Reach-SDP: Reachability Analysis of Closed-Loop Systems with Neural Network Controllers via Semidefinite Programming

Hu, Haimin; Fazlyab, Mahyar; Morari, Manfred; Pappas, George J.

doi:10.1109/cdc42340.2020.9304296

Cited by 65 publications

(74 citation statements)

References 18 publications

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“…[20], [33] introduce conservatism by assuming the NN controller could output its extreme values at every state. Most recently, [21] formulated the problem as a SDP, called Reach-SDP. This work builds on both [20], [21] and makes the latter more scalable by reformulating the SDP as a linear program, introduces sources of uncertainty in the closed-loop dynamics, and shows further improvements by partitioning the input set.…”

Section: Related Workmentioning

confidence: 99%

“…In the special case of Lemma IV.2 where X t = B p (x t , ) for some p ∈ [1, ∞), the following closed-form expressions are equivalent to Eqs. (20) and (21), respectively,…”

Section: E Closed-form Solutionmentioning

confidence: 99%

“…The simplest closed-loop partitioner splits the input set into a uniform grid, as described in Algorithm 2. After splitting X 0 into Π nx k=0 r k cells (Line 2), each cell of X 0 is passed through a closed-loop propagator, CLProp (e.g., CL-CROWN (Algorithm 1) or Reach-SDP [21]). For each timestep, the returned estimate of the reachable set for all of X 0 is thus the union of all reachable sets for cells of the X 0 partition.…”

Section: A Closed-loop Partitionersmentioning

confidence: 99%

“…Several recent works [16]- [21] propose methods that compute forward reachable sets of NFLs. A key challenge is in maintaining computational efficiency while still providing tight bounds on the reachable sets.…”

Section: Introductionmentioning

confidence: 99%

“…Numerical experiments show that the new method provides 5× better accuracy in 150× less computation time compared to [21] and that the method can handle uncertainty sources, high dimensionality, and nonlinear dynamics via applications on quadrotor, 270-state, and polynomial systems. This article extends our conference paper [23] by providing an improved problem formulation, notation, and algorithmic details.…”

Section: Introductionmentioning

confidence: 99%

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Reachability Analysis of Neural Feedback Loops

Everett¹,

Habibi

How

2021

IEEE Access

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Neural Networks (NNs) can provide major empirical performance improvements for closedloop systems, but they also introduce challenges in formally analyzing those systems' safety properties. In particular, this work focuses on estimating the forward reachable set of neural feedback loops (closed-loop systems with NN controllers). Recent work provides bounds on these reachable sets, but the computationally tractable approaches yield overly conservative bounds (thus cannot be used to verify useful properties), and the methods that yield tighter bounds are too intensive for online computation. This work bridges the gap by formulating a convex optimization problem for the reachability analysis of closed-loop systems with NN controllers. While the solutions are less tight than previous (semidefinite program-based) methods, they are substantially faster to compute, and some of those computational time savings can be used to refine the bounds through new input set partitioning techniques, which is shown to dramatically reduce the tightness gap. The new framework is developed for systems with uncertainty (e.g., measurement and process noise) and nonlinearities (e.g., polynomial dynamics), and thus is shown to be applicable to real-world systems. To inform the design of an initial state set when only the target state set is known/specified, a novel algorithm for backward reachability analysis is also provided, which computes the set of states that are guaranteed to lead to the target set. The numerical experiments show that our approach (based on linear relaxations and partitioning) gives a 5× reduction in conservatism in 150× less computation time compared to the state-of-the-art. Furthermore, experiments on quadrotor, 270-state, and polynomial systems demonstrate the method's ability to handle uncertainty sources, high dimensionality, and nonlinear dynamics, respectively.

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Section: Related Workmentioning

confidence: 99%

“…In the special case of Lemma IV.2 where X t = B p (x t , ) for some p ∈ [1, ∞), the following closed-form expressions are equivalent to Eqs. (20) and (21), respectively,…”

Section: E Closed-form Solutionmentioning

confidence: 99%

Section: A Closed-loop Partitionersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Reachability Analysis of Neural Feedback Loops

Everett¹,

Habibi

How

2021

IEEE Access

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LinSyn: Synthesizing Tight Linear Bounds for Arbitrary Neural Network Activation Functions

Paulsen

Wang

2022

Lecture Notes in Computer Science

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The most scalable approaches to certifying neural network robustness depend on computing sound linear lower and upper bounds for the network’s activation functions. Current approaches are limited in that the linear bounds must be handcrafted by an expert, and can be sub-optimal, especially when the network’s architecture composes operations using, for example, multiplication such as in LSTMs and the recently popular Swish activation. The dependence on an expert prevents the application of robustness certification to developments in the state-of-the-art of activation functions, and furthermore the lack of tightness guarantees may give a false sense of insecurity about a particular model. To the best of our knowledge, we are the first to consider the problem of automatically synthesizing tight linear bounds for arbitrary n-dimensional activation functions. We propose the first fully automated method that achieves tight linear bounds while only leveraging the mathematical definition of the activation function itself. Our method leverages an efficient heuristic technique to synthesize bounds that are tight and usually sound, and then verifies the soundness (and adjusts the bounds if necessary) using the highly optimized branch-and-bound SMT solver, dReal. Even though our method depends on an SMT solver, we show that the runtime is reasonable in practice, and, compared with state of the art, our method often achieves 2-5X tighter final output bounds and more than quadruple certified robustness.

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