An official website of the United States government
Gaussian mixtures are commonly used for modeling heavy-tailed error distributions in robust linear regression. Combining the likelihood of a multivariate robust linear regression model with a standard improper prior distribution yields an analytically intractable posterior distribution that can be sampled using a data augmentation algorithm. When the response matrix has missing entries, there are unique challenges to the application and analysis of the convergence properties of the algorithm. Conditions for geometric ergodicity are provided when the incomplete data have a “monotone” structure. In the absence of a monotone structure, an intermediate imputation step is necessary for implementing the algorithm. In this case, we provide sufficient conditions for the algorithm to be Harris ergodic. Finally, we show that, when there is a monotone structure and intermediate imputation is unnecessary, intermediate imputation slows the convergence of the underlying Monte Carlo Markov chain, while post hoc imputation does not. An R package for the data augmentation algorithm is provided.
more »
« less
This content will become publicly available on July 9, 2026
Efficient Construction of the Characteristic Function of the Cauchy Estimator Using Basis Functions
A newly enhanced, recursive, and robust Bayesian state estimation algorithm for linear and nonlinear systems, referred to as the Multivariate Cauchy Estimator (MCE), is presented. The algorithm enables robust state estimation performance for applications with more volatile system noises than the Gaussian distribution suggests. This is achieved by over-bounding realistic process and measurement noises with additive, heavy-tailed Cauchy random variables. The characteristic function (CF) of the un-normalized conditional probability density function (ucpdf) is propagated as a growing sum of terms in the MCE. Here, the CF is simplified by replacing the original with a representation of linear parameter vectors that operate on bases composed of indicator functions. This insight can lead to eliminating over 99% of terms that previously comprised this CF. Using graphical processing units, the MCE can exploit its parallel mathematical structure and achieve a fast execution rate. A target tracking example shows the robustness of the MCE over the Kalman filter in both heavy-tailed and Gaussian noise.
more »
« less
- Award ID(s):
- 2317583
- PAR ID:
- 10616544
- Publisher / Repository:
- American Control Conference (ACC)
- Date Published:
- Subject(s) / Keyword(s):
- control-systems, stochastic estimation, bayesian estimation, CUDA-C programming, GPU linear programming, cell enumeration, Cauchy pdfs
- Format(s):
- Medium: X
- Location:
- 2025 American Control Conference (ACC)
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)The goal of compressed sensing is to estimate a high dimensional vector from an underdetermined system of noisy linear equations. In analogy to classical compressed sensing, here we assume a generative model as a prior, that is, we assume the vector is represented by a deep generative model G:Rk→Rn. Classical recovery approaches such as empirical risk minimization (ERM) are guaranteed to succeed when the measurement matrix is sub-Gaussian. However, when the measurement matrix and measurements are heavy-tailed or have outliers, recovery may fail dramatically. In this paper we propose an algorithm inspired by the Median-of-Means (MOM). Our algorithm guarantees recovery for heavy-tailed data, even in the presence of outliers. Theoretically, our results show our novel MOM-based algorithm enjoys the same sample complexity guarantees as ERM under sub-Gaussian assumptions. Our experiments validate both aspects of our claims: other algorithms are indeed fragile and fail under heavy-tailed and/or corrupted data, while our approach exhibits the predicted robustness.more » « less
-
We consider the task of heavy-tailed statistical estimation given streaming p-dimensional samples. This could also be viewed as stochastic optimization under heavy-tailed distributions, with an additional O(p) space complexity constraint. We design a clipped stochastic gradient descent algorithm and provide an improved analysis, under a more nuanced condition on the noise of the stochastic gradients, which we show is critical when analyzing stochastic optimization problems arising from general statistical estimation problems. Our results guarantee convergence not just in expectation but with exponential concentration, and moreover does so using O(1) batch size. We provide consequences of our results for mean estimation and linear regression. Finally, we provide empirical corroboration of our results and algorithms via synthetic experiments for mean estimation and linear regression.more » « less
-
We study offline reinforcement learning (RL) with heavy-tailed reward distribution and data corruption: (i) Moving beyond subGaussian reward distribution, we allow the rewards to have infinite variances; (ii) We allow corruptions where an attacker can arbitrarily modify a small fraction of the rewards and transitions in the dataset. We first derive a sufficient optimality condition for generalized Pessimistic Value Iteration (PEVI), which allows various estimators with proper confidence bounds and can be applied to multiple learning settings. In order to handle the data corruption and heavy-tailed reward setting, we prove that the trimmed-mean estimation achieves the minimax optimal error rate for robust mean estimation under heavy-tailed distributions. In the PEVI algorithm, we plug in the trimmed mean estimation and the confidence bound to solve the robust offline RL problem. Standard analysis reveals that data corruption induces a bias term in the suboptimality gap, which gives the false impression that any data corruption prevents optimal policy learning. By using the optimality condition for the generalized PEVI, we show that as long as the bias term is less than the ``action gap'', the policy returned by PEVI achieves the optimal value given sufficient data.more » « less
-
Motivated by robust and quantile regression problems, we investigate the stochastic gradient descent (SGD) algorithm for minimizing an objective functionfthat is locally strongly convex with a sub--quadratic tail. This setting covers many widely used online statistical methods. We introduce a novel piecewise Lyapunov function that enables us to handle functionsfwith only first-order differentiability, which includes a wide range of popular loss functions such as Huber loss. Leveraging our proposed Lyapunov function, we derive finite-time moment bounds under general diminishing stepsizes, as well as constant stepsizes. We further establish the weak convergence, central limit theorem and bias characterization under constant stepsize, providing the first geometrical convergence result for sub--quadratic SGD. Our results have wide applications, especially in online statistical methods. In particular, we discuss two applications of our results. 1) Online robust regression: We consider a corrupted linear model with sub--exponential covariates and heavy--tailed noise. Our analysis provides convergence rates comparable to those for corrupted models with Gaussian covariates and noise. 2) Online quantile regression: Importantly, our results relax the common assumption in prior work that the conditional density is continuous and provide a more fine-grained analysis for the moment bounds.more » « less
