A Diffusion Approximation Theory of Momentum Stochastic Gradient Descent in Nonconvex Optimization

Momentum stochastic gradient descent (MSGD) algorithm has been widely applied to many nonconvex optimization problems in machine learning (e.g., training deep neural networks, variational Bayesian inference, etc.). Despite its empirical success, there is still a lack of theoretical understanding of convergence properties of MSGD. To fill this gap, we propose to analyze the algorithmic behavior of MSGD by diffusion approximations for nonconvex optimization problems with strict saddle points and isolated local optima. Our study shows that the momentum helps escape from saddle points but hurts the convergence within the neighborhood of optima (if without the step size annealing or momentum annealing). Our theoretical discovery partially corroborates the empirical success of MSGD in training deep neural networks.

Receding Horizon Control Using Graph Search for Multi-Agent Trajectory Planning

<pre>It is hard to find the global optimum to general nonlinear, nonconvex optimization problems in reasonable time. This paper presents a method to transfer the receding horizon control approach, where nonlinear, nonconvex optimization problems are considered, into graph-search problems. Specifically, systems with symmetries are considered to transfer system dynamics into a finite state automaton. In contrast to traditional graph-search approaches where the search continues until the goal vertex is found, the transfer of a receding horizon control approach to graph-search problems presented in this paper allows to solve them in real-time. We proof that the solutions are recursively feasible by restricting the graph search to end in accepting states of the underlying finite state automaton. The approach is applied to trajectory planning for multiple networked and autonomous vehicles. We evaluate its effectiveness in simulation as well as in experiments in the Cyber-Physical Mobility Lab, an open source platform for networked and autonomous vehicles. We show real-time capable trajectory planning with collision avoidance in experiments on off-the-shelf hardware and code in MATLAB for two vehicles.</pre>

Receding Horizon Control Using Graph Search for Multi-Agent Trajectory Planning

<pre>It is hard to find the global optimum to general nonlinear, nonconvex optimization problems in reasonable time. This paper presents a method to transfer the receding horizon control approach, where nonlinear, nonconvex optimization problems are considered, into graph-search problems. Specifically, systems with symmetries are considered to transfer system dynamics into a finite state automaton. In contrast to traditional graph-search approaches where the search continues until the goal vertex is found, the transfer of a receding horizon control approach to graph-search problems presented in this paper allows to solve them in real-time. We proof that the solutions are recursively feasible by restricting the graph search to end in accepting states of the underlying finite state automaton. The approach is applied to trajectory planning for multiple networked and autonomous vehicles. We evaluate its effectiveness in simulation as well as in experiments in the Cyber-Physical Mobility Lab, an open source platform for networked and autonomous vehicles. We show real-time capable trajectory planning with collision avoidance in experiments on off-the-shelf hardware and code in MATLAB for two vehicles.</pre>