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DeepLearningTutorial,Release0.1
computes the e error on a batch from the testing set
:type datasets: list of pairs of theano.tensor.TensorType
:param datasets: It is a list that contain n all l the datasets;
the has to contain three pairs, ‘train‘,
‘valid‘, ‘test‘ in this order, where each pair
is formed of two Theano variables, one for the
datapoints, the other for the labels
:type batch_size: int
:param batch_size: size of a minibatch
:type learning_rate: float
:param learning_rate: learning rate used during finetune stage
’’’
(train_set_x, train_set_y) datasets[0]
(valid_set_x, valid_set_y) datasets[1]
(test_set_x, test_set_y) datasets[2]
# compute e number r of minibatches for training, validation and testing
n_valid_batches valid_set_x.get_value(borrow=True).shape[0]
n_valid_batches /= batch_size
n_test_batches test_set_x.get_value(borrow=True).shape[0]
n_test_batches /= batch_size
index T.lscalar(’index’)
# index to a [mini]batch
# compute e the e gradients with respect to the e model l parameters
gparams T.grad(self.finetune_cost, self.params)
# compute e list t of fine-tuning updates
updates []
for param, gparam in zip(self.params, gparams):
updates.append((param, param gparam
*
learning_rate))
train_fn theano.function(
inputs=[index],
outputs=self.finetune_cost,
updates=updates,
givens={
self.x: train_set_x[
index
*
batch_size: (index 1)
*
batch_size
],
self.y: train_set_y[
index
*
batch_size: (index 1)
*
batch_size
]
}
)
test_score_i theano.function(
[index],
self.errors,
givens={
10.3. Implementation
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DeepLearningTutorial,Release0.1
self.x: test_set_x[
index
*
batch_size: (index 1)
*
batch_size
],
self.y: test_set_y[
index
*
batch_size: (index 1)
*
batch_size
]
}
)
valid_score_i theano.function(
[index],
self.errors,
givens={
self.x: valid_set_x[
index
*
batch_size: (index 1)
*
batch_size
],
self.y: valid_set_y[
index
*
batch_size: (index 1)
*
batch_size
]
}
)
# Create e a a function that scans the entire validation set
def valid_score():
return [valid_score_i(i) for in xrange(n_valid_batches)]
# Create e a a function that scans the entire test set
def test_score():
return [test_score_i(i) for in xrange(n_test_batches)]
return train_fn, valid_score, test_score
Notethatthereturnedvalid_scoreandtest_scorearenotTheanofunctions, butratherPython
functions. Theseloopovertheentirevalidationsetandtheentiretestsettoproducealistofthelosses
obtainedoverthesesets.
10.4 Puttingitalltogether
Thefewlinesofcodebelowconstructsthedeepbeliefnetwork:
numpy_rng numpy.random.RandomState(123)
print ’... building the model’
# construct the Deep Belief Network
dbn DBN(numpy_rng=numpy_rng, n_ins=28
*
28,
hidden_layers_sizes=[100010001000],
n_outs=10)
Therearetwostagesintrainingthisnetwork:(1)alayer-wisepre-trainingand(2)afine-tuningstage.
Forthepre-trainingstage,weloopoverallthelayersofthenetwork. Foreachlayer,weusethecom-
piledtheanofunctionwhichdeterminestheinputtothei-thlevelRBMandperformsonestepofCD-
116
Chapter10. DeepBeliefNetworks
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DeepLearningTutorial,Release0.1
kwithinthis RBM. Thisfunctionis appliedtothetrainingsetforafixednumberofepochsgivenby
pretraining_epochs.
#########################
# PRETRAINING G THE E MODEL #
#########################
print ’... getting the pretraining functions’
pretraining_fns dbn.pretraining_functions(train_set_x=train_set_x,
batch_size=batch_size,
k=k)
print ’... pre-training the model’
start_time timeit.default_timer()
## Pre-train n layer-wise
for in xrange(dbn.n_layers):
# go through pretraining epochs
for epoch in xrange(pretraining_epochs):
# go o through h the training set
[]
for batch_index in xrange(n_train_batches):
c.append(pretraining_fns[i](index=batch_index,
lr=pretrain_lr))
print ’Pre-training layer %i, epoch %d, cost ’ (i, epoch),
print numpy.mean(c)
end_time timeit.default_timer()
Thefine-tuningloopisverysimilartotheoneintheMultilayerPerceptrontutorial,theonlydifferencebeing
thatwenowusethefunctionsgivenbybuild_finetune_functions.
10.5 RunningtheCode
Theusercanrunthecodebycalling:
python code/DBN.py
Withthedefaultparameters,thecoderunsfor100pre-trainingepochswithmini-batchesofsize10. This
correspondstoperforming500,000unsupervisedparameterupdates.Weuseanunsupervisedlearningrate
of0.01,withasupervisedlearningrateof0.1. TheDBNitselfconsistsofthreehiddenlayerswith1000
unitsperlayer. Withearly-stopping,thisconfigurationachievedaminimalvalidationerrorof1.27with
correspondingtesterrorof1.34after46supervisedepochs.
OnanIntel(R)Xeon(R)CPUX5560runningat2.80GHz,usingamulti-threadedMKLlibrary(runningon
4cores),pretrainingtook615minuteswithanaverageof2.05mins/(layer*epoch).Fine-tuningtookonly
101minutesorapproximately2.20mins/epoch.
Hyper-parameterswereselectedbyoptimizingonthevalidationerror. Wetestedunsupervisedlearning
ratesinf10
1
;:::;10
5
gandsupervisedlearningratesinf10
1
;:::;10
4
g. Wedidnotuseanyformof
regularizationbesidesearly-stopping,nordidweoptimizeoverthenumberofpretrainingupdates.
10.5. RunningtheCode
117
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DeepLearningTutorial,Release0.1
10.6 TipsandTricks
Onewaytoimprovetherunningtimeofyourcode(giventhatyouhavesufficientmemoryavailable),is
tocomputetherepresentationoftheentire datasetatlayeriinasinglepass, oncetheweightsofthe
i 1-thlayershavebeenfixed.Namely,startbytrainingyourfirstlayerRBM.Onceitistrained,youcan
computethehiddenunitsvaluesforeveryexampleinthedatasetandstorethisasanewdatasetwhichis
usedtotrainthe2ndlayerRBM.OnceyoutrainedtheRBMforlayer2,youcompute,inasimilarfashion,
thedatasetforlayer3andsoon. Thisavoidscalculatingtheintermediate(hiddenlayer)representations,
pretraining_epochstimesattheexpenseofincreasedmemoryusage.
118
Chapter10. DeepBeliefNetworks
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CHAPTER
ELEVEN
HYBRIDMONTE-CARLOSAMPLING
Note: Thisisanadvancedtutorial,whichshowshowonecanimplementedHybridMonte-Carlo(HMC)
samplingusingTheano. WeassumethereaderisalreadyfamiliarwithTheanoandenergy-basedmodels
suchastheRBM.
Note: Thecodeforthissectionisavailablefordownloadhere.
11.1 Theory
Maximumlikelihoodlearningofenergy-basedmodelsrequiresarobustalgorithmtosamplenegativephase
particles(seeEq.(4)oftheRestrictedBoltzmannMachines(RBM)tutorial).WhentrainingRBMswithCD
orPCD,thisistypicallydonewithblockGibbssampling,wheretheconditionaldistributionsp(hjv)and
p(vjh)areusedasthetransitionoperatorsoftheMarkovchain.
Incertaincaseshowever,theseconditionaldistributionsmightbedifficulttosamplefrom(i.e. requiring
expensivematrixinversions,asinthecaseofthe“mean-covarianceRBM”).Also,evenifGibbssampling
canbedoneefficiently,itneverthelessoperatesviaarandomwalkwhichmightnotbestatisticallyefficient
forsomedistributions.Inthiscontext,andwhensamplingfromcontinuousvariables,HybridMonteCarlo
(HMC)canprovetobeapowerfultool[Duane87]. Itavoidsrandomwalkbehaviorbysimulatingaphys-
icalsystemgovernedbyHamiltoniandynamics,potentiallyavoidingtrickyconditionaldistributionsinthe
process.
InHMC,modelsamplesareobtainedbysimulatingaphysicalsystem,whereparticlesmoveaboutahigh-
dimensionallandscape, subjecttopotentialandkineticenergies. . Adaptingthenotationfrom[Neal93],
particlesarecharacterizedbyapositionvectororstates2R
D
andvelocityvector2R
D
.Thecombined
stateofaparticleisdenotedas=(s;).TheHamiltonianisthendefinedasthesumofpotentialenergy
E(s)(sameenergyfunctiondefinedbyenergy-basedmodels)andkineticenergyK(),asfollows:
H(s;)=E(s)+K()=E(s)+
1
2
X
i
2
i
Insteadofsamplingp(s)directly,HMCoperatesbysamplingfromthecanonicaldistributionp(s;) =
1
Z
exp( H(s;))=p(s)p(). . Becausethetwovariablesareindependent,marginalizingoveristrivial
andrecoverstheoriginaldistributionofinterest.
HamiltonianDynamics
119
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DeepLearningTutorial,Release0.1
States andvelocityaremodifiedsuchthatH(s;)remainsconstantthroughoutthesimulation. . The
differentialequationsaregivenby:
ds
i
dt
=
@H
@
i
=
i
d
i
dt
@H
@s
i
@E
@s
i
(11.1)
Asshownin[Neal93],theabovetransformationpreservesvolumeandisreversible. Theabovedynamics
canthusbeusedastransitionoperatorsofaMarkovchainandwillleavep(s;)invariant. Thatchainby
itselfisnotergodichowever,sincesimulatingthedynamicsmaintainsafixedHamiltonianH(s;). HMC
thusalternateshamiltoniandynamicsteps,withGibbssamplingofthevelocity.Becausep(s)andp()are
independent,sampling
new
p(js)istrivialsincep(js)=p(),wherep()isoftentakentobethe
uni-variateGaussian.
TheLeap-FrogAlgorithm
Inpractice,wecannotsimulateHamiltoniandynamicsexactlybecauseoftheproblemoftimediscretization.
Thereareseveralwaysonecandothis.TomaintaininvarianceoftheMarkovchainhowever,caremustbe
takentopreservethepropertiesofvolumeconservationandtimereversibility. Theleap-frogalgorithm
maintainsthesepropertiesandoperatesin3steps:
i
(t+=2)=
i
(t) 
2
@
@s
i
E(s(t))
s
i
(t+)=s
i
(t)+
i
(t+=2)
i
(t+)=
i
(t+=2) 
2
@
@s
i
E(s(t+))
(11.2)
Wethusperformahalf-stepupdateofthevelocityattimet+=2,whichisthenusedtocomputes(t+)
and(t+).
Accept/Reject
Inpractice,usingfinitestepsizeswillnotpreserveH(s;)exactlyandwillintroducebiasinthesimulation.
Also,roundingerrorsduetotheuseoffloatingpointnumbersmeansthattheabovetransformationwillnot
beperfectlyreversible.
HMCcancelstheseeffectsexactlybyaddingaMetropolisaccept/rejectstage,afternleapfrogsteps. The
newstate
0
=(s
0
;
0
)isacceptedwithprobabilityp
acc
(;
0
),definedas:
p
acc
(;
0
)=min
1;
exp( H(s
0
;
0
)
exp( H(s;)
HMCAlgorithm
Inthistutorial,weobtainanewHMCsampleasfollows:
1. sampleanewvelocityfromaunivariateGaussiandistribution
2. performnleapfrogstepstoobtainthenewstate
0
3. performaccept/rejectmoveof
0
120
Chapter11. HybridMonte-CarloSampling
DeepLearningTutorial,Release0.1
11.2 ImplementingHMCUsingTheano
InTheano,updatedictionariesandsharedvariablesprovideanaturalwaytoimplementasamplingalgo-
rithm.ThecurrentstateofthesamplercanberepresentedasaTheanosharedvariable,withHMCupdates
beingimplementedbytheupdateslistofaTheanofunction.
WebreakdowntheHMCalgorithmintothefollowingsub-components:
• simulate_dynamics:asymbolicPythonfunctionwhich,givenaninitialpositionandvelocity,will
performn_stepsleapfrogupdatesandreturnthesymbolicvariablesfortheproposedstate
0
.
• hmc_move: asymbolicPythonfunctionwhichgivenastartingposition,generatesbyrandomly
samplingavelocityvector. Itthencallssimulate_dynamicsanddetermineswhetherthetransition
!
0
istobeaccepted.
• hmc_updates:aPythonfunctionwhich,giventhesymbolicoutputsofhmc_move,generatesthelist
ofupdatesforasingleiterationofHMC.
• HMC_sampler:aPythonhelperclasswhichwrapseverythingtogether.
simulate_dynamics
Toperformnleapfrogsteps,wefirstneedtodefineafunctionoverwhichScancaniterateover. Instead
ofimplementingEq.(11.2)verbatim,noticethatwecanobtains(t+n)and(t+n)byperformingan
initialhalf-stepupdatefor,followedbynfull-stepupdatesfors;andonelasthalf-stepupdatefor.In
loopform,thisgives:
i
(t+=2)=
i
(t) 
2
@
@s
i
E(s(t))
s
i
(t+)=s
i
(t)+
i
(t+=2)
Form2[2;n],performfullupdates:
i
(t+(m 1=2))=
i
(t+(m 3=2)) 
@
@s
i
E(s(t+(m 1)))
s
i
(t+m)=s
i
(t)+
i
(t+(m 1=2))
i
(t+n)=
i
(t+(n 1=2)) 
2
@
@s
i
E(s(t+n))
(11.3)
Theinner-loopdefinedaboveisimplementedbythefollowingleapfrogfunction,withpos,velandstep
replacings;andrespectively.
def leapfrog(pos, vel, step):
"""
Inside loop of Scan. Performs one step of leapfrog update, using
Hamiltonian dynamics.
Parameters
----------
pos: theano matrix
in leapfrog update equations, represents pos(t), position at time t
vel: theano matrix
in leapfrog update equations, represents vel(t - stepsize/2),
velocity at time (t - stepsize/2)
11.2. ImplementingHMCUsingTheano
121
DeepLearningTutorial,Release0.1
step: theano scalar
scalar value controlling amount by which to move
Returns
-------
rval1: [theano matrix, theano matrix]
Symbolic theano matrices for new position pos(t + stepsize), and
velocity vel(t + stepsize/2)
rval2: dictionary
Dictionary of updates for the Scan Op
"""
# from pos(t) and vel(t-stepsize/2), compute vel(t+stepsize/2)
dE_dpos TT.grad(energy_fn(pos).sum(), pos)
new_vel vel step
*
dE_dpos
# from vel(t+stepsize/2) compute pos(t+stepsize)
new_pos pos step
*
new_vel
return [new_pos, new_vel], {}
# compute velocity at time-step: t + stepsize/2
Thesimulate
d
ynamicsfunctionperformsthefullalgorithmofEqs.(11.3).Westartwiththeinitialhalf-
stepupdateofandfull-stepofs,andthenscanovertheleapfrogmethodn_steps 1times.
def simulate_dynamics(initial_pos, initial_vel, stepsize, n_steps, energy_fn):
"""
Return final l (position, , velocity) obtained after an ‘n_steps‘ leapfrog
updates, using Hamiltonian dynamics.
Parameters
----------
initial_pos: shared d theano matrix
Initial position at which to start the simulation
initial_vel: shared d theano matrix
Initial velocity of particles
stepsize: shared theano scalar
Scalar value controlling amount by which to o move
energy_fn: python function
Python function, operating on symbolic theano variables, used to
compute the potential energy at a given position.
Returns
-------
rval1: theano o matrix
Final positions obtained after simulation
rval2: theano o matrix
Final velocity obtained after simulation
"""
def leapfrog(pos, vel, step):
"""
Inside loop of Scan. Performs one step of leapfrog update, using
Hamiltonian dynamics.
122
Chapter11. HybridMonte-CarloSampling
DeepLearningTutorial,Release0.1
Parameters
----------
pos: theano matrix
in leapfrog update equations, represents pos(t), position at time t
vel: theano matrix
in leapfrog update equations, represents vel(t - stepsize/2),
velocity at time (t - stepsize/2)
step: theano scalar
scalar value controlling amount by which to move
Returns
-------
rval1: [theano matrix, theano matrix]
Symbolic theano matrices for new position pos(t + stepsize), and
velocity vel(t + stepsize/2)
rval2: dictionary
Dictionary of updates for the Scan Op
"""
# from pos(t) and vel(t-stepsize/2), compute vel(t+stepsize/2)
dE_dpos TT.grad(energy_fn(pos).sum(), pos)
new_vel vel step
*
dE_dpos
# from vel(t+stepsize/2) compute pos(t+stepsize)
new_pos pos step
*
new_vel
return [new_pos, new_vel], {}
# compute velocity at time-step: t + stepsize/2
initial_energy energy_fn(initial_pos)
dE_dpos TT.grad(initial_energy.sum(), initial_pos)
vel_half_step initial_vel 0.5
*
stepsize
*
dE_dpos
# compute position at time-step: t + stepsize
pos_full_step initial_pos stepsize
*
vel_half_step
# perform leapfrog updates: the scan op is used d to o repeatedly compute
# vel(t + (m-1/2)
*
stepsize) and pos(t + m
*
stepsize) for m in [2,n_steps].
(all_pos, all_vel), scan_updates theano.scan(
leapfrog,
outputs_info=[
dict(initial=pos_full_step),
dict(initial=vel_half_step),
],
non_sequences=[stepsize],
n_steps=n_steps 1)
final_pos all_pos[-1]
final_vel all_vel[-1]
# NOTE: Scan n always s returns an updates dictionary, in case the
# scanned function draws samples from a RandomStream. These
# updates must then be used when compiling the e Theano o function, to
# avoid drawing the same random numbers each time the function is
# called. In n this s case however, we consciously y ignore
# "scan_updates" because we know it is empty.
assert not scan_updates
11.2. ImplementingHMCUsingTheano
123
DeepLearningTutorial,Release0.1
# The last velocity returned by scan is vel(t +
# (n_steps - - 1 1 / 2)
*
stepsize) We therefore perform one more half-step
# to return vel(t + n_steps
*
stepsize)
energy energy_fn(final_pos)
final_vel final_vel 0.5
*
stepsize
*
TT.grad(energy.sum(), final_pos)
# return new w proposal l state
return final_pos, final_vel
# start-snippet-1
Afinalhalf-stepisperformedtocompute(t+n),andthefinalproposedstate
0
isreturned.
hmc_move
Thehmc_movefunctionimplementstheremainingsteps(steps1and3)ofanHMCmoveproposal(while
wrappingthesimulate_dynamicsfunction).Givenamatrixofinitialstatess2R
ND
(positions)and
energyfunctionE(s)(energy_fn),itdefinesthesymbolicgraphforcomputingn_stepsofHMC,usinga
givenstepsize.Thefunctionprototypeisasfollows:
def hmc_move(s_rng, positions, energy_fn, stepsize, , n_steps):
"""
This function n performs s one-step of Hybrid Monte-Carlo sampling. We start by
sampling a random velocity from a univariate Gaussian distribution, perform
‘n_steps‘ leap-frog updates using Hamiltonian dynamics and accept-reject
using Metropolis-Hastings.
Parameters
----------
s_rng: theano o shared d random stream
Symbolic random m number generator used to draw random velocity and
perform accept-reject move.
positions: shared theano matrix
Symbolic matrix x whose rows are position vectors.
energy_fn: python function
Python function, operating on symbolic theano variables, used to
compute the potential energy at a given position.
stepsize:
shared theano scalar
Shared variable containing the stepsize to o use e for ‘n_steps‘ of HMC
simulation steps.
n_steps: integer
Number of f HMC C steps to perform before proposing a new position.
Returns
-------
rval1: boolean
True if move is accepted, False otherwise
rval2: theano o matrix
Matrix whose rows contain the proposed "new w position"
"""
Westartbysamplingrandomvelocities,usingtheprovidedsharedRandomStreamobject. Velocitiesare
sampledindependentlyforeachdimensionandforeachparticleundersimulation,yieldingaNDmatrix.
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Chapter11. HybridMonte-CarloSampling
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