Infinitely wide or deep neural networks (NNs) with independent and identically distributed (i.i.d.) parameters have been shown to be equivalent to Gaussian processes. Because of the favorable properties of Gaussian processes, this equivalence is commonly employed to analyze neural networks and has led to various breakthroughs over the years. However, neural networks and Gaussian processes are equivalent only in the limit; in the finite case there are currently no methods available to approximate a trained neural network with a Gaussian model with bounds on the approximation error. In this work, we present an algorithmic framework to approximate a neural network of finite width and depth, and with not necessarily i.i.d. parameters, with a mixture of Gaussian processes with bounds on the approximation error. In particular, we consider the Wasserstein distance to quantify the closeness between probabilistic models and, by relying on tools from optimal transport and Gaussian processes, we iteratively approximate the output distribution of each layer of the neural network as a mixture of Gaussian processes. Crucially, for any NN and ∊ > 0 our approach is able to return a mixture of Gaussian processes that is ∊-close to the NN at a finite set of input points. Furthermore, we rely on the differentiability of the resulting error bound to show how our approach can be employed to tune the parameters of a NN to mimic the functional behavior of a given Gaussian process, e.g., for prior selection in the context of Bayesian inference. We empirically investigate the effectiveness of our results on both regression and classification problems with various neural network architectures. Our experiments highlight how our results can represent an important step towards understanding neural network predictions and formally quantifying their uncertainty.

In this paper, we consider discrete-time nonlinear stochastic dynamical systems with additive process noise in which both the initial state and noise distributions are uncertain. Our goal is to quantify how the uncertainty in these distributions is propagated by the system dynamics for possibly infinite time steps. In particular, we model the uncertainty over input and noise as ambiguity sets of probability distributions close in the ρ-Wasserstein distance and aim to quantify how these sets evolve over time. Our approach relies on results from quantization theory, optimal transport, and stochastic optimization to construct ambiguity sets of distributions centered at mixture of Gaussian distributions that are guaranteed to contain the true sets for both finite and infinite prediction time horizons. We empirically evaluate the effectiveness of our framework in various benchmarks from the control and machine learning literature, showing how our approach can efficiently and formally quantify the uncertainty in linear and non-linear stochastic dynamical systems.

We present a biologically inspired design for swarm foraging based on ant’s pheromone deployment, where the swarm is assumed to have very restricted capabilities. The robots do not require global or relative position measurements and the swarm is fully decentralized and needs no infrastructure in place. Additionally, the system only requires one-hop communication over the robot network, we do not make any assumptions about the connectivity of the communication graph and the transmission of information and computation is scalable versus the number of agents. This is done by letting the agents in the swarm act as foragers or as guiding agents (beacons). We present experimental results computed for a swarm of Elisa-3 robots on a simulator, and show how the swarm self-organizes to solve a foraging problem over an unknown environment, converging to trajectories around the shortest path, and test the approach on a real swarm of Elisa-3 robots. At last, we discuss the limitations of such a system and propose how the foraging efficiency can be increased.

In this paper, we introduce BNN-DP, an efficient algorithmic framework for analysis of adversarial robustness of Bayesian Neural Networks (BNNs). Given a compact set of input points T ⊂ Rⁿ, BNN-DP computes lower and upper bounds on the BNN's predictions for all the points in T. The framework is based on an interpretation of BNNs as stochastic dynamical systems, which enables the use of Dynamic Programming (DP) algorithms to bound the prediction range along the layers of the network. Specifically, the method uses bound propagation techniques and convex relaxations to derive a backward recursion procedure to over-approximate the prediction range of the BNN with piecewise affine functions. The algorithm is general and can handle both regression and classification tasks. On a set of experiments on various regression and classification tasks and BNN architectures, we show that BNN-DP outperforms state-of-the-art methods by up to four orders of magnitude in both tightness of the bounds and computational efficiency.

Neural networks (NNs) are emerging as powerful tools to represent the dynamics of control systems with complicated physics or black-box components. Due to complexity of NNs, however, existing methods are unable to synthesize complex behaviors with guarantees for NN dynamic models (NNDMs). This letter introduces a control synthesis framework for stochastic NNDMs with performance guarantees. The focus is on specifications expressed in linear temporal logic interpreted over finite traces (LTLf), and the approach is based on finite abstraction. Specifically, we leverage recent techniques for convex relaxation of NNs to formally abstract a NNDM into an interval Markov decision process (IMDP). Then, a strategy that maximizes the probability of satisfying a given specification is synthesized over the IMDP and mapped back to the underlying NNDM. We show that the process of abstracting NNDMs to IMDPs reduces to a set of convex optimization problems, hence guaranteeing efficiency. We also present an adaptive refinement procedure that makes the framework scalable. On several case studies, we illustrate that our framework is able to provide non-trivial guarantees of correctness for NNDMs with architectures of up to 5 hidden layers and hundreds of neurons per layer.