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Safety assurance is uncompromisable for safety-critical environments with thepresence of drastic model uncertainties (e.g., distributional shift),especially with humans in the loop. However, incorporating uncertainty in safelearning will naturally lead to a bi-level problem, where at the lower levelthe (worst-case) safety constraint is evaluated within the uncertaintyambiguity set. In this paper, we present a tractable distributionally safereinforcement learning framework to enforce safety under a distributional shiftmeasured by a Wasserstein metric. To improve the tractability, we first useduality theory to transform the lower-level optimization frominfinite-dimensional probability space where distributional shift is measured,to a finite-dimensional parametric space. Moreover, by differentiable convexprogramming, the bi-level safe learning problem is further reduced to asingle-level one with two sequential computationally efficient modules: aconvex quadratic program to guarantee safety followed by a projected gradientascent to simultaneously find the worst-case uncertainty. This end-to-enddifferentiable framework with safety constraints, to the best of our knowledge,is the first tractable single-level solution to address distributional safety.We test our approach on first and second-order systems with varyingcomplexities and compare our results with the uncertainty-agnostic policies,where our approach demonstrates a significant improvement on safety guarantees.

Distributionally Safe Reinforcement Learning under Model Uncertainty: A Single-Level Approach by Differentiable Convex Programming