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1. Catalyst Acceleration for First-order Convex Optimization: from Theory to Practice

   Lin, Hongzhou; Mairal, Julien; Harchaoui, Zaid (April 2018, Journal of machine learning research)

   We introduce a generic scheme for accelerating gradient-based optimization methods in the sense of Nesterov. The approach, called Catalyst, builds upon the inexact accelerated proximal point algorithm for minimizing a convex objective function, and consists of approximately solving a sequence of well-chosen auxiliary problems, leading to faster convergence. One of the keys to achieve acceleration in theory and in practice is to solve these sub-problems with appropriate accuracy by using the right stopping criterion and the right warm-start strategy. We give practical guidelines to use Catalyst and present a comprehensive analysis of its global complexity. We show that Catalyst applies to a large class of algorithms, including gradient descent, block coordinate descent, incremental algorithms such as SAG, SAGA, SDCA, SVRG, MISO/Finito, and their proximal variants. For all of these methods, we establish faster rates using the Catalyst acceleration, for strongly convex and non-strongly convex objectives. We conclude with extensive experiments showing that acceleration is useful in practice, especially for ill-conditioned problems. 

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2. Level constrained first order methods for function constrained optimization

   https://doi.org/10.1007/s10107-024-02057-4  

   Boob, Digvijay; Deng, Qi; Lan, Guanghui (March 2024, Mathematical Programming)

   Abstract We present a new feasible proximal gradient method for constrained optimization where both the objective and constraint functions are given by summation of a smooth, possibly nonconvex function and a convex simple function. The algorithm converts the original problem into a sequence of convex subproblems. Formulating those subproblems requires the evaluation of at most one gradient-value of the original objective and constraint functions. Either exact or approximate subproblems solutions can be computed efficiently in many cases. An important feature of the algorithm is the constraint level parameter. By carefully increasing this level for each subproblem, we provide a simple solution to overcome the challenge of bounding the Lagrangian multipliers and show that the algorithm follows a strictly feasible solution path till convergence to the stationary point. We develop a simple, proximal gradient descent type analysis, showing that the complexity bound of this new algorithm is comparable to gradient descent for the unconstrained setting which is new in the literature. Exploiting this new design and analysis technique, we extend our algorithms to some more challenging constrained optimization problems where (1) the objective is a stochastic or finite-sum function, and (2) structured nonsmooth functions replace smooth components of both objective and constraint functions. Complexity results for these problems also seem to be new in the literature. Finally, our method can also be applied to convex function constrained problems where we show complexities similar to the proximal gradient method. 

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3. Randomized sketch descent methods for non-separable linearly constrained optimization

   https://doi.org/10.1093/imanum/draa018  

   Necoara, Ion; Takáč, Martin (May 2020, IMA Journal of Numerical Analysis)

   null (Ed.)

   Abstract 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 develop new sketch descent methods for solving general smooth linearly constrained problems, in particular, random sketch descent (RSD) and accelerated random sketch descent (A-RSD) methods. To our knowledge, this is the first convergence analysis of RSD algorithms for optimization problems with multiple non-separable linear constraints. For the general case, when the objective function is smooth and non-convex, we prove for the non-accelerated variant sublinear rate in expectation for an appropriate optimality measure. In the smooth convex case, we derive for both algorithms, non-accelerated and A-RSD, sublinear convergence rates in the expected values of the objective function. Additionally, if the objective function satisfies a strong convexity type condition, both algorithms converge linearly in expectation. In special cases, where complexity bounds are known for some particular sketching algorithms, such as coordinate descent methods for optimization problems with a single linear coupled constraint, our theory recovers the best known bounds. Finally, we present several numerical examples to illustrate the performances of our new algorithms. 

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4. On the Exponential Convergence Rate of Proximal Gradient Flow Algorithms

   https://doi.org/10.1109/CDC.2018.8618968  

   Hassan-Moghaddam, Sepideh; Jovanovic, Mihailo R. (December 2018, 2018 IEEE Conference on Decision and Control (CDC))

   Many modern large-scale and distributed optimization problems can be cast into a form in which the objective function is a sum of a smooth term and a nonsmooth regularizer. Such problems can be solved via a proximal gradient method which generalizes standard gradient descent to a nonsmooth setup. In this paper, we leverage the tools from control theory to study global convergence of proximal gradient flow algorithms. We utilize the fact that the proximal gradient algorithm can be interpreted as a variable-metric gradient method on the forward-backward envelope. This continuously differentiable function can be obtained from the augmented Lagrangian associated with the original nonsmooth problem and it enjoys a number of favorable properties. We prove that global exponential convergence can be achieved even in the absence of strong convexity. Moreover, for in-network optimization problems, we provide a distributed implementation of the gradient flow dynamics based on the proximal augmented Lagrangian and prove global exponential stability for strongly convex problems. 

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5. Riemannian Stochastic Proximal Gradient Methods for Nonsmooth Optimization over the Stiefel Manifold

   Wang, B.; Ma, S.; Xue, L. (January 2022, Journal of machine learning research)

   Riemannian optimization has drawn a lot of attention due to its wide applications in practice. Riemannian stochastic first-order algorithms have been studied in the literature to solve large-scale machine learning problems over Riemannian manifolds. However, most of the existing Riemannian stochastic algorithms require the objective function to be differentiable, and they do not apply to the case where the objective function is nonsmooth. In this paper, we present two Riemannian stochastic proximal gradient methods for minimizing nonsmooth function over the Stiefel manifold. The two methods, named R-ProxSGD and R-ProxSPB, are generalizations of proximal SGD and proximal SpiderBoost in Euclidean setting to the Riemannian setting. Analysis on the incremental first-order oracle (IFO) complexity of the proposed algorithms is provided. Specifically, the R-ProxSPB algorithm finds an ϵ -stationary point with O(ϵ−3) IFOs in the online case, and O(n+n‾√ϵ−2) IFOs in the finite-sum case with n being the number of summands in the objective. Experimental results on online sparse PCA and robust low-rank matrix completion show that our proposed methods significantly outperform the existing methods that use Riemannian subgradient information. 

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