Publication

Optimal Affine-Invariant Smooth Minimization Algorithms

2018
Journal Articles

Abstract

We formulate an affine invariant implementation of the accelerated first-order algorithm in [Nesterov, 1983]. Its complexity bound is proportional to an affine invariant regularity constant defined with respect to the Minkowski gauge of the feasible set. We extend these results to more general problems, optimizing Hölder smooth functions using p-uniformly convex prox terms, and derive an algorithm whose complexity better fits the geometry of the feasible set and adapts to both the best Hölder smoothness parameter and the best gradient Lipschitz constant. Finally, we detail matching complexity lower bounds when the feasible set is an ℓp ball. In this setting, our upper bounds on iteration complexity for the algorithm in [Nesterov, 1983] are thus optimal in terms of target precision, smoothness and problem dimension.

Official source

https://infoscience.epfl.ch/entities/publication/4c4c37d0-41c2-42f8-93c4-a2413eaad1c4

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2018
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/4c4c37d0-41c2-42f8-93c4-a2413eaad1c4

About this result

Related concepts (32)

Complexity class

In computational complexity theory, a complexity class is a set of computational problems "of related resource-based complexity". The two most commonly analyzed resources are time and memory. In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time or memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements.

Computational complexity

In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) and memory storage requirements. The complexity of a problem is the complexity of the best algorithms that allow solving the problem. The study of the complexity of explicitly given algorithms is called analysis of algorithms, while the study of the complexity of problems is called computational complexity theory.

Computational complexity theory

In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used.

Related publications (1)

Optimal Affine-Invariant Smooth Minimization Algorithms

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