Blake Woodworth, Nathan Srebro
We provide an explicit construction and direct proof for the lower bound on the number of first order oracle accesses required for a randomized algorithm to minimize a convex Lipschitz function.
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In this paper, we study randomized and cyclic coordinate descent for convex unconstrained optimization problems. We improve the known convergence rates in some cases by using the numerical semidefinite programming performance estimation method. As a spin-off we provide a method to analyse the worst-case performance of the Gauss-Seidel iterative method for linear systems where the coefficient matrix is positive semidefinite with a positive diagonal.
Linear constrained convex programming has many practical applications, including support vector machine and machine learning portfolio problems. We propose the randomized primal-dual coordinate (RPDC) method, a randomized coordinate extension of the first-order primal-dual method by Cohen and Zhu, 1984 and Zhao and Zhu, 2019, to solve linear constrained convex programming. We randomly choose a block of variables based on a uniform distribution, linearize, and apply a Bregman-like function (core function) to the selected block to obtain simple parallel primal-dual decomposition. We then establi…
In the paper we consider an application of mirror descent (dual averaging) to the stochastic online convex optimization problems. We compare classical mirror descent (Nemirovski-Yudin, 1979) with dual averaging (Nesterov, 2005) and Grigoriadis-Khachiyan algorithm (1995). Grigoriadis-Khachiyan algorithm has just proved to be a randomized mirror descent (dual averaging) with randomization in KL-projection of (sub)gradient to a unit simplex. We found out that this randomization is an optimal way of solving sparse matrix games and some other problems arising in convex optimization and experts weig…
We propose STARS, a randomized derivative-free algorithm for unconstrained optimization when the function evaluations are contaminated with random noise. STARS takes dynamic, noise-adjusted smoothing step-sizes that minimize the least-squares error between the true directional derivative of a noisy function and its finite difference approximation. We provide a convergence rate analysis of STARS for solving convex problems with additive or multiplicative noise. Experimental results show that (1) STARS exhibits noise-invariant behavior with respect to different levels of stochastic noise; (2) th…
Randomized gradient-free methods in convex optimization
本文为凸优化无梯度方法综述，介绍仅用目标值（含噪）的这类方法，围绕预言机复杂度等三个指标展开，回应梯度信息难获取的应用需求。
Block coordinate descent methods and stochastic subgradient methods have been extensively studied in optimization and machine learning. By combining randomized block sampling with stochastic subgradient methods based on dual averaging, we present stochastic block dual averaging (SBDA)---a novel class of block subgradient methods for convex nonsmooth and stochastic optimization. SBDA requires only a block of subgradients and updates blocks of variables and hence has significantly lower iteration cost than traditional subgradient methods. We show that the SBDA-based methods exhibit the optimal c…
Theoretical analysis of the randomized subspace regularized Newton method for non-convex optimization
本文提出随机子空间正则化牛顿法用于非凸优化，证明其全局收敛且速率与完整正则化牛顿法相同，并给出局部线性收敛速率及Hessian秩亏时的超线性收敛结果。
Finding a point in the intersection of a collection of closed convex sets, that is the convex feasibility problem, represents the main modeling strategy for many computational problems. In this paper we analyze new stochastic reformulations of the convex feasibility problem in order to facilitate the development of new algorithmic schemes. We also analyze the conditioning problem parameters using certain (linear) regularity assumptions on the individual convex sets. Then, we introduce a general random projection algorithmic framework, which extends to the random settings many existing projecti…
Speeding up the Goemans-Williamson randomized procedure by difference-of-convex optimization
通过凸优化差分方法加速Goemans-Williamson随机化过程，提升QUBO问题求解效率与质量
We propose a new randomized algorithm for solving convex optimization problems that have a large number of constraints (with high probability). Existing methods like interior-point or Newton-type algorithms are hard to apply to such problems because they have expensive computation and storage requirements for Hessians and matrix inversions. Our algorithm is based on nonlinear rescaling (NLR), which is a primal-dual-type algorithm by Griva and Polyak {[{Math. Program., 106(2):237-259, 2006}]}. NLR introduces an equivalent problem through a transformation of the constraint functions, minimizes t…
The Randomized Block Coordinate Descent Method in the Hölder Smooth Setting
本文首次分析随机块坐标下降法在Hölder光滑条件下的收敛性，覆盖非凸、凸和强凸目标，并证明其收敛速率与Lipschitz光滑情形一致。
We consider the problem of minimizing the sum of an average function of a large number of smooth convex components and a general, possibly non-differentiable, convex function. Although many methods have been proposed to solve this problem with the assumption that the sum is strongly convex, few methods support the non-strongly convex case. Adding a small quadratic regularization is a common devise used to tackle non-strongly convex problems; however, it may cause loss of sparsity of solutions or weaken the performance of the algorithms. Avoiding this devise, we propose an accelerated randomize…
Motivated by high-dimensional nonlinear optimization problems as well as ill-posed optimization problems arising in image processing, we consider a bilevel optimization model where we seek among the optimal solutions of the inner level problem, a solution that minimizes a secondary metric. Our goal is to address the high-dimensionality of the bilevel problem, and the nondifferentiability of the objective function. Minimal norm gradient, sequential averaging, and iterative regularization are some of the recent schemes developed for addressing the bilevel problem. But none of them address the hi…
In this paper, we present novel randomized algorithms for solving saddle point problems whose dual feasible region is given by the direct product of many convex sets. Our algorithms can achieve an ${\cal O}(1/N)$ and ${\cal O}(1/N^2)$ rate of convergence, respectively, for general bilinear saddle point and smooth bilinear saddle point problems based on a new prima-dual termination criterion, and each iteration of these algorithms needs to solve only one randomly selected dual subproblem. Moreover, these algorithms do not require strongly convex assumptions on the objective function and/or the …
This paper proposes TriPD, a new primal-dual algorithm for minimizing the sum of a Lipschitz-differentiable convex function and two possibly nonsmooth convex functions, one of which is composed with a linear mapping. We devise a randomized block-coordinate version of the algorithm which converges under the same stepsize conditions as the full algorithm. It is shown that both the original as well as the block-coordinate scheme feature linear convergence rate when the functions involved are either piecewise linear-quadratic, or when they satisfy a certain quadratic growth condition (which is wea…
Given two bounded convex sets $X\subseteq\RR^m$ and $Y\subseteq\RR^n,$ specified by membership oracles, and a continuous convex-concave function $F:X\times Y\to\RR$, we consider the problem of computing an $\eps$-approximate saddle point, that is, a pair $(x^*,y^*)\in X\times Y$ such that $\sup_{y\in Y} F(x^*,y)\le \inf_{x\in X}F(x,y^*)+\eps.$ Grigoriadis and Khachiyan (1995) gave a simple randomized variant of fictitious play for computing an $\eps$-approximate saddle point for matrix games, that is, when $F$ is bilinear and the sets $X$ and $Y$ are simplices. In this paper, we extend their m…
Randomized subspace correction methods for convex optimization
本文提出了一种随机子空间校正方法的抽象框架，统一并推广了多种现有算法，包括域分解、多网格和块坐标下降方法，并分析了不同收敛率。
As the number of samples and dimensionality of optimization problems related to statistics an machine learning explode, block coordinate descent algorithms have gained popularity since they reduce the original problem to several smaller ones. Coordinates to be optimized are usually selected randomly according to a given probability distribution. We introduce an importance sampling strategy that helps randomized coordinate descent algorithms to focus on blocks that are still far from convergence. The framework applies to problems composed of the sum of two possibly non-convex terms, one being s…
We develop a novel unified randomized block-coordinate primal-dual algorithm to solve a class of nonsmooth constrained convex optimization problems, which covers different existing variants and model settings from the literature. We prove that our algorithm achieves optimal $\mathcal{O}(n/k)$ and $\mathcal{O}(n^2/k^2)$ convergence rates (up to a constant factor) in two cases: general convexity and strong convexity, respectively, where $k$ is the iteration counter and n is the number of block-coordinates. Our convergence rates are obtained through three criteria: primal objective residual and p…
Randomized Geometric Algebra Methods for Convex Neural Networks
本文提出将随机算法引入Clifford几何代数，通过凸优化训练神经网络至全局最优，并验证该方法在GPT-4和BERT嵌入的文本分类迁移学习任务中比传统方法更稳定可靠。
Sequential Randomized Algorithms for Convex Optimization in the Presence of Uncertainty
针对存在不确定性的凸优化问题，提出新的顺序随机算法，并分别针对完全约束满足和部分约束满足情况给出解的理论性质分析，通过硬盘驱动器伺服设计仿真验证了方法的有效性。
In this work, we analyze the global convergence property of coordinate gradient descent with random choice of coordinates and stepsizes for non-convex optimization problems. Under generic assumptions, we prove that the algorithm iterate will almost surely escape strict saddle points of the objective function. As a result, the algorithm is guaranteed to converge to local minima if all saddle points are strict. Our proof is based on viewing coordinate descent algorithm as a nonlinear random dynamical system and a quantitative finite block analysis of its linearization around saddle points.
An accelerated randomized Bregman-Kaczmarz method for strongly convex linearly constraint optimization
本文提出一种求解线性约束下强凸优化问题的加速随机Bregman-Kaczmarz方法，利用与坐标下降法的对偶关系实现加速，理论分析与数值实验均证明其比现有方法更高效。
This paper is concerned with multi-agent optimization problem. A distributed randomized gradient-free mirror descent (DRGFMD) method is developed by introducing a randomized gradient-free oracle in the mirror descent scheme where the non-Euclidean Bregman divergence is used. The classical gradient descent method is generalized without using subgradient information of objective functions. The proposed algorithm is the first distributed non-Euclidean zeroth-order method which achieves an $O(1/\sqrt{T})$ convergence rate, recovering the best known optimal rate of distributed compact constrained c…
One of the most attractive recent approaches to processing well-structured large-scale convex optimization problems is based on smooth convex-concave saddle point reformu-lation of the problem of interest and solving the resulting problem by a fast First Order saddle point method utilizing smoothness of the saddle point cost function. In this paper, we demonstrate that when the saddle point cost function is polynomial, the precise gra-dients of the cost function required by deterministic First Order saddle point algorithms and becoming prohibitively computationally expensive in the extremely l…
Near-Optimal Lower Bounds for Randomized Algorithms in Exact Value Zeroth-Order Convex Optimization
研究零阶凸优化中精确函数值访问的预言机限制，针对在\(d\)维欧几里得单位球上最小化非光滑凸函数的自适应随机算法，给出查询次数下限，在不同精度 regime 与已有上界匹配，采用新方法证明。
In this paper, we consider smooth convex optimization problems with simple constraints and inexactness in the oracle information such as value, partial or directional derivatives of the objective function. We introduce a unifying framework, which allows to construct different types of accelerated randomized methods for such problems and to prove convergence rate theorems for them. We focus on accelerated random block-coordinate descent, accelerated random directional search, accelerated random derivative-free method and, using our framework, provide their versions for problems with inexact ora…
Randomized Fast Subspace Descent (RFASD) Methods are developed and analyzed for smooth and non-constraint convex optimization problems. The efficiency of the method relies on a space decomposition which is stable in $A$-norm, and meanwhile, the condition number $κ_A$ measured in $A$-norm is small. At each iteration, the subspace is chosen randomly either uniformly or by a probability proportional to the local Lipschitz constants. Then in each chosen subspace, a preconditioned gradient descent method is applied. RFASD converges sublinearly for convex functions and linearly for strongly convex f…
Consider the problem of minimizing the sum of a smooth (possibly non-convex) and a convex (possibly nonsmooth) function involving a large number of variables. A popular approach to solve this problem is the block coordinate descent (BCD) method whereby at each iteration only one variable block is updated while the remaining variables are held fixed. With the recent advances in the developments of the multi-core parallel processing technology, it is desirable to parallelize the BCD method by allowing multiple blocks to be updated simultaneously at each iteration of the algorithm. In this work, …
In this paper, we consider non-convex stochastic bilevel optimization (SBO) problems that have many applications in machine learning. Although numerous studies have proposed stochastic algorithms for solving these problems, they are limited in two perspectives: (i) their sample complexities are high, which do not match the state-of-the-art result for non-convex stochastic optimization; (ii) their algorithms are tailored to problems with only one lower-level problem. When there are many lower-level problems, it could be prohibitive to process all these lower-level problems at each iteration. To…
The stochastic Frank-Wolfe method has recently attracted much general interest in the context of optimization for statistical and machine learning due to its ability to work with a more general feasible region. However, there has been a complexity gap in the guaranteed convergence rate for stochastic Frank-Wolfe compared to its deterministic counterpart. In this work, we present a new generalized stochastic Frank-Wolfe method which closes this gap for the class of structured optimization problems encountered in statistical and machine learning characterized by empirical loss minimization with …
In this paper we analyze the randomized block-coordinate descent (RBCD) methods proposed in [8,11] for minimizing the sum of a smooth convex function and a block-separable convex function. In particular, we extend Nesterov's technique developed in [8] for analyzing the RBCD method for minimizing a smooth convex function over a block-separable closed convex set to the aforementioned more general problem and obtain a sharper expected-value type of convergence rate than the one implied in [11]. Also, we obtain a better high-probability type of iteration complexity, which improves upon the one in …
We study the question of whether parallelization in the exploration of the feasible set can be used to speed up convex optimization, in the local oracle model of computation. We show that the answer is negative for both deterministic and randomized algorithms applied to essentially any of the interesting geometries and nonsmooth, weakly-smooth, or smooth objective functions. In particular, we show that it is not possible to obtain a polylogarithmic (in the sequential complexity of the problem) number of parallel rounds with a polynomial (in the dimension) number of queries per round. In the ma…
We study decentralized non-convex finite-sum minimization problems described over a network of nodes, where each node possesses a local batch of data samples. In this context, we analyze a single-timescale randomized incremental gradient method, called GT-SAGA. GT-SAGA is computationally efficient as it evaluates one component gradient per node per iteration and achieves provably fast and robust performance by leveraging node-level variance reduction and network-level gradient tracking. For general smooth non-convex problems, we show the almost sure and mean-squared convergence of GT-SAGA to a…
A Randomized Linearly Convergent Frank-Wolfe-type Method for Smooth Convex Minimization over the Spectrahedron
本文提出了一种随机线性收敛的Frank-Wolfe型方法，用于在谱hedron上高效最小化平滑凸函数。
A Unified Zeroth-Order Optimization Framework via Oblivious Randomized Sketching
本文提出基于无关随机投影的统一零阶优化框架，通过选择适当投影矩阵降低RFD估计方差并实现高概率收敛，在凸二次目标上推导出优于线性维度依赖的查询复杂度，并扩展至强凸和Hessian感知优化场景。
In this paper, we consider the problem of minimizing the average of a large number of nonsmooth and convex functions. Such problems often arise in typical machine learning problems as empirical risk minimization, but are computationally very challenging. We develop and analyze a new algorithm that achieves robust linear convergence rate, and both its time complexity and gradient complexity are superior than state-of-art nonsmooth algorithms and subgradient-based schemes. Besides, our algorithm works without any extra error bound conditions on the objective function as well as the common strong…
We propose a new \textit{randomized Bregman (block) coordinate descent} (RBCD) method for minimizing a composite problem, where the objective function could be either convex or nonconvex, and the smooth part are freed from the global Lipschitz-continuous (partial) gradient assumption. Under the notion of relative smoothness based on the Bregman distance, we prove that every limit point of the generated sequence is a stationary point. Further, we show that the iteration complexity of the proposed method is $O(n\varepsilon^{-2})$ to achieve $ε$-stationary point, where $n$ is the number of blocks…
We are interested in solving convex optimization problems with large numbers of constraints. Randomized algorithms, such as random constraint sampling, have been very successful in giving nearly optimal solutions to such problems. In this paper, we combine random constraint sampling with the classical primal-dual algorithm for convex optimization problems with large numbers of constraints, and we give a convergence rate analysis. We then report numerical experiments that verify the effectiveness of this algorithm.
An error bound-based convergence analysis framework for a class of randomized algorithms
本文提出基于统一误差界条件的分析框架，用于研究具有下降性质的随机算法，提供非渐近期望和渐近几乎确定的收敛速率，应用于公共固定点问题和无约束最小化问题。
The Stochastic Primal-Dual Hybrid Gradient (SPDHG) was proposed by Chambolle et al. (2018) and is an efficient algorithm to solve some nonsmooth large-scale optimization problems. In this paper we prove its almost sure convergence for convex but not necessarily strongly convex functionals. We also look into its application to parallel Magnetic Resonance Imaging reconstruction in order to test performance of SPDHG. Our numerical results show that for a range of settings SPDHG converges significantly faster than its deterministic counterpart.
In this paper we consider large-scale smooth optimization problems with multiple linear coupled constraints. Due to the non-separability of the constraints, arbitrary random sketching would not be guaranteed to work. Thus, we first investigate necessary and sufficient conditions for the sketch sampling to have well-defined algorithms. Based on these sampling conditions we developed new sketch descent methods for solving general smooth linearly constrained problems, in particular, random sketch descent and accelerated random sketch descent methods. From our knowledge, this is the first converge…
Random projection (RP) is a classical technique for reducing storage and computational costs. We analyze RP-based approximations of convex programs, in which the original optimization problem is approximated by the solution of a lower-dimensional problem. Such dimensionality reduction is essential in computation-limited settings, since the complexity of general convex programming can be quite high (e.g., cubic for quadratic programs, and substantially higher for semidefinite programs). In addition to computational savings, random projection is also useful for reducing memory usage, and has use…
In this work, we study discrete-time Markov decision processes (MDPs) under constraints with Borel state and action spaces and where all the performance functions have the same form of the expected total reward (ETR) criterion over the infinite time horizon. One of our objective is to propose a convex programming formulation for this type of MDPs. It will be shown that the values of the constrained control problem and the associated convex program coincide and that if there exists an optimal solution to the convex program then there exists a stationary randomized policy which is optimal for th…
In this paper we consider distributed optimization problems in which the cost function is separable (i.e., a sum of possibly non-smooth functions all sharing a common variable) and can be split into a strongly convex term and a convex one. The second term is typically used to encode constraints or to regularize the solution. We propose an asynchronous, distributed optimization algorithm over an undirected topology, based on a proximal gradient update on the dual problem. We show that by means of a proper choice of primal variables, the dual problem is separable and the dual variables can be st…
Based on a preconditioned version of the randomized block-coordinate forward-backward algorithm recently proposed in [Combettes,Pesquet,2014], several variants of block-coordinate primal-dual algorithms are designed in order to solve a wide array of monotone inclusion problems. These methods rely on a sweep of blocks of variables which are activated at each iteration according to a random rule, and they allow stochastic errors in the evaluation of the involved operators. Then, this framework is employed to derive block-coordinate primal-dual proximal algorithms for solving composite convex var…
GeNIOS: an (almost) second-order operator-splitting solver for large-scale convex optimization
GeNIOS是一种用于大规模凸优化的近似二级运算分裂求解器，通过第二阶近似和快速随机求解器加速ADMM，实现高效求解。
Matching Lower Bounds for Randomized First-Order Methods in Hessian-Lipschitz Nonconvex Optimization
本文为非凸优化中寻找 ε-驻点的随机一阶算法建立了匹配下界，证明随机化无法改善高维最坏情况查询速率，并匹配了确定性加速梯度上界。
Minimizing a convex, quadratic objective of the form $f_{\mathbf{A},\mathbf{b}}(x) := \frac{1}{2}x^\top \mathbf{A} x - \langle \mathbf{b}, x \rangle$ for $\mathbf{A} \succ 0 $ is a fundamental problem in machine learning and optimization. In this work, we prove gradient-query complexity lower bounds for minimizing convex quadratic functions which apply to both deterministic and \emph{randomized} algorithms. Specifically, for $κ> 1$, we exhibit a distribution over $(\mathbf{A},\mathbf{b})$ with condition number $\mathrm{cond}(\mathbf{A}) \le κ$, such that any \emph{randomized} algorithm require…
Randomized Solutions to Convex Programs with Multiple Chance Constraints
本文针对带多机会约束的凸规划问题，提出“支撑秩”新概念，改进了场景法的约束违反概率界，在保持概率界的同时提升解的质量并降低计算复杂度。
Lower Bounds for Randomized Linear-Oracle Optimization over Strongly Convex Sets
针对强凸集上随机线性神谕优化的下界问题，现有下界存在适用范围受限的缺陷，本文证明了随机一阶线性神谕方法的分布下界，填补了无限制随机一阶方法下界的研究空白。
In this paper, we consider a class of finite-sum convex optimization problems whose objective function is given by the summation of $m$ ($\ge 1$) smooth components together with some other relatively simple terms. We first introduce a deterministic primal-dual gradient (PDG) method that can achieve the optimal black-box iteration complexity for solving these composite optimization problems using a primal-dual termination criterion. Our major contribution is to develop a randomized primal-dual gradient (RPDG) method, which needs to compute the gradient of only one randomly selected smooth compo…
Zeroth-order methods have become important tools for solving problems where we have access only to function evaluations. However, the zeroth-order methods only using gradient approximations are $n$ times slower than classical first-order methods for solving n-dimensional problems. To accelerate the convergence rate, this paper proposes the zeroth order randomized subspace Newton (ZO-RSN) method, which estimates projections of the gradient and Hessian by random sketching and finite differences. This allows us to compute the Newton step in a lower dimensional subspace, with small computational c…
This paper explores a surprising equivalence between two seemingly-distinct convex optimization methods. We show that simulated annealing, a well-studied random walk algorithms, is directly equivalent, in a certain sense, to the central path interior point algorithm for the the entropic universal barrier function. This connection exhibits several benefits. First, we are able improve the state of the art time complexity for convex optimization under the membership oracle model. We improve the analysis of the randomized algorithm of Kalai and Vempala by utilizing tools developed by Nesterov and …
We propose a novel randomized incremental gradient algorithm, namely, VAriance-Reduced Accelerated Gradient (Varag), for finite-sum optimization. Equipped with a unified step-size policy that adjusts itself to the value of the condition number, Varag exhibits the unified optimal rates of convergence for solving smooth convex finite-sum problems directly regardless of their strong convexity. Moreover, Varag is the first accelerated randomized incremental gradient method that benefits from the strong convexity of the data-fidelity term to achieve the optimal linear convergence. It also establish…
Randomized Coordinate Subgradient Method for Nonsmooth Composite Optimization
针对非光滑复合优化问题，提出首个通用随机坐标次梯度方法RCS，在更宽泛的次梯度有界假设下完成凸与弱凸情形的收敛分析，并通过实验验证其优于传统次梯度方法。
RA-DCA: A Randomized Active-Set DCA for Directional Stationarity in Max-Structured DC Programs
针对减去项为光滑凸函数有限最大值的非光滑DC规划，提出随机活动集DCA（RA-DCA），通过投影活动梯度到采样方向并检查采样顶点残差，以概率1保证聚点的方向稳定性。
Conditional Value at Risk (CVaR) is widely used to account for the preferences of a risk-averse agent in the extreme loss scenarios. To study the effectiveness of randomization in interdiction games with an interdictor that is both risk and ambiguity averse, we introduce a distributionally robust network interdiction game where the interdictor randomizes over the feasible interdiction plans in order to minimize the worst-case CVaR of the flow with respect to both the unknown distribution of the capacity of the arcs and his mixed strategy over interdicted arcs. The flow player, on the contrary,…
We study the first-order convex optimization problem, where we have black-box access to a (not necessarily smooth) function $f:\mathbb{R}^n \to \mathbb{R}$ and its (sub)gradient. Our goal is to find an $ε$-approximate minimum of $f$ starting from a point that is distance at most $R$ from the true minimum. If $f$ is $G$-Lipschitz, then the classic gradient descent algorithm solves this problem with $O((GR/ε)^{2})$ queries. Importantly, the number of queries is independent of the dimension $n$ and gradient descent is optimal in this regard: No deterministic or randomized algorithm can achieve be…
Hamiltonian Descent Algorithms for Optimization: Accelerated Rates via Randomized Integration Time
本文提出随机化积分时间的哈密顿下降算法RHGD，证明其在光滑强凸和弱凸优化中达到与Nesterov加速梯度下降相同的加速收敛速率，并在实验中展现竞争力。
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