Convex interpolation and performance estimation of first-order methods for convex optimization


The goal of this thesis is to show how to derive in a completely automated way exact and global worst-case guarantees for first-order methods in convex optimization. To this end, we formulate a generic optimization problem looking for the worst-case scenarios. The worst-case computation problems, referred to as performance estimation problems (PEPs), are intrinsically infinite-dimensional optimization problems formulated over a given class of objective functions. To render those problems tractable, we develop (smooth and non-smooth) convex interpolation framework, which provides necessary and sufficient conditions to interpolate our objective functions. With this idea, we transform PEPs into solvable finite-dimensional semidefinite programs, from which one obtains worst-case guarantees and worst-case functions, along with the corresponding explicit proofs. PEPs already proved themselves very useful as a tool for developing convergence analyses of first-order optimization methods. Among others, PEPs allow obtaining exact guarantees for gradient methods, along with their inexact, projected, proximal, conditional, decentralized and accelerated versions.

27 Figures and Tables

Cite this paper

@inproceedings{Taylor2017ConvexIA, title={Convex interpolation and performance estimation of first-order methods for convex optimization}, author={Adrien B. Taylor}, year={2017} }