Theano-Deep Learning Tutorials 笔记：Modeling and generating sequences of polyphonic music with the RNN

Posted 2022-11-13 slim1017

教程地址：http://www.deeplearning.net/tutorial/rnnrbm.html#rnnrbm

代码，数据集，论文 见教程。

 

The RNN-RBM

RNN-RBM也是能量模型，用于对时间序列的密度估计，在 time step t 的特征向量   为高维向量。

它可以描述多峰的条件概率分布 , where。

表示在 time t 时刻，历史序列（t 之前所有）。

每一个 time step 就是一个 RBM，而RBM的参数又由 RNN（隐藏层为）决定。

                         (1)

                        (2)

 

RNN隐藏层表示为：                        (3)   最终模型图示：    

The overall probability distribution is given by the sum over the time stepsin agiven sequence:

                    (4)

where the right-hand side multiplicand is the marginalized probability of the RBM.

 

 

Implementation

教程实现了两个函数：一个训练RNN-RBM，另一个生成采样序列。

训练时，有, RNN隐藏层 and 参数 （可以计算出）。参数更新为SGD随机梯度下降，和RBM训练类似，使用 contrastive divergence （CD）算法。

序列的生成和RNN相似，只是在每个time step 都需要按RBM中的Gibbs采样得出。

 

The RBM layer

函数 build_rbm 建立RBM部分（图例中上部）Gibbs链，输入为 mini-batch(a binary matrix)；输入也可以不是 mini-batch（也可以说当mini-batch为1时），a binary vector。

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

函数 build_rnnrbm 融合RNN和RBM，关系如上文图例。

RNN部分（图例中下部）在训练时： 已知，RNN的训练不需要RBM的参数，先把RNN中隐藏层 u0到uT 按公式（3）计算出来，再把 T 个RBM 一次性构建出来。

模型训练完成后：RNN和RBM互相影响， 需在每个 time step  t 通过 t 时刻的 RBM Gibbs采样得到，从而计算 t 时刻的RNN隐藏层 u以及后续 time step 的RBM（公式（2，3））和RNN。

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

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 range(num_epochs):
                numpy.random.shuffle(dataset)
                costs = []

                for s, sequence in enumerate(dataset):
                    for i in range(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

在Nottingham数据集上运行 200 epochs，训练大约 24 小时。

The figures below show the piano-rolls of two sample sequences and we provide the corresponding MIDI files:

Listen to sample1.mid

Listen tosample2.mid

感觉教程对数据集介绍不太清楚，不太知道输入输出是啥，下面介绍下：

不难看出  piano-rolls 为输入，为持续60s的序列（对应 time step）样本，黑色表示1，白色表示0，对应文中提到的输入为 binary ，看下图应该很容易知道 piano-rolls 具体是啥了。

Nottingham数据集中每个样本为一个行数为150左右（代表150个time step），列数为88（代表21-109的midi note number，对应钢琴的音域A0到C8，可以说音域吗。。。）的矩阵，代表一段曲子的 piano-rolls，矩阵元素都为0或1。

目的是通过 序列piano-rolls 预测 后续 piano-rolls。

MIDI note number 60 就是"Middle C"或C5。就是唱歌弹琴乐谱啥的里面那个音高的数值化，范围大概是0到127（钢琴是21-109），数值越大音调越高。