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11 Modeling and generating sequences of polyphonic music with the RNN-RBM¶

Modeling and generating sequences of polyphonic music with the RNN-RBM

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

Need Theano 0.6 or more recent.

The RNN-RBM

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 $u^{(t)}$ :

(1) $b_v^{(t)} = b_v + W_{uv} u^{(t-1)}$

(2) $b_h^{(t)} = b_h + W_{uh} u^{(t-1)}$

and the single-layer RNN recurrence relation is defined by:

(3) $u^{(t)} = \tanh (b_u + W_{uu} u^{(t-1)} + W_{vu} v^{(t)})$

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:

(4) $P(\{v^{(t)}\}) = \sum_{t=1}^T P(v^{(t)} | \mathcal A^{(t)})$

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.

Implementation

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 RBM layer

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, updates

The RNN layer

The 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)

Putting it all together

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')

Results

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

How to improve this code

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.