arXiv subjects

Rajita Chandak

Publications and source records attributed to Rajita Chandak.

5 recordsLinked to original sources

arXiv2026

Global convergence analysis of mixtures of Exponential densities

The theoretical foundations of the EM algorithm are often thought of in the context of Gaussian mixture models, However, the practical use cases of the EM algorithm span beyond Gaussian models. This paper establishes the first step towards understanding the behavior of the EM algorithm under mixtures of non-Gaussian densities. We show that a mixture of two Exponential distributions can be approximated by the EM algorithm at the sub-Exponential rate of convergence in at most $\log(n)$ iterations. The results here show that extending away from Gaussian mixture models does not affect the statistical performance of the EM algorithm. Furthermore, we present generalizations of typical assumptions in the Gaussian setting like minimum mean-separation and signal-to-noise ratio to the sub-Exponential setting. A simulation study is used to highlight the empirical performance of EM for mixtures of exponentials with promising results for the extension of existing theory to a larger class of mixture models.

math.ST↗

arXiv2026

Fast convergence of a Federated Expectation-Maximization Algorithm

Data heterogeneity has been a long-standing bottleneck in studying the convergence rates of Federated Learning algorithms. In order to better understand the issue of data heterogeneity, we study the convergence rate of the Expectation-Maximization (EM) algorithm for the Federated Mixture of $K$ Linear Regressions model (FMLR). We completely characterize the convergence rate of the EM algorithm under all regimes of number of clients and number of data points per client, with partial limits in the number of clients. We show that with a signal-to-noise-ratio (SNR) that is atleast of order $\sqrt{K}$, the well-initialized EM algorithm converges to the ground truth under all regimes. We perform experiments on synthetic data to illustrate our results. In line with our theoretical findings, the simulations show that rather than being a bottleneck, data heterogeneity can accelerate the convergence of iterative federated algorithms.

stat.ML↗

arXiv2025

lpcde: Estimation and Inference for Local Polynomial Conditional Density Estimators

This paper discusses the R package lpcde, which stands for local polynomial conditional density estimation. It implements the kernel-based local polynomial smoothing methods introduced in Cattaneo, Chandak, Jansson, Ma (2024) for statistical estimation and inference of conditional distributions, densities, and derivatives thereof. The package offers mean square error optimal bandwidth selection and associated point estimators, as well as uncertainty quantification based on robust bias correction both pointwise (e.g., confidence intervals) and uniformly (e.g., confidence bands) over evaluation points. The methods implemented are boundary adaptive whenever the data is compactly supported. The package also implements regularized conditional density estimation methods, ensuring the resulting density estimate is non-negative and integrates to one. We contrast the functionalities of lpcde with existing open-source packages for conditional density estimation, and showcase its main features using simulated and real datasets. An abbreviated version of this article is published in Cattaneo, Chandak, Jansson, Ma (2025 JOSS).

stat.CO↗

arXiv2023

Boundary Adaptive Local Polynomial Conditional Density Estimators

We begin by introducing a class of conditional density estimators based on local polynomial techniques. The estimators are boundary adaptive and easy to implement. We then study the (pointwise and) uniform statistical properties of the estimators, offering characterizations of both probability concentration and distributional approximation. In particular, we establish uniform convergence rates in probability and valid Gaussian distributional approximations for the Studentized t-statistic process. We also discuss implementation issues such as consistent estimation of the covariance function for the Gaussian approximation, optimal integrated mean squared error bandwidth selection, and valid robust bias-corrected inference. We illustrate the applicability of our results by constructing valid confidence bands and hypothesis tests for both parametric specification and shape constraints, explicitly characterizing their approximation errors. A companion R software package implementing our main results is provided.

math.ST↗

arXiv2023

Convergence Rates of Oblique Regression Trees for Flexible Function Libraries

We develop a theoretical framework for the analysis of oblique decision trees, where the splits at each decision node occur at linear combinations of the covariates (as opposed to conventional tree constructions that force axis-aligned splits involving only a single covariate). While this methodology has garnered significant attention from the computer science and optimization communities since the mid-80s, the advantages they offer over their axis-aligned counterparts remain only empirically justified, and explanations for their success are largely based on heuristics. Filling this long-standing gap between theory and practice, we show that oblique regression trees (constructed by recursively minimizing squared error) satisfy a type of oracle inequality and can adapt to a rich library of regression models consisting of linear combinations of ridge functions and their limit points. This provides a quantitative baseline to compare and contrast decision trees with other less interpretable methods, such as projection pursuit regression and neural networks, which target similar model forms. Contrary to popular belief, one need not always trade-off interpretability with accuracy. Specifically, we show that, under suitable conditions, oblique decision trees achieve similar predictive accuracy as neural networks for the same library of regression models. To address the combinatorial complexity of finding the optimal splitting hyperplane at each decision node, our proposed theoretical framework can accommodate many existing computational tools in the literature. Our results rely on (arguably surprising) connections between recursive adaptive partitioning and sequential greedy approximation algorithms for convex optimization problems (e.g., orthogonal greedy algorithms), which may be of independent theoretical interest. Using our theory and methods, we also study oblique random forests.

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