Note
This tutorial demonstrates a basic implementation of the RNN-RBM as described in [BoulangerLewandowski12]_ (pdf). We assume the reader is familiar with recurrent neural networks using the scan op and restricted Boltzmann machines (RBM).
Note
The code for this section is available for download here: rnnrbm.py.
You will need the modified Python MIDI package (GPL license) in your $PYTHONPATH or in the working directory in order to convert MIDI files to and from piano-rolls.
The script also assumes that the content of the Nottingham Database of folk tunes has been extracted in the ../data directory.
Alternative MIDI datasets are available here.
Note that both dependencies above can be setup automatically by running the download.sh script in the ../data directory.
Caution!
Depending on your locally installed Theano version, you may have problems running this script. If this is the case, please use the 'bleeding-edge' developer version from github.
The RNN-RBM is an energy-based model for density estimation of temporal sequences, where the feature vector v^{(t)} at time step t may be high-dimensional. It allows to describe multimodal conditional distributions of v^{(t)}|\mathcal A^{(t)}, where \mathcal A^{(t)}\equiv \{v_\tau|\tau<t\} denotes the sequence history at time t, via a series of conditional RBMs (one a each time step) whose parameters b_v^{(t)},b_h^{(t)} depend on the output of a deterministic RNN with hidden units u^{(t)}:
b_v^{(t)} = b_v + W_{uv} u^{(t-1)}
b_h^{(t)} = b_h + W_{uh} u^{(t-1)}
and the single-layer RNN recurrence relation is defined by:
u^{(t)} = \tanh (b_u + W_{uu} u^{(t-1)} + W_{vu} v^{(t)})
The resulting model is unrolled in time in the following figure:
The overall probability distribution is given by the sum over the T time steps in a given sequence:
P(\{v^{(t)}\}) = \sum_{t=1}^T P(v^{(t)} | \mathcal A^{(t)})
where the right-hand side multiplicand is the marginalized probability of the t^\mathrm{th} RBM.
Note that for clarity of the implementation, contrarily to [BoulangerLewandowski12]_, we use the obvious naming convention for weight matrices and we use u^{(t)} instead of \hat h^{(t)} for the recurrent hidden units.
We wish to construct two Theano functions: one to train the RNN-RBM, and one to generate sample sequences from it.
For training, i.e. given \{v^{(t)}\}, the RNN hidden state \{u^{(t)}\} and the associated \{b_v^{(t)}, b_h^{(t)}\} parameters are deterministic and can be readily computed for each training sequence. A stochastic gradient descent (SGD) update on the parameters can then be estimated via contrastive divergence (CD) on the individual time steps of a sequence in the same way that individual training examples are treated in a mini-batch for regular RBMs.
Sequence generation is similar except that the v^{(t)} must be sampled sequentially at each time step with a separate (non-batch) Gibbs chain before being passed down to the recurrence and the sequence history.
The build_rbm function shown below builds a Gibbs chain from an input mini-batch (a binary matrix) via the CD approximation.
Note that it also supports a single frame (a binary vector) in the non-batch case.
def build_rbm(v, W, bv, bh, k):
'''Construct a k-step Gibbs chain starting at v for an RBM.
v : Theano vector or matrix
If a matrix, multiple chains will be run in parallel (batch).
W : Theano matrix
Weight matrix of the RBM.
bv : Theano vector
Visible bias vector of the RBM.
bh : Theano vector
Hidden bias vector of the RBM.
k : scalar or Theano scalar
Length of the Gibbs chain.
Return a (v_sample, cost, monitor, updates) tuple:
v_sample : Theano vector or matrix with the same shape as `v`
Corresponds to the generated sample(s).
cost : Theano scalar
Expression whose gradient with respect to W, bv, bh is the CD-k approximation
to the log-likelihood of `v` (training example) under the RBM.
The cost is averaged in the batch case.
monitor: Theano scalar
Pseudo log-likelihood (also averaged in the batch case).
updates: dictionary of Theano variable -> Theano variable
The `updates` object returned by scan.'''
def gibbs_step(v):
mean_h = T.nnet.sigmoid(T.dot(v, W) + bh)
h = rng.binomial(size=mean_h.shape, n=1, p=mean_h,
dtype=theano.config.floatX)
mean_v = T.nnet.sigmoid(T.dot(h, W.T) + bv)
v = rng.binomial(size=mean_v.shape, n=1, p=mean_v,
dtype=theano.config.floatX)
return mean_v, v
chain, updates = theano.scan(lambda v: gibbs_step(v)[1], outputs_info=[v],
n_steps=k)
v_sample = chain[-1]
mean_v = gibbs_step(v_sample)[0]
monitor = T.xlogx.xlogy0(v, mean_v) + T.xlogx.xlogy0(1 - v, 1 - mean_v)
monitor = monitor.sum() / v.shape[0]
def free_energy(v):
return -(v * bv).sum() - T.log(1 + T.exp(T.dot(v, W) + bh)).sum()
cost = (free_energy(v) - free_energy(v_sample)) / v.shape[0]
return v_sample, cost, monitor, updatesThe build_rnnrbm function defines the RNN recurrence relation to obtain the RBM parameters; the recurrence function is flexible enough to serve both in the training scenario where v^{(t)} is given and the "batch" RBM is constructed at the end on the whole sequence at once, and in the generation scenario where v^{(t)} is sampled separately at each time step using the Gibbs chain defined above.
def build_rnnrbm(n_visible, n_hidden, n_hidden_recurrent):
'''Construct a symbolic RNN-RBM and initialize parameters.
n_visible : integer
Number of visible units.
n_hidden : integer
Number of hidden units of the conditional RBMs.
n_hidden_recurrent : integer
Number of hidden units of the RNN.
Return a (v, v_sample, cost, monitor, params, updates_train, v_t,
updates_generate) tuple:
v : Theano matrix
Symbolic variable holding an input sequence (used during training)
v_sample : Theano matrix
Symbolic variable holding the negative particles for CD log-likelihood
gradient estimation (used during training)
cost : Theano scalar
Expression whose gradient (considering v_sample constant) corresponds to the
LL gradient of the RNN-RBM (used during training)
monitor : Theano scalar
Frame-level pseudo-likelihood (useful for monitoring during training)
params : tuple of Theano shared variables
The parameters of the model to be optimized during training.
updates_train : dictionary of Theano variable -> Theano variable
Update object that should be passed to theano.function when compiling the
training function.
v_t : Theano matrix
Symbolic variable holding a generated sequence (used during sampling)
updates_generate : dictionary of Theano variable -> Theano variable
Update object that should be passed to theano.function when compiling the
generation function.'''
W = shared_normal(n_visible, n_hidden, 0.01)
bv = shared_zeros(n_visible)
bh = shared_zeros(n_hidden)
Wuh = shared_normal(n_hidden_recurrent, n_hidden, 0.0001)
Wuv = shared_normal(n_hidden_recurrent, n_visible, 0.0001)
Wvu = shared_normal(n_visible, n_hidden_recurrent, 0.0001)
Wuu = shared_normal(n_hidden_recurrent, n_hidden_recurrent, 0.0001)
bu = shared_zeros(n_hidden_recurrent)
params = W, bv, bh, Wuh, Wuv, Wvu, Wuu, bu # learned parameters as shared
# variables
v = T.matrix() # a training sequence
u0 = T.zeros((n_hidden_recurrent,)) # initial value for the RNN hidden
# units
# If `v_t` is given, deterministic recurrence to compute the variable
# biases bv_t, bh_t at each time step. If `v_t` is None, same recurrence
# but with a separate Gibbs chain at each time step to sample (generate)
# from the RNN-RBM. The resulting sample v_t is returned in order to be
# passed down to the sequence history.
def recurrence(v_t, u_tm1):
bv_t = bv + T.dot(u_tm1, Wuv)
bh_t = bh + T.dot(u_tm1, Wuh)
generate = v_t is None
if generate:
v_t, _, _, updates = build_rbm(T.zeros((n_visible,)), W, bv_t,
bh_t, k=25)
u_t = T.tanh(bu + T.dot(v_t, Wvu) + T.dot(u_tm1, Wuu))
return ([v_t, u_t], updates) if generate else [u_t, bv_t, bh_t]
# For training, the deterministic recurrence is used to compute all the
# {bv_t, bh_t, 1 <= t <= T} given v. Conditional RBMs can then be trained
# in batches using those parameters.
(u_t, bv_t, bh_t), updates_train = theano.scan(
lambda v_t, u_tm1, *_: recurrence(v_t, u_tm1),
sequences=v, outputs_info=[u0, None, None], non_sequences=params)
v_sample, cost, monitor, updates_rbm = build_rbm(v, W, bv_t[:], bh_t[:],
k=15)
updates_train.update(updates_rbm)
# symbolic loop for sequence generation
(v_t, u_t), updates_generate = theano.scan(
lambda u_tm1, *_: recurrence(None, u_tm1),
outputs_info=[None, u0], non_sequences=params, n_steps=200)
return (v, v_sample, cost, monitor, params, updates_train, v_t,
updates_generate)We now have all the necessary ingredients to start training our network on real symbolic sequences of polyphonic music.
class RnnRbm:
'''Simple class to train an RNN-RBM from MIDI files and to generate sample
sequences.'''
def __init__(self, n_hidden=150, n_hidden_recurrent=100, lr=0.001,
r=(21, 109), dt=0.3):
'''Constructs and compiles Theano functions for training and sequence
generation.
n_hidden : integer
Number of hidden units of the conditional RBMs.
n_hidden_recurrent : integer
Number of hidden units of the RNN.
lr : float
Learning rate
r : (integer, integer) tuple
Specifies the pitch range of the piano-roll in MIDI note numbers, including
r[0] but not r[1], such that r[1]-r[0] is the number of visible units of the
RBM at a given time step. The default (21, 109) corresponds to the full range
of piano (88 notes).
dt : float
Sampling period when converting the MIDI files into piano-rolls, or
equivalently the time difference between consecutive time steps.'''
self.r = r
self.dt = dt
(v, v_sample, cost, monitor, params, updates_train, v_t,
updates_generate) = build_rnnrbm(r[1] - r[0], n_hidden,
n_hidden_recurrent)
gradient = T.grad(cost, params, consider_constant=[v_sample])
updates_train.update(((p, p - lr * g) for p, g in zip(params,
gradient)))
self.train_function = theano.function([v], monitor,
updates=updates_train)
self.generate_function = theano.function([], v_t,
updates=updates_generate)
def train(self, files, batch_size=100, num_epochs=200):
'''Train the RNN-RBM via stochastic gradient descent (SGD) using MIDI
files converted to piano-rolls.
files : list of strings
List of MIDI files that will be loaded as piano-rolls for training.
batch_size : integer
Training sequences will be split into subsequences of at most this size
before applying the SGD updates.
num_epochs : integer
Number of epochs (pass over the training set) performed. The user can
safely interrupt training with Ctrl+C at any time.'''
assert len(files) > 0, 'Training set is empty!' \
' (did you download the data files?)'
dataset = [midiread(f, self.r,
self.dt).piano_roll.astype(theano.config.floatX)
for f in files]
try:
for epoch in xrange(num_epochs):
numpy.random.shuffle(dataset)
costs = []
for s, sequence in enumerate(dataset):
for i in xrange(0, len(sequence), batch_size):
cost = self.train_function(sequence[i:i + batch_size])
costs.append(cost)
print 'Epoch %i/%i' % (epoch + 1, num_epochs),
print numpy.mean(costs)
sys.stdout.flush()
except KeyboardInterrupt:
print 'Interrupted by user.'
def generate(self, filename, show=True):
'''Generate a sample sequence, plot the resulting piano-roll and save
it as a MIDI file.
filename : string
A MIDI file will be created at this location.
show : boolean
If True, a piano-roll of the generated sequence will be shown.'''
piano_roll = self.generate_function()
midiwrite(filename, piano_roll, self.r, self.dt)
if show:
extent = (0, self.dt * len(piano_roll)) + self.r
pylab.figure()
pylab.imshow(piano_roll.T, origin='lower', aspect='auto',
interpolation='nearest', cmap=pylab.cm.gray_r,
extent=extent)
pylab.xlabel('time (s)')
pylab.ylabel('MIDI note number')
pylab.title('generated piano-roll')
if __name__ == '__main__':
model = RnnRbm()
model.train(glob.glob('../data/Nottingham/train/*.mid'))
model.generate('sample1.mid')
model.generate('sample2.mid')
pylab.show()We ran the code on the Nottingham database for 200 epochs; training took approximately 24 hours.
The output was the following:
Epoch 1/200 -15.0308940028
Epoch 2/200 -10.4892606673
Epoch 3/200 -10.2394696138
Epoch 4/200 -10.1431669994
Epoch 5/200 -9.7005382843
Epoch 6/200 -8.5985647524
Epoch 7/200 -8.35115428534
Epoch 8/200 -8.26453580552
Epoch 9/200 -8.21208991542
Epoch 10/200 -8.16847274143
... truncated for brevity ...
Epoch 190/200 -4.74799179994
Epoch 191/200 -4.73488515216
Epoch 192/200 -4.7326138489
Epoch 193/200 -4.73841636884
Epoch 194/200 -4.70255511452
Epoch 195/200 -4.71872634914
Epoch 196/200 -4.7276415885
Epoch 197/200 -4.73497644728
Epoch 198/200 -inf
Epoch 199/200 -4.75554987143
Epoch 200/200 -4.72591935412
The figures below show the piano-rolls of two sample sequences and we provide the corresponding MIDI files:
Listen to sample1.mid
Listen to sample2.mid
The code shown in this tutorial is a stripped-down version that can be improved in the following ways:
- Preprocessing: transposing the sequences in a common tonality (e.g. C major / minor) and normalizing the tempo in beats (quarternotes) per minute can have the most effect on the generative quality of the model.
- Pretraining techniques: initialize the W,b_v,b_h parameters with independent RBMs with fully shuffled frames (i.e. W_{uh}=W_{uv}=W_{uu}=W_{vu}=0); initialize the W_{uv},W_{uu},W_{vu},b_u parameters of the RNN with the auxiliary cross-entropy objective via either SGD or, preferably, Hessian-free optimization [BoulangerLewandowski12]_.
- Optimization techniques: gradient clipping, Nesterov momentum and the use of NADE for conditional density estimation.
- Hyperparameter search: learning rate (separately for the RBM and RNN parts), learning rate schedules, batch size, number of hidden units (recurrent and RBM), momentum coefficient, momentum schedule, Gibbs chain length k and early stopping.
- Learn the initial condition u^{(0)} as a model parameter.
A few samples generated with code including these features are available here.