54 research outputs found
Learning Hierarchically-Structured Concepts II: Overlapping Concepts, and Networks With Feedback
We continue our study from Lynch and Mallmann-Trenn (Neural Networks, 2021),
of how concepts that have hierarchical structure might be represented in
brain-like neural networks, how these representations might be used to
recognize the concepts, and how these representations might be learned.
In Lynch and Mallmann-Trenn (Neural Networks, 2021), we considered simple
tree-structured concepts and feed-forward layered networks. Here we extend the
model in two ways: we allow limited overlap between children of different
concepts, and we allow networks to include feedback edges.
For these more general cases, we describe and analyze algorithms for
recognition and algorithms for learning
Learning Hierarchically-Structured Concepts II: Overlapping Concepts, and Networks With Feedback
How to Color a French Flag--Biologically Inspired Algorithms for Scale-Invariant Patterning
In the French flag problem, initially uncolored cells on a grid must
differentiate to become blue, white or red. The goal is for the cells to color
the grid as a French flag, i.e., a three-colored triband, in a distributed
manner. To solve a generalized version of the problem in a distributed
computational setting, we consider two models: a biologically-inspired version
that relies on morphogens (diffusing proteins acting as chemical signals) and a
more abstract version based on reliable message passing between cellular
agents.
Much of developmental biology research has focused on concentration-based
approaches using morphogens, since morphogen gradients are thought to be an
underlying mechanism in tissue patterning. We show that both our model types
easily achieve a French ribbon - a French flag in the 1D case. However,
extending the ribbon to the 2D flag in the concentration model is somewhat
difficult unless each agent has additional positional information. Assuming
that cells are are identical, it is impossible to achieve a French flag or even
a close approximation. In contrast, using a message-based approach in the 2D
case only requires assuming that agents can be represented as constant size
state machines.
We hope that our insights may lay some groundwork for what kind of message
passing abstractions or guarantees, if any, may be useful in analogy to cells
communicating at long and short distances to solve patterning problems. In
addition, we hope that our models and findings may be of interest in the design
of nano-robots
Palindrome Recognition In The Streaming Model
In the Palindrome Problem one tries to find all palindromes (palindromic
substrings) in a given string. A palindrome is defined as a string which reads
forwards the same as backwards, e.g., the string "racecar". A related problem
is the Longest Palindromic Substring Problem in which finding an arbitrary one
of the longest palindromes in the given string suffices. We regard the
streaming version of both problems. In the streaming model the input arrives
over time and at every point in time we are only allowed to use sublinear
space. The main algorithms in this paper are the following: The first one is a
one-pass randomized algorithm that solves the Palindrome Problem. It has an
additive error and uses ) space. The second algorithm is a two-pass
algorithm which determines the exact locations of all longest palindromes. It
uses the first algorithm as the first pass. The third algorithm is again a
one-pass randomized algorithm, which solves the Longest Palindromic Substring
Problem. It has a multiplicative error using only space. We also
give two variants of the first algorithm which solve other related practical
problems
Crowd Vetting: Rejecting Adversaries via Collaboration--with Application to Multi-Robot Flocking
We characterize the advantage of using a robot's neighborhood to find and
eliminate adversarial robots in the presence of a Sybil attack. We show that by
leveraging the opinions of its neighbors on the trustworthiness of transmitted
data, robots can detect adversaries with high probability. We characterize a
number of communication rounds required to achieve this result to be a function
of the communication quality and the proportion of legitimate to malicious
robots. This result enables increased resiliency of many multi-robot
algorithms. Because our results are finite time and not asymptotic, they are
particularly well-suited for problems with a time critical nature. We develop
two algorithms, \emph{FindSpoofedRobots} that determines trusted neighbors with
high probability, and \emph{FindResilientAdjacencyMatrix} that enables
distributed computation of graph properties in an adversarial setting. We apply
our methods to a flocking problem where a team of robots must track a moving
target in the presence of adversarial robots. We show that by using our
algorithms, the team of robots are able to maintain tracking ability of the
dynamic target
Prophet Inequalities: Separating Random Order from Order Selection
Prophet inequalities are a central object of study in optimal stopping
theory. A gambler is sent values online, sampled from an instance of
independent distributions, in an adversarial, random or selected order,
depending on the model. When observing each value, the gambler either accepts
it as a reward or irrevocably rejects it and proceeds to observe the next
value. The goal of the gambler, who cannot see the future, is maximising the
expected value of the reward while competing against the expectation of a
prophet (the offline maximum). In other words, one seeks to maximise the
gambler-to-prophet ratio of the expectations.
The model, in which the gambler selects the arrival order first, and then
observes the values, is known as Order Selection. Recently it has been shown
that in this model a ratio of can be attained for any instance. If the
gambler chooses the arrival order (uniformly) at random, we obtain the Random
Order model. The worst case ratio over all possible instances has been
extensively studied for at least years. Still, it is not known if
carefully choosing the order, or simply taking it at random, benefits the
gambler. We prove that, in the Random Order model, no algorithm can achieve a
ratio larger than , thus showing for the first time that there is a
real benefit in choosing the order.Comment: 36 pages, 2 figure
Bounds on the Voter Model in Dynamic Networks
In the voter model, each node of a graph has an opinion, and in every round
each node chooses independently a random neighbour and adopts its opinion. We
are interested in the consensus time, which is the first point in time where
all nodes have the same opinion. We consider dynamic graphs in which the edges
are rewired in every round (by an adversary) giving rise to the graph sequence
, where we assume that has conductance at least
. We assume that the degrees of nodes don't change over time as one can
show that the consensus time can become super-exponential otherwise. In the
case of a sequence of -regular graphs, we obtain asymptotically tight
results. Even for some static graphs, such as the cycle, our results improve
the state of the art. Here we show that the expected number of rounds until all
nodes have the same opinion is bounded by , for any
graph with edges, conductance , and degrees at least . In
addition, we consider a biased dynamic voter model, where each opinion is
associated with a probability , and when a node chooses a neighbour with
that opinion, it adopts opinion with probability (otherwise the node
keeps its current opinion). We show for any regular dynamic graph, that if
there is an difference between the highest and second highest
opinion probabilities, and at least nodes have initially the
opinion with the highest probability, then all nodes adopt w.h.p. that opinion.
We obtain a bound on the convergences time, which becomes for
static graphs
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