Sample-Efficient Nonconvex Optimization Algorithms in Machine Learning and Reinforcement Learning

Machine learning and reinforcement learning have achieved tremendous success in solving problems in various real-world applications. Many modern learning problems boil down to a nonconvex optimization problem, where the objective function is the average or the expectation of some loss function over a finite or infinite dataset. Solving such nonconvex optimization problems, in general, can be NP-hard. Thus one often tackles such a problem through incremental steps based on the nature and the goal of the problem: finding a first-order stationary point, finding a second-order stationary point (or a local optimum), and finding a global optimum. With the size and complexity of the machine learning datasets rapidly increasing, it has become a fundamental challenge to design efficient and scalable machine learning algorithms that can improve the performance in terms of accuracy and save computational cost in terms of sample efficiency at the same time. Though many algorithms based on stochastic gradient descent have been developed and widely studied theoretically and empirically for nonconvex optimization, it has remained an open problem whether we can achieve the optimal sample complexity for finding a first-order stationary point and for finding local optima in nonconvex optimization. In this thesis, we start with the stochastic nested variance reduced gradient (SNVRG) algorithm, which is developed based on stochastic gradient descent methods and variance reduction techniques. We prove that SNVRG achieves the near-optimal convergence rate among its type for finding a first-order stationary point of a nonconvex function. We further build algorithms to efficiently find the local optimum of a nonconvex objective function by examining the curvature information at the stationary point found by SNVRG. With the ultimate goal of finding the global optimum in nonconvex optimization, we then provide a unified framework to analyze the global convergence of stochastic gradient Langevin dynamics-based algorithms for a nonconvex objective function. In the second part of this thesis, we generalize the aforementioned sample-efficient stochastic nonconvex optimization methods to reinforcement learning problems, including policy gradient, actor-critic, and Q-learning. For these problems, we propose novel algorithms and prove that they enjoy state-of-the-art theoretical guarantees on the sample complexity. The works presented in this thesis form an incomplete collection of the recent advances and developments of sample-efficient nonconvex optimization algorithms for both machine learning and reinforcement learning