arXiv subjects

Ke Wei

Publications and source records attributed to Ke Wei.

At least 37 records · Page 2Linked to original sources

arXiv2024

Elementary Analysis of Policy Gradient Methods

Projected policy gradient under the simplex parameterization, policy gradient and natural policy gradient under the softmax parameterization, are fundamental algorithms in reinforcement learning. There have been a flurry of recent activities in studying these algorithms from the theoretical aspect. Despite this, their convergence behavior is still not fully understood, even given the access to exact policy evaluations. In this paper, we focus on the discounted MDP setting and conduct a systematic study of the aforementioned policy optimization methods. Several novel results are presented, including 1) global linear convergence of projected policy gradient for any constant step size, 2) sublinear convergence of softmax policy gradient for any constant step size, 3) global linear convergence of softmax natural policy gradient for any constant step size, 4) global linear convergence of entropy regularized softmax policy gradient for a wider range of constant step sizes than existing result, 5) tight local linear convergence rate of entropy regularized natural policy gradient, and 6) a new and concise local quadratic convergence rate of soft policy iteration without the assumption on the stationary distribution under the optimal policy. New and elementary analysis techniques have been developed to establish these results.

math.OC↗

arXiv2024

Global Convergence of Natural Policy Gradient with Hessian-aided Momentum Variance Reduction

Natural policy gradient (NPG) and its variants are widely-used policy search methods in reinforcement learning. Inspired by prior work, a new NPG variant coined NPG-HM is developed in this paper, which utilizes the Hessian-aided momentum technique for variance reduction, while the sub-problem is solved via the stochastic gradient descent method. It is shown that NPG-HM can achieve the global last iterate $ε$-optimality with a sample complexity of $\mathcal{O}(ε^{-2})$, which is the best known result for natural policy gradient type methods under the generic Fisher non-degenerate policy parameterizations. The convergence analysis is built upon a relaxed weak gradient dominance property tailored for NPG under the compatible function approximation framework, as well as a neat way to decompose the error when handling the sub-problem. Moreover, numerical experiments on Mujoco-based environments demonstrate the superior performance of NPG-HM over other state-of-the-art policy gradient methods.

cs.LG↗

arXiv2023

On the Linear Convergence of Policy Gradient under Hadamard Parameterization

The convergence of deterministic policy gradient under the Hadamard parameterization is studied in the tabular setting and the linear convergence of the algorithm is established. To this end, we first show that the error decreases at an $O(\frac{1}{k})$ rate for all the iterations. Based on this result, we further show that the algorithm has a faster local linear convergence rate after $k_0$ iterations, where $k_0$ is a constant that only depends on the MDP problem and the initialization. To show the local linear convergence of the algorithm, we have indeed established the contraction of the sub-optimal probability $b_s^k$ (i.e., the probability of the output policy $π^k$ on non-optimal actions) when $k\ge k_0$.

math.OC↗

arXiv2023

A Zeroth-Order Variance-Reduced Method for Decentralized Stochastic Non-convex Optimization

In this paper, we consider a distributed stochastic non-convex optimization problem, which is about minimizing a sum of $n$ local cost functions over a network with only zeroth-order information. A novel single-loop Decentralized Zeroth-Order Variance Reduction algorithm, called DZOVR, is proposed, which combines two-point gradient estimation, momentum-based variance reduction technique, and gradient tracking. Under mild assumptions, we show that the algorithm is able to achieve $\mathcal{O}(dn^{-1}ε^{-3})$ sampling complexity at each node to reach an $ε$-accurate stationary point and also exhibits network-independent and linear speedup properties. To the best of our knowledge, this is the first stochastic decentralized zeroth-order algorithm that achieves this sampling complexity. Numerical experiments demonstrate that DZOVR outperforms the other state-of-the-art algorithms and has network-independent and linear speedup properties.

math.OC↗

arXiv2023

Improved Complexity Analysis of the Sinkhorn and Greenkhorn Algorithms for Optimal Transport

The Sinkhorn algorithm is a widely used method for solving the optimal transport problem, and the Greenkhorn algorithm is one of its variants. While there are modified versions of these two algorithms whose computational complexities are $O({n^2\|C\|_\infty^2\log n}/{\varepsilon^2})$ to achieve an $\varepsilon$-accuracy, to the best of our knowledge, the existing complexities for the vanilla versions are $O({n^2\|C\|_\infty^3\log n}/{\varepsilon^3})$. In this paper we fill this gap and show that the complexities of the vanilla Sinkhorn and Greenkhorn algorithms are indeed $O({n^2\|C\|_\infty^2\log n}/{\varepsilon^2})$. The analysis relies on the equicontinuity of the dual variables for the discrete entropic regularized optimal transport problem, which is of independent interest.

math.OC↗

arXiv2023

Implicit Regularization and Entrywise Convergence of Riemannian Optimization for Low Tucker-Rank Tensor Completion

This paper is concerned with the low Tucker-rank tensor completion problem, which is about reconstructing a tensor $ T \in\mathbb{R}^{n\times n \times n}$ of low multilinear rank from partially observed entries. Riemannian optimization algorithms are a class of efficient methods for this problem, but the theoretical convergence analysis is still lacking. In this manuscript, we establish the entrywise convergence of the vanilla Riemannian gradient method for low Tucker-rank tensor completion under the nearly optimal sampling complexity $O(n^{3/2})$. Meanwhile, the implicit regularization phenomenon of the algorithm has also been revealed. As far as we know, this is the first work that has shown the entrywise convergence and implicit regularization property of a non-convex method for low Tucker-rank tensor completion. The analysis relies on the leave-one-out technique, and some of the technical results developed in the paper might be of broader interest in investigating the properties of other non-convex methods for this problem.

math.OC↗

arXiv2023

Tensor Completion via Tensor Train Based Low-Rank Quotient Geometry under a Preconditioned Metric

This paper investigates the low-rank tensor completion problem, which is about recovering a tensor from partially observed entries. We consider this problem in the tensor train format and extend the preconditioned metric from the matrix case to the tensor case. The first-order and second-order quotient geometry of the manifold of fixed tensor train rank tensors under this metric is studied in detail. Algorithms, including Riemannian gradient descent, Riemannian conjugate gradient, and Riemannian Gauss-Newton, have been proposed for the tensor completion problem based on the quotient geometry. It has also been shown that the Riemannian Gauss-Newton method on the quotient geometry is equivalent to the Riemannian Gauss-Newton method on the embedded geometry with a specific retraction. Empirical evaluations on random instances as well as on function-related tensors show that the proposed algorithms are competitive with other existing algorithms in terms of recovery ability, convergence performance, and reconstruction quality.

math.OC↗

arXiv2022

Decentralized Natural Policy Gradient with Variance Reduction for Collaborative Multi-Agent Reinforcement Learning

This paper studies a policy optimization problem arising from collaborative multi-agent reinforcement learning in a decentralized setting where agents communicate with their neighbors over an undirected graph to maximize the sum of their cumulative rewards. A novel decentralized natural policy gradient method, dubbed Momentum-based Decentralized Natural Policy Gradient (MDNPG), is proposed, which incorporates natural gradient, momentum-based variance reduction, and gradient tracking into the decentralized stochastic gradient ascent framework. The $\mathcal{O}(n^{-1}ε^{-3})$ sample complexity for MDNPG to converge to an $ε$-stationary point has been established under standard assumptions, where $n$ is the number of agents. It indicates that MDNPG can achieve the optimal convergence rate for decentralized policy gradient methods and possesses a linear speedup in contrast to centralized optimization methods. Moreover, superior empirical performance of MDNPG over other state-of-the-art algorithms has been demonstrated by extensive numerical experiments.

math.OC↗

arXiv2022

Approximation Theory of Total Variation Minimization for Data Completion

Total variation (TV) minimization is one of the most important techniques in modern signal/image processing, and has wide range of applications. While there are numerous recent works on the restoration guarantee of the TV minimization in the framework of compressed sensing, there are few works on the restoration guarantee of the restoration from partial observations. This paper is to analyze the error of TV based restoration from random entrywise samples. In particular, we estimate the error between the underlying original data and the approximate solution that interpolates (or approximates with an error bound depending on the noise level) the given data that has the minimal TV seminorm among all possible solutions. Finally, we further connect the error estimate for the discrete model to the sparse gradient restoration problem and to the approximation to the underlying function from which the underlying true data comes.

math.AP↗

arXiv2021

Is Attention Better Than Matrix Decomposition?

As an essential ingredient of modern deep learning, attention mechanism, especially self-attention, plays a vital role in the global correlation discovery. However, is hand-crafted attention irreplaceable when modeling the global context? Our intriguing finding is that self-attention is not better than the matrix decomposition (MD) model developed 20 years ago regarding the performance and computational cost for encoding the long-distance dependencies. We model the global context issue as a low-rank recovery problem and show that its optimization algorithms can help design global information blocks. This paper then proposes a series of Hamburgers, in which we employ the optimization algorithms for solving MDs to factorize the input representations into sub-matrices and reconstruct a low-rank embedding. Hamburgers with different MDs can perform favorably against the popular global context module self-attention when carefully coping with gradients back-propagated through MDs. Comprehensive experiments are conducted in the vision tasks where it is crucial to learn the global context, including semantic segmentation and image generation, demonstrating significant improvements over self-attention and its variants.

cs.CV↗

arXiv2021

An Inexact Riemannian Proximal Gradient Method

This paper considers the problem of minimizing the summation of a differentiable function and a nonsmooth function on a Riemannian manifold. In recent years, proximal gradient method and its invariants have been generalized to the Riemannian setting for solving such problems. Different approaches to generalize the proximal mapping to the Riemannian setting lead versions of Riemannian proximal gradient methods. However, their convergence analyses all rely on solving their Riemannian proximal mapping exactly, which is either too expensive or impracticable. In this paper, we study the convergence of an inexact Riemannian proximal gradient method. It is proven that if the proximal mapping is solved sufficiently accurately, then the global convergence and local convergence rate based on the Riemannian Kurdyka-Łojasiewicz property can be guaranteed. Moreover, practical conditions on the accuracy for solving the Riemannian proximal mapping are provided. As a byproduct, the proximal gradient method on the Stiefel manifold proposed in~[CMSZ2020] can be viewed as the inexact Riemannian proximal gradient method provided the proximal mapping is solved to certain accuracy. Finally, numerical experiments on sparse principal component analysis are conducted to test the proposed practical conditions.

math.OC↗

arXiv2021

Vectorized Hankel Lift: A Convex Approach for Blind Super-Resolution of Point Sources

We consider the problem of resolving $ r$ point sources from $n$ samples at the low end of the spectrum when point spread functions (PSFs) are not known. Assuming that the spectrum samples of the PSFs lie in low dimensional subspace (let $s$ denote the dimension), this problem can be reformulated as a matrix recovery problem, followed by location estimation. By exploiting the low rank structure of the vectorized Hankel matrix associated with the target matrix, a convex approach called Vectorized Hankel Lift is proposed for the matrix recovery. It is shown that $n\gtrsim rs\log^4 n$ samples are sufficient for Vectorized Hankel Lift to achieve the exact recovery. For the location retrieval from the matrix, applying the single snapshot MUSIC method within the vectorized Hankel lift framework corresponds to the spatial smoothing technique proposed to improve the performance of the MMV MUSIC for the direction-of-arrival (DOA) estimation.

cs.IT↗

arXiv2021

An Extension of Fast Iterative Shrinkage-thresholding to Riemannian Optimization for Sparse Principal Component Analysis

Sparse principal component analysis (PCA), an important variant of PCA, attempts to find sparse loading vectors when conducting dimension reduction. This paper considers the nonsmooth Riemannian optimization problem associated with the ScoTLASS model for sparse PCA which can impose orthogonality and sparsity simultaneously. A Riemannian proximal method is proposed in the work of Chen et al. for the efficient solution of this optimization problem. In this paper, two acceleration schemes are introduced. First and foremost, we extend the FISTA method from the Euclidean space to the Riemannian manifold to solve sparse PCA, leading to the accelerated Riemannian proximal gradient method. Since the Riemannian optimization problem for sparse PCA is essentially non-convex, a restarting technique is adopted to stabilize the accelerated method without sacrificing the fast convergence. Second, a diagonal preconditioner is proposed for the Riemannian proximal subproblem which can further accelerate the convergence of the Riemannian proximal methods. Numerical evaluations establish the computational advantages of the proposed methods over the existing proximal gradient methods on a manifold. Additionally, a short result concerning the convergence of the Riemannian subgradients of a sequence is established, which, together with the result in the work of Chen et al., can show the stationary point convergence of the Riemannian proximal methods.

math.OC↗

arXiv2021

A Novel Strategy for GaN-on-Diamond Device with a High Thermal Boundary Conductance

To achieve high device performance and high reliability for the gallium nitride (GaN)-based high electron mobility transistors (HEMTs), efficient heat dissipation is important but remains challenging. Enormous efforts have been made to transfer a GaN device layer onto a diamond substrate with a high thermal conductivity by bonding. In this work, two GaN-diamond bonded composites are prepared via modified surface activated bonding (SAB) at room temperature with silicon interlayers of different thicknesses (15 nm and 22 nm). Before and after post-annealing process at 800 oC, thermal boundary conductance (TBC) across the bonded interface including the interlayer and the stress of GaN layer are investigated by time-domain thermoreflectance and Raman spectroscopy, respectively. After bonding, the 15 nm Si interlayer achieved a higher TBC. The post-annealing significantly increased the TBC of both interfaces, while the TBC of 22 nm silicon interlayer increased greater and became higher than that of 15 nm. Detailed investigation of the microstructure and composition of the interfaces were carried out to understand the difference in interfacial thermal conduction. The obtained stress was no more than 230 MPa for both before and after the annealing, and this high thermal stability of the bonded composites indicates that the room temperature bonding can realize a GaN-on-diamond template suitable for further epitaxial growth or device process. This work brings a novel strategy of SAB followed by high-temperature annealing to fabricate a GaN-on-diamond device with a high TBC.

physics.app-ph↗

arXiv2021

Riemannian Proximal Gradient Methods (extended version)

In the Euclidean setting, the proximal gradient method and its accelerated variants are a class of efficient algorithms for optimization problems with decomposable objective. In this paper, we develop a Riemannian proximal gradient method (RPG) and its accelerated variant (ARPG) for similar problems but constrained on a manifold. The global convergence of RPG has been established under mild assumptions, and the O(1/k) is also derived for RPG based on the notion of retraction convexity. If assuming the objective function obeys the Rimannian Kurdyka-Lojasiewicz (KL) property, it is further shown that the sequence generated by RPG converges to a single stationary point. As in the Euclidean setting, local convergence rate can be established if the objective function satisfies the Riemannian KL property with an exponent. Moreover, we have shown that the restriction of a semialgebraic function onto the Stiefel manifold satisfies the Riemannian KL property, which covers for example the well-known sparse PCA problem. Numerical experiments on random and synthetic data are conducted to test the performance of the proposed RPG and ARPG.

math.OC↗

arXiv2020

Image Restoration: Structured Low Rank Matrix Framework for Piecewise Smooth Functions and Beyond

Recently, mapping a signal/image into a low rank Hankel/Toeplitz matrix has become an emerging alternative to the traditional sparse regularization, due to its ability to alleviate the basis mismatch between the true support in the continuous domain and the discrete grid. In this paper, we introduce a novel structured low rank matrix framework to restore piecewise smooth functions. Inspired by the total generalized variation to use sparse higher order derivatives, we derive that the Fourier samples of higher order derivatives satisfy an annihilation relation, resulting in a low rank multi-fold Hankel matrix. We further observe that the SVD of a low rank Hankel matrix corresponds to a tight wavelet frame system which can represent the image with sparse coefficients. Based on this observation, we also propose a wavelet frame analysis approach based continuous domain regularization model for the piecewise smooth image restoration. Finally, numerical results on image restoration tasks are presented as a proof-of-concept study to demonstrate that the proposed approach is compared favorably against several popular discrete regularization approaches and structured low rank matrix approaches.

math.NA↗

arXiv2020

Data Driven Tight Frame for Compressed Sensing MRI Reconstruction via Off-the-Grid Regularization

Recently, the finite-rate-of-innovation (FRI) based continuous domain regularization is emerging as an alternative to the conventional on-the-grid sparse regularization for the compressed sensing (CS) due to its ability to alleviate the basis mismatch between the true support of the shape in the continuous domain and the discrete grid. In this paper, we propose a new off-the-grid regularization for the CS-MRI reconstruction. Following the recent works on two dimensional FRI, we assume that the discontinuities/edges of the image are localized in the zero level set of a band-limited periodic function. This assumption induces the linear dependencies among the Fourier samples of the gradient of the image, which leads to a low rank two-fold Hankel matrix. We further observe that the singular value decomposition of a low rank Hankel matrix corresponds to an adaptive tight frame system which can represent the image with sparse canonical coefficients. Based on this observation, we propose a data driven tight frame based off-the-grid regularization model for the CS-MRI reconstruction. To solve the nonconvex and nonsmooth model, a proximal alternating minimization algorithm with a guaranteed global convergence is adopted. Finally, the numerical experiments show that our proposed data driven tight frame based approach outperforms the existing approaches.

math.NA↗

arXiv2020

Fast Cadzow's Algorithm and a Gradient Variant

The Cadzow's algorithm is a signal denoising and recovery method which was designed for signals corresponding to low rank Hankel matrices. In this paper we first introduce a Fast Cadzow's algorithm which is developed by incorporating a novel subspace projection to reduce the high computational cost of the SVD in the Cadzow's algorithm. Then a Gradient method and a Fast Gradient method are proposed to address the non-decreasing MSE issue when applying the Cadzow's or Fast Cadzow's algorithm for signal denoising. Extensive empirical performance comparisons demonstrate that the proposed algorithms can complete the denoising and recovery tasks more efficiently and effectively.

math.NA↗