Intelligent Optimisation and Learning Using Tsallis Statistics

Transcription

1 Intellgent Optmsaton and Learnng Usng Tsalls Statstcs George D. Magoulas

2 Goal of ths presentaton. Sho ho deas from Tsalls statstcs can be ncorporated nto learnng and optmsaton algorthms to enhance exploraton of the search space 2. Present applcatons of ths approach n global optmsaton and neural learnng algorthms 3. Hopefully stmulate the nterest to encourage further research on applcatons of Tsalls statstcs.dcs.bb.ac.u/~gmagoulas/ 2

3 Outlne From Statstcal Mechancs to Tsalls Nonextensve Statstcs Search operators based on the q-dstrbuton Nonextensve Partcle Sarm Dffuson th Nonextensve schedule for neural learnng Conclusons and Future or.dcs.bb.ac.u/~gmagoulas/ 3

4 From Statstcal Mechancs to Tsalls Nonextensve Statstcs Statstcal mechancs has been the frst foundatonal physcal theory n hch probablstc concepts and probablstc explanaton played a fundamental role. A standard assumpton of statstcal mechancs s that quanttes le energy are extensve varables, meanng that the total energy of the system s proportonal to the system sze; smlarly the entropy s also supposed to be extensve. Statstcal Mechancs s dely non through Boltzmann s second la of thermodynamcs. A general assumpton behnd statstcal mechancs s that the systems beng smulated are beleved to be ergodc..dcs.bb.ac.u/~gmagoulas/ 4

5 From Statstcal Mechancs to Tsalls Nonextensve Statstcs Boltzmann s Statstcal Mechancs s based on the Boltzmann- Gbbs BG entropy: W p = = S BG = W = th that provdes exponental las for descrbng statonary states and basc tme-dependent phenomena, here {p } are the probabltes of the mcroscopc confguratons, and > 0. Ths form s non to be the correct entropc form for ergodc systems p ln p.dcs.bb.ac.u/~gmagoulas/ 5

6 From Statstcal Mechancs to Tsalls Nonextensve Statstcs Boltzmann s Statstcal Mechancs s dely used for systems that are n statonary states charactersed by thermal equlbrum consstent th ergodcty. Nonextensve Statstcal Mechancs s an alternatve hch has been proposed as a ay of dealng th anomalous systems e.g. nonergodc systems th statonary states that are metastable and long-lved through mathematcal methods [Tsalls, 988].dcs.bb.ac.u/~gmagoulas/ 6

7 From Statstcal Mechancs to Tsalls defned the nonextensve entropy as Tsalls Nonextensve Statstcs W q p = Sq q q, R here W s the total number of mcroscopc confguratons, hose probabltes are {p }, and s a conventonal postve constant. Ths entropc form possesses the nonextensvty property: for ndependent systems, t s subextensve for q >, superextensve for q <, and for q = recovers to BG entropy, hch s extensve [Tsalls, 988] The normalsed nonextensve entropy [Landsberg and Vedral, 998] W q [ ] / N Sq p q.dcs.bb.ac.u/~gmagoulas/ 7

8 From Statstcal Mechancs to Tsalls Nonextensve Statstcs Reny entropy s related through a monotonc functon to the nonextensve entropy W R q S ln p / q = ln[ + q S ]/ q q q The normalsed entropy has a strong relatonshp th the nonextensve entropy N q S S / p = S /[ + q S ] q q q q.dcs.bb.ac.u/~gmagoulas/ 8

9 Search operators based on the q-dstrbuton Tsalls has shon that the optmsaton of the entropc form S q yelds for the canoncal ensemble q β q E p [ q E ] e β here β s a Lagrange parameter, {Ε } s the energy spectrum, and the q-exponental functon s: e x q q [ + q x] = [ q x] q.dcs.bb.ac.u/~gmagoulas/ 9

10 Search operators based on the q-dstrbuton Operators charactersed by the nonextensve entropc ndex q. Tln 2 q q Q T, = e = [ q Tln 2 ] here T s the temperature and ndcates teratons We can create adaptve operators that ncorporate a schedule for evolvng the nfluence of the parameters T and q q 2 0 q T = T [ ], q > + here T 0 s the ntal temperature, T s the current temperature, s the number of teratons. Ths coolng procedure maes the temperature to decrease as a poerla of tme.dcs.bb.ac.u/~gmagoulas/ 0

11 Nonextensve Partcle Sarm Partcle Sarm method desgned by Kennedy socal psychologst and Eberhart electrcal engneer to smulate socal behavour 995 Basc dea: socal nteracton s able to fnd optmal solutons to hard problems.dcs.bb.ac.u/~gmagoulas/

12 Nonextensve Partcle Sarm Evolutonary Computng - Evoluton motvated - Populaton - Operators nspred by Partcle Sarm - Human socalty -Sarm - Smple nostalga & bologcal systems - Parental nfluence to descendents success of others - Socal nfluence, collaboratve search nformaton sharng.dcs.bb.ac.u/~gmagoulas/ 2

13 Nonextensve Partcle Sarm - An ndvdual n the sarm: - The ndvdual best: p = p, p, K, p 2 n f f x < f p then p = x, x 2, x x =, K - Smple nostalga tendency of organsms to repeat past behavours that have been renforced or return to past successes: v t = v t + c p x t - Emulate the success of others : p = p, p, g g g the best partcle of the sarm 2, K.dcs.bb.ac.u/~gmagoulas/ 3 x n p gn

14 Nonextensve Partcle Sarm x t = here v t = x t v + v t t + c The orgnal algorthm by [Sh & Eberhart, 998] p x t + c r p x t r 2 2 g THE HEURISTICS r, r 2 random values n [0,] Upper lmt to c Inerta eght:, {,2}.dcs.bb.ac.u/~gmagoulas/ V 4 max Maxmum velocty of change:

15 Nonextensve Partcle Sarm Upper lmt of c: any tme the sum of the to coeffcents exceeds the value 4.0, both the veloctes and postons explode toard nfnty [Kennedy & Eberhart, 999] -The golden rule: Maxmum velocty: reducng by too much mpedes the ablty of the Sarm to search [Carlsle & Dozer, 200] - Constrcton factor: [Clerc, 999; Carlsle & Dozer, 200] K = V max c + c2 > 4 c = 2.8 c = 2 0,] c + c 2 c + c 4 c + c Inerta eght: nfluences the trade-off beteen global and local exploraton abltes of the partcle [Sh & Eberhart, 998] and c,.dcs.bb.ac.u/~gmagoulas/ 5

16 Nonextensve Partcle Sarm v d = d v d + c r p x + c r p x d d 2 2 gd d + = + d d T, d x x Q v d =,,D search space dmensonalty ; =,, N sarm sze; s the nerta eght; c postve constants; random numbers By tunng the entropc ndex q and the temperature T, t, provdes an alternatve to usng a fxed constrcton coeffcent for balancng to mportant PSO functons: velocty control and search dversty..dcs.bb.ac.u/~gmagoulas/ 6 Q

17 Nonextensve Partcle Sarm q=2 q=4 q=6 Q T q=.8 q=.5 q=.2 Q q=.8 q=.5 q= dcs.bb.ac.u/~gmagoulas/ 7

18 Nonextensve Partcle Sarm 4.5 x 00 4 Rosenbroc NHPSO SPSO NEPSO Ftness dcs.bb.ac.u/~gmagoulas/ Generatons

19 Nonextensve Partcle Sarm Grean NHPSO SPSO NEPSO 40 Ftness dcs.bb.ac.u/~gmagoulas/ Generatons

20 Nonextensve Partcle Sarm 7 x 04 6 Rastrgrn NHPSO SPSO NEPSO 5 4 Ftness Generatons.dcs.bb.ac.u/~gmagoulas/ 20

21 Nonextensve Partcle Sarm Comparsons made th the Standard PSO PSO, the Fuzzy Partcle Sarm Optmzer FPSO and the Hybrd Partcle Sarm Optmzer th mass extncton HPSO detaled results n IJBC 2006 and TOCSJ Varous sarm szes and space dmensons tested. There s a range of q values.<q<4.0 here the method perform reasonably ell n all cases. Rule of thumb: hen the problem s gettng more complcated hgh dmensonalty, t may be benefcal to apply large values for the q, hle decrease the value of q as the dmenson of the problem reduces..dcs.bb.ac.u/~gmagoulas/ 2

22 Dffuson th Nonextensve schedule for neural learnng Defnton: a netor of nterconnected elements nspred from studes on bologcal nervous systems Functon: Produce an output pattern hen presented th an nput pattern Propertes: Ablty to learn and generalse.dcs.bb.ac.u/~gmagoulas/ 22

23 Dffuson th Nonextensve schedule for neural learnng Supervsed learnng α j j d External sgnal teacher y Comparson E=y-d Supervsed Learnng Algorthm.dcs.bb.ac.u/~gmagoulas/ 23

24 Dffuson th Nonextensve schedule for neural learnng Nose plays a nfluental role n the operaton of real neurons, e.g. neural cells' responses to dentcal stmul have been found to be stochastc n nature The effect of nose on the tranng and operaton of artfcal neural netors has not been nvestgated n great depth..dcs.bb.ac.u/~gmagoulas/ 24

25 Dffuson th Nonextensve schedule for neural learnng Introducng nose durng tranng: Input nose add nose to the tranng set Weght nose Output nose Langevn nose Operatng th nose: The Boltzmann machne Gaussan dstrbutons The Smulated Annealng and ts modfcatons, ncludng hybrd schemes, e.g. + = η E + ρ c 2 d.dcs.bb.ac.u/~gmagoulas/ 25

26 .dcs.bb.ac.u/~gmagoulas/ 26 Dffuson th Nonextensve schedule for neural learnng The perturbed error functon:, ] [ 2 2 ~ = + + = n T Q E E μ, ] [ 2 2 ' ~ T Q g g + + = μ Mnmzaton of the perturbed error requres calculatng the gradent of the error functon th respect to each eght: Employs a form of eght decay that modfes the energy landscape so that smaller eghts are favoured at the begnnng of the tranng, but as learnng progresses the magntude of the eght decay s reduced to favour the groth of large eghts. µ regulates the nfluence of the combned eght decay/nose effect.

27 .dcs.bb.ac.u/~gmagoulas/ 27 mn ~ ~ mn 2 2 ~ ~ ~ ~ max ~ ~, max 0,, 2 max, 0 0, mn 0 Δ = < Δ + = < < = = Δ = > + η η η ρ η η η ρ η η η η η η then g g f T Q c then T Q and g g f then g g f then g g f Dffuson th Nonextensve schedule for neural learnng Memory-based adaptaton of the learnng rate c random number, ρ n 0, Sgn-based eght adjustment: + = η sgn ğ

28 q=.2 W2 0 W2 0 T= W W Weght trajectores of Rprop algorthm left and of the proposed algorthm rght for to dfferent temperatures W dcs.bb.ac.u/~gmagoulas/ W q=.2 T=0.00

29 Dffuson th Nonextensve schedule for neural learnng Error functon value ALS Rprop SARprop Error functon value 0 0 ALS Rprop SARprop Number of epochs Learnng error of the proposed Adaptve Learnng Scheme ALS for the XOR problem Number of epochs Learnng error of the proposed Adaptve Learnng Scheme ALS for the Party-3 problem.dcs.bb.ac.u/~gmagoulas/ 29

30 Dffuson th Nonextensve schedule for neural learnng Algorthm Irs IT n=9 CONV. Cancer IT n=56 CONV. Dabetes IT n=34 CONV. Thyrod IT n=03 CONV. Rprop SARprop ALS Algorthm Party-3 IT n=6 CONV. Party-4 IT n=37 CONV. Party-5 IT n=50 CONV. Rprop SARprop ALS dcs.bb.ac.u/~gmagoulas/ 30 Average performance n the test problems for three algorthms

31 Conclusons and Future or We can explot the nonextensve entropc ndex q and adaptve temperature control to regulate the stochastcty of the nonextensve algorthms. Nonextensve search operators can be used to equp other PSO varants, evolutonary algorthms or learnng algorthms. Approach based on heurstcs hch requres extensve emprcal testng. We need to explore the nfluence of the entropc ndex q and T on the convergence speed and stablty of the method further..dcs.bb.ac.u/~gmagoulas/ 3

32 Bblography Anastasads A., and Magoulas G.D Nonextensve statstcal mechancs for hybrd learnng of neural netors, Physca A: Statstcal Mechancs and ts Applcatons, vol. 344, , Anastasads A. and Magoulas G.D Evolvng Stochastc Learnng Algorthm based on Tsalls Entropc ndex, The European Physcal Journal B, vol. 50, Anastasads A.D., Magoulas G.D Partcle Sarms and Nonextensve Statstcs for Nonlnear Optmsaton, The Open Cybernetcs and Systemcs Journal, vol. 2, Boon J.P. and Tsalls C Nonextensve Statstcal Mechancs - Ne Trends, Ne Perspectves, Europhyscs Nes 36 6, European Physcal Socety. Gell-Mann M. and Tsalls C Nonextensve Entropy - Interdscplnary Applcatons, Oxford Unversty Press, Ne Yor. Magoulas G.D., Anastasads A.D Approaches to Adaptve Stochastc Search Based on the Nonextensve q-dstrbuton, Internatonal Journal of Bfurcaton and Chaos, Vol. 6, No. 7, dcs.bb.ac.u/~gmagoulas/ 32