I will present three new results (no background in optimization will be assumed, all concepts will be defined and motivated): (i) the Cramer transform of the uniform measure on a convex body is a universal self-concordant barrier; (ii) projected gradient descent with Gaussian noise allows to sample from a log-concave measure in polynomial time; and (iii) Thompson sampling combined with a multi-scale exploration solves the Bayesian convex bandit problem. The unifying theme in these results is the interplay between concepts from convex geometry, learning and information theory.
Joint work with Ronen Eldan.