Papers

In two-player zero-sum games, the learning dynamic based on optimistic Hedge achieves one of the best-known regret upper bounds among strongly-uncoupled learning dynamics. With an appropriately chosen learning rate, the social and individual regrets can be bounded by $O(\log(mn))$ in terms of the numbers of actions $m$ and $n$ of the two players. This study investigates the optimality of the dependence on $m$ and $n$ in the regret of optimistic Hedge. To this end, we begin by refining existing regret analysis and show that, in the strongly-uncoupled setting where the opponent's number of actions is known, both the social and individual regret bounds can be improved to $O(\sqrt{\log m \log n})$. In this analysis, we express the regret upper bound as an optimization problem with respect to the learning rates and the coefficients of certain negative terms, enabling refined analysis of the leading constants. We then show that the existing social regret bound as well as these new social and individual regret upper bounds cannot be further improved for optimistic Hedge by providing algorithm-dependent individual regret lower bounds. Importantly, these social regret upper and lower bounds match exactly including the constant factor in the leading term. Finally, building on these results, we improve the last-iterate convergence rate and the dynamic regret of a learning dynamic based on optimistic Hedge, and complement these bounds with algorithm-dependent dynamic regret lower bounds that match the improved bounds.

We study convex composite optimization problems, where the objective function is given by the sum of a prox-friendly function and a convex function whose subgradients are estimated under heavy-tailed noise. Existing work often employs gradient clipping or normalization techniques in stochastic first-order methods to address heavy-tailed noise. %In this paper, we demonstrate that a vanilla stochastic algorithm—without additional modifications such as clipping or normalization—can achieve optimal complexity for these problems. In this paper, we analyze the first-order oracle complexity of vanilla stochastic algorithms—without additional modifications such as clipping or normalization—for solving these problems. In particular, we establish that an accelerated stochastic proximal subgradient method achieves a first-order oracle complexity for finding an approximate optimal solution in expectation that is universally optimal for smooth, weakly smooth, and nonsmooth convex optimization, as well as for stochastic convex optimization under heavy-tailed noise. Moreover, we derive high-probability first-order oracle complexity bounds for the accelerated stochastic proximal subgradient method under heavy-tailed and sub-Weibull noise, respectively. Numerical experiments are further provided to illustrate the numerical behavior of the methods.

We study dynamic measure transport for generative modeling: specifically, flows induced by stochastic processes that bridge a specified source and target distribution. The conditional expectation of the process' velocity defines an ODE whose flow map achieves the desired transport. We ask which processes produce straight-line flows – i.e., flows whose pointwise acceleration vanishes and thus are exactly integrable with a first-order method? We provide a concise PDE characterization of straightness as a balance between conditional acceleration and the divergence of a weighted covariance (Reynolds) tensor. Using this lens, we fully characterize affine-in-time interpolants and show that straightness occurs exactly under deterministic endpoint couplings. We also derive necessary conditions that constrain flow geometry for general processes, offering broad guidance for designing transports that are easier to integrate.

We present StatTestCalculator (STC), a new open-source statistical analysis tool designed for analysis high energy physics experiments. STC provides both asymptotic calculations and Monte Carlo simulations for computing the exact statistical significance of a discovery or for setting upper limits on signal model parameters. We review the underlying statistical formalism, including profile likelihood ratio test statistics for discovery and exclusion hypotheses, and the asymptotic distributions that allow quick significance estimates. We explain the relevant formulas for the likelihood functions, test statistic distributions, and significance metrics (both with and without incorporating systematic uncertainties). The implementation and capabilities of STC are described, and we validate its performance against the widely-used CMS Combine tool. We find excellent agreement in both the expected discovery significances and upper limit calculations. STC is a flexible framework that can accommodate systematic uncertainties and user-defined statistical models, making it suitable for a broad range of analyses.

This paper provides clear and practical guidance on the specification of imputation models when multiple imputation is used in conjunction with doubly robust estimation methods for causal inference. Through theoretical arguments and targeted simulations, we demonstrate that if a confounder has missing data, the corresponding imputation model must include all variables appearing in either the propensity score model or the outcome model, in addition to both the exposure and the outcome, and that these variables must enter the imputation model in the same functional form as in the final analysis. Violating these conditions can lead to biased treatment effect estimates, even when both components of the doubly robust estimator are correctly specified. We present a mathematical framework for doubly robust estimation combined with multiple imputation, establish the theoretical requirements for proper imputation in this setting, and demonstrate the consequences of misspecification through simulation. Based on these findings, we offer concrete recommendations to ensure valid inference when using multiple imputation with doubly robust methods in applied causal analyses.

Unreplicated two-level factorial designs are often used in screening experiments to determine which factors out of a large plausible set are active. A theorem regarding the generalized word count pattern is stated and proved for unreplicated designs. It is shown that a phenomenon regarding optimal designs seen in the recent literature can be explained by the theorem obtained.

Decision-focused learning (DFL) integrates predictive modeling and optimization by training predictors to optimize the downstream decision target rather than merely minimizing prediction error. To date, existing DFL methods typically rely on deterministic point predictions, which are often insufficient to capture the intrinsic stochasticity of real-world environments. To address this challenge, we propose the first diffusion-based DFL approach, which trains a diffusion model to represent the distribution of uncertain parameters and optimizes the decision by solving a stochastic optimization with samples drawn from the diffusion model. Our contributions are twofold. First, we formulate diffusion DFL using the reparameterization trick, enabling end-to-end training through diffusion. While effective, it is memory and compute-intensive due to the need to differentiate through the diffusion sampling process. Second, we propose a lightweight score function estimator that uses only several forward diffusion passes and avoids backpropagation through the sampling. This follows from our results that backpropagating through stochastic optimization can be approximated by a weighted score function formulation. We empirically show that our diffusion DFL approach consistently outperforms strong baselines in decision quality. The source code for all experiments is available at the project repository: https://github.com/GT-KOALA/Diffusion_DFL.

Two-phase sampling is commonly adopted for reducing cost and improving estimation efficiency. In many two-phase studies, the outcome and some cheap covariates are observed for a large sample in Phase I, and expensive covariates are obtained for a selected subset of the sample in Phase II. As a result, the analysis of the association between the outcome and covariates faces a missing data problem. Complete-case analysis, which relies solely on the Phase II sample, is generally inefficient. In this paper, we study a two-step estimation approach, which first obtains an estimator using the complete data, and then updates it using an asymptotically mean-zero estimator obtained from a working model between the outcome and cheap covariates using the full data. This two-step estimator is asymptotically at least as efficient as the complete-data estimator and is robust to misspecification of the working model. We propose a kernel-based method to construct a two-step estimator that achieves optimal efficiency. Additionally, we develop a simple joint update approach based on multiple working models to approximate the optimal estimator when a fully nonparametric kernel approach is infeasible. We illustrate the proposed methods with various outcome models. We demonstrate their advantages over existing approaches through simulation studies and provide an application to a major cancer genomics study.

A key aspect of patient-focused drug development is identifying and measuring outcomes that are important to patients in clinical trials. Many medical conditions affect multiple symptom domains, and a consensus approach to determine the relative importance of the associated multiple outcomes ignores the heterogeneity in individual patient preferences. Patient-selected outcomes offer one way to incorporate individual patient preferences, as proposed in recent regulatory guidance for the treatment for migraine, where each patient selects their most bothersome migraine-associated symptom in addition to pain. Patient-ranked outcomes have also recently been proposed, which go further and consider the full ranking of the relative importance of all the outcomes. This can be assessed using a composite DOOR (Desirability of Outcome Ranking) endpoint. In this paper, we compare the advantages and disadvantages of using patient-selected versus patient-ranked outcomes in the context of a two-arm randomised controlled trial for multiple sclerosis. We compare the power and type I error rate by simulation, and discuss several other important considerations when using the two approaches.

We suppose we are given a list of points $x_1, \dots, x_n \in \mathbb{R}$, a target probability measure $\mu$ and are asked to add additional points $x_{n+1}, \dots, x_{n+m}$ so that $x_1, \dots, x_{n+m}$ is as close as possible to the distribution of $\mu$; additionally, we want this to be true uniformly for all $m$. We propose a simple method that achieves this goal. It selects new points in regions where the existing set is lacking points and avoids regions that are already overly crowded. If we replace $\mu$ by another measure $\mu_2$ in the middle of the computation, the method dynamically adjusts and allows us to keep the original sampling points. $x_{n+1}$ can be computed in $\mathcal{O}(n)$ steps and we obtain state-of-the-art results. It appears to be an interesting dynamical system in its own right; we analyze a continuous mean-field version that reflects much of the same behavior.

High-dimensional regression often suffers from heavy-tailed noise and outliers, which can severely undermine the reliability of least-squares based methods. To improve robustness, we adopt a non-smooth Wilcoxon score based rank objective and incorporate the group sparsity regularization. By extending the tuning-free property originally developed for the rank Lasso, we introduce a simulation-based tuning rule and further establish a finite-sample error bound for the resulting estimator. To solve the associated optimization problem, we develop a proximal augmented Lagrangian method, for which we provide a novel convergence analysis by proving the metric subregularity of the underlying non-polyhedral KKT mapping, while enabling efficient semismooth Newton updates for the subproblems. Extensive numerical experiments demonstrate the robustness and effectiveness of our proposed estimator against several leading alternatives, and showcase the efficiency and scalability of our algorithm compared to the state-of-the-art baseline in both simulated and real-data settings.

With double-truncated lifespans, we test the hypothesis of a parametric distribution family for the lifespan. The typical finding from demography is an instationary behaviour of the life expectancy, and a copula models the resulting weak dependence of lifespan and the age at truncation. Our main example is the Farlie-Gumbel-Morgenststern copula. The test is based on Donsker-class arguments and the functional delta method for empirical processes. The assumptions also allow parametric inference, and proofs slightly simplify due to the compact support of the observations. An algorithm with finitely many operations is given for the computation of the test statistic. Simulations becomes necessary for computing the critical value. With the exponential distribution as an example, and for the application to 55{,}000 German double-truncated enterprise lifespans, the constructed Kolmogorov-Smirnov test rejects clearly an age-homogeneous closure hazard.

In this work, a single-unit multi-state system is considered. The system is subject to internal failures, as well as external shocks with multiple consequences. It also incorporates a preventive maintenance strategy and a Bernoulli vacation policy for the repairperson. It is algorithmically modeled in both continuous and discrete time using Marked Markovian Arrival Processes (MMAP). The system's operation/degradation level is divided into an indeterminate number of levels. Upon returning from a vacation period, the repair technician may initiate corrective repair, perform preventive maintenance, replace the unit, remain idle at the workplace, or begin a new vacation period. The decision in the latter two cases is made probabilistically based on the system's operational level. This methodology allows the model and its associated measures to be algorithmically derived in both transient and stationary regimes, presented in a matrix-algorithmic form. Analytical-matrix methods are used to obtain the system's steady-state behaviour as well as various performance measures. Costs and rewards are introduced to analyze when the system becomes profitable. Measures associated with costs over time and in the stationary regime are defined and considered for optimization studies. A numerical example demonstrates the versatility of the model by solving a probabilistic optimization problem using a multi-objective Pareto analysis approach and performing a comparative evaluation of multiple models. Genetic algorithm is applied to find the optimization results in the reduced solution space. All modeling and associated measures have been computationally implemented in Matlab.

Recent advances in reasoning domains with neural networks have primarily been enabled by a training recipe that optimizes Large Language Models, previously trained to predict the next-token in a sequence, with reinforcement learning algorithms. We introduce a framework to study the success of this paradigm, and we theoretically expose the optimization mechanisms by which reinforcement learning improves over next-token prediction in this setting. We study learning from mixture distributions of short and long “chain-of-thought” sequences encoding a single task. In particular, when the task consists of predicting the parity of $d$ bits and long sequences are rare, we show how reinforcement learning after next-token prediction enables autoregressive transformers to generalize, whereas mere next-token prediction requires extreme statistical or computational resources to do so. We further explain how reinforcement learning leverages increased test-time computation, manifested in longer responses, to facilitate this learning process. In a simplified setting, we theoretically prove that autoregressive linear models following this training recipe can efficiently learn to predict the parity of $d$ bits as long as the proportion of long demonstrations in the data mix is not exponentially small in the input dimension $d$. Finally, we demonstrate these same phenomena in other settings, including the post-training of Llama-series models on mixture variations of common mathematical reasoning benchmarks.

Greenhouse decisions often rely on static thresholds, yet crop output switches among microclimate-driven regimes. We frame daily cucumber yield as transitions among three ordered states and fit a continuous-time, covariate-dependent multistate model. Data come from four greenhouse compartments in Volos, Greece (24 lines, 62 days). States are defined once from control tertiles and applied across compartments. Transition intensities depend on within-compartment z-scores of relative humidity (RH), photosynthetically active radiation (PAR) and CO2, plus fixed effects. Results show an inherent upward drift through the medium state, "sticky" low-yield spells unless conditions improve, and short-horizon persistence once high yield is reached. RH and PAR are dominant levers, accelerating upgrades and damping regressions; day-to-day CO2 deviations show no clear pooled signal. Residual differences between compartments are modest. By mapping intensities to 7–30 day probabilities, the model yields actionable guidance for humidity and lighting and a lightweight, interpretable component for greenhouse digital twins.

Selecting an effective step-size is a fundamental challenge in first-order optimization, especially for problems with non-Euclidean geometries. This paper presents a novel adaptive step-size strategy for optimization algorithms that rely on linear minimization oracles, as used in the Conditional Gradient or non-Euclidean Normalized Steepest Descent algorithms. Using a simple heuristic to estimate a local Lipschitz constant for the gradient, we can determine step-sizes that guarantee sufficient decrease at each iteration. More precisely, we establish convergence guarantees for our proposed Adaptive Conditional Gradient Descent algorithm, which covers as special cases both the classical Conditional Gradient algorithm and non-Euclidean Normalized Steepest Descent algorithms with adaptive step-sizes. Our analysis covers optimization of continuously differentiable functions in non-convex, quasar-convex, and strongly convex settings, achieving convergence rates that match state-of-the-art theoretical bounds. Comprehensive numerical experiments validate our theoretical findings and illustrate the practical effectiveness of Adaptive Conditional Gradient Descent. The results exhibit competitive performance, underscoring the potential of the adaptive step-size for applications.

In recent years, more and more large data sets have become available. Data accuracy, the absence of verifiable errors in data, is crucial for these large materials to enable high-quality research, downstream applications, and model training. This results in the problem of how to curate or improve data accuracy in such large and growing data, especially when the data is too large for manual curation to be feasible. This paper presents a unified procedure for iterative and continuous improvement of data sets. We provide theoretical guarantees that data accuracy tests speed up error reduction and, most importantly, that the proposed approach will, asymptotically, eliminate all errors in data with probability one. We corroborate the theoretical results with simulations and a real-world use case.

Adam is a popular and widely used adaptive gradient method in deep learning, which has also received tremendous focus in theoretical research. However, most existing theoretical work primarily analyzes its full-batch version, which differs fundamentally from the stochastic variant used in practice. Unlike SGD, stochastic Adam does not converge to its full-batch counterpart even with infinitesimal learning rates. We present the first theoretical characterization of how batch size affects Adam's generalization, analyzing two-layer over-parameterized CNNs on image data. Our results reveal that while both Adam and AdamW with proper weight decay $\lambda$ converge to poor test error solutions, their mini-batch variants can achieve near-zero test error. We further prove Adam has a strictly smaller effective weight decay bound than AdamW, theoretically explaining why Adam requires more sensitive $\lambda$ tuning. Extensive experiments validate our findings, demonstrating the critical role of batch size and weight decay in Adam's generalization performance.

We propose a new class of sequential change point tests, both for changes in the mean parameter and in the overall distribution function. The methodology builds on a two-window inspection scheme (TWIN), which aggregates data into symmetric samples and applies strong weighting to enhance statistical performance. The detector yields logarithmic rather than polynomial detection delays, representing a substantial reduction compared to state-of-the-art alternatives. Delays remain short, even for late changes, where existing methods perform worst. Moreover, the new procedure also attains higher power than current methods across broad classes of local alternatives. For mean changes, we further introduce a self-normalized version of the detector that automatically cancels out temporal dependence, eliminating the need to estimate nuisance parameters. The advantages of our approach are supported by asymptotic theory, simulations and an application to monitoring COVID19 data. Here, structural breaks associated with new virus variants are detected almost immediately by our new procedures. This indicates potential value for the real-time monitoring of future epidemics. Mathematically, our approach is underpinned by new exponential moment bounds for the global modulus of continuity of the partial sum process, which may be of independent interest beyond change point testing.

Heavy-tailed probability distributions are extremely useful and play a crucial role in modeling different types of financial data sets. This study presents a two-pronged methodology. First, a mixture probability distribution is created by combining Gaussian and Rayleigh distributions using the arctangent transformation, aimed at producing heavier-tailed features and enhancing alignment with real market data. Some statistical properties of the proposed model are also discussed. Furthermore, essential actuarial risk evaluation instruments, such as value-at-risk (VaR), tail value-at-risk (TVaR) and tail variance (TV) are employed for efficient risk management practices. Lastly, an application is provided using an insurance dataset to demonstrate the applicability of the proposed model. The proposed model demonstrates superior fitting performance compared to current baseline distributions, showcasing its practical value in financial risk evaluation. The combination of Gaussian and Rayleigh distributions through arctangent transformation is particularly successful in representing extreme market behaviour and tail dependencies that are frequently found in real-world financial data.

Directional replicability addresses the question of whether an effect studied across $n$ independent studies is present with the same direction in at least $r$ of them, for $r \geq 2$. When the expected direction of the effect is not specified in advance, the state of the art recommends assessing replicability separately by combining one-sided $p$-values for both directions (left and right), and then doubling the smaller of the two resulting combined $p$-values to account for multiple testing. In this work, we show that this multiplicative correction is not always necessary, and give conditions under which it can be safely omitted.

Spline interpolation is a widely used class of methods for solving interpolation problems by constructing smooth interpolants that minimize a regularized energy functional involving the Laplacian operator. While many existing approaches focus on Euclidean domains or the sphere, relying on the spectral properties of the Laplacian, this work introduces a method for spline interpolation on general manifolds by exploiting its equivalence with kriging. Specifically, the proposed approach uses finite element approximations of random fields defined over the manifold, based on Gaussian Markov Random Fields and a discretization of the Laplace-Beltrami operator on a triangulated mesh. This framework enables the modeling of spatial fields with smooth variations and local anisotropies via domain deformation. The method is first validated on the sphere using both analytical test cases and a pollution-related study, and is compared to the classical spherical harmonics-based method. Additional experiments on the surface of a cylinder further illustrate the generality of the approach.

Multi-type Markov point processes offer a flexible framework for modelling complex multi-type point patterns where it is pertinent to capture both interactions between points as well as large scale trends depending on observed covariates. However, estimation of interaction and covariate effects may be seriously biased in the presence of unobserved spatial confounders. In this paper we introduce a new class of semi-parametric Markov point processes that adjusts for spatial confounding through a non-parametric factor that accommodates effects of latent spatial variables common to all types of points. We introduce a conditional pseudo likelihood for parameter estimation and show that the resulting estimator has desirable asymptotic properties. Our methodology not least has great potential in studies of industry agglomeration and we apply it to study spatial patterns of locations of two types of banks in France.

Climate policy modelling is a key tool for assessing mitigation strategies in complex systems, where uncertainty is inherent and unavoidable. We present a general methodology for extensive uncertainty analysis in this field. While other studies have performed uncertainty analyses, few apply methods from the field of Uncertainty Quantification, which are commonly used in other modelling disciplines. We show how emulators can identify key uncertainties in modelling frameworks and demonstrate a novel policy analysis previously restricted by computational cost and limited representation of uncertainty. We apply this methodology to FTT:Power to explore uncertainties in the electricity system transition both globally and in India to assess the robustness of mitigation strategies to a wide range of policy and techno-economic scenarios. This approach results in much larger uncertainties in transition outcomes than commonly represented, but policy design can be shaped to mitigate this. Globally, our results indicate transition uncertainty is dominated by average rates of renewables cannibalisation, construction times and grid connection lead times, outweighing regional price policies, including policy reversals in the US. Solar PV appears most resilient due to low costs, though still sensitive to infrastructure constraints and cannibalisation. Onshore wind is more exposed to a range of uncertainties. In India, we find evidence that policy packages including partial phase-out instruments have greater robustness to key uncertainties, although longer lead times still hinder policy goals. Our results suggest that enabling policy and regulating fossil fuels are critical for robust power sector transitions.

In the field of spatial data analysis, spatially varying coefficients (SVC) models, which allow regression coefficients to vary by region and flexibly capture spatial heterogeneity, have continued to be developed in various directions. Moreover, the Bayesian generalized fused lasso is often used as a method that efficiently provides estimation under the natural assumption that regression coefficients of adjacent regions tend to take the same value. In most Bayesian methods, the selection of prior distribution is an essential issue, and in the setting of SVC model with the Bayesian generalized fused lasso, determining the complexity of the class of prior distributions is also a challenging aspect, further amplifying the difficulty of the problem. For example, the widely applicable information criterion (WAIC), which has become standard in Bayesian model selection, does not target determining the complexity. Therefore, in this study, we adapted a criterion called the prior intensified information criterion (PIIC) to this setting. Specifically, under an asymptotic setting that retains the influence of the prior distribution, that is, under an asymptotic setting that deliberately does not provide selection consistency, we derived the asymptotic properties of our generalized fused lasso estimator. Then, based on these properties, we constructed an information criterion as an asymptotically bias-corrected estimator of predictive risk. In numerical experiments, we confirmed that PIIC outperforms WAIC in the sense of reducing the predictive risk, and in a real data analysis, we observed that the two criteria give rise to substantially different results.

PAC generalization bounds on the risk, when expressed in terms of the expected loss, are often insufficient to capture imbalances between subgroups in the data. To overcome this limitation, we introduce a new family of risk measures, called constrained f-entropic risk measures, which enable finer control over distributional shifts and subgroup imbalances via f-divergences, and include the Conditional Value at Risk (CVaR), a well-known risk measure. We derive both classical and disintegrated PAC-Bayesian generalization bounds for this family of risks, providing the first disintegratedPAC-Bayesian guarantees beyond standard risks. Building on this theory, we design a self-bounding algorithm that minimizes our bounds directly, yielding models with guarantees at the subgroup level. Finally, we empirically demonstrate the usefulness of our approach.

This paper introduces torchsom, an open-source Python library that provides a reference implementation of the Self-Organizing Map (SOM) in PyTorch. This package offers three main features: (i) dimensionality reduction, (ii) clustering, and (iii) friendly data visualization. It relies on a PyTorch backend, enabling (i) fast and efficient training of SOMs through GPU acceleration, and (ii) easy and scalable integrations with PyTorch ecosystem. Moreover, torchsom follows the scikit-learn API for ease of use and extensibility. The library is released under the Apache 2.0 license with 90% test coverage, and its source code and documentation are available at https://github.com/michelin/TorchSOM.

This study examined the spatial-temporal dynamics of Emergency Examination Order or Authority (EE-O/A) admissions in Far Northern Queensland (FNQ) from 2009 to 2020, using 13,035 unique police records aggregated across 83 postcodes. A two-stage modelling framework was used: Lasso was used to identify a parsimonious set of socio economic and health-service covariates, and a Conditional Autoregressive (CAR) model incorporated these predictors with structured spatial and temporal random effects. This research demonstrates that socio-economic disadvantage and service accessibility drive EE-O/A incidence, underscoring the need for targeted mental-health interventions and resource allocation in impoverished FNQ communities. Limitations include reliance on cross-sectional census data for covariates and potential ecological bias from data fusion.

Conditional sampling is a fundamental task in Bayesian statistics and generative modeling. Consider the problem of sampling from the posterior distribution $P_{X|Y=y^*}$ for some observation $y^*$, where the likelihood $P_{Y|X}$ is known, and we are given $n$ i.i.d. samples $D=\{X_i\}_{i=1}^n$ drawn from an unknown prior distribution $\pi_X$. Suppose that $f(\hat{\pi}_{X^n})$ is the distribution of a posterior sample generated by an algorithm (e.g. a conditional generative model or the Bayes rule) when $\hat{\pi}_{X^n}$ is the empirical distribution of the training data. Although averaging over the randomness of the training data $D$, we have $\mathbb{E}_D\left(\hat{\pi}_{X^n}\right)= \pi_X$, we do not have $\mathbb{E}_D\left\{f(\hat{\pi}_{X^n})\right\}= f(\pi_X)$ due to the nonlinearity of $f$, leading to a bias. In this paper we propose a black-box debiasing scheme that improves the accuracy of such a naive plug-in approach. For any integer $k$ and under boundedness of the likelihood and smoothness of $f$, we generate samples $\hat{X}^{(1)},\dots,\hat{X}^{(k)}$ and weights $w_1,\dots,w_k$ such that $\sum_{i=1}^kw_iP_{\hat{X}^{(i)}}$ is a $k$-th order approximation of $f(\pi_X)$, where the generation process treats $f$ as a black-box. Our generation process achieves higher accuracy when averaged over the randomness of the training data, without degrading the variance, which can be interpreted as improving memorization without compromising generalization in generative models.

This study applies Multiscale Geographically Weighted Regression (MGWR) to examine the spatial determinants of household wealth in Bernalillo County, New Mexico. The model incorporates sociodemographic, environmental, and proximity-based variables to evaluate how locational suitability influences economic outcomes. Key factors considered include income, home value, elevation, PM2.5 concentration, and distances to essential services such as schools, markets, and hospitals. The MGWR model demonstrates strong performance, explaining approximately 63 percent of the variation in household wealth. Results show that proximity to markets, schools, and parks significantly increases wealth in over 40 percent of neighborhoods. In contrast, closeness to hospitals and bus stops is negatively associated with wealth, suggesting that nearby disamenities can reduce housing desirability. Strong spatial autocorrelation (Morans I = 0.53, p < 0.001) indicates that wealthier households are significantly clustered, highlighting the influence of localized factors. Overall, the study reveals that the relationship between locational suitability and household wealth is spatially variable across the county.