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An introduction to search and optimisation using

Stochastic Diffusion Processes

- Stochastic Diffusion Processes define a family

of agent based search and optimisation algorithms

which have been successfully used in a variety of

real-world applications and for which there is a

sound theoretical foundation. - - Mark Bishop
- Goldsmiths, University of London
- This presentation summarises recent research

carried out by the Goldsmiths/Reading/Kings SDP

group Mark Bishop, Slawomir Nasuto, Kris de

Meyer, Darren Myatt Mohammad Majid. - SDP resource pages are maintained at

http//www.cyber.rdg.ac.uk/CIRG/SDP

Some search and optimisation applications

employing Stochastic Diffusion Processes

- 3D computer vision
- Myatt et al.
- Models of attention
- Summers.
- A new connectionist paradigm for cognitive

science - Nasuto, Bishop et al.
- Theoretical
- Nasuto Bishop.
- Sequence detection
- Jones.

- Eye tracking
- Bishop Torr.
- Lip tracking
- Grech-Cini McKee.
- Mobile robot localisation
- Beattie et al.
- Site selection for wireless networks
- Hurley Whitaker.
- Speech recognition
- Nicolaou.

The Restaurant Game a simple Stochastic

Diffusion optimisation

- A group of conference delegates arrive in a

foreign town and want to find a good place to

eat - the search space is the set of all restaurants
- the objective function restaurant quality -

is the sum of numerous independent partial

objective function evaluations - A random independent partial objective function

evaluation is defined by a diners response to a

randomly chosen meal GOOD or BAD. - Usually in a large town a naive exhaustive search

will be impractical as there will be too many

(restaurant ? dish) combinations to evaluate

during the period the delegates are attending the

conference.

Restaurant quality a stochastic dynamic

objective function

- The objective function optimised by the

Restaurant Game - restaurant quality - is a

stochastic variable defined by the sum of mean

diner responses to all the meal combinations

offered by a given restaurant. - Restaurant quality is a stochastic variable as

the perceived quality of each meal may vary - each time a meal is prepared
- with changes in a diners mood
- as each diner is likely to have a different

perception of what tastes good. - In other best-fit searches the partial

evaluation of the objective function is typically

deterministically defined. E.g. - Is a specific coloured-feature red?
- Is a specific numeric-feature 10, etc.

Stochastic Diffusion Search a Swarm

Intelligence metaheuristic

- To find the best restaurant in town each

delegate should - Select a restaurant to visit at random (agents

restaurant hypothesis). - Select meal from the menu at random (partial

hypothesis evaluation). - IF ltmeal goodgt THEN revisit the restaurant and

GOTO (2). - ELSE IF the meal a (randomly chosen) friend ate

was good THEN adopt their restaurant hypothesis

and GOTO (2). - ELSE GOTO (1).
- Unlike an Egon Ronay guide this SDP will

naturally adapt to changing restaurant conditions

and diner taste over time

Stochastic Diffusion Processes in nature tandem

calling

- Consider a search for resource (e.g. food) in a

dynamically changing natural environment. - E.g. Examine the behaviour of the social insects

such as ants (e.g.leptothorax acervorum) honey

bees etc. - Without a-priori information each ant embarks

upon a (random) walk in their environment for a

finite period of time. - Ants that locate the desired resource return

positive. - Ants that didnt locate resource return

negative. - On returning to the nest each positive ant

directly communicates with the next negative

ant it meets, (non stigmergetic communication). - The positive ant communicates the location of

the resource by physically steering the

negative ant towards it in a tandem pair. - Unselected negative ants embark on another

random walk around their environment.

Compositional Objective Functions

- In general SDPs can most easily be applied to

optimisation problems where the objective

function is decomposable into components that can

be evaluated independently - where Fi (x) is defined as the ith partial

evaluation of F (x). - For example SDPs can simply be applied to

best-fit string (pattern) matching. - Such problems can be cast in terms of

optimisation by defining the objective function,

F (x), for a hypothesis x about the location of

the solution, as the similarity between the

target pattern and the corresponding region at x

in the search space and finding x such that F (x)

is maximised.

Partial evidence and inference

- Assuming a compositional structure of the

solution (I.e. objective function decomposable

into components that can be evaluated

independently) agents perform inference on the

basis of partial evidence. - Partial evidence for each agents hypothesis of

the best solution is obtained by a partial

evaluation of the agents current hypothesis. - Every time a person has dinner at a restaurant

the diner selects one meal combination at random

from the entire menu of dishes available. - Partial hypothesis evaluation allows an agent to

quickly form an opinion on the quality of its

hypothesis without exhaustive testing. - E.g. The Restaurant Game will find the best

restaurant in town without delegates exhaustively

sampling all the meals available in each.

Interaction and diffusion

- INTERACTION On the basis of partial knowledge

agents communicate their current hypothesis to

agents whose own current hypothesis is not

supported by recent evidence. - E.g. In the Restaurant Game each diner whose last

meal was BAD asks a randomly chosen member of

the group if their last meal was GOOD - DIFFUSION If the selected diner enjoyed their

last meal then they communicate their current

hypothesis, (e.g. the identity of the restaurant

they last visited). - Conversely, if the selected diner also did not

enjoy their last meal then a new restaurant is

chosen at random from the entire list of those

available.

Stochastic Diffusion Processes as global

optimisation

- Central to the power of a SDP is its ability to

escape local minima. - E.g. Unless all the meals in a restaurant are to

a diners taste, then there is a finite non-zero

probability that a diners randomly chosen meal

will be judged BAD and a new hypothesis adopted. - Hence a Stochastic Diffusion Process achieves

global optimisation by - probabilistic partial hypothesis evaluation -

selecting a meal at random - in combination with dynamic reallocation of

resources (agents/diners) via stochastic

recruitment mechanisms.

Positive feedback mechanisms in SDP

- In a SDP each agent poses a hypothesis about the

possible solution and evaluates it partially. - Successful agents repeatedly test their

hypothesis and recruit unsuccessful agents to it

by direct communication. - This creates a positive feedback mechanism

ensuring rapid convergence of agents onto

promising solutions in the space of all

solutions. - Hence regions of the solution space labelled by

the presence of agent clusters can be interpreted

as good candidate solutions.

Convergence of SDS

- AGENT CLUSTERING a global solution is

constructed from the interaction of many simple,

locally operating agents, forming the largest

cluster. - Such a cluster is dynamic in nature, yet stable,

analogous to, a forest whose contours do not

change but whose individual trees do, (Arthur,

1994). - CONVERGENCE agents posing mutually consistent

hypotheses support each other and over time this

results in the emergence of a stable agent

population identifying the desired solution. - E.g. In the Restaurant Game - at equilibrium - a

stochastically stable group of people with the

same hypothesis rapidly clusters around the

best restaurant in town.

A simple string search

- Target and search space are defined by the sets

of features T, S. - E.g. In a simple string search, component

features of the target, T, and search space, S,

are alpha-numeric characters. - Hypotheses are potential best-fit positions, h,

of T in S. - The solution is compositional as it is defined by

the set of contiguous characters in the search

space that together constitute the best

instantiation of the target. - A population of agents converge on the

hypothesis, h, of the best fit position of T in

S. - Communication between agents is of their

hypothesis of the target mapping position, h. - Feature evaluation is performed by MATCH (a, b)

which identifies if two features (a, b) are

similar (defined by a specified similarity

metric). - Hence the global optimal solution is found at

MAX ? i MATCH (Ti, Shi).

The Stochastic Diffusion Search algorithm

- INITIALISE (agents)
- WHILE NOT TERMINATE (agents) DO
- TEST (agents, T, S)
- DIFFUSE (agents)
- END
- S is the search space the text containing the

target string - T is the target string.
- Each of the agents maintains a hypothesis, (ie.

the best-fit mapping), of the target in S.

The INITIALISE phase

- Assigns each agent a possible hypothesis.
- I.e. A possible mapping of the target string in

the search space text. - In the absence of prior knowledge possible

hypotheses are generated randomly. - A process analogous to
- The initial selection of a restaurant at random.
- An ants initially random walk around their

environment.

TEST agent activity randomised partial

hypothesis evaluation

- Partial information on the accuracy of the

hypothesis maintained by each agent is obtained

by performing a randomised partial hypothesis

evaluation. - In a simple string search we enquire if one

randomly selected letter (i) of the target

string, Ti, is present in the search space,

S, at the position specified by agents current

hypothesis, h i.e. at Shi? - E.g. Is a randomly selected meal good does a

location evaluated by an ant contain resource?

An example TEST

- Target, T c a t
- Component, (i) 0 1 2
- search space, SS T h e c a t s a t
- Hypothesis, (h) 0 1 2 3 4 5 6 7 8
- Consider the agent hypothesis to be, (h 3)
- Decompose the target, T, perform partial

hypothesis evaluation Ti,, (e.g. i 1) - E.g. For (i 1) the target component symbol,

T1,, is the letter a. - Test the agent hypothesis, (h 3) with partial

hypothesis evaluation (i 1) - TEST MATCH (SS 31, T 1) MATCH (SS 4,

T 1) MATCH (a, a) - The result of this partial test of the agents

hypothesis is POSITIVE - As MATCH (a, a) is TRUE.

The DIFFUSION phase stochastic communication

- DIFFUSION Stochastically communicates agent

mappings across the population of agents. - Communication of potentially good restaurants

to friends. - Communication of potentially good resource

locations to other ants. - In a passive recruitment SDP each negative

agent, (one failing its partial hypothesis

test) attempts to communicate with another

randomly selected member of the population - Passive recruitment as used in the Restaurant

Game. - Active recruitment used by leptothorax

acervorum. - Combined recruitment strategies have also been

investigated (Myatt 2006). - If the selected agent is positive then its

mapping is communicated. - Conversely if the selected agent also is

negative then a completely new mapping is

generated at random.

The TERMINATE phase

- SDS is a global search/optimisation algorithm
- SDS converges to the global optimal position of

the target in search space. - A halting criterion examines the activity of

the agent population to determine if target has

been located. - Two such criteria - discussed by Nasuto et al.,

(1999) - are - The Weak Halting Criterion
- Is a function of the total number of positive

agents, (I.e. net activity). - The Strong Halting Criterion
- Is a function of the total number of positive

agents with the same hypothesis, (I.e.

clustered activity).

Algorithm class

- Global optimisation algorithms have been recently

classified in terms of their theoretical

foundations into four distinct classes (Neumaier,

2004) - incomplete methods heuristic searches with no

safeguards against trapping in a local minimum - asymptotically complete methods reaching the

global optimum with probability one if allowed to

run indefinitely long without means to ascertain

when the global optimum has been found - complete methods reaching the global optimum

with probability one in infinite time that know

after a finite time that an approximate solution

has been found within prescribed tolerances - rigorous methods typically reaching the global

solution with certainty and within given

tolerances.

Algorithm class for heuristic multi-agent systems

- In heuristic multi-agent systems Neumaier

characterisation is related to the concept of the

stability of intermediate solutions, because the

probability that any single agent will lose the

best solution is often greater than zero. - This may result in a lack of stability of the

found solutions or in the worst case

non-convergence of the algorithm. - Thus for multi-agent systems it is desirable to

characterise the stability of the discovered

solutions. - For example it is known that many variants of

Genetic Algorithms do not converge hence the

optimal solution may disappear from the next

population. - It has been established that the solutions found

by SDS are exceptionally stable (De Meyer, Nasuto

Bishop). - For example, on SDS convergence in a search with

N1000 agents search space M1000 probability

of a false negative P- 0.2 the mean return time

to the state of all agents inactive is

approximately 10602 iterations.

Convergence criterion

- The convergence of SDS was first rigorously

analysed by Bishop Torr (1992) for the case of

zero noise and ideal target instantiation. - Detailed criteria for SDS convergence under a

variety of noise conditions were first discussed

by Nasuto et al. (1999), in the context of

interacting Markov Chains Ehrenfest Urn models. - However in a recent paper Myatt et al. (2003)

outline a much simpler criterion to estimate the

suitability of employing SDS for a given

search/optimisation problem. - Unlike the earlier analysis by Nasuto, Myatts

analysis employs two key simplifying assumptions - It utilises the mean number transition of agents

between clusters rather than complete probability

distributions. - It assumes homogeneous background noise.
- If a is the quality of the best solution and b an

estimate of homogeneous background noise, then

the minimum quality required for stable

convergence of the algorithm is simply

Time complexity analysis of SDS

- The Time Complexity of SDS was first analysed in

Nasuto et al., (1998) for the case of zero noise

and ideal target instantiation. - The result has also been demonstrated to hold in

the case of convergence under noise. - Given M is the search space size and N is the

number of agents, one can divide the NxM plane

into two distinct regions - Region 1 is Linear M gt M (N)
- Sequential convergence time is O (M).
- Parallel convergence time is O (M/N).
- Region 2 is independent of search space size M
- Sequential convergence time is O (N/log N).
- Parallel convergence time is O (1/log N).

Conclusions

- Stochastic Diffusion Procedures constitute a new

meta heuristic for efficient global search and

optimisation. - In a generic search problem - such as string

search - the worst case time complexity of SDS

compares favourably with the best deterministic

one and two dimensional string search algorithms,

(or their extensions to tree matching). - Further, such performance is achieved without the

use of application specific heuristics. - Unlike many heuristic search methods (such as

Evolutionary techniques Ant Algorithms Particle

Swarm Optimisers etc.), Stochastic Diffusion

Procedures have very thorough mathematical

foundations and correspondingly well

characterised behaviour.

Core references http//www.cyber.rdg.ac.uk/CIRG/S

DP

- Ideal convergence of SDS
- Bishop, J.M. Torr, P.H., (1992), The Stochastic

Search Network, in Lingard, R., Myers, D.

Nightingale, C., (eds), Neural Networks for

Vision Speech Natural Language, pp. 370-388,

Chapman-Hall. - General convergence of SDS
- Nasuto, S.J. Bishop, J.M., (1999), Convergence

Analysis of the Stochastic Diffusion Search,

Parallel Algorithms, 142, pp. 89-107, UK. - Time complexity analysis of SDS
- Nasuto, S.J., Bishop, J.M. Lauria, S., (1998),

Time Complexity Analysis of the Stochastic

Diffusion Search, Neural Computing 98, Vienna. - Simple convergence criteria
- Myatt, D., Bishop, J.M. Nasuto, (2003), Minimum

Stable Criteria for Stochastic Diffusion Search,

Electronics Letters, 402, pp. 112-113, UK. - Change of cognitive metaphor
- Nasuto, S.J., Dautenhahn, K. Bishop, J.M.,

(1999), Communication as an emergent metaphor for

neuronal operation, Lecture Notes in Artificial

Intelligence, 1562, pp. 365-380, Springer-Verlag.