We investigate a projection free method, namely conditional gradient sliding on batched, stochastic and finite-sum non-convex problem. CGS is a smart combination of Nesterov's accelerated gradient method and Frank-Wolfe (FW) method, and outperforms FW in the convex setting by saving gradient computations. However, the study of CGS in the non-convex setting is limited. In this paper, we propose the non-convex conditional gradient sliding (NCGS) which surpasses the non-convex Frank-Wolfe method in batched, stochastic and finite-sum setting.
We present a modification of the conditional gradient sliding (CGS) method that was originally developed in \cite{lan2016conditional}. While the CGS method is a theoretical breakthrough in the theory of projection-free first-order methods since it is the first that reaches the theoretical performance limit, in implementation it requires the knowledge of the Lipschitz constant of the gradient of the objective function $L$ and the number of total gradient evaluations $N$. Such requirements imposes difficulties in the actual implementation, not only because that it can be difficult to choose prop…
In this paper, we present a first-order projection-free method, namely, the universal conditional gradient sliding (UCGS) method, for solving $\varepsilon$-approximate solutions to convex differentiable optimization problems. For objective functions with Hölder continuous gradients, we show that UCGS is able to terminate with $\varepsilon$-solutions with at most $O((M_νD_X^{1+ν}/{\varepsilon})^{2/(1+3ν)})$ gradient evaluations and $O((M_νD_X^{1+ν}/{\varepsilon})^{4/(1+3ν)})$ linear objective optimizations, where $ν\in (0,1]$ and $M_ν>0$ are the exponent and constant of the Hölder condition. Fu…
Nanjing University of Information Science & Technology（南京信息工程大学）；MBZUAI（穆罕默德·本·扎耶德人工智能大学）；JD Finance America Corporation（京东金融美国公司）；University of Pittsburgh（匹兹堡大学）
The conditional gradient algorithm (also known as the Frank-Wolfe algorithm) has recently regained popularity in the machine learning community due to its projection-free property to solve constrained problems. Although many variants of the conditional gradient algorithm have been proposed to improve performance, they depend on first-order information (gradient) to optimize. Naturally, these algorithms are unable to function properly in the field of increasingly popular zeroth-order optimization, where only zeroth-order information (function value) is available. To fill in this gap, we propose…
In this paper, we present a new Hyperfast Second-Order Method with convergence rate $O(N^{-5})$ up to a logarithmic factor for the convex function with Lipshitz the third derivative. This method based on two ideas. The first comes from the superfast second-order scheme of Yu. Nesterov (CORE Discussion Paper 2020/07, 2020). It allows implementing the third-order scheme by solving subproblem using only the second-order oracle. This method converges with rate $O(N^{-4})$. The second idea comes from the work of Kamzolov et al. (arXiv:2002.01004). It is the inexact near-optimal third-order method. …
In this paper we propose new algorithms for solving a class of structured monotone variational inequality (VI) problems over compact feasible sets. By identifying the gradient components existing in the operator of VI, we show that it is possible to skip computations of the gradients from time to time, while still maintaining the optimal iteration complexity for solving these VI problems. Specifically, for deterministic VI problems involving the sum of the gradient of a smooth convex function $\nabla G$ and a monotone operator $H$, we propose a new algorithm, called the mirror-prox sliding met…
We study structured convex optimization problems, with additive objective $r:=p + q$, where $r$ is ($μ$-strongly) convex, $q$ is $L_q$-smooth and convex, and $p$ is $L_p$-smooth, possibly nonconvex. For such a class of problems, we proposed an inexact accelerated gradient sliding method that can skip the gradient computation for one of these components while still achieving optimal complexity of gradient calls of $p$ and $q$, that is, $\mathcal{O}(\sqrt{L_p/μ})$ and $\mathcal{O}(\sqrt{L_q/μ})$, respectively. This result is much sharper than the classic black-box complexity $\mathcal{O}(\sqrt{(…
The goal of this paper is to reduce the total complexity of gradient-based methods for two classes of problems: affine-constrained composite convex optimization and bilinear saddle-point structured non-smooth convex optimization. Our technique is based on a double-loop inexact accelerated proximal gradient (APG) method for minimizing the summation of a non-smooth but proximable convex function and two smooth convex functions with different smoothness constants and computational costs. Compared to the standard APG method, the inexact APG method can reduce the total computation cost if one smoot…
A conditional-gradient-based single-loop augmented Lagrangian method for inequality constrained optimization
School of Mathematical Sciences, South China Normal University（华南师范大学数学科学学院）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）；Department of Mathematics and Statistics, James Madison University（詹姆斯麦迪逊大学数学与统计系）
In this paper we present a new method for solving optimization problems involving the sum of two proper, convex, lower semicontinuous functions, one of which has Lipschitz continuous gradient. The proposed method has a hybrid nature that combines the usual forward-backward and the generalized conditional gradient method. We establish a convergence rate of $o(k^{-1/3})$ under mild assumptions with a specific step-size rule and show an application to a total variation parameter learning problem, which demonstrates its benefits in the context of nonsmooth convex optimization.
University of California, Davis（加州大学戴维斯分校）；University of Waterloo（滑铁卢大学）
We analyze stochastic conditional gradient methods for constrained optimization problems arising in over-parametrized machine learning. We show that one could leverage the interpolation-like conditions satisfied by such models to obtain improved oracle complexities. Specifically, when the objective function is convex, we show that the conditional gradient method requires $\mathcal{O}(ε^{-2})$ calls to the stochastic gradient oracle to find an $ε$-optimal solution. Furthermore, by including a gradient sliding step, we show that the number of calls reduces to $\mathcal{O}(ε^{-1.5})$.
Conditional gradient methods have attracted much attention in both machine learning and optimization communities recently. These simple methods can guarantee the generation of sparse solutions. In addition, without the computation of full gradients, they can handle huge-scale problems sometimes even with an exponentially increasing number of decision variables. This paper aims to significantly expand the application areas of these methods by presenting new conditional gradient methods for solving convex optimization problems with general affine and nonlinear constraints. More specifically, we …
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
We propose two novel conditional gradient-based methods for solving structured stochastic convex optimization problems with a large number of linear constraints. Instances of this template naturally arise from SDP-relaxations of combinatorial problems, which involve a number of constraints that is polynomial in the problem dimension. The most important feature of our framework is that only a subset of the constraints is processed at each iteration, thus gaining a computational advantage over prior works that require full passes. Our algorithms rely on variance reduction and smoothing used in c…
In this paper, we propose a conditional gradient method for solving constrained vector optimization problems with respect to a partial order induced by a closed, convex and pointed cone with nonempty interior. When the partial order under consideration is the one induced by the non-negative orthant, we regain the method for multiobjective optimization recently proposed by Assunção et al. (Comput Optim Appl 78(3):741--768, 2021). In our method, the construction of auxiliary subproblem is based on the well-known oriented distance function. Three different types of step size strategies (Armijio, …
A generalized conditional gradient method for minimizing the sum of two convex functions, one of them differentiable, is presented. This iterative method relies on two main ingredients: First, the minimization of a partially linearized objective functional to compute a descent direction and, second, a stepsize choice based on an Armijo-like condition to ensure sufficient descent in every iteration. We provide several convergence results. Under mild assumptions, the method generates sequences of iterates which converge, on subsequences, towards minimizers. Moreover a sublinear rate of convergen…
ETH Zürich（苏黎世联邦理工学院）；EPFL（洛桑联邦理工学院）；Amazon Web Services（亚马逊云科技）；Umeå University（于默奥大学）
We propose a stochastic conditional gradient method (CGM) for minimizing convex finite-sum objectives formed as a sum of smooth and non-smooth terms. Existing CGM variants for this template either suffer from slow convergence rates, or require carefully increasing the batch size over the course of the algorithm's execution, which leads to computing full gradients. In contrast, the proposed method, equipped with a stochastic average gradient (SAG) estimator, requires only one sample per iteration. Nevertheless, it guarantees fast convergence rates on par with more sophisticated variance reducti…
One fundamental problem in decentralized multi-agent optimization is the trade-off between gradient/sampling complexity and communication complexity. We propose new algorithms whose gradient and sampling complexities are graph topology invariant while their communication complexities remain optimal. For convex smooth deterministic problems, we propose a primal dual sliding (PDS) algorithm that computes an $ε$-solution with $O((\tilde{L}/ε)^{1/2})$ gradient and $O((\tilde{L}/ε)^{1/2}+\|\mathcal{A}\|/ε)$ communication complexities, where $\tilde{L}$ is the smoothness parameter of the objective a…
Efficient Quadratic Corrections for Frank-Wolfe Algorithms
Zuse Institute Berlin（柏林齐泽研究所）；TU Berlin（柏林工业大学）；École nationale des ponts et chaussées（国立桥校）；IP Paris（巴黎理工学院）；Université Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Inria（法国国家信息与自动化研究所）；LIG（信息学实验室）；CNRS（法国国家科学研究中心）
This article deals with multiobjective composite optimization problems that consist of simultaneously minimizing several objective functions, each of which is composed of a combination of smooth and non-smooth functions. To tackle these problems, we propose a generalized version of the conditional gradient method, also known as Frank-Wolfe method. The method is analyzed with three step size strategies, including Armijo-type, adaptive, and diminishing step sizes. We establish asymptotic convergence properties and iteration-complexity bounds, with and without convexity assumptions on the objecti…
The conditional gradient method (CGM) is widely used in large-scale sparse convex optimization, having a low per iteration computational cost for structured sparse regularizers and a greedy approach to collecting nonzeros. We explore the sparsity acquiring properties of a general penalized CGM (P-CGM) for convex regularizers and a reweighted penalized CGM (RP-CGM) for nonconvex regularizers, replacing the usual convex constraints with gauge-inspired penalties. This generalization does not increase the per-iteration complexity noticeably. Without assuming bounded iterates or using line search, …
University of Waterloo（滑铁卢大学）；Vector Institute（向量研究所）；UvA Bosch Delta Lab（阿姆斯特丹大学博世德尔塔实验室）；University of Amsterdam（阿姆斯特丹大学）；Borealis AI（北极光人工智能）
Recently, the conditional SAGE certificate has been proposed as a sufficient condition for signomial positivity over a convex set. In this article, we show that the conditional SAGE certificate is $\textit{complete}$. That is, for any signomial $f(\mathbf{x}) = \sum_{j=1}^{\ell}c_j \exp(\mathbf{A}_j\mathbf{x})$ defined by rational exponents that is positive over a compact convex set $\mathcal{X}$, there is $p \in \mathbb{Z}_+$ and a specific positive definite function $w(\mathbf{x})$ such that $w(\mathbf{x})^p f(\mathbf{x})$ may be verified by the conditional SAGE certificate. The completeness…
Univeristy of California, Berkeley（加州大学伯克利分校）；Georgia Institute of Technology（佐治亚理工学院）；Zuse Institute Berlin（柏林祖斯研究所）；Technische Universität Berlin（柏林工业大学）
