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DeepLearningTutorial,Release0.1
# sample random velocity
initial_vel s_rng.normal(size=positions.shape)
Sincewenowhaveaninitialpositionandvelocity,wecannowcallthesimulate_dynamicstoobtainthe
proposalforthenewstate
0
.
# perform simulation of particles subject to Hamiltonian dynamics
final_pos, final_vel simulate_dynamics(
initial_pos=positions,
initial_vel=initial_vel,
stepsize=stepsize,
n_steps=n_steps,
energy_fn=energy_fn
)
Wethenaccept/rejecttheproposedstatebasedontheMetropolisalgorithm.
# accept/reject the proposed move based on the e joint t distribution
accept metropolis_hastings_accept(
energy_prev=hamiltonian(positions, initial_vel, energy_fn),
energy_next=hamiltonian(final_pos, final_vel, energy_fn),
s_rng=s_rng
)
wheremetropolis_hastings_acceptandhamiltonianarehelperfunctions,definedasfollows.
def metropolis_hastings_accept(energy_prev, energy_next, s_rng):
"""
Performs a Metropolis-Hastings accept-reject move.
Parameters
----------
energy_prev: theano o vector
Symbolic theano o tensor which contains the energy associated with the
configuration at time-step t.
energy_next: theano o vector
Symbolic theano o tensor which contains the energy associated with the
proposed configuration n at time-step t+1.
s_rng: theano.tensor.shared_randomstreams.RandomStreams
Theano shared random stream object used to o generate e the random number
used in proposal.
Returns
-------
return: boolean
True if move is accepted, False otherwise
"""
ediff energy_prev energy_next
return (TT.exp(ediff) s_rng.uniform(size=energy_prev.shape)) >= 0
def hamiltonian(pos, vel, energy_fn):
"""
Returns the Hamiltonian (sum of potential and kinetic energy) for the given
velocity and d position.
11.2. ImplementingHMCUsingTheano
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Parameters
----------
pos: theano matrix
Symbolic matrix x whose rows are position vectors.
vel: theano matrix
Symbolic matrix x whose rows are velocity vectors.
energy_fn: python function
Python function, operating on symbolic theano variables, used tox
compute the potential energy at a given position.
Returns
-------
return: theano vector
Vector whose i-th entry is the Hamiltonian n at t position pos[i] and
velocity vel[i].
"""
# assuming mass is 1
return energy_fn(pos) kinetic_energy(vel)
def kinetic_energy(vel):
"""Returns the kinetic energy associated with the given velocity
and mass of 1.
Parameters
----------
vel: theano matrix
Symbolic matrix x whose rows are velocity vectors.
Returns
-------
return: theano vector
Vector whose i-th entry is the kinetic entry associated with vel[i].
"""
return 0.5
*
(vel
**
2).sum(axis=1)
hmc_movefinallyreturnsthetuple(accept;final_pos).acceptisasymbolicbooleanvariableindicating
whetherornotthenewstatefinal
p
osshouldbeusedornot.
hmc_updates
The purpose ofhmc_updates isto generatethe listofupdatestoperform, wheneverourHMC sam-
plingfunctioniscalled. hmc_updatesthusreceivesasparameters,aseriesofsharedvariablestoupdate
(positions,stepsizeandavg_acceptance_rate),andtheparametersrequiredtocomputetheirnewstate.
def hmc_updates(positions, stepsize, avg_acceptance_rate, final_pos, accept,
target_acceptance_rate, stepsize_inc, stepsize_dec,
stepsize_min, stepsize_max, avg_acceptance_slowness):
"""This function is executed after ‘n_steps‘ of f HMC C sampling
(‘hmc_move‘ function). It creates the updates dictionary used by
the ‘simulate‘ function. It takes care of updating: the position
(if the move e is s accepted), the stepsize (to track a given target
acceptance rate) and the average acceptance rate (computed as a
126
Chapter11. HybridMonte-CarloSampling
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DeepLearningTutorial,Release0.1
moving average).
Parameters
----------
positions: shared variable, theano matrix
Shared theano matrix whose rows contain the e old d position
stepsize: shared variable, theano scalar
Shared theano scalar containing current step size
avg_acceptance_rate: shared variable, theano scalar
Shared theano scalar containing the current t average e acceptance rate
final_pos: shared variable, theano matrix
Shared theano matrix whose rows contain the e new w position
accept: theano scalar
Boolean-type variable representing whether r or r not the proposed HMC move
should be e accepted d or not.
target_acceptance_rate: float
The stepsize is modified in order to track k this s target acceptance rate.
stepsize_inc: float
Amount by y which h to increment stepsize when n acceptance e rate is too high.
stepsize_dec: float
Amount by y which h to decrement stepsize when n acceptance e rate is too low.
stepsize_min: float
Lower-bound on ‘stepsize‘.
stepsize_min: float
Upper-bound on ‘stepsize‘.
avg_acceptance_slowness: float
Average acceptance rate is computed as an exponential moving average.
(1-avg_acceptance_slowness) is the weight given to the newest
observation.
Returns
-------
rval1: dictionary-like
A dictionary of updates to be used by the ‘HMC_Sampler.simulate‘
function. The e updates target the position, , stepsize e and average
acceptance rate.
"""
## POSITION UPDATES ##
# broadcast ‘accept‘ scalar to tensor with the e same e dimensions as
# final_pos.
accept_matrix accept.dimshuffle(0,
*
((’x’,)
*
(final_pos.ndim 1)))
# if accept is True, update to ‘final_pos‘ else e stay y put
new_positions TT.switch(accept_matrix, , final_pos, positions)
Usingtheabovecode, thedictionarypositions:new_positionscanbeusedtoupdatethestateofthe
samplerwitheither(1)thenewstatefinal_posifacceptisTrue,or(2)theoldstateifacceptisFalse.This
conditionalassignmentisperformedbytheswitchop.
switchexpectsasitsfirstargument,abooleanmaskwiththesamebroadcastabledimensionsasthesecond
andthirdargument. Sinceacceptisscalar-valued,wemustfirstusedimshuffletotransformittoatensor
withfinal_pos:ndimbroadcastabledimensions(accept_matrix).
11.2. ImplementingHMCUsingTheano
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DeepLearningTutorial,Release0.1
hmc_updates additionally y implements an adaptive version of f HMC, as implemented in n the accom-
panying code e to [Ranzato10]. We e start by y tracking the e average acceptance e rate of the HMC move
proposals (across many simulations), , using g an exponential moving average with time constant t 1  
avg_acceptance_slowness.
## ACCEPT RATE UPDATES ##
# perform exponential moving average
mean_dtype theano.scalar.upcast(accept.dtype, , avg_acceptance_rate.dtype)
new_acceptance_rate TT.add(
avg_acceptance_slowness
*
avg_acceptance_rate,
(1.0 avg_acceptance_slowness)
*
accept.mean(dtype=mean_dtype))
Iftheaverageacceptancerateislargerthanthetarget_acceptance_rate,weincreasethestepsizebya
factorofstepsize_incinordertoincreasethemixingrateofourchain.Iftheaverageacceptancerateistoo
lowhowever,stepsizeisdecreasedbyafactorofstepsize_dec,yieldingamoreconservativemixingrate.
Theclipopallowsustomaintainthestepsizeintherange[stepsize_min,stepsize_max].
## STEPSIZE UPDATES ##
# if acceptance rate is too low, our sampler is s too o "noisy" and we reduce
# the stepsize. If it is too high, our sampler r is s too conservative, we can
# get away with a larger stepsize (resulting in n better r mixing).
_new_stepsize TT.switch(avg_acceptance_rate target_acceptance_rate,
stepsize
*
stepsize_inc, stepsize
*
stepsize_dec)
# maintain stepsize in [stepsize_min, stepsize_max]
new_stepsize TT.clip(_new_stepsize, , stepsize_min, stepsize_max)
Thefinalupdateslististhenreturned.
return [(positions, new_positions),
(stepsize, new_stepsize),
(avg_acceptance_rate, new_acceptance_rate)]
HMC_sampler
WefinallytieeverythingtogetherusingtheHMC_Samplerclass.Itsmainelementsare:
• new_from_shared_positions:aconstructormethodwhichallocatesvarioussharedvariablesand
strings togetherthe e calls to hmc_move e and d hmc_updates. . It t also builds s the theano o function
simulate,whosesolepurposeistoexecutetheupdatesgeneratedbyhmc_updates.
• draw: aconveniencemethodwhichcallstheTheanofunctionsimulateandreturnsacopyofthe
contentsofthesharedvariableself:positions.
class HMC_sampler(object):
"""
Convenience wrapper for performing Hybrid Monte e Carlo o (HMC). It creates the
symbolic graph for performing an HMC simulation n (using g ‘hmc_move‘ and
‘hmc_updates‘). The graph is then compiled into o the e ‘simulate‘ function, a
theano function which runs the simulation and updates the required shared
variables.
Users should d interface e with the sampler thorugh h the e ‘draw‘ function which
advances the e markov v chain and returns the current sample by calling
‘simulate‘ and ‘get_position‘ in sequence.
128
Chapter11. HybridMonte-CarloSampling
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DeepLearningTutorial,Release0.1
The hyper-parameters are the same as those used d by y Marc’Aurelio’s
’train_mcRBM.py’ file (available on his personal home page).
"""
def __init__(self,
**
kwargs):
self.__dict__.update(kwargs)
@classmethod
def new_from_shared_positions(
cls,
shared_positions,
energy_fn,
initial_stepsize=0.01,
target_acceptance_rate=.9,
n_steps=20,
stepsize_dec=0.98,
stepsize_min=0.001,
stepsize_max=0.25,
stepsize_inc=1.02,
# used in n geometric c avg. 1.0 would be not moving at all
avg_acceptance_slowness=0.9,
seed=12345
):
"""
:param shared_positions: theano ndarray shared var with
many particle e [initial] positions
:param energy_fn:
callable such that energy_fn(positions)
returns theano vector of energies.
The len of this vector is the batchsize.
The sum of this energy vector must be differentiable (with
theano.tensor.grad) with respect to the e positions s for HMC
sampling to work.
"""
# allocate shared variables
stepsize sharedX(initial_stepsize, ’hmc_stepsize’)
avg_acceptance_rate sharedX(target_acceptance_rate,
’avg_acceptance_rate’)
s_rng TT.shared_randomstreams.RandomStreams(seed)
# define e graph h for an ‘n_steps‘ HMC simulation
accept, final_pos hmc_move(
s_rng,
shared_positions,
energy_fn,
stepsize,
n_steps)
# define e the e dictionary of updates, to apply on every ‘simulate‘ call
simulate_updates hmc_updates(
11.2. ImplementingHMCUsingTheano
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DeepLearningTutorial,Release0.1
shared_positions,
stepsize,
avg_acceptance_rate,
final_pos=final_pos,
accept=accept,
stepsize_min=stepsize_min,
stepsize_max=stepsize_max,
stepsize_inc=stepsize_inc,
stepsize_dec=stepsize_dec,
target_acceptance_rate=target_acceptance_rate,
avg_acceptance_slowness=avg_acceptance_slowness)
# compile e theano o function
simulate function([], , [], updates=simulate_updates)
# create e HMC_sampler r object with the following attributes ...
return cls(
positions=shared_positions,
stepsize=stepsize,
stepsize_min=stepsize_min,
stepsize_max=stepsize_max,
avg_acceptance_rate=avg_acceptance_rate,
target_acceptance_rate=target_acceptance_rate,
s_rng=s_rng,
_updates=simulate_updates,
simulate=simulate)
def draw(self,
**
kwargs):
"""
Returns a a new w position obtained after ‘n_steps‘ of HMC simulation.
Parameters
----------
kwargs: dictionary
The ‘kwargs‘ dictionary is passed to the shared variable
(self.positions) ‘get_value()‘ function.
For example, to avoid
copying the shared variable value, consider passing ‘borrow=True‘.
Returns
-------
rval: numpy matrix
Numpy matrix x whose of dimensions similar to ‘initial_position‘.
"""
self.simulate()
return self.positions.get_value(borrow=False)
11.3 TestingourSampler
WetestourimplementationofHMCbysamplingfromamulti-variateGaussiandistribution. Westartby
generatingarandommeanvectormuandcovariancematrixcov,whichallowsustodefinetheenergy
functionofthecorrespondingGaussiandistribution:gaussian_energy. Wetheninitializethestateofthe
130
Chapter11. HybridMonte-CarloSampling
DeepLearningTutorial,Release0.1
samplerbyallocatingapositionsharedvariable. ItispassedtotheconstructorofHMC_sampleralong
withourtargetenergyfunction.
Followingaburn-inperiod,wethengeneratealargenumberofsamplesandcomparetheempiricalmean
andcovariancematrixtotheirtruevalues.
def sampler_on_nd_gaussian(sampler_cls, burnin, n_samples, dim=10):
batchsize 3
rng numpy.random.RandomState(123)
# Define a covariance and mu for a gaussian
mu numpy.array(rng.rand(dim)
*
10, dtype=theano.config.floatX)
cov numpy.array(rng.rand(dim, dim), dtype=theano.config.floatX)
cov (cov cov.T) 2.
cov[numpy.arange(dim), numpy.arange(dim)] 1.0
cov_inv linalg.inv(cov)
# Define energy function for a multi-variate Gaussian
def gaussian_energy(x):
return 0.5
*
(theano.tensor.dot((x mu), cov_inv)
*
(x mu)).sum(axis=1)
# Declared shared random variable for positions
position rng.randn(batchsize, dim).astype(theano.config.floatX)
position theano.shared(position)
# Create HMC C sampler
sampler sampler_cls(position, gaussian_energy,
initial_stepsize=1e-3, stepsize_max=0.5)
# Start with h a a burn-in process
garbage [sampler.draw() for in xrange(burnin)]
# burn-in Draw
# ‘n_samples‘: result is a 3D tensor of dim [n_samples, batchsize,
# dim]
_samples numpy.asarray([sampler.draw() for in xrange(n_samples)])
# Flatten to o [n_samples
*
batchsize, dim]
samples _samples.T.reshape(dim, -1).T
print 
******
TARGET VALUES
******
print ’target mean:’, mu
print ’target cov:\n, cov
print 
******
EMPIRICAL MEAN/COV USING HMC
******
print ’empirical mean: ’, samples.mean(axis=0)
print ’empirical_cov:\n, numpy.cov(samples.T)
print 
******
HMC INTERNALS
******
print ’final stepsize’, sampler.stepsize.get_value()
print ’final acceptance_rate’, sampler.avg_acceptance_rate.get_value()
return sampler
11.3. TestingourSampler
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DeepLearningTutorial,Release0.1
def test_hmc():
sampler sampler_on_nd_gaussian(HMC_sampler.new_from_shared_positions,
burnin=1000, n_samples=1000, dim=5)
assert abs(sampler.avg_acceptance_rate.get_value() -
sampler.target_acceptance_rate) < .1
assert sampler.stepsize.get_value() >= sampler.stepsize_min
assert sampler.stepsize.get_value() <= sampler.stepsize_max
Theabovecodecanberunusingthecommand: “nosetests-scode/hmc/test_hmc.py”. . Theoutputisas
follows:
[desjagui@atchoum hmc]python test_hmc.py
******
TARGET VALUES
******
target mean: 6.96469186
2.86139335
2.26851454 5.51314769
7.1946897 ]
target cov:
[[ 1.
0.66197111
0.71141257
0.55766643 0.35753822]
0.66197111
1.
0.31053199
0.45455485 0.37991646]
0.71141257
0.31053199
1.
0.62800335 0.38004541]
0.55766643
0.45455485
0.62800335
1.
0.50807871]
0.35753822
0.37991646
0.38004541
0.50807871 1.
]]
******
EMPIRICAL MEAN/COV V USING HMC
******
empirical mean: 6.94155164
2.81526039
2.26301715
5.46536853
7.19414496]
empirical_cov:
[[ 1.05152997
0.68393537
0.76038645
0.59930252 0.37478746]
0.68393537
0.97708159
0.37351422
0.48362404 0.3839558 ]
0.76038645
0.37351422
1.03797111
0.67342957 0.41529132]
0.59930252
0.48362404
0.67342957
1.02865056 0.53613649]
0.37478746
0.3839558
0.41529132
0.53613649 0.98721449]]
******
HMC INTERNALS
******
final stepsize 0.460446628091
final acceptance_rate 0.922502043428
Ascanbeseenabove,thesamplesgeneratedbyourHMCsampleryieldanempiricalmeanandcovariance
matrix,whichareveryclosetothetrueunderlyingparameters.Theadaptivealgorithmalsoseemedtowork
wellasthefinalacceptancerateisclosetoourtargetof0:9.
11.4 References
132
Chapter11. HybridMonte-CarloSampling
CHAPTER
TWELVE
RECURRENTNEURALNETWORKSWITHWORDEMBEDDINGS
12.1 Summary
Inthistutorial,youwilllearnhowto:
• learnWordEmbeddings
• usingRecurrentNeuralNetworksarchitectures
• withContextWindows
inordertoperformSemanticParsing/Slot-Filling(SpokenLanguageUnderstanding)
12.2 Code-Citations-Contact
12.2.1 Code
Directlyrunningexperimentsisalsopossibleusingthisgithubrepository.
12.2.2 Papers
Ifyouusethistutorial,citethefollowingpapers:
• [pdf]GrégoireMesnil,XiaodongHe,LiDengandYoshuaBengio.InvestigationofRecurrent-Neural-
NetworkArchitecturesandLearningMethodsforSpokenLanguageUnderstanding. Interspeech,
2013.
• [pdf]GokhanTur,DilekHakkani-TurandLarryHeck.WhatislefttobeunderstoodinATIS?
• [pdf]ChristianRaymondandGiuseppeRiccardi.Generativeanddiscriminativealgorithmsforspoken
languageunderstanding.Interspeech,2007.
• [pdf]Bastien,Frédéric,Lamblin,Pascal,Pascanu,Razvan,Bergstra,James,Goodfellow,Ian,Berg-
eron,Arnaud,Bouchard,Nicolas,andBengio,Yoshua. Theano: : newfeaturesandspeedimprove-
ments.NIPSWorkshoponDeepLearningandUnsupervisedFeatureLearning,2012.
• [pdf]Bergstra,James,Breuleux,Olivier,Bastien,Frédéric,Lamblin,Pascal,Pascanu,Razvan,Des-
jardins,Guillaume,Turian,Joseph,Warde-Farley,David,andBengio,Yoshua. Theano:aCPUand
133
DeepLearningTutorial,Release0.1
GPUmathexpressioncompiler. InProceedingsofthePythonforScientificComputingConference
(SciPy),June2010.
Thankyou!
12.2.3 Contact
PleaseemailtoGrégoireMesnilforanyproblemreportorfeedback.Wewillbegladtohearfromyou.
12.3 Task
TheSlot-Filling(SpokenLanguageUnderstanding)consistsinassigningalabeltoeachwordgivenasen-
tence.It’saclassificationtask.
12.4 Dataset
AnoldandsmallbenchmarkforthistaskistheATIS(AirlineTravelInformationSystem)datasetcollected
byDARPA.Hereisasentence(orutterance)exampleusingtheInsideOutsideBeginning(IOB)represen-
tation.
Input(words)
show
flights
from
Boston
to
New
York
today
Output(labels)
O
O
O
B-dept
O
B-arr
I-arr
B-date
TheATISofficalsplitcontains4,978/893sentencesforatotalof56,590/9,198words(averagesentence
lengthis15)inthetrain/testset.Thenumberofclasses(differentslots)is128includingtheOlabel(NULL).
AsMicrosoftResearchpeople,wedealwithunseenwordsinthetestsetbymarkinganywordswithonly
onesingleoccurrenceinthetrainingsetas<UNK>andusethistokentorepresentthoseunseenwordsinthe
testset.AsRonanCollobertandcolleagues,weconvertedsequencesofnumberswiththestringDIGITi.e.
1984isconvertedtoDIGITDIGITDIGITDIGIT.
Wesplittheofficialtrainsetintoatrainingandvalidationsetthatcontainrespectively80%and20%ofthe
officialtrainingsentences.Significantperformanceimprovementdifferencehastobegreaterthan0.6%in
F1measureatthe95%levelduetothesmallsizeofthedataset.Forevaluationpurpose,experimentshave
toreportthefollowingmetrics:
• Precision
• Recall
• F1score
WewillusetheconllevalPERLscripttomeasuretheperformanceofourmodels.
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Chapter12. RecurrentNeuralNetworkswithWordEmbeddings
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