Acceleration in optimization.

Summary In many fields, such as optimization, the performance of a method is often characterized by its convergence rate. However, accelerating an algorithm requires some knowledge of the problem structure and such improvements are the result of a case-by-case study. Many acceleration techniques have been developed in the last decades and are now massively used. Despite their simplicity, these methods are often based on purely algebraic arguments and generally lack intuitive explanations. Recently, a lot of work has been done to make connections between accelerated algorithms and other scientific domains, such as control theory or differential equations. However, these explanations are often based on complex arguments and most of them use unconventional tools in their analysis. One of the contributions of this thesis is an attempt to explain accelerated algorithms using the theory of integration methods, which has been extensively studied and enjoys a strong theoretical analysis. In particular, we show that optimization methods are in fact instances of integration methods, when integrating the gradient flow equation. With standard arguments, we intuitively explain the origin of the acceleration. On the other hand, accelerated methods need additional parameters, compared to slower methods, which are usually difficult to estimate. Moreover, these schemes are built for a particular configuration and cannot be used elsewhere. Here, we explore another approach to accelerating optimization algorithms, which uses generic acceleration arguments. In numerical analysis, these tools have been developed to accelerate sequences of scalars or vectors, by building in parallel another sequence with a better convergence rate. These methods can be combined with an iterative algorithm, accelerating it in most cases. In practice, these extrapolation methods are not so much used, notably because of their lack of convergence guarantees and their instability. We extend these methods by regularizing them, which will allow a deeper theoretical analysis and stronger convergence results, especially when applied to optimization methods.