaluation on synthetic graphs validates the accuracy of our approach at small and large scales, and we show in a case study on the Sachs protein dataset how our method can be used to aid scientific exploration by providing pathway-level summaries.

Language model classifiers with explanations are used for moderation, routing, topic triage, and low-resource annotation. We study black-box auditing when the defender has only clean calibration data without trigger information but can ask the classifier for a label plus a short rationale or quoted evidence. We introduce Groundedness Drift, a lightweight score measuring whether the answer summary remains grounded in the input. Across two 7B backbones, five datasets, and four common non-adaptive OpenBackdoor-style attack families, Groundedness Drift achieves higher AUROC and lower residual target ASR than every compared detector in all cases at a nominal 5% clean-FPR budget. We then evaluate Unsupported Groundedness, a multi-probe escalation for explanation-camouflage stress cases. Unsupported Groundedness improves signals but does not close the adaptive gap.

Factor-MIDAS regressions forecast a low-frequency target by extracting common factors from a large panel of high-frequency predictors via principal component analysis (PCA). While PCA mitigates the curse of dimensionality, it relies on factor pervasiveness, an assumption often violated when factors are weak, as is common in macro-financial forecasting. We propose SsPCA-MIDAS, which integrates supervised scaled PCA (SsPCA) into the mixed-data sampling framework. We establish consistency and asymptotic normality under weak factors, permitting inference on the prediction target. Simulations show that SsPCA-MIDAS outperforms competing PCA-based and supervised methods, especially when weak factors are prevalent. Applying machine-learning techniques such as boosting to the cleaner factors it extracts yields further gains. An extensive application to U.S. macro-financial forecasting shows that SsPCA-MIDAS selects economically meaningful predictors and improves forecasts of GDP, inflation, unemployment, asset prices, and volatility.

Organizations decide whom to treat under a budget and want to know what a targeting rule would have earned before deploying it. Off-policy evaluation promises this from logged data, but the deployable rule is a deterministic top-k policy: it removes all averaging over actions, so weak overlap hits the estimate directly. We benchmark six estimators across five datasets and two known-effect sweeps, and validate the mechanisms against a non-simulated paired reference. First, weak overlap is governed by logger-target action alignment, not by logging sharpness alone: what governs support is the logger's probability of the target's actions. Sharpening a logger built from the target's own score barely moves overlap over the tested range; action-level disagreement collapses it. Effective sample size ranks this risk across logging environments, but is weak at ranking candidates within the single log a practitioner holds, and its cut point does not transfer. Second, the optimizer's curse is not fixed by cross-fitting the outcome nuisance. When the rule is fit on the data used to evaluate it, cross-fitting the nuisance alone leaves the reuse bias in place and makes it worse. Honest policy-level splitting avoids the reuse by targeting the learning procedure's value – a change of estimand, not a de-biasing of the full-sample policy. Third, propensity-estimation error is the largest degradation we measure: an out-of-fold estimate hurts IPS more than any other stress we apply, leaves doubly-robust estimation almost unchanged, and can invert the overlap diagnostic itself. Logging is synthesized and propensities floored at 0.02, so every failure occurs with bounded weights; the floor also reduces the two tuned hybrids to their untuned parents, leaving four practically distinct estimators, and all exact-value surfaces are synthetic or semi-synthetic. We release the benchmark; public data only.

Large astrophysical simulation campaigns often generate training data by sampling parameters across a Uniform prior box. Due to the proposal's sharp edge, neural posterior estimators struggle to learn accurate approximations near the boundaries. We propose Tailed-Uniform, a family of hybrid proposal distributions for sampling training simulations for robust simulation-based inference. By padding the original hard-truncated training box with decaying tails, Tailed-Uniform-trained networks yield more accurate posteriors near and beyond the edges. We demonstrate these improvements on a family of tail shapes, including a widened Uniform box as a control. Our results suggest that additional simulations near the prior boundary better constrain the networks as it approaches the edge of the training box, even for Uniform assumed priors. We show these advantages on a toy problem and cosmological parameter inference from the matter power spectrum. These benefits increase in high dimensions, where boundaries dominate parameter space volume.

Neural combinatorial optimization (NCO) relies on parallel solution sampling for training, yet existing methods fail to fully exploit the rich information latent in a co-sampled solution group. Preference-optimization methods anchor on the single best solution and discard fine-grained quality and structural signal from all other peers-a failure we term gradient signal polarization. Mean-based baselines instead weight peers uniformly, so structurally near-identical peers flood the baseline with redundant information and keep gradient variance high-a failure we term baseline redundancy. We propose SSPO (Structure-Aware Similarity-Weighted Preference Optimization), which scores all $B$ sampled solutions jointly through a dissimilarity-weighted leave-one-out baseline: structurally distinct peers receive higher weight, resolving both failures in a single mechanism. The baseline uses zero-parameter, problem-adaptive solution embeddings built from the encoder's existing node representations. Experiments on TSP, EFL, and JSP benchmarks show consistent gains over prior best-anchor and uniform-weight baselines. A direct comparison against uniform RLOO on TSP and EFL confirms that structure-aware weighting is the primary driver of improvement. The SSPO-trained EFL policy has been deployed in a production facility-location system at JD$\mathord{.}$com, confirming practical viability at scale.

We formulate generative modeling as a path integral in which flow-based, diffusion-based, variational, and adversarial models arise as different evaluation principles for a single master action. Its Martin-Siggia-Rose-Janssen-de~Dominicis (MSRJD) form separates free from interacting probability flows and opens them to diagrammatic perturbation theory. The expansion yields a one-loop correction to deterministic samplers at no stochastic-sampling cost, which we validate on solvable and nonlinear drifts, where it reduces a 53 % tree-level error to 1.6 %. Imperfect learned scores enter as insertions and yield a response-weighted score-matching objective, and symmetry-equivariant drift design becomes an operator expansion with EFT power counting.

In high-stakes applications, reliable confidence estimates are as important as the predictions themselves. Confidence calibration ensures that predicted probabilities reflect the likelihood of correctness, making it essential for safe deployment of deep learning models. However, existing methods typically assume access to clean validation data, which is often unrealistic due to label noise and domain shifts. This thesis develops methods for improving calibration under these conditions. First, we address calibration under label noise. Standard methods can produce misleading confidence estimates when labels are unreliable. We propose a framework that uses an estimated noise model to reconstruct noise-free confidence estimates by modeling the relationship between noisy and clean label distributions. We extend this approach to Conformal Prediction (CP), which provides set-valued predictions with guaranteed coverage. Our noise-aware CP method estimates clean conformity scores despite label noise, enabling reliable uncertainty quantification. Next, we study calibration in unsupervised domain adaptation, where a model trained on a labeled source domain is adapted to an unlabeled target domain. Since labeled target data are unavailable, we estimate target-domain accuracy from source performance and domain discrepancies, enabling calibration without target labels. We also consider privacy-preserving settings in which user labels and model outputs must remain protected. We propose a locally differentially private conformal prediction framework that provides valid uncertainty quantification while maintaining privacy guarantees and balancing privacy, computational feasibility, and prediction reliability. Our results bridge calibration theory and practical deployment in safety-critical applications, contributing to reliable, privacy-preserving, and noise-resilient neural network predictions.

Learning mappings between infinite-dimensional objects is a central challenge in scientific machine learning. We introduce a general kernel-based encoder-decoder framework for operator learning that separates observation, representation, learning, and reconstruction. We develop this framework for multi-input, multi-output operator learning, where operators map between products of potentially distinct function spaces. Our approximation theory shows that, although the number of inputs and outputs can increase, the convergence rate is governed by the most challenging constituent approximation problem rather than the overall problem dimension. The framework leads to practical kernel methods with closed-form training and inference, combining mathematical tractability with computational efficiency. We further specialize the approach to multiple operator learning by introducing KernelMO, a family of kernel methods with complementary operator-valued and product-space formulations. Across five families of parametric partial differential equations, the proposed methods achieve competitive or state-of-the-art predictive accuracy while reducing training and inference costs relative to neural operator architectures and deep learning based models, offering an efficient and lightweight alternative.

Outcomes are regressed on a calibrated probability vector for unobserved class membership. Under a structural conditional mean excluding the score and conditional calibration, the observed-data model reduces to a partially linear regression. The probability vector is a Berkson-type surrogate for membership, so the effect vector $\tau$ is identified without attenuation. In practice the vector is often coarsened to a hard label - an argmax, a confidence threshold - and that label need not retain the Berkson property. For any coarsening the plug-in estimator converges to $\mathcal{A}\tau$, where $\mathcal{A}-I$ is determined by the regression of the discarded part of the score on the retained part. Coarsening therefore leaves $\tau$ undistorted exactly when that regression vanishes, and otherwise distorts some contrasts far more than others. The same operator determines the bias that drives coverage loss. Where that bias is of the order of the standard error, the Wald interval has limiting coverage $\Phi(z-\nu)-\Phi(-z-\nu)$, with $\nu$ their ratio. A fixed bias sends coverage to zero. Operator, standard error, and - through the uncoarsened estimator - the bias are estimable from observed data, so the implied coverage can be approximated before the interval is reported. Simulations show severe coverage loss after argmax coarsening. Three real-data audits exhibit the direction-specific distortion.

We revisit the Sinkhorn-Knopp (SK) algorithm for the matrix scaling problem. Despite extensive literature on the global convergence of SK and its variants, its local linear convergence behavior remains less understood. We address this gap by providing the first nonasymptotic local analysis of SK that matches the rate obtained from existing asymptotic Jacobian-based arguments. We show that under certain connectivity conditions, SK is a polynomial-time algorithm for doubly stochastic matrix scaling. With the developed tools, we showcase the local suboptimality of SK and provide accelerated variants. Finally, for dense matrices, we improve the complexity of existing first-order matrix scaling algorithms from $O(\tfrac{n^{7/3}}{\varepsilon^{2/3}})$ to $O(\tfrac{n^{9/4}}{\sqrt{\varepsilon}})$.

Multi-objective optimization (MOO) has demonstrated significant success in multi-task learning by mitigating task conflicts through gradient manipulation. However, most existing methods flatten model parameters into vectors and perform gradient manipulation under Euclidean geometry, thereby overlooking the matrix structure prevalent in modern architectures such as Transformers. In this paper, we show that gradient manipulation in Euclidean space does not generally yield the steepest descent direction under matrix geometry, potentially limiting optimization efficiency. Drawing from the theory of steepest descent for matrix-valued parameters, we propose MOON (Multi-Objective OrthoNormalized Updates), which performs gradient manipulation under spectral–nuclear norm geometry and uses the orthonormalized manipulated gradient for parameter updates. Theoretically, for smooth non-convex objectives, we establish convergence of the averaged Pareto-stationarity measure at rates of $\mathcal{O}(T^{-1/2})$ in the deterministic setting and $\mathcal{O}(T^{-1/4})$ under stochastic gradients. Empirical results across various benchmarks show that MOON consistently improves both optimization efficiency and final multi-task performance. Our code is available at https://github.com/KunlinLyu/MOON.

Continual learning must absorb new tasks without erasing old ones, and replay—mixing a small buffer of past examples into current training—is among the most effective remedies for catastrophic forgetting. Yet its generalization behavior is shaped by two coupled effects that existing analyses fold into a single hypothesis-level quantity: finite memory replaces each past distribution with an empirical proxy, and repeated reuse couples the buffer, the current data, and the final hypothesis through a shared optimization trajectory. We develop a layer-wise information-theoretic framework that separates these effects at every depth. Our main result decomposes the expected generalization gap into a replay-induced representation drift and an optimization-dependence term, the latter further resolved into stability, plasticity, interaction, and residual-coupling components. Two refinements make the framework operational. A Wasserstein relaxation of the drift term, valid under support mismatch, yields a depth-dependent drift–sensitivity trade-off whose minimizer identifies which interior layer to stabilize. An SGLD instantiation of the optimization term reduces it to a trajectory-level log-determinant budget, exposing a curvature-aware gradient-alignment statistic that serves as an online diagnostic of task-wise forgetting. Controlled and benchmark experiments confirm the predicted memory scaling, the interior funnel, and the alignment signal's link to forgetting.

Consider two PAC learning algorithms, both having access to quantum computation, but differing in the types of examples they obtain: one is provided with classical samples, while the other is given quantum samples. Are there any learning tasks that can be efficiently performed by the latter, but not by the former? This question, the focus of our work, is surprisingly still open. Our main result is to show that relative to an oracle, there are distributions that can be efficiently generated by a quantum learner with access to quantum samples, but not by a quantum learner with access to only classical samples, making progress to answering this question in the affirmative.

In many high-stakes domains, extreme events carry substantial consequences, yet learning the heavy-tailed distributions that govern them from finite samples remains challenging: the quantities of interest are driven by a few extreme observations, so the tail is under-sampled relative to its importance. Even generative models tailored for heavy tails capture the tail region inadequately in practice. We propose the Conditional Value-at-Risk (CVaR)-penalized Generative Particle Algorithm (CVaR-GPA), a tail-agnostic algorithm for fine-tuning generative models toward heavy-tailed targets, built as a time discretization of the Wasserstein gradient flow of the Lipschitz-regularized KL divergence penalized by a CVaR discrepancy term. Such a flow can be initialized from the output samples of the pre-trained model, which are then transported along the gradient descent of the loss functional, without requiring access to the pre-trained model's internal architecture. The Lipschitz-regularized KL divergence requires minimal assumptions on the target, while the CVaR penalty focuses the flow on the tail. The CVaR penalty depends on the target only through a scalar tail statistic, inducing a velocity field that remains active in the under-sampled tail region at a dimension-free estimation cost. The resulting velocity field, and hence the depth of the transport map, implicitly adapts to the target, without target-specific modifications. Across four targets, including two real-world, high-dimensional datasets (daily streamflow in the Ohio River basin ($d=64$) and the Fama-French portfolios ($d=25$) with tail indices ranging from $1.05$ to $3.34$), fine-tuning with CVaR-GPA reduces global and tail errors by geometric-mean factors of $14.0 \times$ and $9.8 \times$, respectively, across seven pre-trained models spanning GANs, diffusion models, and other generative flows, with a single set of hyperparameters.

Machine Learning and more specifically Deep Learning involves solving large scale nonconvex optimization problems. Several algorithms have been proposed in the literature, that seem to achieve satisfactory practical efficiency for difficult instances, the Stochastic Gradient Method being the most rudimentary, while still outperforming more recent algorithms at a number of learning tasks. A major open question about the current methods used in deep learning is to understand their convergence properties. Following a line of previous works about the long time behavior of gradient-type algorithms, %and in particular the recent contributions from Azizian et al., we present a new approach for studying the asymptotic properties of a wide family of methods from an ergodic theoretical viewpoint. Our main results include a study of the expected time for a stochastic optimisation algorithm to reach a certain small neighborhood of a minimizer and show that this reaching time distributes exponentially around its average, which is given by the inverse of the stationary measure of the target. The assumptions on the Stochastic Gradient noise include the Gaussian and the Sub-Exponential assumptions.

We study the multiplayer multi-armed bandit problem with information asymmetry under Bernoulli rewards, for three information structures: asymmetry in actions, in rewards, and in both. Replacing the Hoeffding-style confidence intervals of prior work with Kullback–Leibler (KL) divergence-based bounds gives strictly tighter regret guarantees in each case. We propose mKL-UCB, mKL-UCB-Intervals and mKL-DSEE, and show that the improvement factor is at least two by Pinsker's inequality and far larger when reward means are near zero or one. For asymmetry in rewards we prove that two arms' KL intervals separate after a deterministic number of samples, and that $M$ independent players accelerate elimination further.

We study the implementable finite-batch particle algorithm for mean-field variational inference as a fully discrete stochastic approximation of the projected Wasserstein dynamics. The target potential is globally smooth but need not be strongly convex. The departure from contractivity is quantified by the curvature defect \[ \mathfrak d_\alpha(x,y) = \bigl[\alpha\|x-y\|^2- \langle\nabla V(x)-\nabla V(y),x-y\rangle\bigr]_+, \] which is the additive loss in the one-step Euler contraction estimate. We prove a non-asymptotic Wasserstein stability bound that separates initialization, product-empirical approximation, finite-batch drift error, time discretization, and the defects accumulated along the coupled trajectories. Under the uniform bound $\mathfrak d_\alpha\leq\beta$, the particle iterates remain within $O(\sqrt{\beta/\alpha})$ of any MFVI minimizer, up to explicit errors in the particle number, batch size, and step size. The proof uses a stationary comparison array whose population law is an MFVI minimizer but whose particle-level law is a random product empirical measure, and it controls the resulting projected-drift discrepancy explicitly. We also give coordinatewise defect estimates and structural conditions for dimension-independent projected-drift sensitivity, construct an arbitrary-dimensional smooth nonconvex benchmark with a closed-form MFVI minimizer, and explain why polynomially growing drifts require a modification of the untamed explicit scheme.

We study spectral clustering in the presence of a confounding latent geometry. The leading eigenvectors may then be dominated by the latent geometry rather than by the communities. Nevertheless, we show in a block latent-space model that communities can be recovered from eigenvectors deeper in the spectrum. We analyze the spectral properties of the adjacency matrix through a limiting integral operator and use its structure to develop DBSPEC, a density-based spectral clustering algorithm that requires only approximate localization of the informative eigenvalue and is robust to poor eigenvalue separation. Crucially, this approach handles general latent geometries, overcoming restrictions to homogeneous toroidal models in prior works. Our theoretical predictions for the location of the informative eigenvalue notably align with observations in real-world experiments.

Conditional Independence (CI) tests are the statistical engine of constraint-based causal discovery: in algorithms such as PC (Peter-Clark) and FCI (Fast Causal Inference), skeleton pruning and key orientations follow directly from CI decisions. This survey reviews CI testing with emphasis on assumptions, robustness, and scalability in high-dimensional and mixed-type settings common in biomedical domains. The survey organizes widely used CI methods into six families: partial-correlation, contingency-table, regression, nearest-neighbor, kernel, and machine-learning-based. Special emphasis is provided on the robustness layers that address the limitations of these families. For each family, the survey examines when CI decisions reflect the data-generating distribution and when they fail. By this, we link test-level properties, including power decay with conditioning set size and asymmetric type I/II error consequences, to graph-level errors in skeleton recovery and v-structure orientation. The survey also compares adoption across major R and Python libraries and summarizes open challenges, including mixed-type CI testing without discretization, small-sample error control, and strategies for improving scalability of CI-testing.

Assessing the goodness-of-fit of a logistic regression model is a critical prerequisite before the model is used for inference. However, goodness-of-fit (GOF) tests such as the chi-square and deviance tests often give invalid results when the data are "sparse" – a common issue with continuous predictors like age or weight, where the asymptotic distributional assumptions are not satisfied. This thesis studies classical GOF tests for binary logistic regression under both grouped and sparse data, comparing about 30 statistical tests and machine-learning calibration algorithms. These span the classical chi-square and Hosmer-Lemeshow variants, standardized Pearson statistics, covariate-space partitioning, smoothing-based methods, and contemporary calibration machine-learning and bootstrap procedures. At a fixed size, the GiViTI calibration test (2016), McCullagh (1989), Osius-Rojek (1992), le Cessie (1995) and Stute-Zhu (2002) proved empirically powerful, balancing correct identification of bad models (high empirical power) against not raising false alarms on good models (correct empirical Type I error). Relying on formal methods alone is insufficient: visual diagnostics such as calibration plots are a vital exploratory step for detecting model deficiencies that formal tests often overlook. An application to real data (the Low Birth Weight dataset) shows that many of these tests fail to give valid conclusions when exposed to the complexities of actual datasets. The main conclusion is that model assessment requires a combination of several powerful statistical tests alongside careful visual inspection of model calibration.

We study posterior contraction in positive-order Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families. We embed the natural parameter in a Hilbert scale and model it via a Gaussian series prior expanded in the eigenbasis generating the scale. Under a two-sided link condition on the Fisher information and suitable local regularity assumptions, we show that smoothness-matching priors achieve minimax-optimal posterior contraction rates in any Hilbert scale norm up to the regularity of the ground truth. Our analysis builds on the novel approach to posterior contraction based on the Wasserstein distance recently introduced by Dolera et al. (2024, Probab. Theory Relat. Fields). It combines Laplace-type approximations for infinite-dimensional integrals associated to the posterior kernels with a mixed-geometry estimate controlling their stability under fluctuations in the data, itself resting on a tailored Poincaré inequality for posterior distributions conditioned on neighbourhoods of the truth. We apply the general theory to density estimation under the logistic parametrisation, Poisson intensity estimation under the exponential link, and mildly ill-posed linear inverse problems observed in Gaussian white noise. The resulting minimax rates yield optimal recovery of density score functions, derivatives of Poisson intensities and, in a concrete elliptic inverse problem, the unknown source function.

Inverse reinforcement learning (IRL) aims to recover a reward function under which the resulting policy reproduces the behavior observed in expert demonstrations. A natural approach is to formulate IRL as a bilevel optimization problem, in which the inner level corresponds to policy optimization under the learned reward and the outer level measures the discrepancy between the induced policy and expert data. However, this formulation is computationally challenging in practice because the outer update requires a hypergradient involving an inverse-Hessian-vector product for the inner objective. We address this challenge by showing that, at the inner optimum, the Hessian of the inner objective is proportional to the Fisher information matrix of the policy, yielding a structured Fisher-based hypergradient closely related to Natural Hypergradient Descent. To address the resulting scalability bottleneck associated with large Fisher matrices, we approximate the required inverse-Fisher-vector product using a streaming spectral sketch, avoiding explicit construction of the Fisher matrix. We evaluate our approach against a first-order stochastic bilevel baseline across discrete- and continuous-control environments. The results demonstrate competitive policy performance and strong reward-ranking quality, while Fisher sketching reduces curvature-storage complexity and can improve computational efficiency relative to an explicit Fisher solver.

Constant-stepsize temporal-difference (TD) learning is attractive for policy evaluation, but inference from a single Markov trajectory must account for serial dependence and a stepsize-dependent stationary target. For fixed-stepsize linear TD, we establish a functional central limit theorem whose covariance retains the multiplicative component induced by the random TD matrix and the stationary iterate error. We then derive a joint functional limit for parallel Richardson–Romberg (RR) recursions driven by the same trajectory. A Brownian-bridge self-normalizer yields asymptotically pivotal confidence regions for prespecified state-value contrasts without estimating the long-run covariance or selecting a bandwidth or batch length. For such a contrast, the procedure admits a one-pass implementation whose memory does not grow with the trajectory length. At a fixed stepsize, the inferential center is the RR stationary target. We also study horizon-indexed designs in which the stepsize remains constant within each run and decreases across longer horizons. Under an explicit RR-dependent rate window, the residual RR target shift, multiplicative remainder, and initialization effect are negligible at the root-$n$ scale, yielding inference for the projected Bellman solution. Experiments on FrozenLake and Garnet illustrate stationary-target coverage, RR target correction, and the finite-sample behavior of the horizon-indexed design.