Design of First-Order Optimization Algorithms via Sum-of-Squares Programming

In this paper, we propose a framework based on sum-of-squares programming to design iterative first-order optimization algorithms for smooth and strongly convex problems. Our starting point is to develop a polynomial matrix inequality as a sufficient condition for exponential convergence of the algorithm. The entries of this matrix are polynomial functions of the unknown parameters (exponential decay rate, stepsize, momentum coefficient, etc.). We then formulate a polynomial optimization, in which the objective is to optimize the exponential decay rate over the parameters of the algorithm. Finally, we use sum-of-squares programming as a tractable relaxation of the proposed polynomial optimization problem. We illustrate the utility of the proposed framework by designing a first-order algorithm that shares the same structure as Nesterov's accelerated gradient method

Automated Reachability Analysis of Neural Network-Controlled Systems via Adaptive Polytopes

Over-approximating the reachable sets of dynamical systems is a fundamental problem in safety verification and robust control synthesis. The representation of these sets is a key factor that affects the computational complexity and the approximation error. In this paper, we develop a new approach for over-approximating the reachable sets of neural network dynamical systems using adaptive template polytopes. We use the singular value decomposition of linear layers along with the shape of the activation functions to adapt the geometry of the polytopes at each time step to the geometry of the true reachable sets. We then propose a branch-and-bound method to compute accurate over-approximations of the reachable sets by the inferred templates. We illustrate the utility of the proposed approach in the reachability analysis of linear systems driven by neural network controllers

ReachLipBnB: A branch-and-bound method for reachability analysis of neural autonomous systems using Lipschitz bounds

We propose a novel Branch-and-Bound method for reachability analysis of neural networks in both open-loop and closed-loop settings. Our idea is to first compute accurate bounds on the Lipschitz constant of the neural network in certain directions of interest offline using a convex program. We then use these bounds to obtain an instantaneous but conservative polyhedral approximation of the reachable set using Lipschitz continuity arguments. To reduce conservatism, we incorporate our bounding algorithm within a branching strategy to decrease the over-approximation error within an arbitrary accuracy. We then extend our method to reachability analysis of control systems with neural network controllers. Finally, to capture the shape of the reachable sets as accurately as possible, we use sample trajectories to inform the directions of the reachable set over-approximations using Principal Component Analysis (PCA). We evaluate the performance of the proposed method in several open-loop and closed-loop settings

Effect of Dexamethasone Intraligamentary Injection on Post-Endodontic Pain in Patients with Symptomatic Irreversible Pulpitis: A Randomized Controlled Clinical Trial

Introduction: The aim of this randomized-controlled clinical trial was to assess the effect of intraligamentary (PDL) injection of dexamethasone on onset and severity of post-treatment pain in patients with symptomatic irreversible pulpitis. Methods and Materials: A total number of 60 volunteers were included according to the inclusion criteria and were assigned to three groups (n=20). After administration of local anesthesia and before treatment, group 1 (control) PDL injection was done with syringe containing empty cartridge, while in groups 2 and 3 the PDL injection was done with 0.2 mL of 2% lidocaine or dexamethasone (8 mg/2 mL), respectively. Immediately after endodontic treatment patients were requested to mark their level of pain on a visual analogue scale (VAS) during the next 48 h (on 6, 12, 24 and 48-h intervals). They were also asked to mention whether analgesics were taken and its dosage. Considering the 0-170 markings on the VAS ruler, the level of pain was scored as follows: score 0 (mild pain; 0-56), score 1 (moderate pain; 57-113) and score 3 (severe pain; 114-170). The data were analyzed using the Kruskal-Wallis and the Chi-square tests and the level of significance was set at 0.05. Results: After 6 and 12 h, group 1 and group 3 had the highest and lowest pain values, respectively (P<0.01 and P<0.001 for 6 and 12 h, respectively). However, after 24 and 48 h the difference in the pain was not significant between groups 1 and 2 (P<0.6) but group 3 had lower pain levels (P<0.01 and P<0.8 for 24 and 48 h, respectively). Conclusion: Pretreatment PDL injection of dexamethasone can significantly reduce the post-treatment endodontic pain in patients with symptomatic irreversible pulpitis.Keywords: Dexamethasone; Endodontic Treatment; Intraligamentary Injection; Post endodontic Pain; Symptomatic Irreversible Pulpiti

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

There has been an increasing interest in using neural networks in closed-loop control systems to improve performance and reduce computational costs for on-line implementation. However, providing safety and stability guarantees for these systems is challenging due to the nonlinear and compositional structure of neural networks. In this paper, we propose a novel forward reachability analysis method for the safety verification of linear time-varying systems with neural networks in feedback interconnection. Our technical approach relies on abstracting the nonlinear activation functions by quadratic constraints, which leads to an outer-approximation of forward reachable sets of the closed-loop system. We show that we can compute these approximate reachable sets using semidefinite programming. We illustrate our method in a quadrotor example, in which we first approximate a nonlinear model predictive controller via a deep neural network and then apply our analysis tool to certify finite-time reachability and constraint satisfaction of the closed-loop system