Luke Snow

Luke Snow, Vikram Krishnamurthy, Brian M. Sadler

Consider a target being tracked by a cognitive radar network. If the target can intercept some radar network emissions, how can it detect coordination among the radars? By 'coordination' we mean that the radar emissions satisfy Pareto optimality with respect to multi-objective optimization over each radar's utility. This paper provides a novel multi-objective inverse reinforcement learning approach which allows for both detection of such Pareto optimal ('coordinating') behavior and subsequent reconstruction of each radar's utility function, given a finite dataset of radar network emissions. The method for accomplishing this is derived from the micro-economic setting of Revealed Preferences, and also applies to more general problems of inverse detection and learning of multi-objective optimizing systems.

Luke Snow, Vikram Krishnamurthy

This paper provides a finite-sample analysis of a passive stochastic gradient Langevin dynamics (PSGLD) algorithm. This algorithm is designed to achieve adaptive inverse reinforcement learning (IRL). Adaptive IRL aims to estimate the cost function of a forward learner performing a stochastic gradient algorithm (e.g., policy gradient reinforcement learning) by observing their estimates in real-time. The PSGLD algorithm is considered passive because it incorporates noisy gradients provided by an external stochastic gradient algorithm (forward learner), of which it has no control. The PSGLD algorithm acts as a randomized sampler to achieve adaptive IRL by reconstructing the forward learner's cost function nonparametrically from the stationary measure of a Langevin diffusion. This paper analyzes the non-asymptotic (finite-sample) performance; we provide explicit bounds on the 2-Wasserstein distance between PSGLD algorithm sample measure and the stationary measure encoding the cost function, and provide guarantees for a kernel density estimation scheme which reconstructs the cost function from empirical samples. Our analysis uses tools from the study of Markov diffusion operators. The derived bounds have both practical and theoretical significance. They provide finite-time guarantees for an adaptive IRL mechanism, and substantially generalize the analytical framework of a line of research in passive stochastic gradient algorithms.

Vikram Krishnamurthy, Luke Snow

Inverse reinforcement learning (IRL) recovers the loss function of a forward learner from its observed responses adaptive IRL aims to reconstruct the loss function of a forward learner by passively observing its gradients as it performs reinforcement learning (RL). This paper proposes a novel passive Langevin-based algorithm that achieves adaptive IRL. The key difficulty in adaptive IRL is that the required gradients in the passive algorithm are counterfactual, that is, they are conditioned on events of probability zero under the forward learner's trajectory. Therefore, naive Monte Carlo estimators are prohibitively inefficient, and kernel smoothing, though common, suffers from slow convergence. We overcome this by employing Malliavin calculus to efficiently estimate the required counterfactual gradients. We reformulate the counterfactual conditioning as a ratio of unconditioned expectations involving Malliavin quantities, thus recovering standard estimation rates. We derive the necessary Malliavin derivatives and their adjoint Skorohod integral formulations for a general Langevin structure, and provide a concrete algorithmic approach which exploits these for counterfactual gradient estimation.

Luke Snow, Vikram Krishnamurthy

We derive a minimax distributionally robust inverse reinforcement learning (IRL) algorithm to reconstruct the utility functions of a multi-agent sensing system. Specifically, we construct utility estimators which minimize the worst-case prediction error over a Wasserstein ambiguity set centered at noisy signal observations. We prove the equivalence between this robust estimation and a semi-infinite optimization reformulation, and we propose a consistent algorithm to compute solutions. We illustrate the efficacy of this robust IRL scheme in numerical studies to reconstruct the utility functions of a cognitive radar network from observed tracking signals.

Luke Snow, Vikram Krishnamurthy

A neural stochastic differential equation (SDE) is an SDE with drift and diffusion terms parametrized by neural networks. The training procedure for neural SDEs consists of optimizing the SDE vector field (neural network) parameters to minimize the expected value of an objective functional on infinite-dimensional path-space. Existing training techniques focus on methods to efficiently compute path-wise gradients of the objective functional with respect to these parameters, then pair this with Monte-Carlo simulation to estimate the gradient expectation. In this work we introduce a novel training technique which bypasses and improves upon this Monte-Carlo simulation; we extend results in the theory of Wiener space cubature to approximate the expected objective functional value by a weighted sum of functional evaluations of deterministic ODE solutions. Our main mathematical contribution enabling this approximation is an extension of cubature bounds to the setting of Lipschitz-nonlinear functionals acting on path-space. Our resulting constructive algorithm allows for more computationally efficient training along several lines. First, it circumvents Brownian motion simulation and enables the use of efficient parallel ODE solvers, thus decreasing the complexity of path-functional evaluation. Furthermore, and more surprisingly, we show that the number of paths required to achieve a given (expected loss functional oracle value) approximation can be reduced in this deterministic cubature regime. Specifically, we show that under reasonable regularity assumptions we can observe a O(1/n) convergence rate, where n is the number of path evaluations; in contrast with the standard O(1/sqrt(n)) rate of naive Monte-Carlo or the O(log(n)^d /n) rate of quasi-Monte-Carlo.

Vikram Krishnamurthy, Luke Snow

We study counterfactual stochastic optimization of conditional loss functionals under misspecified and noisy gradient information. The difficulty is that when the conditioning event has vanishing or zero probability, naive Monte Carlo estimators are prohibitively inefficient; kernel smoothing, though common, suffers from slow convergence. We propose a two-stage kernel-free methodology. First, we show using Malliavin calculus that the conditional loss functional of a diffusion process admits an exact representation as a Skorohod integral, yielding variance comparable to classical Monte-Carlo variance. Second, we establish that a weak derivative estimate of the conditional loss functional with respect to model parameters can be evaluated with constant variance, in contrast to the widely used score function method whose variance grows linearly in the sample path length. Together, these results yield an efficient framework for counterfactual conditional stochastic gradient algorithms in rare-event regimes.