The Frank-Wolfe algorithm is a popular method in structurally constrained machine learning applications, due to its fast per-iteration complexity. However, one major limitation of the method is a slow rate of convergence that is difficult to accelerate due to erratic, zig-zagging step directions, even asymptotically close to the solution. We view this as an artifact of discretization; that is to say, the Frank-Wolfe \emph{flow}, which is its trajectory at asymptotically small step sizes, does not zig-zag, and reducing discretization error will go hand-in-hand in producing a more stabilized method, with better convergence properties. We propose two improvements: a multistep Frank-Wolfe method that directly applies optimized higher-order discretization schemes; and an LMO-averaging scheme with reduced discretization error, and whose local convergence rate over general convex sets accelerates from a rate of $O(1/k)$ to up to $O(1/k^{3/2})$.

Learning continuous-time stochastic dynamics is a fundamental and essential problem in modeling sporadic time series, whose observations are irregular and sparse in both time and dimension. For a given system whose latent states and observed data are high-dimensional, it is generally impossible to derive a precise continuous-time stochastic process to describe the system behaviors. To solve the above problem, we apply Variational Bayesian method and propose a flexible continuous-time stochastic recurrent neural network named Variational Stochastic Differential Networks (VSDN), which embeds the complicated dynamics of the sporadic time series by neural Stochastic Differential Equations (SDE). VSDNs capture the stochastic dependency among latent states and observations by deep neural networks. We also incorporate two differential Evidence Lower Bounds to efficiently train the models. Through comprehensive experiments, we show that VSDNs outperform state-of-the-art continuous-time deep learning models and achieve remarkable performance on prediction and interpolation tasks for sporadic time series.