We show that parametric models trained by a stochastic gradient method (SGM) with few iterations have vanishing generalization error. We prove our results by arguing that SGM is algorithmically stable in the sense of Bousquet and Elisseeff. Our analysis only employs elementary tools from convex and continuous optimization. We derive stability bounds for both convex and non-convex optimization u...

We propose a new technique to perform unsupervised data classification (clustering) based on density induced metric and non-smooth optimization. Our goal is to automatically recognize multidimensional clusters of non-convex shape. We present a modification of the fuzzy c-means algorithm, which uses the data induced metric, defined with the help of Delaunay triangulation. We detail computation o...

Due to their axiomatic foundation and their favorable computational properties convex risk measures are becoming a powerful tool in financial risk management. In this paper we will review the fundamental structural concepts of convex risk measures within the framework of convex analysis. Then we will exploit it for deriving strong duality relations in a generic portfolio optimization context. I...

Semidefinite relaxation methods transform a variety of non-convex optimization problems into convex problems, but square the number of variables. We study a new type of convex relaxation for phase retrieval problems, called PhaseMax, that convexifies the underlying problem without lifting. The resulting problem formulation can be solved using standard convex optimization routines, while still w...

In this paper, we give some observation of applying modern optimization methods for functionals describing digital predistortion (DPD) signals with orthogonal frequency division multiplexing (OFDM) modulation. The considered family model is determined by the class cascade Wiener--Hammerstein models, which can be represented as a computational graph consisting various nonlinear blocks. To assess...

The non-negative matrix factorization (NMF) model with an additional orthogonality constraint on one of the factor matrices, called orthogonal NMF (ONMF), has been found a promising clustering and can outperform classical K-means. However, solving ONMF is challenging optimization problem because coupling non-negativity constraints introduces mixed combinatorial aspect into due to determination ...

Recently, the centroidal momentum dynamics has received substantial attention to plan dynamically consistent motions for robots with arms and legs in multi-contact scenarios. However, it is also non convex which renders any optimization approach difficult and timing is usually kept fixed in most trajectory optimization techniques to not introduce additional non convexities to the problem. But t...