A Stochastic Smoothing Algorithm for Semidefinite Programming

d’Aspremont, Alexandre; Karoui, Noureddine El

doi:10.1137/12088728x

Cited by 6 publications

(3 citation statements)

References 31 publications

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“…Instead, we leverage a stochastic smoothing approximation borrowed from D’Aspremont and Karoui (2014), whereby one approximates the maximum eigenvalue of G ( θ ) as…”

Section: Solution Algorithmmentioning

confidence: 99%

“…where σ > 0 is a small noise parameter, and z i are independent and identically distributed samples from N ( 0 , I d ) . In particular, λ ¯ ( G false( θ false) + ε n z i z i normalT ) is unique with probability one, for any z i ~ N ( 0 , I d ) (see D’Aspremont and Karoui, 2014: Proposition 3.3 and Lemma 3.4). While the algorithm in D’Aspremont and Karoui (2014) leverages a finite-sample expectation of the randomized function above and its gradient, for our implementation, we simply leverage the uniqueness property induced by the use of the random rank-one perturbation, and compute the gradient and Hessian expressions using the eigenvalue decomposition of the matrix G ( θ ) + ( ε / d ) z z T , where z …”

Section: Solution Algorithmmentioning

confidence: 99%

“…In particular, λ ¯ ( G false( θ false) + ε n z i z i normalT ) is unique with probability one, for any z i ~ N ( 0 , I d ) (see D’Aspremont and Karoui, 2014: Proposition 3.3 and Lemma 3.4). While the algorithm in D’Aspremont and Karoui (2014) leverages a finite-sample expectation of the randomized function above and its gradient, for our implementation, we simply leverage the uniqueness property induced by the use of the random rank-one perturbation, and compute the gradient and Hessian expressions using the eigenvalue decomposition of the matrix G ( θ ) + ( ε / d ) z z T , where z is a single Gaussian vector sample. In addition, we employ early termination of the Newton iterations if the loop termination condition is met by the current iterate.…”

Section: Solution Algorithmmentioning

confidence: 99%

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Learning stabilizable nonlinear dynamics with contraction-based regularization

Singh

Richards

Sindhwani

et al. 2020

The International Journal of Robotics Research

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We propose a novel framework for learning stabilizable nonlinear dynamical systems for continuous control tasks in robotics. The key contribution is a control-theoretic regularizer for dynamics fitting rooted in the notion of stabilizability, a constraint which guarantees the existence of robust tracking controllers for arbitrary open-loop trajectories generated with the learned system. Leveraging tools from contraction theory and statistical learning in reproducing kernel Hilbert spaces, we formulate stabilizable dynamics learning as a functional optimization with a convex objective and bi-convex functional constraints. Under a mild structural assumption and relaxation of the functional constraints to sampling-based constraints, we derive the optimal solution with a modified representer theorem. Finally, we utilize random matrix feature approximations to reduce the dimensionality of the search parameters and formulate an iterative convex optimization algorithm that jointly fits the dynamics functions and searches for a certificate of stabilizability. We validate the proposed algorithm in simulation for a planar quadrotor, and on a quadrotor hardware testbed emulating planar dynamics. We verify, both in simulation and on hardware, significantly improved trajectory generation and tracking performance with the control-theoretic regularized model over models learned using traditional regression techniques, especially when learning from small supervised datasets. The results support the conjecture that the use of stabilizability constraints as a form of regularization can help prune the hypothesis space in a manner that is tailored to the downstream task of trajectory generation and feedback control. This produces models that are not only dramatically better conditioned, but also data efficient.

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“…Instead, we leverage a stochastic smoothing approximation borrowed from D’Aspremont and Karoui (2014), whereby one approximates the maximum eigenvalue of G ( θ ) as…”

Section: Solution Algorithmmentioning

confidence: 99%