A learning based method for obtaining feedback laws for nonlinear optimal control problems is proposed. The learning problem is posed such that the open loop value function is its optimal solution. This infinite dimensional, function space, problem, is approximated by a polynomial ansatz and its convergence is analyzed. An $\ell_1$ penalty term is employed, which combined with the proximal point method, allows to find sparse solutions for the learning problem. The approach requires multiple evaluations of the elements of the polynomial basis and of their derivatives. In order to do this efficiently a graph-theoretic algorithm is devised. Several examples underline that the proposed methodology provides a promising approach for mitigating the curse of dimensionality which would be involved in case the optimal feedback law was obtained by solving the Hamilton Jacobi Bellman equation.

Image classification has advanced significantly in recent years with the availability of large-scale image sets. However, fine-grained classification remains a major challenge due to the annotation cost of large numbers of fine-grained categories. This project shows that compelling classification performance can be achieved on such categories even without labeled training data. Given image and class embeddings, we learn a compatibility function such that matching embeddings are assigned a higher score than mismatching ones; zero-shot classification of an image proceeds by finding the label yielding the highest joint compatibility score. We use state-of-the-art image features and focus on different supervised attributes and unsupervised output embeddings either derived from hierarchies or learned from unlabeled text corpora. We establish a substantially improved state-of-the-art on the Animals with Attributes and Caltech-UCSD Birds datasets. Most encouragingly, we demonstrate that purely unsupervised output embeddings (learned from Wikipedia and improved with finegrained text) achieve compelling results, even outperforming the previous supervised state-of-the-art. By combining different output embeddings, we further improve results.