Many important problems in contemporary machine learning involve solving highly non- convex problems in sampling, optimization, or games. The absence of convexity poses significant challenges to convergence analysis of most training algorithms, and in some cases (such as min-max games) it is not even known whether common training algorithms converge or not. In this thesis, we aim to partially bridge the gap by

1. Proposing a new sampling framework to transform non-convex problems in to convex ones.
2. Characterizing the convergent sets of a wide family of popular algorithms for min-max optimization.
3. Devising provably convergent algorithms for finding mixed Nash Equilibria of infinite-dimensional bi-affine games.

Our theory has several important implications. First, we resolve a decade-old open problem in Bayesian learning via our non-convex sampling framework. Second, our algorithms for bi-affine games apply to the formidably difficult training of generative adversarial networks and robust reinforcement learning, and on both examples we demonstrate promising empirical performance. Finally, our results on min-max optimization lead to a series of negative results for state-of-the-art algorithms, suggesting that one requires fundamentally new tools to advance the theory.