Acceleration in optimization.

Summary In many different fields such as optimization, the performance of a method is often characterized by its rate of convergence. However, accelerating an algorithm requires a lot of knowledge about the problem’s structure, and such improvement is done on a case-by-case basis. Many accelerated schemes have been developed in the past few decades and are massively used in practice. Despite their simplicity, such methods are usually based on purely algebraic arguments and often do not have an intuitive explanation. Recently, heavy work has been done to link accelerated algorithms with other fields of science, such as control theory or differential equations. However, these explanations often rely on complex arguments, usually using non-conventional tools in their analysis. One of the contributions of this thesis is a tentative explanation of optimization algorithms using the theory of integration methods, which has been well studied and enjoys a solid theoretical analysis. In particular, we will show that optimization scheme are special instance of integration methods when integrating the classical gradient flow. With standard arguments, we intuitively explain the origin of acceleration. On the other hand, accelerated methods usually need additional parameters in comparison with slower one, which are in most cases difficult to estimate. In addition, these schemes are designed for one particular setting and cannot be used elsewhere. In this thesis, we explore a new approach for accelerating optimization algorithms, which uses generic acceleration arguments. In numerical analysis, these tools have been developed for accelerating sequences of scalars or vectors, by building on the side another sequence with a better convergence rate. These methods can be combined with an iterative algorithm, speeding it up in most cases. In practice, extrapolation schemes are not widely used due to their lack of theoretical guarantees and their instability. We will extend these methods by regularizing them, allowing a deeper theoretical analysis and stronger convergence results, especially when applied to optimization methods.