Presenter: 楊皓鈞

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 diner’s 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.

Stochastic Diffusion Search: a ‘Swarm Intelligence’ metaheuristic To find the ‘best’ restaurant in town each delegate should:

1. Select a restaurant to visit at random (agent’s ‘restaurant hypothesis’).

2. Select meal from the menu at random (partial hypothesis evaluation).

3. IF <meal good> THEN revisit the restaurant and GOTO (2).

4. ELSE IF the meal a (randomly chosen) friend ate was good THEN adopt their restaurant hypothesis and GOTO (2).

5. ELSE GOTO (1).

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 didn’t 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

n

ii xFxF

1)()(

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).

Interaction and diffusionINTERACTION: 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 diner’s 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;

Convergence of SDSAGENT 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 (T[i], S[h+i]).

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 phaseAssigns each agent a ‘possible hypothesis’.

I.e. A possible mapping of the target string in the search space text.

TEST agent activity: randomised partial hypothesis evaluation

Test Function: return a boolean result.Return true if the hypothesis is ‘good’ and

false otherwise.

Each agent applies the Test Function to its current hypothesis. If Test Function return true then this agent becomes

‘positive’ and otherwise becomes ‘negative’.

An example TESTTarget, T: c [a] tComponent, (i): 0 1 2

search space, SS: T h e c a t s a tHypothesis, (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 [3+1], 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 Each negative agent, A select another agent B at random.

If B is positive then the hypothesis of B is copied to A.

If B is also inactive then A selects another hypothesis over the entire search space.

The TERMINATE phaseAs iterations progress clusters of agents with the

same hypothesis form.

At convergence of the largest cluster agents is associated with the optimal solution.

Some search and optimisation applications employing Stochastic Diffusion ProcessesEye tracking

Bishop & Torr.

Lip tracking Grech-Cini & McKee.

Mobile robot localisation Beattie et al.

Site selection for wireless networks Hurley & Whitaker.

Speech recognition Nicolaou.

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.

ConclusionsStochastic Diffusion Procedures constitute a new

‘meta heuristic’ for efficient global search and optimisation.

The problem should have ‘Compositional’ Objective Functions.