Research Talks at the Zuse Institute Berlin

The Frank-Wolfe method is a classical first-order algorithm for constrained optimization that avoids projections by relying on linear minimization oracles. While extensively studied over polytopes, its behavior over strongly convex domains remains less understood, especially near the boundary. In this talk, we will explore the challenges of optimizing over such sets, with a particular focus on the difficulty of finding optima located on the boundary. Using the simple example of projections onto the L2-ball, we will provide geometric insights into convergence behavior of Frank-Wolfe. This example illustrates fundamental limitations of the method in boundary regimes and motivates further analysis of step size strategies and alternative update directions.

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Riemannian optimization refers to the optimization of functions defined over Riemannian manifolds. Such problems arise when the constraints of Euclidean optimization problems can be viewed as Riemannian manifolds, such as the symmetric positive-definite cone, the sphere, or the set of orthogonal linear layers for a neural network. This Riemannian formulation enables us to leverage the geometric structure of such problems by viewing them as unconstrained problems on a manifold. The convergence rates of Riemannian optimization algorithms often rely on geometric quantities depending on the sectional curvature and the distance between iterates and an optimizer. Numerous previous works bound the latter only by assumption, resulting in incomplete analysis and unquantified rates. In this talk, I will discuss how to remove this limitation for multiple algorithms and as a result quantify their rates of convergence.