15-382 C OLLECTIVE I NTELLIGENCE – S19 L ECTURE 20: S WARM I NTELLIGENCE 1 / P ARTICLE S WARM O PTIMIZATION 1 T EACHER : G IANNI A. D I C ARO
L IMITATIONS OF THE ( CLASSIC ) CA MODEL 2D Environment tized according to an ! × # regular lattice § Sp Spac ace is di discreti The CA is equivalent to $ × % coupled it § iterated m maps § Coupling is determined by the definition of neigh ghbor orhood oods , and by bou oundary con ondition ons and synchron onization on § Neighborhoods are statically defined based on the lattice’s topology, and capture some notion of meaningful spatial prox oximity § The CA is useful as a simulation on mod odel for a dynamical system, but has some obvious limitations for different types of use 2
F ROM CA TO PSO / SI Discrete à Con ontinuou ous ! " ! # Spatially-fixed cell state ~ FSM %(', ) ' ) à Age gent-mod odel: § internal state ! $ § mobile Spatially-related neighborhood (static) Physical topology induced by the lattice à Relation onal neigh ghbor orhood ood § Agents form a network § Logical topology § Can be dynamic 3
R EFERENCE TASK : G LOBAL F UNCTION O PTIMIZATION obal maximum of the function ! " Find the gl glob (and the point " where it happens) Find the gl obal minimum of the function glob 4
O PTIMIZATION P ROBLEMS § Optimization problems expressed in mathematical form: min $ % $ subject to $ ∈ ℱ § %: ℝ * ⟼ ℝ , is the ob objective function on (for now, m =1) § $ ∈ - * ⊆ ℝ * is the op optimization on vector or variable § ℱ ⊆ - * is the fe feasibl ble set (constraints = values the variables can feasibly take) § $ ∗ ∈ - * is an op optimal sol olution on (gl glob obal minimum) if $ ∗ ∈ ℱ and % $ ∗ ≤ %($) for all $ ∈ ℱ § Mathematical programming prob oblem 5
B ASIC PROPERTIES Given an optimization problem: min $ % $ $ ∈ ) * ⊆ ℝ * subject to $ ∈ ℱ, § min $ % $ is equivalent to m-. −% $ $ § If ℱ = ∅ the problem has no solution (unfeasible) § If ℱ is an open set, only the inf (sup) is guaranteed but not min (max) § The problem is unbounded if f → −∞ 6
U NCONSTRAINED VS . C ONSTRAINED O PT § Sublevel sets (isolines): {" ∈ ℝ % : ' " = )} ü Any constrained optimization problem can be formulated as an unconstrained one by including constraint violations as penalty terms in the objective function 7
G LOBAL AND LOCAL OPTIMALITY § A point ! ∈ # $ ⊆ ℝ $ is gl optimal (global minimum) if ! ∈ ℱ globa obally opt and for all ( ∈ ℱ , ) ! ≤ )(() § A point ! ∈ ℝ $ is loc optimal (local minimum) if ! ∈ ℱ and ocally opt there exists ε > 0 small such that for all ( ∈ ℱ with ! − ( 1 ≤ ε , ) ! ≤ )(() 8
G LOBAL AND LOCAL OPTIMALITY § What about discrete spaces? ! ∈ # $ ⊆ ℤ $ min Z ILP = x 2 s.t. 2 x 1 + x 2 > 13 5 x 1 + 2 x 2 6 30 − x 1 + x 2 > 5 x 1 , x 2 ∈ Z + 9
B LACK - BOX O PTIMIZATION § The function to optimize is not given in algebraic form , or § The function is given, but it’s not amenable to analytical treatment in terms of using its derivatives for finding min/max § All we can do is to query the black box and observe (", $$ " ) pairs… § Bl Black-box box / / Derivative-free opt optimization on vs. Whi hite box box opt optimization on 10
H OW DO WE FIND MINIMA / MAXIMA ? Objective global maximum function +(%) shoulder local maximum “saddle” “flat” local maximum state space % ∈ ! % Using rates of change: derivatives / gradients / Jacobians / Hessians / … § Sampling / Searching in the ! " ⊆ ℝ " domain, the input state space: § Figure out where to search / sample next, from the values that are § returned from the function, without generating a model of the function § Using the sampled data to generate a model of the function and in turn, using it to iteratively direct the search Iteratively constructing a solution, by adding / trying out assignment to § solution components % & , % ( , … % " … many, many variants and combinations of these two basic approaches... § 11
P ARTICLE S WARM O PTIMIZATION (PSO) J. Kennedy, R. Eberhart, Particle Swarm Optimization . Proc. 4th IEEE Int. Conf. on Neural Networks, 1995. & ( & ' § Mu Multi-age gent black-box ox op optimization on inspired by soc ocial and roos oosting g behavior or of of floc ocking g birds: § Each agent (a particle ) encodes a solution point ! § Agents move in " # ⊆ ℝ # , searching for the spots regions/points where the objective function gets its max (min) values § Individual swarm members establish a soc ocial networ ork and can profit from the discoveries and previous experience of the other members of the swarm: § Each agent iteratively changes its position (i.e., decides how to move) using information from personal past experience and from its social neighborhood 12
C OMPLEX SYSTEMS ↔ S WARM I NTELLIGENCE /PSO Com omplex systems (definition adapted from Lecture 1): ü Multi-agent / Multi-component ü Distributed: each agent/component is situated in the embedding environment and acts autonomously ü Decentralized: neither central controller, nor representation of global patterns/goals ü Possibly (not necessarily) with a large number of components ü Localized interactions (allowing propagation of information) ü Emerging and / or Self-Organizing properties ü Agents do not need to be “complex” ü Dynamic: Time and space evolution of the system 13
PSO IS A FIRST EXAMPLE OF S WARM I NTELLIGENCE Swarm intellige gence: Study and design gn of complex systems that: Are potentially made of a large number of components, a swa § swarm § Each component has purpos ose(s) (as for animals, or artificially designed agents), that implicitly contributes to the “ performance ” of the whole Under certain conditions, the system displays forms of swarm intelligence § in terms of generation at the system-level, of effective spatio-temporal patterns and/or optimized decision-making and action-making Modeling: Mod g: study of natural complex systems with the above characteristics in order to identify the local rules that give raise to complex system-level behaviors and self-organization, make formal models Mimicking nature: Engi gineering: g: bottom-up design of artificial systems that display useful system-level behaviors, possibly, but not Bi Bio-mi mimet metic c necessarily, taking inspiration from the natural systems (algor gorithms, rob obot ots) 14
PSO IS A FIRST EXAMPLE OF S WARM I NTELLIGENCE Swarm intellige gence: Study and design gn of complex systems that: § Are potentially made of a large number of components, a swa swarm § Each component is “ sentient ” and has purpos ose(s) (s) (as for animals, or artificially designed agents), that implicitly contributes to the “ performance ” of the whole § Under certain conditions, the system displays forms of swarm intelligence in terms of generation at the system-level of effective spatio-temporal patterns and/or optimized decision-making and action-making Mod Modeling: g: study of natural complex systems with the above characteristics in order to identify the local rules that give raise to complex system-level behaviors and self-organization, make formal models Mimicking nature: Engi gineering: g: bottom-up design of artificial systems that display useful system-level behaviors, possibly, but not Bi Bio-mi mimet metic c necessarily, taking inspiration from the natural systems (algor gorithms, rob obot ots) 15
B OTTOM - UP VS . T OP - DOWN DESIGN Ontogenetic and phylogenetic evolution has (necessarily) followed a bot ottom om-up up approach (grassroots) to “ design ” systems: § Instantiation on of of the basic units (atoms, cells, organs, organisms, individuals) composing the system and let them (se (self lf-)or orga ganize to generate more complex/organized system-level behaviors, structures, and functions § Pop opulation on + Interaction on prot otoc ocol ols are more important than single modules § System-level structural patterns and behaviors are emerging properties From an engineering point of view we can also choose a top op-dow own approach: § Acquisition of comprehensive knowledge about the problem/system, make analysis, decomposition, definition of a possibly optimal strategy § Amenable to formal analysis, “ predictable ” response 16
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