Venkatraman Renganathan

Lekan Molu, Venkatraman Renganathan, Namhoon Cho

Backward reachable tubes (BRTs), computed via viscous Hamilton-Jacobi (HJ) partial differential equations, provide principled safety certificates for learned controllers and planning algorithms in trustworthy machine learning. However, classical grid-based HJ solvers require $O(M^n)$ memory footprint for $M$ grid points per $n$ state dimension. This renders them impractical for high-dimensional systems. We address this bottleneck with a local PDE linearization that enables a frozen-coefficient sampling scheme for the viscous HJ PDE: a generalized Cole-Hopf-type transformation reduces the nonlinear HJ equation to a sequence of linear heat equations whose solutions admit Gaussian heat-kernel representations. The value function and its spatial gradient are then recovered via roll-outs of Monte Carlo expectations on Gaussian densities, yielding a storage and grid-free algorithm that scales as $N\cdot n$ for $N$ samples. This decoupling of memory from dimensionality enables reachability analysis on problems where grid-based methods are simply impossible. We prove a finite-sample concentration bound $O(N^{-1/2})$ error and conditional linear convergence for the introduced Monte-Carlo Picard iterative scheme. Numerical validation on pursuit-evasion games demonstrates relative $L^2_{\text{rel}}$ errors of $0.03 - 0.20$, with $14-26$ second wall-clock times per 2D slice on a CPU. Crucially, the method scales with validation on up to (but not limited to) $n=45$-dimensional multi-agent games.

Adrien Banse, Venkatraman Renganathan, Raphaël M. Jungers

We extend the notion of Cantor-Kantorovich distance between Markov chains introduced by (Banse et al., 2023) in the context of Markov Decision Processes (MDPs). The proposed metric is well-defined and can be efficiently approximated given a finite horizon. Then, we provide numerical evidences that the latter metric can lead to interesting applications in the field of reinforcement learning. In particular, we show that it could be used for forecasting the performance of transfer learning algorithms.

Sleiman Safaoui, Benjamin J. Gravell, Venkatraman Renganathan et al.

We propose a two-phase risk-averse architecture for controlling stochastic nonlinear robotic systems. We present Risk-Averse Nonlinear Steering RRT* (RANS-RRT*) as an RRT* variant that incorporates nonlinear dynamics by solving a nonlinear program (NLP) and accounts for risk by approximating the state distribution and performing a distributionally robust (DR) collision check to promote safe planning. The generated plan is used as a reference for a low-level tracking controller. We demonstrate three controllers: finite horizon linear quadratic regulator (LQR) with linearized dynamics around the reference trajectory, LQR with robustness-promoting multiplicative noise terms, and a nonlinear model predictive control law (NMPC). We demonstrate the effectiveness of our algorithm using unicycle dynamics under heavy-tailed Laplace process noise in a cluttered environment.