Learning Control Lyapunov Functions from Counterexamples and Demonstrations

We present a technique for learning control Lyapunov-like functions, which are used in turn to synthesize controllers for nonlinear dynamical systems that can stabilize the system, or satisfy specifications such as remaining inside a safe set, or eventually reaching a target set while remaining inside a safe set. The learning framework uses a demonstrator that implements a black-box, untrusted strategy presumed to solve the problem of interest, a learner that poses finitely many queries to the demonstrator to infer a candidate function, and a verifier that checks whether the current candidate is a valid control Lyapunov function. The overall learning framework is iterative, eliminating a set of candidates on each iteration using the counterexamples discovered by the verifier and the demonstrations over these counterexamples. We prove its convergence using ellipsoidal approximation techniques from convex optimization. We also implement this scheme using nonlinear MPC controllers to serve as demonstrators for a set of state and trajectory stabilization problems for nonlinear dynamical systems. We show how the verifier can be constructed efficiently using convex relaxations of the verification problem for polynomial systems to semi-definite programming (SDP) problem instances. Our approach is able to synthesize relatively simple polynomial control Lyapunov functions, and in that process replace the MPC using a guaranteed and computationally less expensive controller

Perturbed Message Passing for Constraint Satisfaction Problems

We introduce an efficient message passing scheme for solving Constraint Satisfaction Problems (CSPs), which uses stochastic perturbation of Belief Propagation (BP) and Survey Propagation (SP) messages to bypass decimation and directly produce a single satisfying assignment. Our first CSP solver, called Perturbed Blief Propagation, smoothly interpolates two well-known inference procedures; it starts as BP and ends as a Gibbs sampler, which produces a single sample from the set of solutions. Moreover we apply a similar perturbation scheme to SP to produce another CSP solver, Perturbed Survey Propagation. Experimental results on random and real-world CSPs show that Perturbed BP is often more successful and at the same time tens to hundreds of times more efficient than standard BP guided decimation. Perturbed BP also compares favorably with state-of-the-art SP-guided decimation, which has a computational complexity that generally scales exponentially worse than our method (wrt the cardinality of variable domains and constraints). Furthermore, our experiments with random satisfiability and coloring problems demonstrate that Perturbed SP can outperform SP-guided decimation, making it the best incomplete random CSP-solver in difficult regimes

Revisiting Algebra and Complexity of Inference in Graphical Models

This paper studies the form and complexity of inference in graphical models using the abstraction offered by algebraic structures. In particular, we broadly formalize inference problems in graphical models by viewing them as a sequence of operations based on commutative semigroups. We then study the computational complexity of inference by organizing various problems into an "inference hierarchy". When the underlying structure of an inference problem is a commutative semiring -- i.e. a combination of two commutative semigroups with the distributive law -- a message passing procedure called belief propagation can leverage this distributive law to perform polynomial-time inference for certain problems. After establishing the NP-hardness of inference in any commutative semiring, we investigate the relation between algebraic properties in this setting and further show that polynomial-time inference using distributive law does not (trivially) extend to inference problems that are expressed using more than two commutative semigroups. We then extend the algebraic treatment of message passing procedures to survey propagation, providing a novel perspective using a combination of two commutative semirings. This formulation generalizes the application of survey propagation to new settings

A Class of Control Certificates to Ensure Reach-While-Stay for Switched Systems

In this article, we consider the problem of synthesizing switching controllers for temporal properties through the composition of simple primitive reach-while-stay (RWS) properties. Reach-while-stay properties specify that the system states starting from an initial set I, must reach a goal (target) set G in finite time, while remaining inside a safe set S. Our approach synthesizes switched controllers that select between finitely many modes to satisfy the given RWS specification. To do so, we consider control certificates, which are Lyapunov-like functions that represent control strategies to achieve the desired specification. However, for RWS problems, a control Lyapunov-like function is often hard to synthesize in a simple polynomial form. Therefore, we combine control barrier and Lyapunov functions with an additional compatibility condition between them. Using this approach, the controller synthesis problem reduces to one of solving quantified nonlinear constrained problems that are handled using a combination of SMT solvers. The synthesis of controllers is demonstrated through a set of interesting numerical examples drawn from the related work, and compared with the state-of-the-art tool SCOTS. Our evaluation suggests that our approach is computationally feasible, and adds to the growing body of formal approaches to controller synthesis.Comment: In Proceedings SYNT 2017, arXiv:1711.1022

Learning Lyapunov (Potential) Functions from Counterexamples and Demonstrations

We present a technique for learning control Lyapunov (potential) functions, which are used in turn to synthesize controllers for nonlinear dynamical systems. The learning framework uses a demonstrator that implements a black-box, untrusted strategy presumed to solve the problem of interest, a learner that poses finitely many queries to the demonstrator to infer a candidate function and a verifier that checks whether the current candidate is a valid control Lyapunov function. The overall learning framework is iterative, eliminating a set of candidates on each iteration using the counterexamples discovered by the verifier and the demonstrations over these counterexamples. We prove its convergence using ellipsoidal approximation techniques from convex optimization. We also implement this scheme using nonlinear MPC controllers to serve as demonstrators for a set of state and trajectory stabilization problems for nonlinear dynamical systems. Our approach is able to synthesize relatively simple polynomial control Lyapunov functions, and in that process replace the MPC using a guaranteed and computationally less expensive controller

Counterexample Guided Synthesis of Switched Controllers for Reach-While-Stay Properties

We introduce a counter-example guided inductive synthesis (CEGIS) framework for synthesizing continuous-time switching controllers that guarantee reach while stay (RWS) properties of the closed loop system. The solution is based on synthesizing specially defined class of control Lyapunov functions (CLFs) for switched systems, that yield switching controllers with a guaranteed minimum dwell time in each mode. Next, we use a CEGIS-based approach to iteratively solve the resulting quantified exists-forall constraints, and find a CLF. We introduce relaxations to guarantee termination, as well as heuristics to increase convergence speed. Finally, we evaluate our approach on a set of benchmarks ranging from two to six state variables. Our evaluation includes a preliminary comparison with related tools. The proposed approach shows the promise of nonlinear SMT solvers for the synthesis of provably correct switching control laws

Equivariant Entity-Relationship Networks

The relational model is a ubiquitous representation of big-data, in part due to its extensive use in databases. In this paper, we propose the Equivariant Entity-Relationship Network (EERN), which is a Multilayer Perceptron equivariant to the symmetry transformations of the Entity-Relationship model. To this end, we identify the most expressive family of linear maps that are exactly equivariant to entity relationship symmetries, and further show that they subsume recently introduced equivariant maps for sets, exchangeable tensors, and graphs. The proposed feed-forward layer has linear complexity in the data and can be used for both inductive and transductive reasoning about relational databases, including database embedding, and the prediction of missing records. This provides a principled theoretical foundation for the application of deep learning to one of the most abundant forms of data. Empirically, EERN outperforms different variants of coupled matrix tensor factorization in both synthetic and real-data experiments

Equivariance Through Parameter-Sharing

We propose to study equivariance in deep neural networks through parameter symmetries. In particular, given a group $\mathcal{G}$ that acts discretely on the input and output of a standard neural network layer $\phi_{W}: \Re^{M} \to \Re^{N}$, we show that $\phi_{W}$ is equivariant with respect to $\mathcal{G}$-action iff $\mathcal{G}$ explains the symmetries of the network parameters $W$. Inspired by this observation, we then propose two parameter-sharing schemes to induce the desirable symmetry on $W$. Our procedures for tying the parameters achieve $\mathcal{G}$-equivariance and, under some conditions on the action of $\mathcal{G}$, they guarantee sensitivity to all other permutation groups outside $\mathcal{G}$.Comment: icml'1

Incidence Networks for Geometric Deep Learning

Sparse incidence tensors can represent a variety of structured data. For example, we may represent attributed graphs using their node-node, node-edge, or edge-edge incidence matrices. In higher dimensions, incidence tensors can represent simplicial complexes and polytopes. In this paper, we formalize incidence tensors, analyze their structure, and present the family of equivariant networks that operate on them. We show that any incidence tensor decomposes into invariant subsets. This decomposition, in turn, leads to a decomposition of the corresponding equivariant linear maps, for which we prove an efficient pooling-and-broadcasting implementation.Comment: Last revised August 10, 202

Training Restricted Boltzmann Machine by Perturbation

A new approach to maximum likelihood learning of discrete graphical models and RBM in particular is introduced. Our method, Perturb and Descend (PD) is inspired by two ideas (I) perturb and MAP method for sampling (II) learning by Contrastive Divergence minimization. In contrast to perturb and MAP, PD leverages training data to learn the models that do not allow efficient MAP estimation. During the learning, to produce a sample from the current model, we start from a training data and descend in the energy landscape of the "perturbed model", for a fixed number of steps, or until a local optima is reached. For RBM, this involves linear calculations and thresholding which can be very fast. Furthermore we show that the amount of perturbation is closely related to the temperature parameter and it can regularize the model by producing robust features resulting in sparse hidden layer activation