This section proposes additional regularization methods for learning stable deep Markov models. The most direct approach is to include the stability conditions as extra penalties in the DMM loss function.

In this paper, we provide sufficient conditions of DMM's stochastic stability as defined in the context of dynamical systems

This section proposes additional regularization methods for learning stable deep Markov models. The most direct approach is to include the stability conditions as extra penalties in the DMM loss function.

In this paper, we provide sufficient conditions of DMM's stochastic stability as defined in the context of dynamical systems

Artificial intelligence (AI) models are becoming key components in an autonomous vehicle (AV), especially in handling complicated perception tasks. However, closing the loop through AI-based feedback may pose significant risks on reliability of autonomous driving due to very limited understanding about the mechanism of AI-driven perception processes. To overcome it, this paper aims to develop tools for modeling, analysis, and synthesis for a class of AI-based AV; in particular, their closed-loop properties, e.g., stability, robustness, and performance, are rigorously studied in the statistical sense. First, we provide a novel modeling means for the AI-driven perception processes by looking at their error characteristics. Specifically, three fundamental AI-induced perception uncertainties are recognized and modeled by Markov chains, Gaussian processes, and bounded disturbances, respectively. By means of that, the closed-loop stochastic stability (SS) is established in the sense of mean square, and then, an SS control synthesis method is presented within the framework of linear matrix inequalities (LMIs). Besides the SS properties, the robustness and performance of AI-based AVs are discussed in terms of a stochastic guaranteed cost, and criteria are given to test the robustness level of an AV when in the presence of AI-induced uncertainties. Furthermore, the stochastic optimal guaranteed cost control is investigated, and an efficient design procedure is developed innovatively based on LMI techniques and convex optimization. Finally, to illustrate the effectiveness, the developed results are applied to an example of car following control, along with extensive simulation.

Deep Markov models (DMM) are generative models which are scalable and expressive generalization of Markov models for representation, learning, and inference problems. However, the fundamental stochastic stability guarantees of such models have not been thoroughly investigated. In this paper, we present a novel stability analysis method and provide sufficient conditions of DMM's stochastic stability. The proposed stability analysis is based on the contraction of probabilistic maps modeled by deep neural networks. We make connections between the spectral properties of neural network's weights and different types of used activation function on the stability and overall dynamic behavior of DMMs with Gaussian distributions.

Recent findings by Cohen et al., 2021, demonstrate that when training neural networks with full-batch gradient descent at a step size of $\eta$, the sharpness--defined as the largest eigenvalue of the full batch Hessian--consistently stabilizes at $2/\eta$. These results have significant implications for convergence and generalization. Unfortunately, this was observed not to be the case for mini-batch stochastic gradient descent (SGD), thus limiting the broader applicability of these findings. We show that SGD trains in a different regime we call Edge of Stochastic Stability. In this regime, what hovers at $2/\eta$ is, instead, the average over the batches of the largest eigenvalue of the Hessian of the mini batch (MiniBS) loss--which is always bigger than the sharpness. This implies that the sharpness is generally lower when training with smaller batches or bigger learning rate, providing a basis for the observed implicit regularization effect of SGD towards flatter minima and a number of well established empirical phenomena. Additionally, we quantify the gap between the MiniBS and the sharpness, further characterizing this distinct training regime.

We consider two nonlinear state estimation problems in a setting where an extended Kalman filter receives measurements from two sets of sensors via two channels (2C). In the stochastic-2C problem, the channels drop measurements stochastically, whereas in 2C scheduling, the estimator chooses when to read each channel. In the first problem, we generalize linear-case 2C analysis to obtain -- for a given pair of channel arrival rates -- boundedness conditions for the trace of the error covariance, as well as a worst-case upper bound. For scheduling, an optimization problem is solved to find arrival rates that balance low channel usage with low trace bounds, and channels are read deterministically with the expected periods corresponding to these arrival rates. We validate both solutions in simulations for linear and nonlinear dynamics; as well as in a real experiment with an underwater robot whose position is being intermittently found in a UAV camera image.

We consider the problem of synthesizing resilient and stochastically stable strategies for systems of cooperating agents striving to minimize the expected time between consecutive visits to selected locations in a known environment. A strategy profile is resilient if it retains its functionality even if some of the agents fail, and stochastically stable if the visiting time variance is small. We design a novel specification language for objectives involving resilience and stochastic stability, and we show how to efficiently compute strategy profiles (for both autonomous and coordinated agents) optimizing these objectives. Our experiments show that our strategy synthesis algorithm can construct highly non-trivial and efficient strategy profiles for environments with general topology.

A specific type of neural network, the Restricted Boltzmann Machine (RBM), is implemented for classification and feature detection in machine learning. RBM is characterized by separate layers of visible and hidden units, which are able to learn efficiently a generative model of the observed data. We study a "hybrid" version of RBM's, in which hidden units are analog and visible units are binary, and we show that thermodynamics of visible units are equivalent to those of a Hopfield network, in which the N visible units are the neurons and the P hidden units are the learned patterns. We apply the method of stochastic stability to derive the thermodynamics of the model, by considering a formal extension of this technique to the case of multiple sets of stored patterns, which may act as a benchmark for the study of correlated sets. Our results imply that simulating the dynamics of a Hopfield network, requiring the update of N neurons and the storage of N(N-1)/2 synapses, can be accomplished by a hybrid Boltzmann Machine, requiring the update of N+P neurons but the storage of only NP synapses. In addition, the well known glass transition of the Hopfield network has a counterpart in the Boltzmann Machine: It corresponds to an optimum criterion for selecting the relative sizes of the hidden and visible layers, resolving the trade-off between flexibility and generality of the model. The low storage phase of the Hopfield model corresponds to few hidden units and hence a overly constrained RBM, while the spin-glass phase (too many hidden units) corresponds to unconstrained RBM prone to overfitting of the observed data.