Neural Networks. Adapted from slides by Tim Finin and Marie desjardins. Some material adapted from lecture notes by Lise Getoor and Ron Parr

Transcription

1 Neural Networks Adapted from sldes by Tm Fnn and Mare desjardns. Some materal adapted from lecture notes by Lse Getoor and Ron Parr

2 Neural func1on Bran func1on (thought) occurs as the result of the frng of neurons Neurons connect to each other through synapses, whch propagate ac+on poten+al (electrcal mpulses) by releasng neurotransm1ers Synapses can be exctatory (poten1al- ncreasng) or nhbtory (poten1al- decreasng), and have varyng ac+va+on thresholds Learnng occurs as a result of the synapses plas+ccty: They exhbt long- term changes n connec1on strength There are about neurons and about synapses n the human bran! 2

3 Bology of a neuron 3

4 Bran structure Dfferent areas of the bran have dfferent func1ons Some areas seem to have the same func1on n all humans (e.g., Broca s regon for motor speech); the overall layout s generally consstent Some areas are more plas1c, and vary n ther func1on; also, the lower- level structure and func1on vary greatly We don t know how dfferent func1ons are assgned or acqured Partly the result of the physcal layout / connec1on to nputs (sensors) and outputs (effectors) Partly the result of experence (learnng) We really don t understand how ths neural structure leads to what we perceve as conscousness or thought Ar1fcal neural networks are not nearly as complex or ntrcate as the actual bran structure 4

5 Comparson of compu1ng power INFORMATION CIRCA 2012 Computer Human Bran Computaton Unts 10-core Xeon: 10 9 Gates Neurons Storage Unts 10 9 bts RAM, bts dsk neurons, synapses Cycle tme 10-9 sec 10-3 sec Bandwdth 10 9 bts/sec bts/sec Computers are way faster than neurons But there are a lot more neurons than we can reasonably model n modern dgtal computers, and they all fre n parallel Neural networks are desgned to be massvely parallel The bran s effec1vely a bllon 1mes faster 5

6 Neural networks Output unts Hdden unts Layered feed-forward network Neural networks are made up of nodes or unts, connected by lnks Each lnk has an assocated weght and ac+va+on level Each node has an nput func+on (typcally summng over weghted nputs), an ac+va+on func+on, and an output 6 Input unts

7 Model of a neuron Neuron modeled as a unt weghts on nput unt to, w net nput to unt s: n = " Ac1va1on func1on g() determnes the neuron s output g() s typcally a sgmod output s ether 0 or 1 (no par1al ac1va1on) 7 w! o

8 Execu1ng neural networks Input unts are set by some exteror func1on (thnk of these as sensors), whch causes ther output lnks to be ac+vated at the specfed level Workng forward through the network, the nput func+on of each unt s appled to compute the nput value Usually ths s ust the weghted sum of the ac1va1on on the lnks feedng nto ths node The ac+va+on func+on transforms ths nput func1on nto a fnal value Typcally ths s a nonlnear func1on, o_en a sgmod func1on correspondng to the threshold of that node 8

9 Learnng rules Rosenblab (1959) suggested that f a target output value s provded for a sngle neuron wth fxed nputs, can ncrementally change weghts to learn to produce these outputs usng the perceptron learnng rule assumes bnary valued nput/outputs assumes a sngle lnear threshold unt 9

10 Perceptron learnng rule If the target output for unt s t w = w + η( t o ) o Equvalent to the ntu1ve rules: If output s correct, don t change the weghts If output s low (o =0, t =1), ncrement weghts for all the nputs whch are 1 If output s hgh (o =1, t =0), decrement weghts for all nputs whch are 1 Must also adust threshold. Or equvalently assume there s a weght w 0 for an extra nput unt that has an output of 1. 10

11 Perceptron learnng algorthm Repeatedly terate through examples adus1ng weghts accordng to the perceptron learnng rule un1l all outputs are correct In1alze the weghts to all zero (or random) Un1l outputs for all tranng examples are correct for each tranng example e do compute the current output o compare t to the target t and update weghts Each execu1on of outer loop s called an epoch For mul1ple category problems, learn a separate perceptron for each category and assgn to the class whose perceptron most exceeds ts threshold 11

12 Representa1on lmta1ons of a perceptron Perceptrons can only represent lnear threshold func1ons and can therefore only learn func1ons whch lnearly separate the data..e., the pos1ve and nega1ve examples are separable by a hyperplane n n- dmensonal space <W,X> - θ = 0 > 0 on ths sde < 0 on ths sde 12

13 Perceptron learnablty Perceptron Convergence Theorem: If there s a set of weghts that s consstent wth the tranng data (.e., the data s lnearly separable), the perceptron learnng algorthm wll converge (Mncksy & Papert, 1969) Unfortunately, many func1ons (lke party) cannot be represented by LTU 13

14 Learnng: Backpropaga1on Smlar to perceptron learnng algorthm, we cycle through our examples f the output of the network s correct, no changes are made f there s an error, the weghts are adusted to reduce the error The trck s to assess the blame for the error and dvde t among the contrbu1ng weghts 14

15 Output layer As n perceptron learnng algorthm, we want to mnmze dfference between target output and the output actually computed W = W + α a Err g ʹ (n ) Δ W = = Err actvaton of hdden unt (T O ) dervatve of actvaton functon W g ʹ (n ) + α a 15 Δ

16 Hdden layers Need to defne error; we do error backpropaga1on. Intu1on: Each hdden node s responsble for some frac1on of the error Δ I n each of the output nodes to whch t connects. Δ I dvded accordng to the strength of the connec1on between hdden node and the output node and propagated back to provde the Δ values for the hdden layer: update rule: Δ W k = = gʹ (n W k 16 ) + α W I k Δ Δ

17 Backproga1on algorthm Compute the Δ values for the output unts usng the observed error Star1ng wth output layer, repeat the followng for each layer n the network, un1l earlest hdden layer s reached: propagate the Δ values back to the prevous layer update the weghts between the two layers 17

18 Backprop ssues Backprop s the cockroach of machne learnng. It s ugly, and annoyng, but you ust can t get rd of t. - Geoff Hnton Problems: black box local mnma 18

19 Restrcted Boltzmann Machnes and Deep Belef Networks Sldes from: Geoffrey Hnton, Sue Becker, Yann Le Cun, Yoshua Bengo, Frank Wood

20 Mo1va1ons Supervsed tranng of deep models (e.g. many- layered NNets) s dffcult (op1mza1on problem) Shallow models (SVMs, one- hdden- layer NNets, boos1ng, etc ) are unlkely canddates for learnng hgh- level abstrac1ons needed for AI Unsupervsed learnng could do local- learnng (each module tres ts best to model what t sees) Inference (+ learnng) s ntractable n drected graphcal models wth many hdden varables Current unsupervsed learnng methods don t easly extend to learn mul1ple levels of representa1on

21 Belef Nets A belef net s a drected acyclc graph composed of stochas1c varables. Can observe some of the varables and we would lke to solve two problems: The nference problem: Infer the states of the unobserved varables. The learnng problem: Adust the nterac1ons between varables to make the network more lkely to generate the observed data. stochas1c hdden cause vsble effect Use nets composed of layers of stochas1c bnary varables wth weghted connec1ons. Later, we wll generalze to other types of varable.

22 Stochas1c bnary neurons These have a state of 1 or 0 whch s a stochas1c func1on of the neuron s bas, b, and the nput t receves from other neurons. p( s = 1 1 ) = 1+ exp( b 1 s w ) p( = 1) s b + s 0 w

23 Stochas1c unts Replace the bnary threshold unts by bnary stochas1c unts that make based random decsons. The temperature controls the amount of nose. Decreasng all the energy gaps between confgura1ons s equvalent to rasng the nose level. ) ( ) ( ) ( = = Δ = = Δ = + = + = T E T w s s E s E E gap Energy e e s p temperature

24 The Energy of a ont confgura1on bnary state of unt n ont confgura1on v, h E( v, h) = vh s unts b < s vh s vh w Energy wth confgura1on v on the vsble unts and h on the hdden unts bas of unt weght between unts and ndexes every non- den1cal par of and once

25 Weghts à Energes à Probabl1es Each possble ont confgura1on of the vsble and hdden unts has an energy The energy s determned by the weghts and bases (as n a Hopfeld net). The energy of a ont confgura1on of the vsble and hdden unts determnes ts probablty: p( v, h) E( v, h) e The probablty of a confgura1on over the vsble unts s found by summng the probabl1es of all the ont confgura1ons that contan t.

26 Restrcted Boltzmann Machnes Restrct the connec1vty to make learnng easer. Only one layer of hdden unts. Deal wth more layers later No connec1ons between hdden unts. In an RBM, the hdden unts are cond1onally ndependent gven the vsble states. So can quckly get an unbased sample from the posteror dstrbu1on when gven a data- vector. Ths s a bg advantage over drected belef nets hdden vsble

27 Restrcted Boltzmann Machnes In an RBM, the hdden unts are cond1onally ndependent gven the vsble states. It only takes one step to reach thermal equlbrum when the vsble unts are clamped. Can quckly get the exact value of : < s s > v hdden vsble

28 A pcture of the Boltzmann machne learnng algorthm for an RBM 0 < s s > 1 < s s > < s s > a fantasy t = 0 t = 1 t = 2 t = nfnty Start wth a tranng vector on the vsble unts. Then alternate between upda1ng all the hdden unts n parallel and upda1ng all the vsble unts n parallel. Δw = ε ( < s s > 0 < ss > )

29 Contras1ve dvergence learnng: A quck way to learn an RBM 0 < s s > 1 < s s > t = 0 t = 1 data reconstruc1on Start wth a tranng vector on the vsble unts. Update all the hdden unts n parallel Update the all the vsble unts n parallel to get a reconstruc1on. Update the hdden unts agan. Δw = ε ( < s s > 0 < s s > 1 ) Ths s not followng the gradent of the log lkelhood. But t works well. When we consder nfnte drected nets t wll be easy to see why t works.

30 How to learn a set of features that are good for reconstruc1ng mages of the dgt 2 50 bnary feature neurons Increment weghts between an ac1ve pxel and an ac1ve feature 50 bnary feature neurons Decrement weghts between an ac1ve pxel and an ac1ve feature 16 x 16 pxel mage data (realty) 16 x 16 pxel mage reconstruc1on (beber than realty)

31 Usng an RBM to learn a model of a dgt class Reconstruc1ons by model traned on 2 s Data Reconstruc1ons by model traned on 3 s 0 < s s > 1 < s s > 100 hdden unts (features) data reconstruc1on 256 vsble unts (pxels)

32 The fnal 50 x 256 weghts Each neuron grabs a dfferent feature.

33 How well can we reconstruct the dgt mages from the bnary feature ac1va1ons? Data Reconstruc1on from ac1vated bnary features Data Reconstruc1on from ac1vated bnary features New test mages from the dgt class that the model was traned on Images from an unfamlar dgt class (the network tres to see every mage as a 2)

34 Deep Belef Networks

35 Dvde and conquer mul1layer learnng Re- represen1ng the data: Each 1me the base learner s called, t passes a transformed verson of the data to the next learner. Can we learn a deep, dense DAG one layer at a 1me, star1ng at the bobom, and s1ll guarantee that learnng each layer mproves the overall model of the tranng data? Ths seems very unlkely. Surely we need to know the weghts n hgher layers to learn lower layers?

36 Mul1layer contras1ve dvergence Start by learnng one hdden layer. Then re- present the data as the ac1v1es of the hdden unts. The same learnng algorthm can now be appled to the re- presented data. Can we prove that each step of ths greedy learnng mproves the log probablty of the data under the overall model? What s the overall model?

37 Learnng a deep drected network etc. W T Frst learn wth all the weghts 1ed Ths s exactly equvalent to learnng an RBM Contras1ve dvergence learnng s equvalent to gnorng the small derva1ves contrbuted by the 1ed weghts between deeper layers. h0 v0 W h2 v2 h1 v1 h0 v0 W W W W W T T

38 Then freeze the frst layer of weghts n both drec1ons and learn the remanng weghts (s1ll 1ed together). Ths s equvalent to learnng another RBM, usng the aggregated posteror dstrbu1on of h0 as the data. v1 h0 W T W frozen etc. h2 v2 h1 v1 h0 v0 W W W W W T T T W frozen

39 A smplfed verson wth all hdden layers the same sze The RBM at the top can be vewed as shorthand for an nfnte drected net. When learnng W1 we can vew the model n two qute dfferent ways: The model s an RBM composed of the data layer and h1. The model s an nfnte DAG wth 1ed weghts. A_er learnng W1 we un1e t from the other weght matrces. We then learn W2 whch s s1ll 1ed to all the matrces above t. T W 2 T W 1 h3 h2 h1 data W 3 W 2 W 1

40 A neural network model of dgt recogn1on The top two layers form a restrcted Boltzmann machne whose free energy landscape models the low dmensonal manfolds of the dgts. The valleys have names: 10 label unts 2000 top- level unts 500 unts The model learns a ont densty for labels and mages. To perform recogn1on we can start wth a neutral state of the label unts and do one or two tera1ons of the top- level RBM. Or we can ust compute the free energy of the RBM wth each of the 10 labels 500 unts 28 x 28 pxel mage

41 Show the move of the network genera1ng dgts (avalable at