Daniel Liberzon, Roberto Tempo
This paper is concerned with the problem of finding a quadratic common Lyapunov function for a family of stable linear systems. We present gradient iteration algorithms which give deterministic convergence for finite system families and probabilistic convergence for infinite families.
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Quadratic Lyapunov functions are prevalent in stability analysis of linear consensus systems. In this paper we show that weighted sums of convex functions of the different coordinates are Lyapunov functions for irreducible consensus systems. We introduce a particular class of additive convex Lyapunov functions that includes as particular instances the stored electric energy in RC circuits, Kullback-Leiber's information divergence between two probability measures, and Gibbs free energy in chemical reaction networks. On that basis we establish a general gradient formalism that allows to represen…
We present a unified convergence analysis for first order convex optimization methods using the concept of strong Lyapunov conditions. Combining this with suitable time scaling factors, we are able to handle both convex and strong convex cases, and establish contraction properties of Lyapunov functions for many existing ordinary differential equation models. Then we derive prevailing first order optimization algorithms, such as proximal gradient methods, heavy ball methods (also known as momentum methods), Nesterov accelerated gradient methods, and accelerated proximal gradient methods from nu…
(Adaptive) Scaled gradient methods beyond locally Holder smoothness: Lyapunov analysis, convergence rate and complexity
本文针对梯度局部Hölder连续的光滑凸函数无约束最小化问题，分析SGA的收敛性与复杂度，提出可自动调参的自适应变体AdaSGA并证明其收敛性质。
Composite Lyapunov Criteria for Stability and Convergence with Applications to Optimization Dynamics
提出复合Lyapunov框架，通过微分不等式对实现非线性系统的严格衰减与收敛，统一收敛到点和集合，应用于惯性梯度与原始-对偶梯度流。
We develop a theory of accelerated first-order optimization from the viewpoint of differential equations and Lyapunov functions. Building upon the previous work of many researchers, we consider differential equations which model the behavior of accelerated gradient descent. Our main contributions are to provide a general framework for discretizating the differential equations to produce accelerated methods, and to provide physical intuition which helps explain the optimal damping rate. An important novelty is the generality of our approach, which leads to a unified derivation of a wide variety…
A unified Euler--Lagrange system for analyzing continuous-time accelerated gradient methods
本文构建统一 Euler–Lagrange 系统与 Lyapunov 分析框架，涵盖多种连续时间加速梯度 ODE，并推导参数条件、改进收敛速率及不可微情形证明。
This paper considers the problem of designing accelerated gradient-based algorithms for optimization and saddle-point problems. The class of objective functions is defined by a generalized sector condition. This class of functions contains strongly convex functions with Lipschitz gradients but also non-convex functions, which allows not only to address optimization problems but also saddle-point problems. The proposed design procedure relies on a suitable class of Lyapunov functions and on convex semi-definite programming. The proposed synthesis allows the design of algorithms that reach the p…
A Concise Lyapunov Analysis of Nesterov's Accelerated Gradient Method
本文针对Nesterov加速梯度法，利用Lyapunov分析方法给出了其在一般凸和强凸情形下收敛速率的简明直接证明。
Understanding Accelerated Gradient Methods: Lyapunov Analyses and Hamiltonian Assisted Interpretations
本文构建两类通用一阶算法，通过离散Lyapunov分析给出匹配Nesterov方法的加速条件，分析极限ODE收敛差距，并提出基于Hamiltonian函数的Hamiltonian辅助梯度法以统一解释加速机制。
This manuscript aims to develop and describe gain scheduling control concept for a gas turbine engine which drives a variable pitch propeller. An architecture for gain-scheduling control is developed that controls the turboshaft engine for large thrust commands in stable fashion with good performance. Fuel ow and propeller pitch angle are the two control inputs of the system. New stability proof has been developed for gain scheduling control of gas turbine engines using global linearization and LMI techniques. This approach guarantees absolute stability of the closed loop gas turbine engine sy…
For a class of nonsmooth composite optimization problems with linear equality constraints, we utilize a Lyapunov-based approach to establish the global exponential stability of the primal-dual gradient flow dynamics based on the proximal augmented Lagrangian. The result holds when the differentiable part of the objective function is strongly convex with a Lipschitz continuous gradient; the non-differentiable part is proper, lower semi-continuous, and convex; and the matrix in the linear constraint is full row rank. Our quadratic Lyapunov function generalizes recent result from strongly convex …
A Refined Parameter Condition in the Lyapunov Analysis of IGAHD
该研究改进了IGAHD的李雅普诺夫分析中的参数条件，使其适用于更小的步长，还针对凸二次函数得到了更大的参数区域，并提出了关于非二次函数的待研究问题。
We present a family of algorithms, called descent algorithms, for optimizing convex and non-convex functions. We also introduce a new first-order algorithm, called rescaled gradient descent (RGD), and show that RGD achieves a faster convergence rate than gradient descent provided the function is strongly smooth -- a natural generalization of the standard smoothness assumption on the objective function. When the objective function is convex, we present two novel frameworks for "accelerating" descent methods, one in the style of Nesterov and the other in the style of Monteiro and Svaiter, using …
Convergence Rate Analysis of Continuous- and Discrete-Time Smoothing Gradient Algorithms
本文针对不可微目标函数最小化，分析采用递减平滑参数的连续与离散时间平滑梯度算法，并借助统一 Lyapunov 函数证明二者收敛速率等价。
Inexact projected preconditioned gradient methods with variable metrics: a Lyapunov convergence theory (extended version)
本文针对不精确预条件投影梯度下降（IPPGD）方法的理论空白，提出不精确投影算子构造策略，结合李雅普诺夫分析证明其收敛性，经数值实验验证了方法的鲁棒性与计算效率。
A Tutorial on a Lyapunov-Based Approach to the Analysis of Iterative Optimization Algorithms
本文以教程形式介绍了一种基于李雅普诺夫函数的分析方法，用于分析迭代梯度优化算法的性能，可恢复与积分二次约束相同的性能界，并额外构造李雅普诺夫函数。
Stochastic Gradient Tracking over Time-Varying Networks: One-Step Lyapunov Analysis
该研究针对时变网络下的去中心化随机梯度跟踪问题，通过构造时变二次范数将窗口压缩转化为一步李雅普诺夫恒等式，得到误差递推关系，在两类凸目标下均匹配集中式小批量性能且实现线性加速。
We provide algorithms for computing a Lyapunov function for a class of systems where the state trajectories are constrained to evolve within a closed convex set. The dynamical systems that we consider comprise a differential equation which ensures continuous evolution within the domain, and a normal cone inclusion which ensures that the state trajectory remains within a prespecified set at all times. Finding a Lyapunov function for such a system boils down to finding a function which satisfies certain inequalities on the admissible set of state constraints. It is well-known that this problem, …
Langevin Dynamics: A Unified Perspective on Optimization via Lyapunov Potentials
本文提出基于Lyapunov势函数的新策略分析SGLD在非凸优化中的全局收敛性，给出改进的收敛速率、首个有限梯度复杂度保证，并建立了连续时间Langevin动力学与离散时间SGLD之间的成功性联系。
We provide general methods for explicitly constructing strict Lyapunov functions for fully nonlinear slowly time-varying systems. Our results apply to cases where the given dynamics and corresponding frozen dynamics are not necessarily exponentially stable. This complements our previous Lyapunov function constructions for rapidly time-varying dynamics. We also explicitly construct input-to-state stable Lyapunov functions for slowly time-varying control systems. We illustrate our findings by constructing explicit Lyapunov functions for a pendulum model, an example from identification theory, an…
This paper studies the projected saddle-point dynamics associated to a convex-concave function, which we term saddle function. The dynamics consists of gradient descent of the saddle function in variables corresponding to convexity and (projected) gradient ascent in variables corresponding to concavity. We examine the role that the local and/or global nature of the convexity-concavity properties of the saddle function plays in guaranteeing convergence and robustness of the dynamics. Under the assumption that the saddle function is twice continuously differentiable, we provide a novel character…
Momentum methods play a significant role in optimization. Examples include Nesterov's accelerated gradient method and the conditional gradient algorithm. Several momentum methods are provably optimal under standard oracle models, and all use a technique called estimate sequences to analyze their convergence properties. The technique of estimate sequences has long been considered difficult to understand, leading many researchers to generate alternative, "more intuitive" methods and analyses. We show there is an equivalence between the technique of estimate sequences and a family of Lyapunov fun…
Certifying the stability of dynamical systems is a central and challenging task in control theory and systems analysis. To tackle these problems we present an algorithmic approach to finding polynomial Lyapunov functions. Our method relies on sums of nonnegative circuit functions (SONC), a certificate of nonnegativity of real polynomials. We show that both the problem of verifying as well as the more difficult task of finding Lyapunov functions can be carried out via relative entropy programming when using SONC certificates. This approach is analogue yet independent to finding Lyapunov functio…
Lyapunov Constructions for System Interconnections Arising from Adaptation in Some Optimization Methods
该研究针对自适应梯度方法中的互联系统，采用不同技术构造李雅普诺夫函数，证明自适应梯度优化器全局渐近稳定，提供了验证该性质的李雅普诺夫函数构造方法。
In a Hilbert framework, for convex differentiable optimization, we consider accelerated gradient methods obtained by combining temporal scaling and averaging techniques with Tikhonov regularization. We start from the continuous steepest descent dynamic with an additional Tikhonov regularization term whose coefficient vanishes asymptotically. We provide an extensive Lyapunov analysis of this first-order evolution equation. Then we apply to this dynamic the method of time scaling and averaging recently introduced by Attouch, Bot and Nguyen. We thus obtain an inertial dynamic which involves visco…
Modern Theory of Gradient-Based Optimization
综述基于梯度的优化技术，重点阐述常微分方程视角与离散Lyapunov分析的结合，揭示加速机制，并统一分析多种优化算法。
Score-Aware Policy-Gradient and Performance Guarantees using Local Lyapunov Stability
本文提出了适用于基于模型的强化学习的分数感知梯度估计器SAGE，无需值函数估计即可估计策略梯度，证明了其局部收敛性与遗憾界，实验显示其比actor-critic方法更快找到近优策略。
In this paper, we develop a unified framework able to certify both exponential and subexponential convergence rates for a wide range of iterative first-order optimization algorithms. To this end, we construct a family of parameter-dependent nonquadratic Lyapunov functions that can generate convergence rates in addition to proving asymptotic convergence. Using Integral Quadratic Constraints (IQCs) from robust control theory, we propose a Linear Matrix Inequality (LMI) to guide the search for the parameters of the Lyapunov function in order to establish a rate bound. Based on this result, we for…
We present a framework to transform the problem of finding a Lyapunov function of a Chemical Reaction Network (CRN) in concentration coordinates with arbitrary monotone kinetics into finding a common Lyapunov function for a linear parameter varying system in reaction coordinates. Alternative formulations of the proposed Lyapunov function is presented also. This is applied to reinterpret previous results by the authors on Piecewise Linear in Rates Lyapunov functions, and to establish a link with contraction analysis. Persistence and uniqueness of equilibria are discussed also.
In this paper, a primal-dual gradient flow algorithm for distributed support vector machines (DSVM) is proposed. A network of computing nodes, each carrying a subset of horizontally partitioned large dataset is considered. The nodes are represented as dynamical systems with Arrow-Hurwicz-Uzawa gradient flow dynamics, derived from the Lagrangian function of the DSVM problem. It is first proved that the nodes are passive dynamical systems. Then, by employing the Krasovskii type candidate Lyapunov functions, it is proved that the computing nodes asymptotically converge to the optimal primal-dual …
For many problems, some of which are reviewed in the paper, popular algorithms like Douglas--Rachford (DR), ADMM, and FISTA produce approximating sequences that show signs of spiraling toward the solution. We present a meta-algorithm that exploits such dynamics to potentially enhance performance. The strategy of this meta-algorithm is to iteratively build and minimize surrogates for the Lyapunov function that captures those dynamics. As a first motivating application, we show that for prototypical feasibility problems the circumcentered-reflection method (CRM), subgradient projections, and New…
A Rate Transfer Method: Convergence Rates in Stochastic Gradient Descent via Stochastic Calculus
该研究提出速率转移方法，通过随机微积分将SDE的李雅普诺夫估计转化为随机梯度算法的收敛界，可复现经典方法速率并用于新优化算法的收敛速率分析。
Adaptive control strategies usually are designed based on gradient methods for the sake of simplicity in Lyapunov analysis. However, least squares (LS)-based parameter identifiers, with proper selection of design parameters, exhibit better transient performance than the gradient-based ones, from the aspects of convergence speed and robustness to measurement noise. On the other hand, most of the LS-based adaptive control procedures are designed via the indirect adaptive control approaches, due to the difficulty in integrating an LS-based adaptive law within the direct approaches starting with a…
Another approach to build Lyapunov functions for the first order methods in the quadratic case
本文提出利用 Schur 分解将迭代矩阵上三角化，通过状态向量末坐标构造适用于二次函数一阶方法的通用 Lyapunov 函数，并证明其适用于 Heavy Ball 等多种方法。
While training error of most deep neural networks degrades as the depth of the network increases, residual networks appear to be an exception. We show that the main reason for this is the Lyapunov stability of the gradient descent algorithm: for an arbitrarily chosen step size, the equilibria of the gradient descent are most likely to remain stable for the parametrization of residual networks. We then present an architecture with a pair of residual networks to approximate a large class of functions by decomposing them into a convex and a concave part. Some parameters of this model are shown to…
Bias-Optimal Bounds for SGD: A Computer-Aided Lyapunov Analysis
本文通过计算机辅助的李雅普诺夫分析，推导出SGD在偏差最优意义上的收敛界限，适用于常数步长范围，包括之前未探索的临界和大步长区域。
Toward a Systematic Understanding and Interactive Search of Lyapunov-Style Proofs in Optimization
提出系统框架将优化算法的紧致解析收敛证明转化为Lyapunov函数证明，结合PEP工具箱与线性代数实现交互式搜索，发现四个新Lyapunov风格证明。
A feasible smoothing accelerated projected gradient method for nonsmooth convex optimization
针对非光滑凸优化，提出一种保证变量可行性的光滑加速投影梯度方法，通过Lyapunov函数证明$O(k^{-1}\log k)$收敛率，并在桁架鲁棒柔顺度优化中验证。
In this paper, a distributed optimization problem with general differentiable convex objective functions is studied for single-integrator and double-integrator multi-agent systems. Two distributed adaptive optimization algorithm is introduced which uses the relative information to construct the gain of the interaction term. The analysis is performed based on the Lyapunov functions, the analysis of the system solution and the convexity of the local objective functions. It is shown that if the gradients of the convex objective functions are continuous, the team convex objective function can be m…
A Unified Lyapunov-IQC Framework for Uniform Stability of Smooth Quadratic First-Order Accelerated Optimizers
本文提出统一的Lyapunov积分二次约束框架，用于在β-光滑和γ-强凸条件下建立一阶加速优化算法的均匀稳定性，通过Lyapunov函数和Lur'e型反馈互联模型，结合半定规划解决稳定性问题。
In 1964, Polyak showed that the Heavy-ball method, the simplest momentum technique, accelerates convergence of strongly-convex problems in the vicinity of the solution. While Nesterov later developed a globally accelerated version, Polyak's original algorithm remains simpler and more widely used in applications such as deep learning. Despite this popularity, the question of whether Heavy-ball is also globally accelerated or not has not been fully answered yet, and no convincing counterexample has been provided. This is largely due to the difficulty in finding an effective Lyapunov function: in…
Performance analysis for linear time-invariant (LTI) systems has been closely tied to quadratic Lyapunov functions ever since it was shown that LTI system stability is equivalent to the existence of such a Lyapunov function. Some metrics for LTI systems, however, have resisted treatment via means of quadratic Lyapunov functions. Among these, point-wise-in-time metrics, such as peak norms, are not captured accurately using these techniques, and this shortcoming has prevented the development of tools to analyze system behavior by means other than e.g. time-domain simulations. This work demonstra…
On continuation and convex Lyapunov functions
该研究证明了在特定条件下，不同向量场之间可通过同伦保持原点稳定性，并探讨了其在稳定性证书和切换系统中的应用。
Symbolic Reduction for Formal Synthesis of Global Lyapunov Functions
本文提出符号化简规则，结合SMT求解与拉萨尔原理，高效综合全局李雅普诺夫函数，并验证或反驳系统稳定性。
Convergence and Regret of the Policy Gradient for Multi-Armed Bandits in Diffusion Environment
本文针对扩散环境中多臂老虎机的策略梯度算法，证明其收敛性并推导O(log T)阶悔憾上界，改进了现有分析且可推广至离散时间算法。
This paper presents a non-linear stability analysis for dc-microgrids in both, interconnected mode and island operation with primary control. The proposed analysis is based on the fact that the dynamical model of the grid is a gradient system generated by a strongly convex function. The stability analysis is thus reduced to a series of convex optimization problems. The proposed method allows to: i) demonstrate the existence and uniqueness of the equilibrium ii) calculate this equilibrium numerically iii) give conditions for global stability using a Lyapunov function iv) estimate the attraction…
Lyapunov Analysis For Monotonically Forward-Backward Accelerated Algorithms
本文通过Lyapunov分析证明单调加速算法M-NAG和M-FISTA在强凸条件下线性收敛，无需完整NAG迭代，深化了理论理解。
Stability Analysis and Data-driven Verification via Recurrent Lyapunov Functions
该研究提出循环李雅普诺夫函数（RLF）概念，放松李雅普诺夫函数的不变性要求，开发GPU算法仅用轨迹数据即可验证系统稳定性，揭示了性能与成本间的固有权衡。
We present a coupled system of ODEs which, when discretized with a constant time step/learning rate, recovers Nesterov's accelerated gradient descent algorithm. The same ODEs, when discretized with a decreasing learning rate, leads to novel stochastic gradient descent (SGD) algorithms, one in the convex and a second in the strongly convex case. In the strongly convex case, we obtain an algorithm superficially similar to momentum SGD, but with additional terms. In the convex case, we obtain an algorithm with a novel order $k^{3/4}$ learning rate. We prove, extending the Lyapunov function approa…
In this work, we develop a new method to design energy minimum low-thrust missions (L2-minimization). In the Circular Restricted Three Body Problem, the knowledge of invariant manifolds helps us initialize an indirect method solving a transfer mission between periodic Lyapunov orbits. Indeed, using the PMP, the optimal control problem is solved using Newton-like algorithms finding the zero of a shooting function. To compute a Lyapunov to Lyapunov mission, we first compute an admissible trajectory using a heteroclinic orbit between the two periodic orbits. It is then used to initialize a multip…
For backward differentiation formulae (BDF) applied to gradient flows of semiconvex functions, quadratic stability implies the existence of a Lyapunov functional. We compute the maximum time step which can be derived from quadratic stability for the 3-step BDF method (BDF3). Applications to the asymptotic behaviour of sequences generated by the BDF3 scheme are given.
State-Dependent Lyapunov Analysis of Rank-1 Matrix Factorization
本文通过状态依赖Lyapunov视角研究梯度下降在秩1矩阵分解中的收敛性，提出参数化二次证书I(δ;·)，证明在临界步长以下收敛到全局极小值，临界步长以上则进入平衡终端状态并表现出周期2行为。
Neural-NPV Control: Learning Parameter-Dependent Controllers and Lyapunov Functions
本文提出Neural-NPV框架，通过神经网络联合合成参数依赖的控制器和李雅普诺夫函数，解决传统SOS方法在可扩展性和保守性方面的不足，通过实验验证其在应用性和性能上的优势。
The gradient descent-ascent (GDA) algorithm has been widely applied to solve minimax optimization problems. In order to achieve convergent policy parameters for minimax optimization, it is important that GDA generates convergent variable sequences rather than convergent sequences of function values or gradient norms. However, the variable convergence of GDA has been proved only under convexity geometries, and there lacks understanding for general nonconvex minimax optimization. This paper fills such a gap by studying the convergence of a more general proximal-GDA for regularized nonconvex-stro…
We consider the Linear-Quadratic-Regulator (LQR) problem in terms of optimizing a real-valued matrix function over the set of feedback gains. Such a setup facilitates examining the implications of a natural initial-state independent formulation of LQR in designing first order algorithms. It is shown that this cost function is smooth and coercive, and provide an alternate means of noting its gradient dominated property. In the process, we provide a number of analytic observations on the LQR cost when directly analyzed in terms of the feedback gain. We then examine three types of well-posed flow…
Stochastic Approximation Beyond Gradient for Signal Processing and Machine Learning
本文提出基于Lyapunov函数的统一随机近似理论框架，涵盖非渐近与渐近收敛，并用于分析压缩随机梯度、随机EM和强化学习等非随机梯度算法。
A Converse Sum of Squares Lyapunov Function for Outer Approximation of Minimal Attractor Sets of Nonlinear Systems
针对非线性ODE不稳定系统最小吸引集未知的问题，提出一种非保守的吸引集SOS Lyapunov新刻画，结合行列式最大化SOS规划得到体积最小的1-子水平集，实现最小吸引集的最优外逼近。
Stochastic Gradient Descent (SGD) has been the method of choice for learning large-scale non-convex models. While a general analysis of when SGD works has been elusive, there has been a lot of recent progress in understanding the convergence of Gradient Flow (GF) on the population loss, partly due to the simplicity that a continuous-time analysis buys us. An overarching theme of our paper is providing general conditions under which SGD converges, assuming that GF on the population loss converges. Our main tool to establish this connection is a general converse Lyapunov like theorem, which impl…
Near-optimal Closed-loop Method via Lyapunov Damping for Convex Optimization
针对凸优化中最优收敛速率多依赖开环阻尼非自治方法的问题，提出带闭环阻尼的自治系统，借Lyapunov函数耦合阻尼与收敛速率实现近最优速率，离散化得到算法并经数值实验验证。
We consider the continuous-time Linear-Quadratic-Regulator (LQR) problem in terms of optimizing a real-valued matrix function over the set of feedback gains. The results developed are in parallel to those in Bu et al. [1] for discrete-time LTI systems. In this direction, we characterize several analytical properties (smoothness, coerciveness, quadratic growth) that are crucial in the analysis of gradient-based algorithms. We also point out similarities and distinctive features of the continuous time setup in comparison with its discrete time analogue. First, we examine three types of well-pose…
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