Local search algorithms

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Local Search Algorithms
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Chapter 12: Local Search
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 What is Local Search Algorithms
 Different algorithmic methods within the
Local Search Algorithm
1. Gradient Descent
2. The Metropolis Algorithm
3. Simulated Annealing Implementation
4. Hopfield Neural Networks
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 A local search algorithm takes the neighbor relation, and
working according to the following high level scheme. At all
times, it maintains a current solution. Throughout the
execution of the algorithms, it remembers the minimum cost
solution that it has seen so far. So for as long as it runs, it will
gradually finds better and better solutions to the same
problem. Thus one can think of a neighbor relation as defining
a graphs on the set of all possible solutions, with edges joining
neighboring pairs of solutions.
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 Local Search is a very technique, it describes any algorithms that
explores the space of possible solution in a sequential fashion,
moving in one step from a current solution to a nearby one.
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 Advantage:
Local search is that it offers flexibility to design it to almost any
computationally hard problem that would involve an algorithmic
solution to solve.
 Disadvantage:
Local search is difficult to determine if it is precise or provable
about the quality of the solutions that a local search algorithm finds.
It would be hard to tell if one is performing proper the local search or
not.

1. Gradient Descent
Gradient descent is an optimization algorithm. To find
a local minimum of a function using gradient
descent, one takes steps proportional to
the negative of the gradient of the function at the
current point.
This function describes the error the neural network
makes in approximating or classifying the training
data, as a function of the weights of the network.
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“Gradient descent is a function optimization method
which uses the derivative of the function and the
idea of steepest descent. The derivative of a
function is simply the slope. So if we know the
slope of a function, then it stands to reason that all
we have to do is somehow move the function in the
negative direction of the slope, and that will reduce
the value of the function.” (Dan Simon)
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Gradient descent. Let S denote current solution. If
there is a neighbor S' of S with strictly lower cost,
replace S with the neighbor whose cost is as small
as possible. Otherwise, terminate the algorithm.
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2. The Metropolis Algorithm
[Metropolis, Rosenbluth, Rosenbluth, Teller, Teller 1953]
– Step by Step Simulation behavior of a physical
system according to principles of statistical
mechanics.
– Simulation maintains a current state of the
system and tries to produce a new state by
applying a perturbation to this state.

“A different class of Monte Carlo methods is based on Markov chains and is
known as Markov Chain Monte Carlo. The basic difference from the methods
described above is that the sequence of generated points takes a kind of
random walk in parameter space, instead of each point being generated, one
independently from another. Moreover, the probability of jumping from one
point to an other depends only on the last point and not on the entire previous
history (this is the peculiar property of a Markov chain). There are several
MCMC algorithms. One of the most popular and simple algorithms, applicable
to a wide class of problems, is the Metropolis algorithm.”
Giulio D'Agostini
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 Metropolis Algorithms does not always behave the way one
would want, even in some very simple situations. Gradient
descent solves this instance with no trouble, deleting nodes in
sequence until none are left. But, while the Metropolis
Algorithms will start out this way, it begins to go astray as it
nears the global optimum.
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3. Simulated Annealing Implementation
Simulated annealing is a generic probabilistic meta-algorithm for
the global optimization problem, namely locating a good
approximation to the global minimum of a given function in a
large search space.
Simulated annealing is a generalization of a Monte Carlo method for
examining the equations of state and frozen states of n-body
systems [Metropolis et al. 1953]
Simulated annealing has been used in various combinatorial
optimization problems and has been particularly successful in circuit
design problems.
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By analogy with this physical process, each step of the SA algorithm
replaces the current solution by a random "nearby" solution, chosen
with a probability that depends on the difference between the
corresponding function values and on a global
parameter T (temperature), that is gradually decreased during the
process.
The dependency is such that the current solution changes almost
randomly when T is large, but increasingly "downhill" as T goes to
zero. The allowance for "uphill" moves saves the method from
becoming stuck at local minima.
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4. Hopfield Neural Networks
Hopfield networks have been proposed as a simple model of an
associative memory, in which a large collection of units are
connected by the underlying network, and neighboring units try to
correlate their states.
 Goal. Find a stable configuration, if such a configuration exists.
 State-flipping algorithm. Repeated flip state of an unsatisfied node
Hopfield-Flip(G, w) {
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S  arbitrary configuration
while (current configuration is
not stable) {
u  unsatisfied node
su = -su
}
return S
}

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unsatisfied node
10 - 8 > 0
unsatisfied
node
8 - 4 - 1 -
1 > 0
stable
State-Flipping Algorithms

The State Flipping Algorithms used for Hopfield networks provides a
local search algorithm to approximate the Maximum Cut objective.
A configuration S of the network is an assignment of the value -1 or +1
to each node. The meaning of the configuration is that each node u,
representing a unit of the neutral network, is trying to choose
between one of two possible states, and its choice is influenced by
those of its neighbors.
If u is joined to v by an edge of negative weight, then u and v want to
have the same state, while if u is joined to v by an edge of positive
weight, then u and v want to have opposite states.
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Relationship among the algorithms.
Minor analogy when looking into the methods and
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rules for each of the algorithms.
Maximum Cut Problem and Hopfield Neural Network
uses the rules and procedure of the State Flipping
Algorithms
Metropolis and Gradient Descent uses the simulated
annealing algorithm

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Resources:
1. Algorithms Design By Jon Kleinberg & Eva Tardos
2. http://ccp.uchicago.edu/~tpregier/hopfield/
Matt Hill, at IBM's TJ Watson Research Center
3. http://www.innovatia.com/software/papers/gradient.htm
Dan Simon Copyright 1998–2007 Innovatia Software
4. http://simone.neuro.kuleuven.ac.be/lab_session/img2.png
Temujin Gautama & Karl Pauwels
5. http://wwwcs.uni-paderborn.de/cs/sensen/Scheduling/icpp/img33.gif
Norbert Sensen University of Paderborn, Germany
6. http://www.roma1.infn.it/~dagos/rpp/node42.html
Giulio D'Agostini 2003-05-13

Local search algorithms

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