3,975 research outputs found
Inertial game dynamics and applications to constrained optimization
Aiming to provide a new class of game dynamics with good long-term
rationality properties, we derive a second-order inertial system that builds on
the widely studied "heavy ball with friction" optimization method. By
exploiting a well-known link between the replicator dynamics and the
Shahshahani geometry on the space of mixed strategies, the dynamics are stated
in a Riemannian geometric framework where trajectories are accelerated by the
players' unilateral payoff gradients and they slow down near Nash equilibria.
Surprisingly (and in stark contrast to another second-order variant of the
replicator dynamics), the inertial replicator dynamics are not well-posed; on
the other hand, it is possible to obtain a well-posed system by endowing the
mixed strategy space with a different Hessian-Riemannian (HR) metric structure,
and we characterize those HR geometries that do so. In the single-agent version
of the dynamics (corresponding to constrained optimization over simplex-like
objects), we show that regular maximum points of smooth functions attract all
nearby solution orbits with low initial speed. More generally, we establish an
inertial variant of the so-called "folk theorem" of evolutionary game theory
and we show that strict equilibria are attracting in asymmetric
(multi-population) games - provided of course that the dynamics are well-posed.
A similar asymptotic stability result is obtained for evolutionarily stable
strategies in symmetric (single- population) games.Comment: 30 pages, 4 figures; significantly revised paper structure and added
new material on Euclidean embeddings and evolutionarily stable strategie
Generalized Opinion Dynamics from Local Optimization Rules
We study generalizations of the Hegselmann-Krause (HK) model for opinion
dynamics, incorporating features and parameters that are natural components of
observed social systems. The first generalization is one where the strength of
influence depends on the distance of the agents' opinions. Under this setup, we
identify conditions under which the opinions converge in finite time, and
provide a qualitative characterization of the equilibrium. We interpret the HK
model opinion update rule as a quadratic cost-minimization rule. This enables a
second generalization: a family of update rules which possess different
equilibrium properties. Subsequently, we investigate models in which a external
force can behave strategically to modulate/influence user updates. We consider
cases where this external force can introduce additional agents and cases where
they can modify the cost structures for other agents. We describe and analyze
some strategies through which such modulation may be possible in an
order-optimal manner. Our simulations demonstrate that generalized dynamics
differ qualitatively and quantitatively from traditional HK dynamics.Comment: 20 pages, under revie
Past, Present, and Future of Simultaneous Localization And Mapping: Towards the Robust-Perception Age
Simultaneous Localization and Mapping (SLAM)consists in the concurrent
construction of a model of the environment (the map), and the estimation of the
state of the robot moving within it. The SLAM community has made astonishing
progress over the last 30 years, enabling large-scale real-world applications,
and witnessing a steady transition of this technology to industry. We survey
the current state of SLAM. We start by presenting what is now the de-facto
standard formulation for SLAM. We then review related work, covering a broad
set of topics including robustness and scalability in long-term mapping, metric
and semantic representations for mapping, theoretical performance guarantees,
active SLAM and exploration, and other new frontiers. This paper simultaneously
serves as a position paper and tutorial to those who are users of SLAM. By
looking at the published research with a critical eye, we delineate open
challenges and new research issues, that still deserve careful scientific
investigation. The paper also contains the authors' take on two questions that
often animate discussions during robotics conferences: Do robots need SLAM? and
Is SLAM solved
Asymptotic behavior of gradient-like dynamical systems involving inertia and multiscale aspects
In a Hilbert space , we study the asymptotic behaviour, as time
variable goes to , of nonautonomous gradient-like dynamical
systems involving inertia and multiscale features.
Given a general Hilbert space, and two convex
differentiable functions, a positive damping parameter, and a function of which tends to zero as goes to , we
consider the second-order differential equation This
system models the emergence of various collective behaviors in game theory, as
well as the asymptotic control of coupled nonlinear oscillators. Assuming that
tends to zero moderately slowly as goes to infinity, we show
that the trajectories converge weakly in . The limiting equilibria
are solutions of the hierarchical minimization problem which consists in
minimizing over the set of minimizers of . As key assumptions,
we suppose that and that, for
every belonging to a convex cone depending on the data
and where is
the Fenchel conjugate of , and is the support function of
. An application is given to coupled oscillators
Fast convergence of dynamical ADMM via time scaling of damped inertial dynamics
In this paper, we propose in a Hilbertian setting a second-order
time-continuous dynamic system with fast convergence guarantees to solve
structured convex minimization problems with an affine constraint. The system
is associated with the augmented Lagrangian formulation of the minimization
problem. The corresponding dynamics brings into play three general time-varying
parameters, each with specific properties, and which are respectively
associated with viscous damping, extrapolation and temporal scaling. By
appropriately adjusting these parameters, we develop a Lyapunov analysis which
provides fast convergence properties of the values and of the feasibility gap.
These results will naturally pave the way for developing corresponding
accelerated ADMM algorithms, obtained by temporal discretization
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