Distributionally Robust Learning with Weakly Convex Losses: Convergence Rates and Finite-Sample Guarantees

We consider a distributionally robust stochastic optimization problem and formulate it as a stochastic two-level composition optimization problem with the use of the mean--semideviation risk measure. In this setting, we consider a single time-scale algorithm, involving two versions of the inner function value tracking: linearized tracking of a continuously differentiable loss function, and SPIDER tracking of a weakly convex loss function. We adopt the norm of the gradient of the Moreau envelope as our measure of stationarity and show that the sample complexity of $\mathcal{O}(\varepsilon^{-3})$ is possible in both cases, with only the constant larger in the second case. Finally, we demonstrate the performance of our algorithm with a robust learning example and a weakly convex, non-smooth regression example

Primary singularities of vector fields on surfaces

Unless another thing is stated one works in the C∞ category and manifolds have empty boundary. Let X and Y be vector fields on a manifold M. We say that Y tracks X if [Y, X] = fX for some continuous function f: M→ R. A subset K of the zero set Z(X) is an essential block for X if it is non-empty, compact, open in Z(X) and its Poincaré-Hopf index does not vanishes. One says that X is non-flat at p if its ∞-jet at p is non-trivial. A point p of Z(X) is called a primary singularity of X if any vector field defined about p and tracking X vanishes at p. This is our main result: consider an essential block K of a vector field X defined on a surface M. Assume that X is non-flat at every point of K. Then K contains a primary singularity of X. As a consequence, if M is a compact surface with non-zero characteristic and X is nowhere flat, then there exists a primary singularity of X

Deep Network Flow for Multi-Object Tracking

Data association problems are an important component of many computer vision applications, with multi-object tracking being one of the most prominent examples. A typical approach to data association involves finding a graph matching or network flow that minimizes a sum of pairwise association costs, which are often either hand-crafted or learned as linear functions of fixed features. In this work, we demonstrate that it is possible to learn features for network-flow-based data association via backpropagation, by expressing the optimum of a smoothed network flow problem as a differentiable function of the pairwise association costs. We apply this approach to multi-object tracking with a network flow formulation. Our experiments demonstrate that we are able to successfully learn all cost functions for the association problem in an end-to-end fashion, which outperform hand-crafted costs in all settings. The integration and combination of various sources of inputs becomes easy and the cost functions can be learned entirely from data, alleviating tedious hand-designing of costs.Comment: Accepted to CVPR 201