arXiv subjects

Ya-xiang Yuan

Publications and source records attributed to Ya-xiang Yuan.

At least 37 records · Page 2Linked to original sources

arXiv2024

A Symplectic Discretization Based Proximal Point Algorithm for Convex Minimization

The proximal point algorithm plays a central role in non-smooth convex programming. The Augmented Lagrangian Method, one of the most famous optimization algorithms, has been found to be closely related to the proximal point algorithm. Due to its importance, accelerated variants of the proximal point algorithm have received considerable attention. In this paper, we first study an Ordinary Differential Equation (ODE) system, which provides valuable insights into proving the convergence rate of the desired algorithm. Using the Lyapunov function technique, we establish the convergence rate of the ODE system. Next, we apply the Symplectic Euler Method to discretize the ODE system to derive a new proximal point algorithm, called the Symplectic Proximal Point Algorithm (SPPA). By utilizing the proof techniques developed for the ODE system, we demonstrate the convergence rate of the SPPA. Additionally, it is shown that existing accelerated proximal point algorithm can be considered a special case of the SPPA in a specific manner. Furthermore, under several additional assumptions, we prove that the SPPA exhibits a finer convergence rate.

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arXiv2024

Desingularization of bounded-rank tensor sets

Low-rank tensors appear to be prosperous in many applications. However, the sets of bounded-rank tensors are non-smooth and non-convex algebraic varieties, rendering the low-rank optimization problems to be challenging. To this end, we delve into the geometry of bounded-rank tensor sets, including Tucker and tensor train formats. We propose a desingularization approach for bounded-rank tensor sets by introducing slack variables, resulting in a low-dimensional smooth manifold embedded in a higher-dimensional space while preserving the structure of low-rank tensor formats. Subsequently, optimization on tensor varieties can be reformulated to optimization on smooth manifolds, where the methods and convergence are well explored. We reveal the relationship between the landscape of optimization on varieties and that of optimization on manifolds. Numerical experiments on tensor completion illustrate that the proposed methods are in favor of others under different rank parameters.

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arXiv2024

Symplectic Discretization Approach for Developing New Proximal Point Algorithm

The rapid advancements in high-dimensional statistics and machine learning have increased the use of first-order methods. Many of these methods can be regarded as instances of the proximal point algorithm. Given the importance of the proximal point algorithm, there has been growing interest in developing its accelerated variants. However, some existing accelerated proximal point algorithms exhibit oscillatory behavior, which can impede their numerical convergence rate. In this paper, we first introduce an ODE system and demonstrate its $ o(1/t^2) $ convergence rate and weak convergence property. Next, we apply the Symplectic Euler Method to discretize the ODE and obtain a new accelerated proximal point algorithm, which we call the Symplectic Proximal Point Algorithm. The reason for using the Symplectic Euler Method is its ability to preserve the geometric structure of the ODEs. Theoretically, we demonstrate that the Symplectic Proximal Point Algorithm achieves an $ o(1/k^2) $ convergence rate and that the sequences generated by our method converge weakly to the solution set. Practically, our numerical experiments illustrate that the Symplectic Proximal Point Algorithm significantly reduces oscillatory behavior, leading to improved long-time behavior and faster numerical convergence rate.

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arXiv2024

Low-rank optimization on Tucker tensor varieties

In the realm of tensor optimization, the low-rank Tucker decomposition is crucial for reducing the number of parameters and for saving storage. We explore the geometry of Tucker tensor varieties -- the set of tensors with bounded Tucker rank -- which is notably more intricate than the well-explored matrix varieties. We give an explicit parametrization of the tangent cone of Tucker tensor varieties and leverage its geometry to develop provable gradient-related line-search methods for optimization on Tucker tensor varieties. To the best of our knowledge, this is the first work concerning geometry and optimization on Tucker tensor varieties. In practice, low-rank tensor optimization suffers from the difficulty of choosing a reliable rank parameter. To this end, we incorporate the established geometry and propose a Tucker rank-adaptive method that aims to identify an appropriate rank with guaranteed convergence. Numerical experiments on tensor completion reveal that the proposed methods are in favor of recovering performance over other state-of-the-art methods. The rank-adaptive method performs the best across various rank parameter selections and is indeed able to find an appropriate rank.

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arXiv2024

Linear convergence of forward-backward accelerated algorithms without knowledge of the modulus of strong convexity

A significant milestone in modern gradient-based optimization was achieved with the development of Nesterov's accelerated gradient descent (NAG) method. This forward-backward technique has been further advanced with the introduction of its proximal generalization, commonly known as the fast iterative shrinkage-thresholding algorithm (FISTA), which enjoys widespread application in image science and engineering. Nonetheless, it remains unclear whether both NAG and FISTA exhibit linear convergence for strongly convex functions. Remarkably, these algorithms demonstrate convergence without requiring any prior knowledge of strongly convex modulus, and this intriguing characteristic has been acknowledged as an open problem in the comprehensive review [Chambolle and Pock, 2016, Appendix B]. In this paper, we address this question by utilizing the high-resolution ordinary differential equation (ODE) framework. Expanding upon the established phase-space representation, we emphasize the distinctive approach employed in crafting the Lyapunov function, which involves a dynamically adapting coefficient of kinetic energy that evolves throughout the iterations. Furthermore, we highlight that the linear convergence of both NAG and FISTA is independent of the parameter $r$. Additionally, we demonstrate that the square of the proximal subgradient norm likewise advances towards linear convergence.

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arXiv2024

A New Two-dimensional Model-based Subspace Method for Large-scale Unconstrained Derivative-free Optimization: 2D-MoSub

This paper proposes the method 2D-MoSub (2-dimensional model-based subspace method), which is a novel derivative-free optimization (DFO) method based on the subspace method for general unconstrained optimization and especially aims to solve large-scale DFO problems. Our method combines 2-dimensional quadratic interpolation models and trust-region techniques to iteratively update the points and explore the 2-dimensional subspace. Its framework includes initialization, constructing the interpolation set, building the quadratic interpolation model, performing trust-region trial steps, and updating the trust-region radius and subspace. We introduce the framework and computational details of 2D-MoSub, and discuss the poisedness and quality of the interpolation set in the corresponding 2-dimensional subspace. We also analyze some properties of our method, including the model's approximation error with projection property and the algorithm's convergence. Numerical results demonstrate the effectiveness and efficiency of 2D-MoSub for solving a variety of unconstrained optimization problems.

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arXiv2023

Derivative-Free Optimization with Transformed Objective Functions (DFOTO) and the Algorithm Based on the Least Frobenius Norm Updating Quadratic Model

Derivative-free optimization problems are optimization problems where derivative information is unavailable. The least Frobenius norm updating quadratic interpolation model function is one of the essential under-determined model functions for model-based derivative-free trust-region methods. This article proposes derivative-free optimization with transformed objective functions and gives a trust-region method with the least Frobenius norm model. The model updating formula is based on Powell's formula. The method shares the same framework with those for problems without transformations, and its query scheme is given. We propose the definitions related to optimality-preserving transformations to understand the interpolation model in our method. We prove the existence of model optimality-preserving transformations beyond translation transformation. The necessary and sufficient condition for such transformations is given. The affine transformation with a positive multiplication coefficient is not model optimality-preserving. We also analyze the corresponding least Frobenius norm updating model and its interpolation error when the objective function is affinely transformed. Convergence property of a provable algorithmic framework containing our model is given. Numerical results of solving test problems and a real-world problem with the implementation NEWUOA-Trans show that our method can successfully solve most problems with objective optimality-preserving transformations, even though such transformations will change the optimality of the model function. To our best knowledge, this is the first work providing the model-based derivative-free algorithm and analysis for transformed problems with the function evaluation oracle (not the function-value comparison oracle). This article also proposes the "moving-target" optimization problem.

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arXiv2023

Riemannian preconditioned algorithms for tensor completion via tensor ring decomposition

We propose Riemannian preconditioned algorithms for the tensor completion problem via tensor ring decomposition. A new Riemannian metric is developed on the product space of the mode-2 unfolding matrices of the core tensors in tensor ring decomposition. The construction of this metric aims to approximate the Hessian of the cost function by its diagonal blocks, paving the way for various Riemannian optimization methods. Specifically, we propose the Riemannian gradient descent and Riemannian conjugate gradient algorithms. We prove that both algorithms globally converge to a stationary point. In the implementation, we exploit the tensor structure and adopt an economical procedure to avoid large matrix formulation and computation in gradients, which significantly reduces the computational cost. Numerical experiments on various synthetic and real-world datasets -- movie ratings, hyperspectral images, and high-dimensional functions -- suggest that the proposed algorithms are more efficient and have better reconstruction ability than other candidates.

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arXiv2023

Analysis Accelerated Mirror Descent via High-resolution ODEs

Mirror descent plays a crucial role in constrained optimization and acceleration schemes, along with its corresponding low-resolution ordinary differential equations (ODEs) framework have been proposed. However, the low-resolution ODEs are unable to distinguish between Polyak's heavy-ball method and Nesterov's accelerated gradient method. This problem also arises with accelerated mirror descent. To address this issue, we derive high-resolution ODEs for accelerated mirror descent and propose a general Lyapunov function framework to analyze its convergence rate in both continuous and discrete time. Furthermore, we demonstrate that accelerated mirror descent can minimize the squared gradient norm at an inverse cubic rate.

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arXiv2023

On Underdamped Nesterov's Acceleration

The high-resolution differential equation framework has been proven to be tailor-made for Nesterov's accelerated gradient descent method~(\texttt{NAG}) and its proximal correspondence -- the class of faster iterative shrinkage thresholding algorithms (FISTA). However, the systems of theories is not still complete, since the underdamped case ($r < 2$) has not been included. In this paper, based on the high-resolution differential equation framework, we construct the new Lyapunov functions for the underdamped case, which is motivated by the power of the time $t^γ$ or the iteration $k^γ$ in the mixed term. When the momentum parameter $r$ is $2$, the new Lyapunov functions are identical to the previous ones. These new proofs do not only include the convergence rate of the objective value previously obtained according to the low-resolution differential equation framework but also characterize the convergence rate of the minimal gradient norm square. All the convergence rates obtained for the underdamped case are continuously dependent on the parameter $r$. In addition, it is observed that the high-resolution differential equation approximately simulates the convergence behavior of~\texttt{NAG} for the critical case $r=-1$, while the low-resolution differential equation degenerates to the conservative Newton's equation. The high-resolution differential equation framework also theoretically characterizes the convergence rates, which are consistent with that obtained for the underdamped case with $r=-1$.

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arXiv2023

Least $H^2$ Norm Updating Quadratic Interpolation Model Function for Derivative-free Trust-region Algorithms

Derivative-free optimization methods are numerical methods for optimization problems in which no derivative information is used. Such optimization problems are widely seen in many real applications. One particular class of derivative-free optimization algorithms is trust-region algorithms based on quadratic models given by under-determined interpolation. Different techniques in updating the quadratic model from iteration to iteration will give different interpolation models. In this paper, we propose a new way to update the quadratic model by minimizing the $H^2$ norm of the difference between neighbouring quadratic models. Motivation for applying the $H^2$ norm is given, and theoretical properties of our new updating technique are also presented. Projection in the sense of $H^2$ norm and interpolation error analysis of our model function are proposed. We obtain the coefficients of the quadratic model function by using the KKT conditions. Numerical results show advantages of our model, and the derivative-free algorithms based on our least $H^2$ norm updating quadratic model functions can solve test problems with fewer function evaluations than algorithms based on least Frobenius norm updating.

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arXiv2023

Linear Convergence of ISTA and FISTA

In this paper, we revisit the class of iterative shrinkage-thresholding algorithms (ISTA) for solving the linear inverse problem with sparse representation, which arises in signal and image processing. It is shown in the numerical experiment to deblur an image that the convergence behavior in the logarithmic-scale ordinate tends to be linear instead of logarithmic, approximating to be flat. Making meticulous observations, we find that the previous assumption for the smooth part to be convex weakens the least-square model. Specifically, assuming the smooth part to be strongly convex is more reasonable for the least-square model, even though the image matrix is probably ill-conditioned. Furthermore, we improve the pivotal inequality tighter for composite optimization with the smooth part to be strongly convex instead of general convex, which is first found in [Li et al., 2022]. Based on this pivotal inequality, we generalize the linear convergence to composite optimization in both the objective value and the squared proximal subgradient norm. Meanwhile, we set a simple ill-conditioned matrix which is easy to compute the singular values instead of the original blur matrix. The new numerical experiment shows the proximal generalization of Nesterov's accelerated gradient descent (NAG) for the strongly convex function has a faster linear convergence rate than ISTA. Based on the tighter pivotal inequality, we also generalize the faster linear convergence rate to composite optimization, in both the objective value and the squared proximal subgradient norm, by taking advantage of the well-constructed Lyapunov function with a slight modification and the phase-space representation based on the high-resolution differential equation framework from the implicit-velocity scheme.

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arXiv2022

Revisiting the acceleration phenomenon via high-resolution differential equations

Nesterov's accelerated gradient descent (NAG) is one of the milestones in the history of first-order algorithms. It was not successfully uncovered until the high-resolution differential equation framework was proposed in [Shi et al., 2022] that the mechanism behind the acceleration phenomenon is due to the gradient correction term. To deepen our understanding of the high-resolution differential equation framework on the convergence rate, we continue to investigate NAG for the $μ$-strongly convex function based on the techniques of Lyapunov analysis and phase-space representation in this paper. First, we revisit the proof from the gradient-correction scheme. Similar to [Chen et al., 2022], the straightforward calculation simplifies the proof extremely and enlarges the step size to $s=1/L$ with minor modification. Meanwhile, the way of constructing Lyapunov functions is principled. Furthermore, we also investigate NAG from the implicit-velocity scheme. Due to the difference in the velocity iterates, we find that the Lyapunov function is constructed from the implicit-velocity scheme without the additional term and the calculation of iterative difference becomes simpler. Together with the optimal step size obtained, the high-resolution differential equation framework from the implicit-velocity scheme of NAG is perfect and outperforms the gradient-correction scheme.

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arXiv2022

Proximal Subgradient Norm Minimization of ISTA and FISTA

For first-order smooth optimization, the research on the acceleration phenomenon has a long-time history. Until recently, the mechanism leading to acceleration was not successfully uncovered by the gradient correction term and its equivalent implicit-velocity form. Furthermore, based on the high-resolution differential equation framework with the corresponding emerging techniques, phase-space representation and Lyapunov function, the squared gradient norm of Nesterov's accelerated gradient descent (\texttt{NAG}) method at an inverse cubic rate is discovered. However, this result cannot be directly generalized to composite optimization widely used in practice, e.g., the linear inverse problem with sparse representation. In this paper, we meticulously observe a pivotal inequality used in composite optimization about the step size $s$ and the Lipschitz constant $L$ and find that it can be improved tighter. We apply the tighter inequality discovered in the well-constructed Lyapunov function and then obtain the proximal subgradient norm minimization by the phase-space representation, regardless of gradient-correction or implicit-velocity. Furthermore, we demonstrate that the squared proximal subgradient norm for the class of iterative shrinkage-thresholding algorithms (ISTA) converges at an inverse square rate, and the squared proximal subgradient norm for the class of faster iterative shrinkage-thresholding algorithms (FISTA) is accelerated to convergence at an inverse cubic rate.

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arXiv2022

Some Sharp Error Bounds for Multivariate Linear Interpolation and Extrapolation

We study in this paper the function approximation error of linear interpolation and extrapolation. Several upper bounds are presented along with the conditions under which they are sharp. All results are under the assumptions that the function has Lipschitz continuous gradient and is interpolated on an affinely independent sample set. Errors for quadratic functions and errors of bivariate linear extrapolation are analyzed in depth.

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arXiv2022

Gradient Norm Minimization of Nesterov Acceleration: $o(1/k^3)$

In the history of first-order algorithms, Nesterov's accelerated gradient descent (NAG) is one of the milestones. However, the cause of the acceleration has been a mystery for a long time. It has not been revealed with the existence of gradient correction until the high-resolution differential equation framework proposed in [Shi et al., 2021]. In this paper, we continue to investigate the acceleration phenomenon. First, we provide a significantly simplified proof based on precise observation and a tighter inequality for $L$-smooth functions. Then, a new implicit-velocity high-resolution differential equation framework, as well as the corresponding implicit-velocity version of phase-space representation and Lyapunov function, is proposed to investigate the convergence behavior of the iterative sequence $\{x_k\}_{k=0}^{\infty}$ of NAG. Furthermore, from two kinds of phase-space representations, we find that the role played by gradient correction is equivalent to that by velocity included implicitly in the gradient, where the only difference comes from the iterative sequence $\{y_{k}\}_{k=0}^{\infty}$ replaced by $\{x_k\}_{k=0}^{\infty}$. Finally, for the open question of whether the gradient norm minimization of NAG has a faster rate $o(1/k^3)$, we figure out a positive answer with its proof. Meanwhile, a faster rate of objective value minimization $o(1/k^2)$ is shown for the case $r > 2$.

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arXiv2021

On the Complexity of an Augmented Lagrangian Method for Nonconvex Optimization

In this paper we study the worst-case complexity of an inexact Augmented Lagrangian method for nonconvex constrained problems. Assuming that the penalty parameters are bounded, we prove a complexity bound of $\mathcal{O}(|\log(ε)|)$ outer iterations for the referred algorithm to generate an $ε$-approximate KKT point, for $ε\in (0,1)$. When the penalty parameters are unbounded, we prove an outer iteration complexity bound of $\mathcal{O}\left(ε^{-2/(α-1)}\right)$, where $α>1$ controls the rate of increase of the penalty parameters. For linearly constrained problems, these bounds yield to evaluation complexity bounds of $\mathcal{O}(|\log(ε)|^{2}ε^{-2})$ and $\mathcal{O}\left(ε^{-\left(\frac{2(2+α)}{α-1}+2\right)}\right)$, respectively, when appropriate first-order methods ($p=1$) are used to approximately solve the unconstrained subproblems at each iteration. In the case of problems having only linear equality constraints, the latter bounds are improved to $\mathcal{O}(|\log(ε)|^{2}ε^{-(p+1)/p})$ and $\mathcal{O}\left(ε^{-\left(\frac{4}{α-1}+\frac{p+1}{p}\right)}\right)$, respectively, when appropriate $p$-order methods ($p\geq 2$) are used as inner solvers.

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arXiv2021

A Penalty-free Infeasible Approach for a Class of Nonsmooth Optimization Problems over the Stiefel Manifold

Transforming into an exact penalty function model with convex compact constraints yields efficient infeasible approaches for optimization problems with orthogonality constraints. For smooth and $\ell_{2,1}$-norm regularized cases, these infeasible approaches adopt simple and orthonormalization-free updating scheme and show their high efficiency in the test examples. However, to avoid orthonormalization while enforcing the feasibility of the final solution, these infeasible approaches introduce a quadratic penalty term, where an inappropriate penalty parameter can lead to numerical inefficiency. Inspired by penalty-free approaches for smooth optimization problems, we proposed a proximal first-order algorithm for a class of optimization problems with orthogonality constraints and nonsmooth regularization term. The consequent algorithm, named sequential linearized proximal gradient method (SLPG), alternatively takes tangential steps and normal steps to improve the optimality and feasibility respectively. In SLPG, the orthonormalization process is invoked only once at the last step if high precision in feasibility is needed, showing that main iterations in SLPG are orthonormalization-free. Besides, both the tangential steps and normal steps do not involve the penalty parameter, and thus SLPG is penalty-free and avoids the inefficiency by inappropriate penalty parameter. We analyze the global convergence properties of SLPG where the tangential steps are inexactly computed. By inexactly computing tangential steps, for smooth cases and $\ell_{2,1}$-norm regularized cases, SLPG has a closed-form updating scheme, which leads to its cheap tangential steps. Numerical experiments illustrate the numerical advantages of SLPG when compared with existing first-order methods.

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