Learning the Structure of Deep Sparse Graphical Models

Deep belief networks are a powerful way to model complex probability distributions. However, learning the structure of a belief network, particularly one with hidden units, is difficult. The Indian buffet process has been used as a nonparametric Bayesian prior on the directed structure of a belief network with a single infinitely wide hidden layer. In this paper, we introduce the cascading Indian buffet process (CIBP), which provides a nonparametric prior on the structure of a layered, directed belief network that is unbounded in both depth and width, yet allows tractable inference. We use the CIBP prior with the nonlinear Gaussian belief network so each unit can additionally vary its behavior between discrete and continuous representations. We provide Markov chain Monte Carlo algorithms for inference in these belief networks and explore the structures learned on several image data sets.Comment: 20 pages, 6 figures, AISTATS 2010, Revise

Learning Nonlinear Loop Invariants with Gated Continuous Logic Networks (Extended Version)

Verifying real-world programs often requires inferring loop invariants with nonlinear constraints. This is especially true in programs that perform many numerical operations, such as control systems for avionics or industrial plants. Recently, data-driven methods for loop invariant inference have shown promise, especially on linear invariants. However, applying data-driven inference to nonlinear loop invariants is challenging due to the large numbers of and magnitudes of high-order terms, the potential for overfitting on a small number of samples, and the large space of possible inequality bounds. In this paper, we introduce a new neural architecture for general SMT learning, the Gated Continuous Logic Network (G-CLN), and apply it to nonlinear loop invariant learning. G-CLNs extend the Continuous Logic Network (CLN) architecture with gating units and dropout, which allow the model to robustly learn general invariants over large numbers of terms. To address overfitting that arises from finite program sampling, we introduce fractional sampling---a sound relaxation of loop semantics to continuous functions that facilitates unbounded sampling on real domain. We additionally design a new CLN activation function, the Piecewise Biased Quadratic Unit (PBQU), for naturally learning tight inequality bounds. We incorporate these methods into a nonlinear loop invariant inference system that can learn general nonlinear loop invariants. We evaluate our system on a benchmark of nonlinear loop invariants and show it solves 26 out of 27 problems, 3 more than prior work, with an average runtime of 53.3 seconds. We further demonstrate the generic learning ability of G-CLNs by solving all 124 problems in the linear Code2Inv benchmark. We also perform a quantitative stability evaluation and show G-CLNs have a convergence rate of $97.5\%$ on quadratic problems, a $39.2\%$ improvement over CLN models

Stability of Neural Ordinary Differential Equations with Power Nonlinearities

The article presents a study of solutions of ODEs system with a specialnonlinear part, which is a continuous analogue of an arbitrary recurrent neural network(neural ODEs). As a nonlinear part of the mentioned system of differential equations, weused sums of piecewise continuous functions, where each term is a power function. (Theseare activation functions.) The use of power activation functions (PAF) in neural networksis a generalization of well-known the rectified linear units (ReLU). In the present timeReLU are commonly used to increase the depth of trained of a neural network. Therefore,the introduction of PAF into neural networks significantly expands the possibilities ofReLU. Note that the purpose of introducing power activation functions is that theyallow one to obtain verifiable Lyapunov stability conditions for solutions of the systemdifferential equations simulating the corresponding dynamic processes. In turn, Lyapunovstability is one of the guarantees of the adequacy of the neural network model for theprocess under study. In addition, from the global stability (or at least the boundedness)of continuous analog solutions it follows that learning process of the corresponding neuralnetwork will not diverge for any training sample.The article presents a study of solutions of ODEs system with a special nonlinear part, which is a continuous analogue of an arbitrary recurrent neural network (neural ODEs). As a nonlinear part of the mentioned system of differential equations, we used sums of piecewise continuous functions, where each term is a power function. (These are activation functions.) The use of power activation functions (PAF) in neural networks is a generalization of well-known the rectified linear units (ReLU). In the present time ReLU are commonly used to increase the depth of trained of a neural network. Therefore, the introduction of PAF into neural networks significantly expands the possibilities ofReLU. Note that the purpose of introducing power activation functions is that they allow one to obtain verifiable Lyapunov stability conditions for solutions of the system differential equations simulating the corresponding dynamic processes. In turn, Lyapunov stability is one of the guarantees of the adequacy of the neural network model for the process under study. In addition, from the global stability (or at least the boundedness) of continuous analog solutions it follows that learning process of the corresponding neural network will not diverge for any training sample

Neural Network-based Model Predictive Control with Input-to-State Stability

Learning-based controllers, and especially learning-based model predictive controllers, have been used for a number of different applications with great success. In spite of good performance, a lot of these cases lack stability guarantees. In this paper we consider a scenario where the dynamics of a nonlinear system are unknown, but where input and output data are available. A prediction model is learned from data using a neural network, which in turn is used in a nonlinear model predictive control scheme. The closed-loop system is shown to be input-to-state stable with respect to the prediction error of the learned model. The approach is tested and verified in simulations, by employing the controller to a benchmark system, namely a continuous stirred tank reactor plant. Simulations show that the proposed controller successfully drives the system from random initial conditions, to a reference equilibrium point, even in the presence of noise. The results also verify the theoretical stability result.acceptedVersio

Control-Consistent Learning of Koopman Embedding Models

A learning method is proposed for Koopman operator-based models with the goal of improving closed-loop control behavior. A neural network-based approach is used to discover a space of observables in which nonlinear dynamics is linearly embedded. While accurate state predictions can be expected with the use of such complex, high-dimensional state-to-observable maps, undesirable side-effects may be introduced when the model is deployed in a closed-loop environment. This is because of modeling or residual error in the linear embedding process, which can manifest itself in a different manner compared to the state prediction. To this end, a technique is proposed to refine the originally trained model with the goal of improving the closed-loop behavior of the model while retaining the state-prediction accuracy obtained in the initial learning. Finally, a simple data sampling strategy is proposed to use input signals deterministically sampled from continuous functions, leading to additional improvements in the controller performance for nonlinear dynamical systems. Several numerical examples are provided to show the efficacy of the proposed method

Neural Stochastic Contraction Metrics for Learning-based Control and Estimation

We present Neural Stochastic Contraction Metrics (NSCM), a new design framework for provably-stable learning-based control and estimation for a class of stochastic nonlinear systems. It uses a spectrally-normalized deep neural network to construct a contraction metric and its differential Lyapunov function, sampled via simplified convex optimization in the stochastic setting. Spectral normalization constrains the state-derivatives of the metric to be Lipschitz continuous, thereby ensuring exponential boundedness of the mean squared distance of system trajectories under stochastic disturbances. The trained NSCM model allows autonomous systems to approximate optimal stable control and estimation policies in real-time, and outperforms existing nonlinear control and estimation techniques including the state-dependent Riccati equation, iterative LQR, EKF, and the deterministic NCM, as shown in simulation results