In this work we show that various algorithms, ubiquitous in convex optimization (e.g. proximal-gradient, alternating projections and averaged projections) generate self-contracted sequences $\{x_{k}\}_{k\in\mathbb{N}}$. As a consequence, a novel universal bound for the \emph{length} ($\sum_{k\ge 0}\Vert x_{k+1}-x_k\Vert$) can be deduced. In addition, this bound is independent of both the concrete data of the problem (sets, functions) as well as the stepsize involved, and only depends on the dimension of the space.
In this paper, we propose new accelerated methods for smooth convex optimization, called contracting proximal methods. At every step of these methods, we need to minimize a contracted version of the objective function augmented by a regularization term in the form of Bregman divergence. We provide global convergence analysis for a general scheme admitting inexactness in solving the auxiliary subproblem. In the case of using for this purpose high-order tensor methods, we demonstrate an acceleration effect for both convex and uniformly convex composite objective functions. Thus, our construction…
The task of projecting onto $\ell_p$ norm balls is ubiquitous in statistics and machine learning, yet the availability of actionable algorithms for doing so is largely limited to the special cases of $p = \left\{ 0, 1,2, \infty \right\}$. In this paper, we introduce novel, scalable methods for projecting onto the $\ell_p$ ball for general $p>0$. For $p \geq1 $, we solve the univariate Lagrangian dual via a dual Newton method. We then carefully design a bisection approach for $p<1$, presenting theoretical and empirical evidence of zero or a small duality gap in the non-convex case. The success …
Convexification based on convex envelopes is ubiquitous in the non-linear optimization literature. Thanks to considerable efforts of the optimization community for decades, we are able to compute the convex envelopes of a considerable number of functions that appear in practice, and thus obtain tight and tractable approximations to challenging problems. We contribute to this line of work by considering a family of functions that, to the best of our knowledge, has not been considered before in the literature. We call this family ray-concave functions. We show sufficient conditions that allow us…
We consider the differentiation of the value function for parametric optimization problems. Such problems are ubiquitous in Machine Learning applications such as structured support vector machines, matrix factorization and min-min or minimax problems in general. Existing approaches for computing the derivative rely on strong assumptions of the parametric function. Therefore, in several scenarios there is no theoretical evidence that a given algorithmic differentiation strategy computes the true gradient information of the value function. We leverage a well known result from convex duality theo…
Convex nonsmooth optimization problems, whose solutions live in very high dimensional spaces, have become ubiquitous. To solve them, the class of first-order algorithms known as proximal splitting algorithms is particularly adequate: they consist of simple operations, handling the terms in the objective function separately. In this overview, we demystify a selection of recent proximal splitting algorithms: we present them within a unified framework, which consists in applying splitting methods for monotone inclusions in primal-dual product spaces, with well-chosen metrics. Along the way, we ea…
George Mason University（乔治梅森大学）；Boston University（波士顿大学）
Convex-concave min-max problems are ubiquitous in machine learning, and people usually utilize first-order methods (e.g., gradient descent ascent) to find the optimal solution. One feature which separates convex-concave min-max problems from convex minimization problems is that the best known convergence rates for min-max problems have an explicit dependence on the size of the domain, rather than on the distance between initial point and the optimal solution. This means that the convergence speed does not have any improvement even if the algorithm starts from the optimal solution, and hence, i…
Non-convex optimization is ubiquitous in modern machine learning. Researchers devise non-convex objective functions and optimize them using off-the-shelf optimizers such as stochastic gradient descent and its variants, which leverage the local geometry and update iteratively. Even though solving non-convex functions is NP-hard in the worst case, the optimization quality in practice is often not an issue -- optimizers are largely believed to find approximate global minima. Researchers hypothesize a unified explanation for this intriguing phenomenon: most of the local minima of the practically-u…
Hebrew University of Jerusalem（耶路撒冷希伯来大学）；Tel Aviv University（特拉维夫大学）；Google Research（谷歌研究院）
We consider regression problems with binary weights. Such optimization problems are ubiquitous in quantized learning models and digital communication systems. A natural approach is to optimize the corresponding Lagrangian using variants of the gradient ascent-descent method. Such maximin techniques are still poorly understood even in the concave-convex case. The non-convex binary constraints may lead to spurious local minima. Interestingly, we prove that this approach is optimal in linear regression with low noise conditions as well as robust regression with a small number of outliers. Practic…
Stochastic convex optimization over an $\ell_1$-bounded domain is ubiquitous in machine learning applications such as LASSO but remains poorly understood when learning with differential privacy. We show that, up to logarithmic factors the optimal excess population loss of any $(\varepsilon,δ)$-differentially private optimizer is $\sqrt{\log(d)/n} + \sqrt{d}/\varepsilon n.$ The upper bound is based on a new algorithm that combines the iterative localization approach of~\citet{FeldmanKoTa20} with a new analysis of private regularized mirror descent. It applies to $\ell_p$ bounded domains for $p\…
Microsoft（微软）；Johns Hopkins University（约翰斯·霍普金斯大学）；Zhejiang University（浙江大学）；Georgia Institute of Technology（佐治亚理工学院）；University of Washington（华盛顿大学）；University of Denver（丹佛大学）
Sparsity-inducing regularization problems are ubiquitous in machine learning applications, ranging from feature selection to model compression. In this paper, we present a novel stochastic method -- Orthant Based Proximal Stochastic Gradient Method (OBProx-SG) -- to solve perhaps the most popular instance, i.e., the l1-regularized problem. The OBProx-SG method contains two steps: (i) a proximal stochastic gradient step to predict a support cover of the solution; and (ii) an orthant step to aggressively enhance the sparsity level via orthant face projection. Compared to the state-of-the-art met…
The Chinese University of Hong Kong（香港中文大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Cainiao Network（菜鸟网络）；University of Michigan（密歇根大学）；University of Denver（丹佛大学）
We consider a class of nonsmooth optimization problems over the Stiefel manifold, in which the objective function is weakly convex in the ambient Euclidean space. Such problems are ubiquitous in engineering applications but still largely unexplored. We present a family of Riemannian subgradient-type methods -- namely Riemannain subgradient, incremental subgradient, and stochastic subgradient methods -- to solve these problems and show that they all have an iteration complexity of ${\cal O}(\varepsilon^{-4})$ for driving a natural stationarity measure below $\varepsilon$. In addition, we establ…
Kurdyka-Lojasiewicz (KL) exponent plays an important role in estimating the convergence rate of many contemporary first-order methods. In particular, a KL exponent of $\frac12$ for a suitable potential function is related to local linear convergence. Nevertheless, KL exponent is in general extremely hard to estimate. In this paper, we show under mild assumptions that KL exponent is preserved via inf-projection. Inf-projection is a fundamental operation that is ubiquitous when reformulating optimization problems via the lift-and-project approach. By studying its operation on KL exponent, we sho…
Decentralized optimization, particularly the class of decentralized composite convex optimization (DCCO) problems, has found many applications. Due to ubiquitous communication congestion and random dropouts in practice, it is highly desirable to design decentralized algorithms that can handle stochastic communication networks. However, most existing algorithms for DCCO only work in networks that are deterministically connected during bounded communication rounds, and therefore cannot be extended to stochastic networks. In this paper, we propose a new decentralized dual averaging (DDA) algorith…
The Hong Kong University of Science and Technology（香港科技大学）
Differentially private (DP) stochastic convex optimization (SCO) is ubiquitous in trustworthy machine learning algorithm design. This paper studies the DP-SCO problem with streaming data sampled from a distribution and arrives sequentially. We also consider the continual release model where parameters related to private information are updated and released upon each new data, often known as the online algorithms. Despite that numerous algorithms have been developed to achieve the optimal excess risks in different $\ell_p$ norm geometries, yet none of the existing ones can be adapted to the str…
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}…
Certified bounds on optimization problems in quantum theory
ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（光子科学研究所，巴塞罗那科学与技术学院）；LAAS-CNRS - Laboratoire d’Analyse et d’Architecture des Systèmes, Centre National de la Recherche Scientifique（系统分析与架构实验室，法国国家科学研究中心）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（数学科学学院国家重点实验室，中国科学院数学与系统科学研究院）；Institute of Mathematics, Toulouse（图卢兹数学研究所）；ICREA - Institució Catalana de Recerca i Estudis Avançats（加泰罗尼亚高级研究学院）
