Ya-xiang Yuan

Shuo Chen, Bin Shi, Ya-xiang Yuan

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 $r > 2$.

Bowen Li, Bin Shi, Ya-xiang Yuan

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.

Bowen Li, Bin Shi, Ya-xiang Yuan

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.

Shuo Chen, Bin Shi, Ya-xiang Yuan

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.

Bowen Li, Bin Shi, Ya-xiang Yuan

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.

Shuo Chen, Bin Shi, Ya-xiang Yuan

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 $r=-1$.

Bin Gao, Renfeng Peng, Ya-xiang Yuan

Provably finding stationary points on bounded-rank tensors turns out to be an open problem [E. Levin, J. Kileel, and N. Boumal, Math. Program., 199 (2023), pp. 831--864] due to the inherent non-smoothness of the set of bounded-rank tensors. In contrast with bounded-rank matrices, tensors where some but not all modes are of full rank render essential difficulties in developing provable first-order methods. We resolve this problem by proposing two first-order methods with guaranteed convergence to stationary points. Specifically, we revisit the variational geometry of bounded-rank tensors and explicitly characterize its normal cones. Moreover, we propose gradient-related approximate projection methods that are provable to find stationary points, where the decisive ingredients are gradient-related vectors from tangent cones, line search along approximate projections, and rank-decreasing mechanisms near rank-deficient points. Numerical experiments on tensor completion validate that the proposed methods converge to stationary points across various rank parameters.

Yan Yang, Bin Gao, Ya-xiang Yuan

Optimization over the intersection of two manifolds arises in a broad range of applications, but is hindered by the coupled geometry of the feasible region. In this paper, we prove that the regularities -- clean intersection and intrinsic transversality -- are equivalent, which yields a tractable projection onto the tangent space of the intersection. Therefore, we propose a geometric method that employs a retraction on only one manifold and updates the iterate along two orthogonal directions. Specifically, the iterates stay on one manifold, and the two directions are responsible for asymptotically approaching the other manifold and decreasing the objective function, respectively. Under intrinsic transversality, we derive the convergence rate for both the feasibility and optimality measures, and show that every accumulation point is first-order stationary. Numerical experiments on problems stemming from sparse and low-rank optimization, including fitting spherical data, approximating hyperbolic embeddings on real data, and computing compressed modes, demonstrate the effectiveness of the proposed method.

Yan Yang, Bin Gao, Ya-xiang Yuan

Bilevel optimization, with broad applications in machine learning, has an intricate hierarchical structure. Gradient-based methods have emerged as a common approach to large-scale bilevel problems. However, the computation of the hyper-gradient, which involves a Hessian inverse vector product, confines the efficiency and is regarded as a bottleneck. To circumvent the inverse, we construct a sequence of low-dimensional approximate Krylov subspaces with the aid of the Lanczos process. As a result, the constructed subspace is able to dynamically and incrementally approximate the Hessian inverse vector product with less effort and thus leads to a favorable estimate of the hyper-gradient. Moreover, we propose a provable subspace-based framework for bilevel problems where one central step is to solve a small-size tridiagonal linear system. To the best of our knowledge, this is the first time that subspace techniques are incorporated into bilevel optimization. This successful trial not only enjoys $\mathcal{O}(ε^{-1})$ convergence rate but also demonstrates efficiency in a synthetic problem and two deep learning tasks.

Yan Yang, Bin Gao, Ya-xiang Yuan

Imposing additional constraints on low-rank optimization has garnered growing interest. However, the geometry of coupled constraints hampers the well-developed low-rank structure and makes the problem intricate. To this end, we propose a space-decoupling framework for optimization on bounded-rank matrices with orthogonally invariant constraints. The "space-decoupling" is reflected in several ways. We show that the tangent cone of coupled constraints is the intersection of tangent cones of each constraint. Moreover, we decouple the intertwined bounded-rank and orthogonally invariant constraints into two spaces, leading to optimization on a smooth manifold. Implementing Riemannian algorithms on this manifold is painless as long as the geometry of additional constraints is known. In addition, we unveil the equivalence between the reformulated problem and the original problem. Numerical experiments on real-world applications -- spherical data fitting, graph similarity measuring, low-rank SDP, model reduction of Markov processes, reinforcement learning, and deep learning -- validate the superiority of the proposed framework.

Yan Yang, Bin Gao, Ya-xiang Yuan

Determinantal varieties -- the sets of bounded-rank matrices or tensors -- have attracted growing interest in low-rank optimization. The tangent cone to low-rank sets is widely studied and underpins a range of geometric methods. The second-order geometry, which encodes curvature information, is more intricate. In this work, we develop a unified framework to derive explicit formulas for both first- and second-order tangent sets to various low-rank sets, including low-rank matrices, tensors, symmetric matrices, and positive semidefinite matrices. The framework also accommodates the intersection of a low-rank set and another set satisfying mild assumptions, thereby yielding a tangent intersection rule. Through the lens of tangent sets, we establish a necessary and sufficient condition under which a nonsmooth problem and its smooth parameterization share equivalent second-order stationary points. Moreover, we exploit tangent sets to characterize optimality conditions for low-rank optimization and prove that verifying second-order optimality is NP-hard. In a separate line of analysis, we investigate variational geometry of the graph of the normal cone to matrix varieties, deriving the explicit Bouligand tangent cone, Fréchet and Mordukhovich normal cones to the graph. These results are further applied to develop optimality conditions for low-rank bilevel programs.