Safe Approximate Dynamic Programming via Kernelized Lipschitz Estimation

Chakrabarty, Ankush; Jha, Devesh K.; Buzzard, Gregery T.; Wang, Yebin; Vamvoudakis, Kyriakos G.

doi:10.1109/tnnls.2020.2978805

Cited by 15 publications

(4 citation statements)

References 49 publications

(70 reference statements)

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“…Broadly, there are two ways for estimating Lipschitz constants of general nonlinear functions, either sampling-based as in [16] and [17], or using optimization techniques [7], [18]. A naive approach is to calculate the product of the norm of the weights of each individual layer.…”

Section: A Related Workmentioning

confidence: 99%

Chordal Sparsity for Lipschitz Constant Estimation of Deep Neural Networks

Xue¹,

Lindemann²,

Robey³

et al. 2022

Preprint

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Lipschitz constants of neural networks allow for guarantees of robustness in image classification, safety in controller design, and generalizability beyond the training data. As calculating Lipschitz constants is NP-hard, techniques for estimating Lipschitz constants must navigate the trade-off between scalability and accuracy. In this work, we significantly push the scalability frontier of a semidefinite programming technique known as LipSDP while achieving zero accuracy loss. We first show that LipSDP has chordal sparsity, which allows us to derive a chordally sparse formulation that we call Chordal-LipSDP. The key benefit is that the main computational bottleneck of LipSDP, a large semidefinite constraint, is now decomposed into an equivalent collection of smaller onesallowing Chordal-LipSDP to outperform LipSDP particularly as the network depth grows. Moreover, our formulation uses a tunable sparsity parameter that enables one to gain tighter estimates without incurring a significant computational cost. We illustrate the scalability of our approach through extensive numerical experiments.

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Section: A Related Workmentioning

confidence: 99%

Chordal Sparsity for Lipschitz Constant Estimation of Deep Neural Networks

Xue¹,

Lindemann²,

Robey³

et al. 2022

Preprint

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“…For basis functions whose Lipschitz constant are not analytically computable, one could use the sampling-based kernelized learning method discussed in [27,Section III] to obtain an overestimate of the Lipschitz constant with high probability. With the estimate Lφ , we can use LMIs to obtain a redesigned gain L. The following theorem encapsulates these redesign conditions.…”

Section: Observer Gain Redesignmentioning

confidence: 99%

“…then the redesigned observer (27) with gain L " P ´1K makes the error dynamics (3) L-ISS with respect to e p , with an improvement of the convergence rate compared with (11), quantified by a Lyapunov function decrease bound as…”

Section: Observer Gain Redesignmentioning

confidence: 99%

“…With the learned nonlinearity in a neural approximator/basis expansion form p J 8 ψpqq, we select Q " r´5, 5s 2 and use the kernelized Lipschitz estimation method in [27] to compute Lφ " 24.537, with which we can solve (29) to get L " " 39.3221 958.7488 ı J . As seen from Figure 2, the redesign results in further reduction of state estimation error.…”

Section: Numerical Examplementioning

confidence: 99%

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Safe Learning-based Observers for Unknown Nonlinear Systems using Bayesian Optimization

Chakrabarty¹,

Benosman²

2020

Preprint

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Data generated from dynamical systems with unknown dynamics enable the learning of state observers that are: robust to modeling error, computationally tractable to design, and capable of operating with guaranteed performance. In this paper, a modular design methodology is formulated, that consists of three design phases: (i) an initial robust observer design that enables one to learn the dynamics without allowing the state estimation error to diverge (hence, safe); (ii) a learning phase wherein the unmodeled components are estimated using Bayesian optimization and Gaussian processes; and, (iii) a re-design phase that leverages the learned dynamics to improve convergence rate of the state estimation error. The potential of our proposed learning-based observer is demonstrated on a benchmark nonlinear system. Additionally, certificates of guaranteed estimation performance are provided.

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