61 research outputs found
Functional Large Deviations for Cox Processes and Queues, with a Biological Application
We consider an infinite-server queue into which customers arrive according to
a Cox process and have independent service times with a general distribution.
We prove a functional large deviations principle for the equilibrium queue
length process. The model is motivated by a linear feed-forward gene regulatory
network, in which the rate of protein synthesis is modulated by the number of
RNA molecules present in a cell. The system can be modelled as a tandem of
infinite-server queues, in which the number of customers present in a queue
modulates the arrival rate into the next queue in the tandem. We establish
large deviation principles for this queueing system in the asymptotic regime in
which the arrival process is sped up, while the service process is not scaled.Comment: 36 pages, 2 figures, to appear in Annals of Applied Probabilit
Connectivity in One-Dimensional Soft Random Geometric Graphs
In this paper, we study the connectivity of a one-dimensional soft random
geometric graph (RGG). The graph is generated by placing points at random on a
bounded line segment and connecting pairs of points with a probability that
depends on the distance between them. We derive bounds on the probability that
the graph is fully connected by analysing key modes of disconnection. In
particular, analytic expressions are given for the mean and variance of the
number of isolated nodes, and a sharp threshold established for their
occurrence. Bounds are also derived for uncrossed gaps, and it is shown
analytically that uncrossed gaps have negligible probability in the scaling at
which isolated nodes appear. This is in stark contrast to the hard RGG in which
uncrossed gaps are the most important factor when considering network
connectivity
The Gossiping Insert-Eliminate Algorithm for Multi-Agent Bandits
We consider a decentralized multi-agent Multi Armed Bandit (MAB) setup
consisting of agents, solving the same MAB instance to minimize individual
cumulative regret. In our model, agents collaborate by exchanging messages
through pairwise gossip style communications on an arbitrary connected graph.
We develop two novel algorithms, where each agent only plays from a subset of
all the arms. Agents use the communication medium to recommend only arm-IDs
(not samples), and thus update the set of arms from which they play. We
establish that, if agents communicate times through any
connected pairwise gossip mechanism, then every agent's regret is a factor of
order smaller compared to the case of no collaborations. Furthermore, we
show that the communication constraints only have a second order effect on the
regret of our algorithm. We then analyze this second order term of the regret
to derive bounds on the regret-communication tradeoffs. Finally, we empirically
evaluate our algorithm and conclude that the insights are fundamental and not
artifacts of our bounds. We also show a lower bound which gives that the regret
scaling obtained by our algorithm cannot be improved even in the absence of any
communication constraints. Our results thus demonstrate that even a minimal
level of collaboration among agents greatly reduces regret for all agents.Comment: To Appear in AISTATS 2020. The first two authors contributed equall
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