Linear Associator Linear Layer

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1 Hebbia Learig opic 6 Note: lecture otes by Michael Negevitsky (uiversity of asmaia) Bob Keller (Harvey Mudd College CA) ad Marti Haga (Uiversity of Colorado) are used Mai idea: learig based o associatio betwee euros he mai property of a eural etwork is a ability to lear from its eviromet ad to improve its performace through learig. So far we have cosidered supervised or active learig learig with a exteral teacher or a supervisor who presets a traiig set to the etwork. But aother type of learig also exists: usupervised learig.. I cotrast to supervised learig usupervised or self-orgaised learig does ot require a exteral teacher. Durig the traiig sessio the eural etwork receives a umber of differet iput patters discovers sigificat features i these patters ad lears how to classify iput data ito appropriate categories. Usupervised learig teds to follow the euro-biological orgaisatio of the brai. Usupervised learig algorithms aim to lear rapidly ad ca be used i real-time. Hebbia learig I 99 Doald Hebb proposed oe of the key ideas i biological learig commoly kow as Hebb s Law. Hebb s Law states that if euro i is ear eough to excite euro j ad repeatedly participates i its activatio the syaptic coectio betwee these two euros is stregtheed ad euro j becomes more sesitive to stimuli from euro i. Hebb s Postulate Whe a axo of cell A is ear eough to excite a cell B ad repeatedly or persistetly takes part i firig it some growth process or metabolic chage takes place i oe or both cells such that A s efficiecy as oe of the cells firig B is icreased. D. O. Hebb 99 Dedrites B Syapse Cell Body Axo A Iputs R Liear Associator p R x W Liear Layer S x R a S a S x S x a = pureli (Wp) = Wp raiig Set: R a = i w p j j = { p t } { p t } { p Q t } Q Hebb Rule w ew = Simplified Form: Supervised Form: Matrix Form: wold + α f i ( a iq )g j ( p jq ) Postsyaptic Sigal w ew = w old+ αa iq p jq w ew = w old + t iq p jq W ew = W old + t q p q Presyaptic Sigal

2 Matrix Form: Batch Operatio Q W t p t p = t Q p Q = t q p q q = p W = p t t t Q = P p Q P = p p p Q = t t t Q (Zero Iitial Weights) Hebb s Law ca be represeted i the form of two rules:. If two euros o either side of a coectio are activated sychroously the the weight of that coectio is icreased.. If two euros o either side of a coectio are activated asychroously the the weight of that coectio is decreased (added later) Hebb s Law provides the basis for learig without a teacher. Learig here is a local pheomeo occurrig without feedback from the eviromet. I p u t S i g a l s Hebbia learig i a eural etwork i j O u t p u t S i g a l s Hebbia learig implies that weights ca oly icrease. o resolve this problem we might impose a limit o the growth of syaptic weights. It ca be doe by itroducig a o-liear forgettig factor ito Hebb s Law: w = α y j xi ϕ y j w where ϕ is the forgettig factor. Forgettig factor usually falls i the iterval betwee ad typically betwee. ad. to allow oly a little forgettig while limitig the weight growth. Simple Associative Network Iputs p w Hard Limit Neuro a Σ b = -.5 a = hardlim (wp + b) a = hardlim( wp + b) = hardlim( wp.5 ) stimulus respose p = a = o stimulus o respose Shape Fruit Network Baaa? Baaa Associator Smell Sight of baaa p Smell of baaa p Ucoditioed Stimulus Iputs Hard Limit Neuro w = Σ w = p shape detected = p = shape ot detected b = -.5 a = hardlim (w p + w p + b) Coditioed Stimulus smell detected smell ot detected a Baaa?

3 Usupervised Hebb Rule w q ( ) = w ( q ) + αa i ( q)p j ( q) Vector Form: W( q) = W ( q ) + αa( q )p ( q) raiig Sequece: p()p () p () Q Baaa Recogitio Example Iitial Weights: w = w( ) = raiig Sequece: { p ( ) = p( ) = } { p ( ) = p( ) = } α = wq () = wq ( ) + aq ( )pq ( ) First Iteratio (sight fails): a( ) = hardlim( w p ( ) + w( )p( ).5) = hardlim( +.5) = (o respose) w( ) = w ( ) + a( )p( ) = + = Example Secod Iteratio (sight works): a( ) = hardlim( w p ( ) + w( )p( ).5) = hardlim( +.5) = (baaa) w( ) = w ( ) + a( )p( ) = + = hird Iteratio (sight fails): a( ) = hardlim( w p ( ) + w ( )p( ).5) = hardlim( +.5) = (baaa) w( ) = w( ) + a( )p( ) = + = Baaa will ow be detected if either sesor works. Problems with Hebb Rule Weights ca become arbitrarily large here is o mechaism for weights to decrease Hebb Rule with Decay W ( q) = W ( q ) + αa( q)p( q) γw( q ) W ( q ) = ( γ )W ( q ) + αa( q)p( q) his keeps the weight matrix from growig without boud which ca be demostrated by settig both a i ad p j to : wmax = ( γ )wmax + α a p i j wmax = ( γ )wmax + α wmax = α -- γ Example: Baaa Associator α = γ =. First Iteratio (sight fails): a( ) = hardlim( w p ( ) + w( )p( ).5) = hardlim( +.5) = (o respose) w ( ) = w ( ) + a( )p( ). w( ) = +. ( ) = Secod Iteratio (sight works): a( ) = hardlim( w p ( ) + w( )p( ).5) = hardlim( +.5) = (baaa) w() = w( ) + a( )p( ).w( ) = +.( ) =

4 Example hird Iteratio (sight fails): a( ) = hardlim( w p ( ) + w ( )p( ).5) = hardlim( +.5) = (baaa) w( ) = w( ) + a( )p( ).w( ) = +.( ) =.9 Hebb Rule 8 6 max α w = -- = ---- = γ. Hebb with Decay Problem of Hebb with Decay Associatios will decay away if stimuli are ot occasioally preseted. If a i = the w ( q) = ( γ )w ( q ) If γ = this becomes w ( q) = (.9)w ( q ) herefore the weight decays by % at each iteratio where there is o stimulus. Usig Hebb s Law we ca express the adjustmet applied to the weight w at iteratio p i the followig form: w ( p ) = F [ y j ( p ) x i ( p )] As a special case we ca represet Hebb s Law as follows: w = α y x j i where α is the learig rate parameter. his equatio is referred to as the activity product rule. Hebbia learig algorithm Step : Iitialisatio. Set iitial syaptic weights ad thresholds to small radom values say i a iterval [ ]. Step : Activatio. Compute the euro output at iteratio p y j = xi w θ j i= where is the umber of euro iputs ad θ j is the threshold value of euro j. Step : Learig. Update the weights i the etwork: w( p + ) = w( p) + w( p) where w (p)) is the weight correctio at iteratio p. he weight correctio is determied by the geeralised activity product rule: w = ϕ y j [ λ xi w ] Step : Iteratio. Icrease iteratio p by oe go back to Step. Hebbia learig examplee xample o illustrate Hebbia learig cosider a fully coected feedforward etwork with a sigle layer of five computatio euros. Each euro is represeted by a McCulloch ad Pitts model with the sig activatio fuctio. he etwork is traied o the followig set of iput vectors: X = X = X = X = X 5 =

5 Iitial ad fial states of the etwork x x x x x 5 5 Iput layer y y y y 5 y 5 Output layer x x x x x 5 5 Iput layer y y y y 5 y 5 Output layer I p u t l a y e r Iitial ad fial weight matrices O u t p u t l a y e r 5 5 I p u t l a y e r O u t p u t l a y e r A test iput vector or probe is defied as X = Whe this probe is preseted to the etwork we obtai: Y = sig = Variatios of Hebbia Learig Basic Rule: W ew W old = + t q p q Learig Rate: W ew W old = + α t q p q Smoothig: W ew W old + α t q p q γ W old ( γ )W old = = + αt q p q Delta Rule: W ew = W old + α( t q a q )p q Usupervised: W ew = W old + αa q p q