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This paper presents a novel strategy for constructing body source terms in the high-order lattice Boltzmann method (LBM), designed to efficiently introduce various physical phenomena by modifying the non-equilibrium distribution function. The source term, expressed as a Hermite polynomial, provides a flexible framework for simulating complex fluid flows. Three typical source terms are given: a body force source for gravity-driven flows, a thermal dissipation source for controlling the Prandtl number, and a pressure tensor source for modeling multiphase flows. Chapman-Enskog analysis confirms that the source terms recover the expected macroscopic equations. Notably, the proposed strategy eliminates the need for explicit construction of the collision operator, a challenge in conventional approaches for handling diverse physical scenarios. Furthermore, the method is compatible with the traditional BGK model, ensuring its applicability to various high-order lattices. The model's accuracy and versatility are validated through a series of benchmark tests, showing excellent agreement with existing literature results.

Recently, reinforcement learning has gained prominence in modern statistics, with policy evaluation being a key component. Unlike traditional machine learning literature on this topic, our work places emphasis on statistical inference for the parameter estimates computed using reinforcement learning algorithms. While most existing analyses assume random rewards to follow standard distributions, limiting their applicability, we embrace the concept of robust statistics in reinforcement learning by simultaneously addressing issues of outlier contamination and heavy-tailed rewards within a unified framework. In this paper, we develop an online robust policy evaluation procedure, and establish the limiting distribution of our estimator, based on its Bahadur representation. Furthermore, we develop a fully-online procedure to efficiently conduct statistical inference based on the asymptotic distribution. This paper bridges the gap between robust statistics and statistical inference in reinforcement learning, offering a more versatile and reliable approach to policy evaluation. Finally, we validate the efficacy of our algorithm through numerical experiments conducted in real-world reinforcement learning experiments., Comment: 63 pages, 32 figures

The debiased estimator is a crucial tool in statistical inference for high-dimensional model parameters. However, constructing such an estimator involves estimating the high-dimensional inverse Hessian matrix, incurring significant computational costs. This challenge becomes particularly acute in distributed setups, where traditional methods necessitate computing a debiased estimator on every machine. This becomes unwieldy, especially with a large number of machines. In this paper, we delve into semi-supervised sparse statistical inference in a distributed setup. An efficient multi-round distributed debiased estimator, which integrates both labeled and unlabelled data, is developed. We will show that the additional unlabeled data helps to improve the statistical rate of each round of iteration. Our approach offers tailored debiasing methods for $M$-estimation and generalized linear models according to the specific form of the loss function. Our method also applies to a non-smooth loss like absolute deviation loss. Furthermore, our algorithm is computationally efficient since it requires only one estimation of a high-dimensional inverse covariance matrix. We demonstrate the effectiveness of our method by presenting simulation studies and real data applications that highlight the benefits of incorporating unlabeled data., Comment: IEEE Transactions on Information Theory, 2023

Privacy-preserving data analysis has become more prevalent in recent years. In this study, we propose a distributed group differentially private Majority Vote mechanism, for the sign selection problem in a distributed setup. To achieve this, we apply the iterative peeling to the stability function and use the exponential mechanism to recover the signs. For enhanced applicability, we study the private sign selection for mean estimation and linear regression problems, in distributed systems. Our method recovers the support and signs with the optimal signal-to-noise ratio as in the non-private scenario, which is better than contemporary works of private variable selections. Moreover, the sign selection consistency is justified by theoretical guarantees. Simulation studies are conducted to demonstrate the effectiveness of the proposed method., Comment: 41 pages, 5 figures

This paper develops an efficient distributed inference algorithm, which is robust against a moderate fraction of Byzantine nodes, namely arbitrary and possibly adversarial machines in a distributed learning system. In robust statistics, the median-of-means (MOM) has been a popular approach to hedge against Byzantine failures due to its ease of implementation and computational efficiency. However, the MOM estimator has the shortcoming in terms of statistical efficiency. The first main contribution of the paper is to propose a variance reduced median-of-means (VRMOM) estimator, which improves the statistical efficiency over the vanilla MOM estimator and is computationally as efficient as the MOM. Based on the proposed VRMOM estimator, we develop a general distributed inference algorithm that is robust against Byzantine failures. Theoretically, our distributed algorithm achieves a fast convergence rate with only a constant number of rounds of communications. We also provide the asymptotic normality result for the purpose of statistical inference. To the best of our knowledge, this is the first normality result in the setting of Byzantine-robust distributed learning. The simulation results are also presented to illustrate the effectiveness of our method., Comment: 64 pages, 3 figures