Safe-by-Repair: A Convex Optimization Approach for Repairing Unsafe Two-Level Lattice Neural Network Controllers

Abstract

In this paper, we consider the problem of repairing a data-trained Rectified Linear Unit (ReLU) Neural Network (NN) controller for a discrete-time, input-affine system. That is we assume that such a NN controller is available, and we seek to repair unsafe closed-loop behavior at one known "counterexample" state while simultaneously preserving a notion of safe closed-loop behavior on a separate, verified set of states. To this end, we further assume that the NN controller has a Two-Level Lattice (TLL) architecture, and exhibit an algorithm that can systematically and efficiently repair such an network. Facilitated by this choice, our approach uses the unique semantics of the TLL architecture to divide the repair problem into two significantly decoupled sub-problems, one of which is concerned with repairing the un-safe counterexample -and hence is essentially of local scope -and the other of which ensures that the repairs are realized in the output of the network -and hence is essentially of global scope. We then show that one set of sufficient conditions for solving each these sub-problems can be cast as a convex feasibility problem, and this allows us to formulate the TLL repair problem as two separate, but significantly decoupled, convex optimization problems. Finally, we evaluate our algorithm on a TLL controller on a simple dynamical model of a four-wheel-car.

Constrained Feedforward Neural Network Training via Reachability Analysis

“…These methods augment their networks with constrained optimization, but are unable to guarantee constraint satisfaction upon convergence of the training. Alternatively, [17] uses a systematic process of small changes to conform a "mostly-correct" network to constraints. However the method only works for networks with a Two-Level Lattice (TLL) architecture, requires an already-trained network, and again does not guarantee a provably safe solution.…”

“…We can then solve (17) for the Jacobian of v * with respect to any entry of the zonotope centers or generators by setting the right-hand side appropriately (see [22] for details). That is, we can now differentiate (14) with respect to the elements of c 1 , G 1 , c 2 , or G 2 .…”