  The problem of synthesizing stochastic explicit model predictive control
policies is known to be quickly intractable even for systems of modest
complexity when using classical control-theoretic methods. To address this
challenge, we present a scalable alternative called stochastic parametric
differentiable predictive control (SP-DPC) for unsupervised learning of neural
control policies governing stochastic linear systems subject to nonlinear
chance constraints. SP-DPC is formulated as a deterministic approximation to
the stochastic parametric constrained optimal control problem. This formulation
allows us to directly compute the policy gradients via automatic
differentiation of the problem's value function, evaluated over sampled
parameters and uncertainties. In particular, the computed expectation of the
SP-DPC problem's value function is backpropagated through the closed-loop
system rollouts parametrized by a known nominal system dynamics model and
neural control policy which allows for direct model-based policy optimization.
We provide theoretical probabilistic guarantees for policies learned via the
SP-DPC method on closed-loop stability and chance constraints satisfaction.
Furthermore, we demonstrate the computational efficiency and scalability of the
proposed policy optimization algorithm in three numerical examples, including
systems with a large number of states or subject to nonlinear constraints.
