KERDREUX Thomas

We review various characterizations of uniform convexity and smoothness on norm balls in finite-dimensional spaces and connect results stemming from the geometry of Banach spaces with \textit{scaling inequalities} used in analysing the convergence of optimization methods. In particular, we establish local versions of these conditions to provide sharper insights on a recent body of complexity results in learning theory, online learning, or offline optimization, which rely on the strong convexity of the feasible set. While they have a significant impact on complexity, these strong convexity or uniform convexity properties of feasible sets are not exploited as thoroughly as their functional counterparts, and this work is an effort to correct this imbalance. We conclude with some practical examples in optimization and machine learning where leveraging these conditions and localized assumptions lead to new complexity results.

Frank-Wolfe algorithms are methods for optimizing problems under constraints. They decompose a non-linear problem into a series of linear problems. This makes them the methods of choice for high-dimensional optimization and explains their use in many applied fields. Here we propose new Frank-Wolfe algorithms that converge more quickly to the solution of the optimization problem under certain fairly generic structural assumptions. In particular, we show that, unlike other types of algorithms, this family adapts to these assumptions without having to specify the parameters that control them.

The Frank-Wolfe algorithms, a.k.a. conditional gradient algorithms, solve constrained optimization problems. They break down a non-linear problem into a series of linear minimization on the constraint set. This contributes to their recent revival in many applied domains, in particular those involving large-scale optimization problems. In this dissertation, we design or analyze versions of the Frank-Wolfe algorithms. We notably show that, contrary to other types of algorithms, this family is adaptive to a broad spectrum of structural assumptions, without the need to know and specify the parameters controlling these hypotheses.