Machine learning has seen an explosive growth recently, driven mostly by breakthroughs in classification and generative models. However ML applications in decision making settings are much more limited, where data collection is much higher and ML models must be sufficiently robust and accurate to deal with unforeseen consequences and avoid worst case scenarios. In this talk we will introduce some classical results for online decision making in stochastic linear spaces, with applications to active learning, bandit/bayesian optimization and deep learning. Starting from a rigorous analysis of the noise propagation we can formulate provably robust (i.e. no-regret) algorithms, and then create variants that can scale to modern ML data regimes without sacrificing safety. And if time suffice, we will highlight how these approaches inspired a new wave of exploration techniques to enable reinforcement learning agents to solve extremely long horizon tasks.

We study ferromagnetic Ising models on finite graphs with an inhomogeneous external field. We show that the influence of boundary conditions on any given spin is maximised when the external field is identically 0. One corollary is that spin-spin correlation is maximised when the external field vanishes. In particular, the random field Ising model on Z^d, d ≥ 3, exhibits exponential decay of correlations in the entire high temperature regime of the pure Ising model. Another corollary is that the pure Ising model on Z^d, d ≥ 3, satisfies the conjectured strong spatial mixing property in the entire high temperature regime. Based on joint work with Jian Ding and Jian Song.