Philipp Braun

Philipp Braun, Timothy L. Molloy, Gal Barkai et al.

Solutions to pursuit-evasion and surveillance-evasion differential games are typically computed and expressed using open-loop representations, with the synthesis of feedback strategies significantly less common. We propose a numerical scheme for obtaining feedback strategies for the recently introduced prying-pedestrian surveillance-evasion differential game. The scheme involves computing feedback strategies as input-output maps approximated via neural networks trained using data obtained from open-loop representations of solutions. Simulations show the effectiveness of neural networks trained with an appropriate learning-loss function. Since optimal feedback strategies are discontinuous, as a second contribution, the potential loss/gain of individual players is subsequently studied for players using sample-and-hold feedback compared to continuous-time feedback.

Amir Ali Farzin, Philipp Braun, Iman Shames

While it is generally understood that zeroth-order (ZO) algorithms have an extra dependency on their number of iterations for any choice of parameters, compared to their first-order (FO) counterparts, in this work, we show that under several conditions, in expectation, ZO methods do not suffer from extra dimension dependencies in their convergence rates with respect to their FO counterparts. We look at optimisation algorithms from the dynamical systems perspective and analyse the conditions under which one can formulate the average of a ZO algorithm as the average of its FO counterpart with bounded perturbations with values dependent on design parameters. Then, using input-to-state stability properties, we show ZO methods follow the same decay rate as their FO counterparts and converge to a neighbourhood of the fixed point of FO methods, where its radius depends on the bound of the norm of the perturbations, which can be made arbitrarily small. The theoretical findings are illustrated via numerical examples.

Amir Ali Farzin, Yuen-Man Pun, Philipp Braun et al.

This work establishes a rigorous connection between stability properties of discrete-time algorithms (DTAs) and corresponding continuous-time dynamical systems derived through $ O(s^r) $-resolution ordinary differential equations (ODEs). We show that for discrete- and continuous-time dynamical systems satisfying a mild error assumption, exponential stability of a common equilibrium with respect to the continuous time dynamics implies exponential stability of the corresponding equilibrium for the discrete-time dynamics, provided that the step size is chosen sufficiently small. We extend this result to common compact invariant sets. We prove that if an equilibrium is exponentially stable for the $ O(s^r) $-resolution ODE, then it is also exponentially stable for the associated DTA. We apply this framework to analyse the limit point properties of several prominent optimisation algorithms, including Two-Timescale Gradient Descent--Ascent (TT-GDA), Generalised Extragradient (GEG), Two-Timescale Proximal Point (TT-PPM), Damped Newton (DN), Regularised Damped Newton (RDN), and the Jacobian method (JM), by studying their $ O(1) $- and $ O(s) $-resolution ODEs. We show that under a proper choice of hyperparameters, the set of saddle points of the objective function is a subset of the set of exponentially stable equilibria of GEG, TT-PPM, DN, and RDN. We relax the common Hessian invariance assumption through direct analysis of the resolution ODEs, broadening the applicability of our results. Numerical examples illustrate the theoretical findings.