前言：

　　本文主要是bengio的deep learning tutorial教程主頁中最后一個sample：rnn-rbm in polyphonic music.?即用RNN-RBM來model復調音樂，訓練過程中采用的是midi格式的音頻文件，接著用建好的model來產生復調音樂。對音樂建模的難點在與每首樂曲中幀間是高度時間相關的（這樣樣本的維度會很高），用普通的網絡模型是不能搞定的（普通設計網絡模型沒有考慮時間維度，圖模型中的HMM有這方面的能力），這種情況下可以采用RNN來處理，這里的RNN為recurrent neural network中文為循環神經網絡，另外還有一種RNN為recursive neural network翻為遞歸神經網絡。本文中指的是循環神經網絡。

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　　RNN簡單介紹：

　　首先來看RNN和普通的feed-forward網絡有什么不同。RNN的網絡框架如下：

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　　由結構圖可以知道，RNN和feed-forward相比只是中間隱含層多了一個循環的圈而已，這個圈表示上一次隱含層的輸出作為這一次隱含層的輸入，當然此時的輸入是需要乘以一個權值矩陣的，這樣的話RNN模型參數只多了個權值矩陣。更形象的RNN圖可以參考：

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　　以及圖：

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　　按照上圖所示，可知道RNN網絡前向傳播過程中滿足下面的公式（參考文獻Learning Recurrent Neural Networks with Hessian-Free Optimization）：

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　　其代價函數可以是重構的誤差：

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　　也可以是交叉熵：

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　　相信熟悉普通深究網絡的同學看懂這些應該不難。

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　　RNN-RBM簡單介紹：

　　RNN-RBM來自ICML2012的論文：Modeling Temporal Dependencies in High-Dimensional Sequences: Application to Polyphonic Music Generation and Transcription，它由一個單層的RBM網絡和單層的RNN網絡構成，且由RNN網絡的輸出作為最終網絡的輸出。RBM部分當生成模型的功能，比如這里的音樂生成，RNN部分當判別模型作用，比如它的輸出當值可當做提取的特征。RNN-RBM模型的結構如下：

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　　模型上面是RBM部分，下面是RNN部分，對應的公式可以參考論文。模型中一共有9個參數：

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　　整個模型的代價函數為-P(v)，其中：

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　　對該loss function求導，然后采用SGD算法就可以求出模型中的各個參數了。當然了，其中的RBM部分還需要用Gibbs采樣完成CD-k算法。

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　　實驗結果：

　　實驗部分參考http://deeplearning.net/tutorial/rnnrbm.html，實驗須用的數據和paper對應的見http://www-etud.iro.umontreal.ca/~boulanni/icml2012. 由于本人對樂理方面的知識不是很懂，很多實驗代碼細節沒有去深究，只是看下算法的大概流程。由RNN-RBM生成的兩個pinao roll數據如下（程序跑了20個小時左右）：?

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　　迭代200次后的cost為：

...... Epoch 197/200 -4.7050858655 Epoch 198/200 -4.69198161366 Epoch 199/200 -4.66586797348 Epoch 200/200 -4.68651185036

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　　代碼如下：?　　

# Author: Nicolas Boulanger-Lewandowski # University of Montreal (2012) # RNN-RBM deep learning tutorial # More information at http://deeplearning.net/tutorial/rnnrbm.htmlimport glob import os import sysimport numpy try:import pylab except ImportError:print "pylab isn't available, if you use their fonctionality, it will crash"print "It can be installed with 'pip install -q Pillow'"from midi.utils import midiread, midiwrite import theano import theano.tensor as T from theano.tensor.shared_randomstreams import RandomStreams#Don't use a python long as this don't work on 32 bits computers. numpy.random.seed(0xbeef) rng = RandomStreams(seed=numpy.random.randint(1 << 30)) theano.config.warn.subtensor_merge_bug = False#給定rbm的3個參數w,bv,bh，輸入端數據v，以及gibbs采用長度k #返回的tuple元素依次是:v_samples(k次gibbs采用得到的輸入端數據,01化后的),cost(rbm模型中的-log(v)),monitor(監控用變量), #updates(保留每次迭代的中間過程，如果是shared變量的話) def build_rbm(v, W, bv, bh, k):'''Construct a k-step Gibbs chain starting at v for an RBM.v : Theano vector or matrixIf a matrix, multiple chains will be run in parallel (batch). W : Theano matrixWeight matrix of the RBM. bv : Theano vectorVisible bias vector of the RBM. bh : Theano vectorHidden bias vector of the RBM. k : scalar or Theano scalarLength 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 scalarExpression whose gradient with respect to W, bv, bh is the CD-k approximationto the log-likelihood of `v` (training example) under the RBM.The cost is averaged in the batch case. monitor: Theano scalarPseudo log-likelihood (also averaged in the batch case). updates: dictionary of Theano variable -> Theano variableThe `updates` object returned by scan.'''def gibbs_step(v): #該函數功能是一次gibbs采樣后得到的mean_v,vmean_h = T.nnet.sigmoid(T.dot(v, W) + bh)h = rng.binomial(size=mean_h.shape, n=1, p=mean_h, #產生二項分布,隱含層節點01化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, #反向傳播，輸入層節點也01化dtype=theano.config.floatX)return mean_v, v #一次Gibbs采樣后輸入層01化前后的值#輸入的是v，輸出的是每一次Gibbs采樣后的v構成的list，一共進行k次Gibbs采樣chain, updates = theano.scan(lambda v: gibbs_step(v)[1], outputs_info=[v], n_steps=k) #updates里面裝的是每次的輸入值v_sample = chain[-1] #k次Gibbs采樣后輸入端的值（01化過后的） mean_v = gibbs_step(v_sample)[0] #再次Gibbs前進一次,得到沒有01化的輸入端數碼，用于監控的變量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): #公式4,能量的計算公式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] #代價函數return v_sample, cost, monitor, updatesdef shared_normal(num_rows, num_cols, scale=1):'''Initialize a matrix shared variable with normally distributed elements.'''return theano.shared(numpy.random.normal(scale=scale, size=(num_rows, num_cols)).astype(theano.config.floatX))def shared_zeros(*shape):'''Initialize a vector shared variable with zero elements.'''return theano.shared(numpy.zeros(shape, dtype=theano.config.floatX))def build_rnnrbm(n_visible, n_hidden, n_hidden_recurrent):'''Construct a symbolic RNN-RBM and initialize parameters.n_visible : integerNumber of visible units. n_hidden : integerNumber of hidden units of the conditional RBMs. n_hidden_recurrent : integerNumber of hidden units of the RNN.Return a (v, v_sample, cost, monitor, params, updates_train, v_t,updates_generate) tuple:v : Theano matrixSymbolic variable holding an input sequence (used during training) v_sample : Theano matrixSymbolic variable holding the negative particles for CD log-likelihoodgradient estimation (used during training) cost : Theano scalarExpression whose gradient (considering v_sample constant) corresponds to theLL gradient of the RNN-RBM (used during training) monitor : Theano scalarFrame-level pseudo-likelihood (useful for monitoring during training) params : tuple of Theano shared variablesThe parameters of the model to be optimized during training. updates_train : dictionary of Theano variable -> Theano variableUpdate object that should be passed to theano.function when compiling thetraining function.v_t : Theano matrixSymbolic variable holding a generated sequence (used during sampling) updates_generate : dictionary of Theano variable -> Theano variableUpdate object that should be passed to theano.function when compiling thegeneration 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 sequenceu0 = 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.# 如果給定t時刻的v和t-1時刻的u，那么返回t時刻的u，bv,bh,含有25次Gibbs采樣過程# 如果只給定t-1時刻的u（即沒有t時刻的v),則表示的是由rbm來產生v了，所以這時候返回的是t時刻的v和u，以及# 迭代過程中輸入端的變換過程updatesdef 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 Noneif generate:v_t, _, _, updates = build_rbm(T.zeros((n_visible,)), W, bv_t, #第一個參數應該是v,因此這里的v是0bh_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( #訓練rbm過程的符號表達式(每次只包括25步的Gibbs采樣)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),#進行generate產生過程的符號表達式，迭代200次outputs_info=[None, u0], non_sequences=params, n_steps=200)return (v, v_sample, cost, monitor, params, updates_train, v_t, #cost在build_rbm()中產生 updates_generate)class RnnRbm: #兩個功能，訓練RNN-RBM模型和用訓練好的RNN-RBM模型來產生樣本'''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 : integerNumber of hidden units of the conditional RBMs. n_hidden_recurrent : integerNumber of hidden units of the RNN. lr : floatLearning rate r : (integer, integer) tupleSpecifies the pitch range of the piano-roll in MIDI note numbers, includingr[0] but not r[1], such that r[1]-r[0] is the number of visible units of theRBM at a given time step. The default (21, 109) corresponds to the full rangeof piano (88 notes). dt : floatSampling period when converting the MIDI files into piano-rolls, orequivalently the time difference between consecutive time steps.'''self.r = rself.dt = dt(v, v_sample, cost, monitor, params, updates_train, v_t,updates_generate) = build_rnnrbm(r[1] - r[0], n_hidden, #在該函數里面有設置迭代次數等參數 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))) #sgd算法,利用公式4的cost公式搞定8個參數的更新self.train_function = theano.function([v], monitor,updates=updates_train)self.generate_function = theano.function([], v_t, #updates_generate步驟在build_rnnrbm()中產生，音樂的產生主要在那函數中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 stringsList of MIDI files that will be loaded as piano-rolls for training. batch_size : integerTraining sequences will be split into subsequences of at most this sizebefore applying the SGD updates. num_epochs : integerNumber of epochs (pass over the training set) performed. The user cansafely 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] #讀取midi文件try:for epoch in xrange(num_epochs): #訓練200次numpy.random.shuffle(dataset) #將訓練樣本打亂costs = []for s, sequence in enumerate(dataset): #返回的s是序號，sequence是dataset對應序號下的值for i in xrange(0, len(sequence), batch_size):cost = self.train_function(sequence[i:i + batch_size]) #train_function在init()函數中 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 : stringA MIDI file will be created at this location. show : booleanIf True, a piano-roll of the generated sequence will be shown.'''piano_roll = self.generate_function() #直接生成piano roll文件midiwrite(filename, piano_roll, self.r, self.dt)#將piano_roll文件轉換成midi文件并保存if show:extent = (0, self.dt * len(piano_roll)) + self.rpylab.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')def test_rnnrbm(batch_size=100, num_epochs=200):model = RnnRbm()#os.path.dirname(__file__)為獲得當前文件的目錄,os.path.split(path)是將path按照最后一個斜線分成父和子的部分re = os.path.join(os.path.split(os.path.dirname(__file__))[0], #該代碼完成的功能是，找到當前文件的上級目錄下的/data/Nottinghan/train/*.mid文件'data', 'Nottingham', 'train', '*.mid') #re得到該目錄下的所有.mid文件model.train(glob.glob(re),#glob.glob()只是將文件路徑名等弄成linux的格式batch_size=batch_size, num_epochs=num_epochs)return modelif __name__ == '__main__':model = test_rnnrbm() #該函數主要用來訓練RNN-RBM參數model.generate('sample1.mid') #產生數據的v_t初始化都是0model.generate('sample2.mid')pylab.show()

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　　實驗總結：

　　關于bp算法：由于RNN-RBM中對loss函數求導用到了BPTT(back propgation through time)算法：BP算法加入了時間維度。為了加深對BP算法的理解，重新看了一遍推導過程。bp算法的推導過程是主要是由求導中的鏈式法則得到的。具體算法可參考Martin T.Hagan 的《神經網絡設計》第11章（這本書寫得不錯，翻譯得也還可以）。其思想大概為：損失函數F對第m層wij(連接第m層第i個節點和第m-1層第j個節點之間的權值)的導數等于F對第m層第i個節點輸入值的導數，乘上該輸入值對wij的導數(很容易知道這個導數等于第m-1層第j個節點的輸出值)。而F對第m層第i個節點輸入值的導數值又等于F對第m+1層輸入值的導數（這時需要考慮第m+1中所有的節點）乘以第m+1層輸入值對第m層第i個輸入值的導數（這個導數值很容易由激發函數的導函數求得）,并且我們通常說的bp算法是誤差方向傳播，這里的第m層誤差指的就是F對第m層輸入值的導數。由此可知，可以從最后一層依次往前求解，這就是bp算法的思想，本質上是高數里面的鏈式求導法則。

? ? ? 另外，實驗中關于樂理對應的具體細節沒有深究。

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　　參考資料：

? ? ? ?http://deeplearning.net/tutorial/rnnrbm.html（教程主頁）

? ? ?《神經網絡設計》，Martin T.Hagan.

? ? ? ?http://www.cse.unsw.edu.au/~waleed/phd/html/node37.html(RNN圖片來源1)

? ? ? ?Recurrent Neural Networks in Ruby.（RNN圖片來源2）

　　Learning Recurrent Neural Networks with Hessian-Free Optimization, James Martens，Ilya Sutskever.

　　Modeling Temporal Dependencies in High-Dimensional Sequences: Application to Polyphonic Music Generation and Transcription, Nicolas Boulanger-Lewandowski，Yoshua Bengio，Pascal Vincent.

? ? ? ?http://www-etud.iro.umontreal.ca/~boulanni/icml2012(rnn-rbm項目主頁)

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by：tornadomeet? from：http://www.cnblogs.com/tornadomeet

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