Xudong Li

Yukuan Wei, Xudong Li, Lin F. Yang

Recent advances have significantly improved our understanding of the sample complexity of learning in average-reward Markov decision processes (AMDPs) under the generative model. However, much less is known about the constrained average-reward MDP (CAMDP), where policies must satisfy long-run average constraints. In this work, we address this gap by studying the sample complexity of learning an $ε$-optimal policy in CAMDPs under a generative model. We propose a model-based algorithm that operates under two settings: (i) relaxed feasibility, which allows small constraint violations, and (ii) strict feasibility, where the output policy satisfies the constraint. We show that our algorithm achieves sample complexities of $\tilde{O}\left(\frac{S A (B+H)}{ ε^2}\right)$ and $\tilde{O} \left(\frac{S A (B+H)}{ε^2 ζ^2} \right)$ under the relaxed and strict feasibility settings, respectively. Here, $ζ$ is the Slater constant indicating the size of the feasible region, $H$ is the span bound of the bias function, and $B$ is the transient time bound. Moreover, a matching lower bound of $\tildeΩ\left(\frac{S A (B+H)}{ ε^2ζ^2}\right)$ for the strict feasibility case is established, thus providing the first minimax-optimal bounds for CAMDPs. Our results close the theoretical gap in understanding the complexity of constrained average-reward MDPs.

Shuoguang Yang, Xudong Li, Guanghui Lan

Attention to data-driven optimization approaches, including the well-known stochastic gradient descent method, has grown significantly over recent decades, but data-driven constraints have rarely been studied, because of the computational challenges of projections onto the feasible set defined by these hard constraints. In this paper, we focus on the non-smooth convex-concave stochastic minimax regime and formulate the data-driven constraints as expectation constraints. The minimax expectation constrained problem subsumes a broad class of real-world applications, including two-player zero-sum game and data-driven robust optimization. We propose a class of efficient primal-dual algorithms to tackle the minimax expectation-constrained problem, and show that our algorithms converge at the optimal rate of $\mathcal{O}(\frac{1}{\sqrt{N}})$. We demonstrate the practical efficiency of our algorithms by conducting numerical experiments on large-scale real-world applications.

Yuetian Luo, Xudong Li, Anru R. Zhang

In this paper, we propose a general procedure for establishing the geometric landscape connections of a Riemannian optimization problem under the embedded and quotient geometries. By applying the general procedure to the fixed-rank positive semidefinite (PSD) and general matrix optimization, we establish an exact Riemannian gradient connection under two geometries at every point on the manifold and sandwich inequalities between the spectra of Riemannian Hessians at Riemannian first-order stationary points (FOSPs). These results immediately imply an equivalence on the sets of Riemannian FOSPs, Riemannian second-order stationary points (SOSPs), and strict saddles of fixed-rank matrix optimization under the embedded and the quotient geometries. To the best of our knowledge, this is the first geometric landscape connection between the embedded and the quotient geometries for fixed-rank matrix optimization and it provides a concrete example of how these two geometries are connected in Riemannian optimization. In addition, the effects of the Riemannian metric and quotient structure on the landscape connection are discussed. We also observe an algorithmic connection between two geometries with some specific Riemannian metrics in fixed-rank matrix optimization: there is an equivalence between gradient flows under two geometries with shared spectra of Riemannian Hessians. A number of novel ideas and technical ingredients including a unified treatment for different Riemannian metrics, novel metrics for the Stiefel manifold, and new horizontal space representations under quotient geometries are developed to obtain our results. The results in this paper deepen our understanding of geometric and algorithmic connections of Riemannian optimization under different Riemannian geometries and provide a few new theoretical insights to unanswered questions in the literature.

Yuetian Luo, Xudong Li, Anru R. Zhang

In this paper, we consider the geometric landscape connection of the widely studied manifold and factorization formulations in low-rank positive semidefinite (PSD) and general matrix optimization. We establish a sandwich relation on the spectrum of Riemannian and Euclidean Hessians at first-order stationary points (FOSPs). As a result of that, we obtain an equivalence on the set of FOSPs, second-order stationary points (SOSPs) and strict saddles between the manifold and the factorization formulations. In addition, we show the sandwich relation can be used to transfer more quantitative geometric properties from one formulation to another. Similarities and differences in the landscape connection under the PSD case and the general case are discussed. To the best of our knowledge, this is the first geometric landscape connection between the manifold and the factorization formulations for handling rank constraints, and it provides a geometric explanation for the similar empirical performance of factorization and manifold approaches in low-rank matrix optimization observed in the literature. In the general low-rank matrix optimization, the landscape connection of two factorization formulations (unregularized and regularized ones) is also provided. By applying these geometric landscape connections, in particular, the sandwich relation, we are able to solve unanswered questions in literature and establish stronger results in the applications on geometric analysis of phase retrieval, well-conditioned low-rank matrix optimization, and the role of regularization in factorization arising from machine learning and signal processing.

Xudong Li, Li Feng, Lei Li et al.

The promotion of construction robots can solve the problem of human resource shortage and improve the quality of decoration. To help the construction robots obtain environmental information, we need to use 3D point cloud, which is widely used in robotics, autonomous driving, and so on. With a good understanding of environmental information, construction robots can work better. However, the dynamic changes of 3D point cloud data may bring difficulties for construction robots to understand environmental information, such as when construction robots renovate houses. The paper proposes a semantic segmentation method for point cloud based on meta-learning. The method includes a basic learning module and a meta-learning module. The basic learning module is responsible for learning data features and evaluating the model, while the meta-learning module is responsible for updating the parameters of the model and improving the model generalization ability. In our work, we pioneered the method of producing datasets for meta-learning in 3D scenes, as well as demonstrated that the Model-Agnostic Meta-Learning (MAML) algorithm can be applied to process 3D point cloud data. At the same time, experiments show that our method can allow the model to be quickly applied to new environments with a few samples. Our method has important applications.

Yuetian Luo, Wen Huang, Xudong Li et al.

In this paper, we propose {\it \underline{R}ecursive} {\it \underline{I}mportance} {\it \underline{S}ketching} algorithm for {\it \underline{R}ank} constrained least squares {\it \underline{O}ptimization} (RISRO). The key step of RISRO is recursive importance sketching, a new sketching framework based on deterministically designed recursive projections, which significantly differs from the randomized sketching in the literature \citep{mahoney2011randomized,woodruff2014sketching}. Several existing algorithms in the literature can be reinterpreted under this new sketching framework and RISRO offers clear advantages over them. RISRO is easy to implement and computationally efficient, where the core procedure in each iteration is to solve a dimension-reduced least squares problem. We establish the local quadratic-linear and quadratic rate of convergence for RISRO under some mild conditions. We also discover a deep connection of RISRO to the Riemannian Gauss-Newton algorithm on fixed rank matrices. The effectiveness of RISRO is demonstrated in two applications in machine learning and statistics: low-rank matrix trace regression and phase retrieval. Simulation studies demonstrate the superior numerical performance of RISRO.

Ziwei Zhu, Xudong Li, Mengdi Wang et al.

Modeling unknown systems from data is a precursor of system optimization and sequential decision making. In this paper, we focus on learning a Markov model from a single trajectory of states. Suppose that the transition model has a small rank despite of having a large state space, meaning that the system admits a low-dimensional latent structure. We show that one can estimate the full transition model accurately using a trajectory of length that is proportional to the total number of states. We propose two maximum likelihood estimation methods: a convex approach with nuclear-norm regularization and a nonconvex approach with rank constraint. We explicitly derive the statistical rates of both estimators in terms of the Kullback-Leiber divergence and the $\ell_2$ error and also establish a minimax lower bound to assess the tightness of these rates. For computing the nonconvex estimator, we develop a novel DC (difference of convex function) programming algorithm that starts with the convex M-estimator and then successively refines the solution till convergence. Empirical experiments demonstrate consistent superiority of the nonconvex estimator over the convex one.

Xudong Li, Mengdi Wang, Anru Zhang

This paper studies the estimation of low-rank Markov chains from empirical trajectories. We propose a non-convex estimator based on rank-constrained likelihood maximization. Statistical upper bounds are provided for the Kullback-Leiber divergence and the $\ell_2$ risk between the estimator and the true transition matrix. The estimator reveals a compressed state space of the Markov chain. We also develop a novel DC (difference of convex function) programming algorithm to tackle the rank-constrained non-smooth optimization problem. Convergence results are established. Experiments show that the proposed estimator achieves better empirical performance than other popular approaches.

Xudong Li, Defeng Sun, Kim-Chuan Toh

For a symmetric positive semidefinite linear system of equations $\mathcal{Q} {\bf x} = {\bf b}$, where ${\bf x} = (x_1,\ldots,x_s)$ is partitioned into $s$ blocks, with $s \geq 2$, we show that each cycle of the classical block symmetric Gauss-Seidel (block sGS) method exactly solves the associated quadratic programming (QP) problem but added with an extra proximal term of the form $\frac{1}{2} \| {\bf x}-{\bf x}^k \|_{\mathcal T}^2$, where ${\mathcal T}$ is a symmetric positive semidefinite matrix related to the sGS decomposition and ${\bf x}^k$ is the previous iterate. By leveraging on such a connection to optimization, we are able to extend the result (which we name as the block sGS decomposition theorem) for solving a convex composite QP (CCQP) with an additional possibly nonsmooth term in $x_1$, i.e., $\min\{ p(x_1) + \frac{1}{2}\langle {\bf x},\, \mathcal{Q} {\bf x} \rangle -\langle {\bf b},\, {\bf x}\rangle\}$, where $p(\cdot)$ is a proper closed convex function. Based on the block sGS decomposition theorem, we are able to extend the classical block sGS method to solve a CCQP. In addition, our extended block sGS method has the flexibility of allowing for inexact computation in each step of the block sGS cycle. At the same time, we can also accelerate the inexact block sGS method to achieve an iteration complexity of $O(1/k^2)$ after performing $k$ block sGS cycles. As a {fundamental} building block, the block sGS decomposition theorem has played a key role in various recently developed algorithms such as the inexact semiproximal {ALM/ADMM} for linearly constrained multi-block convex composite conic programming (CCCP), and the accelerated block coordinate descent method for multi-block CCCP.