29 research outputs found
A Contracting Dynamical System Perspective toward Interval Markov Decision Processes
Interval Markov decision processes are a class of Markov models where the
transition probabilities between the states belong to intervals. In this paper,
we study the problem of efficient estimation of the optimal policies in
Interval Markov Decision Processes (IMDPs) with continuous action-space. Given
an IMDP, we show that the pessimistic (resp. the optimistic) value iterations,
i.e., the value iterations under the assumption of a competitive adversary
(resp. cooperative agent), are monotone dynamical systems and are contracting
with respect to the -norm. Inspired by this dynamical system
viewpoint, we introduce another IMDP, called the action-space relaxation IMDP.
We show that the action-space relaxation IMDP has two key features: (i) its
optimal value is an upper bound for the optimal value of the original IMDP, and
(ii) its value iterations can be efficiently solved using tools and techniques
from convex optimization. We then consider the policy optimization problems at
each step of the value iterations as a feedback controller of the value
function. Using this system-theoretic perspective, we propose an
iteration-distributed implementation of the value iterations for approximating
the optimal value of the action-space relaxation IMDP
Non-Euclidean Contraction Theory for Robust Nonlinear Stability
We study necessary and sufficient conditions for contraction and incremental
stability of dynamical systems with respect to non-Euclidean norms. First, we
introduce weak pairings as a framework to study contractivity with respect to
arbitrary norms, and characterize their properties. We introduce and study the
sign and max pairings for the and norms, respectively.
Using weak pairings, we establish five equivalent characterizations for
contraction, including the one-sided Lipschitz condition for the vector field
as well as matrix measure and Demidovich conditions for the corresponding
Jacobian. Third, we extend our contraction framework in two directions: we
prove equivalences for contraction of continuous vector fields and we formalize
the weaker notion of equilibrium contraction, which ensures exponential
convergence to an equilibrium. Finally, as an application, we provide (i)
incremental input-to-state stability and finite input-state gain properties for
contracting systems, and (ii) a general theorem about the Lipschitz
interconnection of contracting systems, whereby the Hurwitzness of a gain
matrix implies the contractivity of the interconnected system
A Toolbox for Fast Interval Arithmetic in numpy with an Application to Formal Verification of Neural Network Controlled Systems
In this paper, we present a toolbox for interval analysis in numpy, with an
application to formal verification of neural network controlled systems. Using
the notion of natural inclusion functions, we systematically construct interval
bounds for a general class of mappings. The toolbox offers efficient
computation of natural inclusion functions using compiled C code, as well as a
familiar interface in numpy with its canonical features, such as n-dimensional
arrays, matrix/vector operations, and vectorization. We then use this toolbox
in formal verification of dynamical systems with neural network controllers,
through the composition of their inclusion functions
Forward Invariance in Neural Network Controlled Systems
We present a framework based on interval analysis and monotone systems theory
to certify and search for forward invariant sets in nonlinear systems with
neural network controllers. The framework (i) constructs localized first-order
inclusion functions for the closed-loop system using Jacobian bounds and
existing neural network verification tools; (ii) builds a dynamical embedding
system where its evaluation along a single trajectory directly corresponds with
a nested family of hyper-rectangles provably converging to an attractive set of
the original system; (iii) utilizes linear transformations to build families of
nested paralleletopes with the same properties. The framework is automated in
Python using our interval analysis toolbox , in
conjunction with the symbolic arithmetic toolbox , demonstrated
on an -dimensional leader-follower system