非光滑非凸全局优化在各类应用中构成了重大挑战，标准的基于梯度的方法往往难以应对。我们提出球邻近点方法（Ball-Proximal Point Method），又称Broximal点方法，简称球点方法（BPM）——这一全新算法框架受经典邻近点方法（Proximal Point Method, PPM）（Rockafellar, 1976）启发，我们证明它为若干基础优化范式与现象提供了新的视角，包括非凸非光滑优化、加速、平滑、自适应步长选择以及信赖域方法。BPM的核心是球邻近（"broximal"）算子，它由经典邻近算子将二次距离惩罚替换为球约束推导而来。令人意外的是，与PPM在非光滑凸场景下的次线性收敛速率形成鲜明对比，我们证明BPM在相同场景下可线性收敛，且能在有限步数内收敛。此外，通过引入球凸性（ball-convexity）概念，我们证明BPM在更弱的假设下仍能保持相同的全局收敛保证，使其成为求解范围更广的一类潜在非凸优化问题的有力工具。正如PPM作为概念性方法，启发了诸多实用高效算法与算法要素的发展，例如梯度下降、自适应步长、加速方法（Ahn & Sra, 2020）以及AdamW中的"W"项（Zhuang et al., 2022），我们认为BPM也应被以同样的方式看待：作为后续研究发展的蓝图与灵感来源。
英文摘要：
Non-smooth and non-convex global optimization poses significant challenges across various applications, where standard gradient-based methods often struggle. We propose the Ball-Proximal Point Method, Broximal Point Method, or Ball Point Method (BPM) for short - a novel algorithmic framework inspired by the classical Proximal Point Method (PPM) (Rockafellar, 1976), which, as we show, sheds new light on several foundational optimization paradigms and phenomena, including non-convex and non-smooth optimization, acceleration, smoothing, adaptive stepsize selection, and trust-region methods. At the core of BPM lies the ball-proximal ("broximal") operator, which arises from the classical proximal operator by replacing the quadratic distance penalty by a ball constraint. Surprisingly, and in sharp contrast with the sublinear rate of PPM in the nonsmooth convex regime, we prove that BPM converges linearly and in a finite number of steps in the same regime. Furthermore, by introducing the concept of ball-convexity, we prove that BPM retains the same global convergence guarantees under weaker assumptions, making it a powerful tool for a broader class of potentially non-convex optimization problems. Just like PPM plays the role of a conceptual method inspiring the development of practically efficient algorithms and algorithmic elements, e.g., gradient descent, adaptive step sizes, acceleration (Ahn & Sra, 2020), and "W" in AdamW (Zhuang et al., 2022), we believe that BPM should be understood in the same manner: as a blueprint and inspiration for further development.
National University of Defense Technology（国防科技大学）；Hunan University（湖南大学）
Nonconvex optimization algorithms with random initialization have attracted increasing attention recently. It has been showed that many first-order methods always avoid saddle points with random starting points. In this paper, we answer a question: can the nonconvex heavy-ball algorithms with random initialization avoid saddle points? The answer is yes! Direct using the existing proof technique for the heavy-ball algorithms is hard due to that each iteration of the heavy-ball algorithm consists of current and last points. It is impossible to formulate the algorithms as iteration like xk+1= g(x…
The University of Edinburgh（爱丁堡大学）；King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；Moscow Institute of Physics and Technology (MIPT)（莫斯科物理技术学院）
In this paper we study several classes of stochastic optimization algorithms enriched with heavy ball momentum. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic dual subspace ascent. This is the first time momentum variants of several of these methods are studied. We choose to perform our analysis in a setting in which all of the above methods are equivalent. We prove global nonassymptotic linear convergence rates for all methods and various measures of success, including primal function values, primal iterates (in L2 sense…
Tracking the Median of Gradients with a Stochastic Proximal Point Method
Technical University of Munich and Inria Paris, ENS, PSL Research University（慕尼黑技术大学和巴黎Inria、ENS、PSL研究大学）；Université Paris Cité and Sorbonne Université, CNRS（巴黎国立大学和索邦大学、CNRS）；Inria, CNRS, ENS, PSL Research University Paris（Inria、CNRS、ENS、PSL研究大学巴黎）；CCM, Flatiron Institute, Simons Foundation（CCM、Flatiron研究所、Simons基金会）
Zuse Institute Berlin（柏林祖斯研究所）；Technische Universität Berlin（柏林工业大学）
We propose a globally-accelerated, first-order method for the optimization of smooth and (strongly or not) geodesically-convex functions in a wide class of Hadamard manifolds. We achieve the same convergence rates as Nesterov's accelerated gradient descent, up to a multiplicative geometric penalty and log factors. Crucially, we can enforce our method to stay within a compact set we define. Prior fully accelerated works \emph{resort to assuming} that the iterates of their algorithms stay in some pre-specified compact set, except for two previous methods of limited applicability. For our manifol…
Institute for Mathematical and Computational Engineering（数学与计算工程研究所）；Faculty of Mathematics（理学院）；School of Engineering（工学院）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
We propose a novel data-driven method to learn a mixture of multiple kernels with random features that is certifiabaly robust against adverserial inputs. Specifically, we consider a distributionally robust optimization of the kernel-target alignment with respect to the distribution of training samples over a distributional ball defined by the Kullback-Leibler (KL) divergence. The distributionally robust optimization problem can be recast as a min-max optimization whose objective function includes a log-sum term. We develop a mini-batch biased stochastic primal-dual proximal method to solve the…
A local convergence result for abstract descent methods is proved. The sequence of iterates is attracted by a local (or global) minimum, stays in its neighborhood and converges within this neighborhood. This result allows algorithms to exploit local properties of the objective function. In particular, the abstract theory in this paper applies to the inertial forward--backward splitting method: iPiano---a generalization of the Heavy-ball method. Moreover, it reveals an equivalence between iPiano and inertial averaged/alternating proximal minimization and projection methods. Key for this equival…
Optimization is at the heart of machine learning, statistics and many applied scientific disciplines. It also has a long history in physics, ranging from the minimal action principle to finding ground states of disordered systems such as spin glasses. Proximal algorithms form a class of methods that are broadly applicable and are particularly well-suited to nonsmooth, constrained, large-scale, and distributed optimization problems. There are essentially five proximal algorithms currently known: Forward-backward splitting, Tseng splitting, Douglas-Rachford, alternating direction method of multi…
University of Freiburg（弗莱堡大学）；BIOSS Centre for Biological Signalling Studies（BIOSS生物信号研究中心）；Graz University of Technology（格拉茨工业大学）
In this paper we study an algorithm for solving a minimization problem composed of a differentiable (possibly non-convex) and a convex (possibly non-differentiable) function. The algorithm iPiano combines forward-backward splitting with an inertial force. It can be seen as a non-smooth split version of the Heavy-ball method from Polyak. A rigorous analysis of the algorithm for the proposed class of problems yields global convergence of the function values and the arguments. This makes the algorithm robust for usage on non-convex problems. The convergence result is obtained based on the \KL ine…
Composite minimization involves a collection of smooth functions which are aggregated in a nonsmooth manner. In the convex setting, we design an algorithm by linearizing each smooth component in accordance with its main curvature. The resulting method, called the Multiprox method, consists in solving successively simple problems (e.g. constrained quadratic problems) which can also feature some proximal operators. To study the complexity and the convergence of this method we are led to study quantitative qualification conditions to understand the impact of multipliers on the complexity bounds. …
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…
Huawei Belgium Research Center（华为比利时研究中心）；HCMC University of Education（胡志明市教育大学）；University of Mons（蒙斯大学）
In this paper, we introduce TITAN, a novel inerTIal block majorizaTion minimizAtioN framework for non-smooth non-convex optimization problems. To the best of our knowledge, TITAN is the first framework of block-coordinate update method that relies on the majorization-minimization framework while embedding inertial force to each step of the block updates. The inertial force is obtained via an extrapolation operator that subsumes heavy-ball and Nesterov-type accelerations for block proximal gradient methods as special cases. By choosing various surrogate functions, such as proximal, Lipschitz gr…
We give an effective solution to the regularized optimization problem $g (\boldsymbol{x}) + h (\boldsymbol{x})$, where $\boldsymbol{x}$ is constrained on the unit sphere $\Vert \boldsymbol{x} \Vert_2 = 1$. Here $g (\cdot)$ is a smooth cost with Lipschitz continuous gradient within the unit ball $\{\boldsymbol{x} : \Vert \boldsymbol{x} \Vert_2 \le 1 \}$ whereas $h (\cdot)$ is typically non-smooth but convex and absolutely homogeneous, \textit{e.g.,}~norm regularizers and their combinations. Our solution is based on the Riemannian proximal gradient, using an idea we call \textit{proxy step-size}…
Motivated by some applications in signal processing and machine learning, we consider two convex optimization problems where, given a cone $K$, a norm $\|\cdot\|$ and a smooth convex function $f$, we want either 1) to minimize the norm over the intersection of the cone and a level set of $f$, or 2) to minimize over the cone the sum of $f$ and a multiple of the norm. We focus on the case where (a) the dimension of the problem is too large to allow for interior point algorithms, (b) $\|\cdot\|$ is "too complicated" to allow for computationally cheap Bregman projections required in the first-orde…
Microsecond-Class Powered-Descent Optimization via Exact Condensation and Strong Convex Regularization
School of Aerospace Engineering, Tsinghua University（清华大学航天航空学院）；School of Astronautics, Beihang University（北京航空航天大学宇航学院）；State Key Laboratory of High-Efficiency Reusable Aerospace Transportation Technology（高效可重复使用航天运输技术国家重点实验室）
In a Hilbertian framework, for the minimization of a general convex differentiable function $f$, we introduce new inertial dynamics and algorithms that generate trajectories and iterates that converge fastly towards the minimizer of $f$ with minimum norm. Our study is based on the non-autonomous version of the Polyak heavy ball method, which, at time $t$, is associated with the strongly convex function obtained by adding to $f$ a Tikhonov regularization term with vanishing coefficient $ε(t)$. In this dynamic, the damping coefficient is proportional to the square root of the Tikhonov regulariza…
We consider stochastic convex optimization problems with affine constraints and develop several methods using either primal or dual approach to solve it. In the primal case, we use a special penalization technique to make the initial problem more convenient for using optimization methods. We propose algorithms to solve it based on Similar Triangles Method with Inexact Proximal Step for the convex smooth and strongly convex smooth objective functions and methods based on Gradient Sliding algorithm to solve the same problems in the non-smooth case. We prove the convergence guarantees in the smoo…
Prox-DBRO-VR: A Unified Analysis on Byzantine-Resilient Decentralized Stochastic Composite Optimization with Variance Reduction and Non-Asymptotic Convergence Rates
Central South University（中南大学）；City University of Hong Kong（香港城市大学）；University of New South Wales（新南威尔士大学）；Southwest University（西南大学）
