2,807 research outputs found
Parametric Polynomial Time Perceptron Rescaling Algorithm
Let us consider a linear feasibility problem with a possibly infinite number of inequality constraints posed in an on-line setting: an algorithm suggests a candidate solution, and the oracle either confirms its feasibility, or outputs a violated constraint vector. This model can be solved by subgradient optimisation algorithms for non-smooth functions, also known as the perceptron algorithms in the machine learning community, and its solvability depends on the problem dimension and the radius of the constraint set. The classical perceptron algorithm may have an exponential complexity in the worst case when the radius is infinitesimal [1]. To overcome this difficulty, the space dilation technique was exploited in the ellipsoid algorithm to make its running time polynomial [3]. A special case of the space dilation, the rescaling procedure is utilised in the perceptron rescaling algorithm [2] with a probabilistic approach to choosing the direction of dilation. A parametric version of the perceptron rescaling algorithm is the focus of this work. It is demonstrated that some fixed parameters of the latter algorithm (the initial estimate of the radius and the relaxation parameter) may be modified and adapted for particular problems. The generalised theoretical framework allows to determine convergence of the algorithm with any chosen set of values of these parameters, and suggests a potential way of decreasing the complexity of the algorithm which remains the subject of current research
Moment-Sum-Of-Squares Approach For Fast Risk Estimation In Uncertain Environments
In this paper, we address the risk estimation problem where one aims at
estimating the probability of violation of safety constraints for a robot in
the presence of bounded uncertainties with arbitrary probability distributions.
In this problem, an unsafe set is described by level sets of polynomials that
is, in general, a non-convex set. Uncertainty arises due to the probabilistic
parameters of the unsafe set and probabilistic states of the robot. To solve
this problem, we use a moment-based representation of probability
distributions. We describe upper and lower bounds of the risk in terms of a
linear weighted sum of the moments. Weights are coefficients of a univariate
Chebyshev polynomial obtained by solving a sum-of-squares optimization problem
in the offline step. Hence, given a finite number of moments of probability
distributions, risk can be estimated in real-time. We demonstrate the
performance of the provided approach by solving probabilistic collision
checking problems where we aim to find the probability of collision of a robot
with a non-convex obstacle in the presence of probabilistic uncertainties in
the location of the robot and size, location, and geometry of the obstacle.Comment: 57th IEEE Conference on Decision and Control 201
Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization
We describe methods for proving upper and lower bounds on infinite-time
averages in deterministic dynamical systems and on stationary expectations in
stochastic systems. The dynamics and the quantities to be bounded are assumed
to be polynomial functions of the state variables. The methods are
computer-assisted, using sum-of-squares polynomials to formulate sufficient
conditions that can be checked by semidefinite programming. In the
deterministic case, we seek tight bounds that apply to particular local
attractors. An obstacle to proving such bounds is that they do not hold
globally; they are generally violated by trajectories starting outside the
local basin of attraction. We describe two closely related ways past this
obstacle: one that requires knowing a subset of the basin of attraction, and
another that considers the zero-noise limit of the corresponding stochastic
system. The bounding methods are illustrated using the van der Pol oscillator.
We bound deterministic averages on the attracting limit cycle above and below
to within 1%, which requires a lower bound that does not hold for the unstable
fixed point at the origin. We obtain similarly tight upper and lower bounds on
stochastic expectations for a range of noise amplitudes. Limitations of our
methods for certain types of deterministic systems are discussed, along with
prospects for improvement.Comment: 25 pages; Added new Section 7.2; Added references; Corrected typos;
Submitted to SIAD
Computing Approximate Nash Equilibria in Polymatrix Games
In an -Nash equilibrium, a player can gain at most by
unilaterally changing his behaviour. For two-player (bimatrix) games with
payoffs in , the best-known achievable in polynomial time is
0.3393. In general, for -player games an -Nash equilibrium can be
computed in polynomial time for an that is an increasing function of
but does not depend on the number of strategies of the players. For
three-player and four-player games the corresponding values of are
0.6022 and 0.7153, respectively. Polymatrix games are a restriction of general
-player games where a player's payoff is the sum of payoffs from a number of
bimatrix games. There exists a very small but constant such that
computing an -Nash equilibrium of a polymatrix game is \PPAD-hard.
Our main result is that a -Nash equilibrium of an -player
polymatrix game can be computed in time polynomial in the input size and
. Inspired by the algorithm of Tsaknakis and Spirakis, our
algorithm uses gradient descent on the maximum regret of the players. We also
show that this algorithm can be applied to efficiently find a
-Nash equilibrium in a two-player Bayesian game
Multicast Multigroup Beamforming under Per-antenna Power Constraints
Linear precoding exploits the spatial degrees of freedom offered by
multi-antenna transmitters to serve multiple users over the same frequency
resources. The present work focuses on simultaneously serving multiple groups
of users, each with its own channel, by transmitting a stream of common symbols
to each group. This scenario is known as physical layer multicasting to
multiple co-channel groups. Extending the current state of the art in
multigroup multicasting, the practical constraint of a maximum permitted power
level radiated by each antenna is tackled herein. The considered per antenna
power constrained system is optimized in a maximum fairness sense. In other
words, the optimization aims at favoring the worst user by maximizing the
minimum rate. This Max-Min Fair criterion is imperative in multicast systems,
where the performance of all the receivers listening to the same multicast is
dictated by the worst rate in the group. An analytic framework to tackle the
Max-Min Fair multigroup multicasting scenario under per antenna power
constraints is therefore derived. Numerical results display the accuracy of the
proposed solution and provide insights to the performance of a per antenna
power constrained system.Comment: Presented in IEEE ICC 2014, Sydney, AUS. arXiv admin note:
substantial text overlap with arXiv:1406.755
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