--In this article, we consider the infinite-horizon reach-avoid (RA) and stabilize-avoid (SA) zero-sum game problems for general nonlinear continuous-time systems, where the goal is to find the set of states that can be controlled to reach or stabilize to a target set, without violating constraints even under the worst-case disturbance. Based on the Hamilton-Jacobi reachability method, we address the RA problem by designing a new Lipschitz continuous RA value function, whose zero sublevel set exactly characterizes the RA set. We establish that the associated Bellman backup operator is contractive and that the RA value function is the unique viscosity solution of a Hamilton-Jacobi variational inequality. Finally, we develop a two-step framework for the SA problem by integrating our RA strategies with a recently proposed Robust Control Lyapunov-V alue Function, thereby ensuring both target reachability and long-term stability. We numerically verify our RA and SA frameworks on a 3D Dubins car system to demonstrate the efficacy of the proposed approach.

In this paper, we study a learning problem in which a forecaster only observes partial information. By properly rescaling the problem, we heuristically derive a limiting PDE on Wasserstein space which characterizes the asymptotic behavior of the regret of the forecaster. Using a verification type argument, we show that the problem of obtaining regret bounds and efficient algorithms can be tackled by finding appropriate smooth sub/supersolutions of this parabolic PDE.

The widespread application of modern machine learning has increased the need for robust statistical algorithms. This work studies one such fundamental statistical measure known as the Tukey depth. We study the problem in the continuum (population) limit. In particular, we derive the associated necessary conditions, which take the form of a first-order partial differential equation. We discuss the classical interpretation of this necessary condition as the viscosity solution of a Hamilton-Jacobi equation, but with a non-classical Hamiltonian with discontinuous dependence on the gradient at zero. We prove that this equation possesses a unique viscosity solution and that this solution always bounds the Tukey depth from below. In certain cases, we prove that the Tukey depth is equal to the viscosity solution, and we give some illustrations of standard numerical methods from the optimal control community which deal directly with the partial differential equation. We conclude by outlining several promising research directions both in terms of new numerical algorithms and theoretical challenges.

We study the approximation of backward stochastic differential equations (BSDEs for short) with a constraint on the gains process. We first discretize the constraint by applying a so-called facelift operator at times of a grid. We show that this discretely constrained BSDE converges to the continuously constrained one as the mesh grid converges to zero. We then focus on the approximation of the discretely constrained BSDE. For that we adopt a machine learning approach. We show that the facelift can be approximated by an optimization problem over a class of neural networks under constraints on the neural network and its derivative. We then derive an algorithm converging to the discretely constrained BSDE as the number of neurons goes to infinity. We end by numerical experiments. Mathematics Subject Classification (2010): 65C30, 65M75, 60H35, 93E20, 49L25.

For a finite function class we describe the large sample limit of the sequential Rademacher complexity in terms of the viscosity solution of a $G$-heat equation. In the language of Peng's sublinear expectation theory, the same quantity equals to the expected value of the largest order statistics of a multidimensional $G$-normal random variable. We illustrate this result by deriving upper and lower bounds for the asymptotic sequential Rademacher complexity.