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An introduction to search and optimisation using Stochastic Diffusion Processes

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Title: An introduction to search and optimisation using Stochastic Diffusion Processes


1
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

2
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.

3
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.

4
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.

5
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

6
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.

7
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.

8
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.

9
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.

10
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.

11
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.

12
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.

13
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).

14
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.

15
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.

16
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?

17
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.

18
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.

19
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).

20
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.

21
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.

22
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

23
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).

24
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.

25
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.
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