layers.py cs231n

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 import numpy as np


def affine_forward(x, w, b):       第一个最简单的 affine_forward简单的前向传递,返回 out,cache 
  """
  Computes the forward pass for an affine (fully-connected) layer.

  The input x has shape (N, d_1, ..., d_k) and contains a minibatch of N
  examples, where each example x[i] has shape (d_1, ..., d_k). We will
  reshape each input into a vector of dimension D = d_1 * ... * d_k, and
  then transform it to an output vector of dimension M.

  Inputs:
  - x: A numpy array containing input data, of shape (N, d_1, ..., d_k)
  - w: A numpy array of weights, of shape (D, M)
  - b: A numpy array of biases, of shape (M,)
  
  Returns a tuple of:
  - out: output, of shape (N, M)
  - cache: (x, w, b)
  """
  out = None
  tx=x.reshape((x.shape[0],-1))
  out=tx.dot(w)+b.T
  cache = (x, w, b)
  return out, cache


def affine_backward(dout, cache):     后向传递,BP的时候梯度向前传递的实现可以想想。 
dx的计算,db的计算由于前向传递的时候出现过broadcast,所以后向传递时就是求和 """ Computes the backward pass for an affine layer. Inputs: - dout: Upstream derivative, of shape (N, M) - cache: Tuple of: - x: Input data, of shape (N, d_1, ... d_k) - w: Weights, of shape (D, M) Returns a tuple of: - dx: Gradient with respect to x, of shape (N, d1, ..., d_k) - dw: Gradient with respect to w, of shape (D, M) - db: Gradient with respect to b, of shape (M,) """ x, w, b = cache dx, dw, db = None, None, None dx=dout.dot(w.T) dx=dx.reshape(x.shape)
dw=x.reshape(x.shape[0],-1).T.dot(dout) db=np.sum(dout,axis=0)
# # print "db is"+str(db.shape) return dx, dw, db def relu_forward(x): relu激励函数,后向传递的时候要截住一部分梯度 """ Computes the forward pass for a layer of rectified linear units (ReLUs). Input: - x: Inputs, of any shape Returns a tuple of: - out: Output, of the same shape as x - cache: x """ out = None out=np.maximum(0,x) cache = x return out, cache def relu_backward(dout, cache): """ Computes the backward pass for a layer of rectified linear units (ReLUs). Input: - dout: Upstream derivatives, of any shape - cache: Input x, of same shape as dout Returns: - dx: Gradient with respect to x """ dx, x = None, cache dx=dout dx[x<0]=0 return dx def batchnorm_forward(x, gamma, beta, bn_param): BN层,有了BN层就不太容易出现vanish gradient """ Forward pass for batch normalization. During training the sample mean and (uncorrected) sample variance are computed from minibatch statistics and used to normalize the incoming data. During training we also keep an exponentially decaying running mean of the mean and variance of each feature, and these averages are used to normalize data at test-time. At each timestep we update the running averages for mean and variance using an exponential decay based on the momentum parameter: running_mean = momentum * running_mean + (1 - momentum) * sample_mean running_var = momentum * running_var + (1 - momentum) * sample_var Note that the batch normalization paper suggests a different test-time behavior: they compute sample mean and variance for each feature using a large number of training images rather than using a running average. For this implementation we have chosen to use running averages instead since they do not require an additional estimation step; the torch7 implementation of batch normalization also uses running averages. Input: - x: Data of shape (N, D) - gamma: Scale parameter of shape (D,) - beta: Shift paremeter of shape (D,) - bn_param: Dictionary with the following keys: - mode: ‘train‘ or ‘test‘; required
- eps: Constant for numeric stability - momentum: Constant for running mean / variance. - running_mean: Array of shape (D,) giving running mean of features - running_var Array of shape (D,) giving running variance of features
后两个是在动态更新的,因为py传list是传引用 Returns a tuple of: - out: of shape (N, D) - cache: A tuple of values needed in the backward pass """ mode = bn_param[‘mode‘] eps = bn_param.get(‘eps‘, 1e-5)
dict的 get 方法 momentum = bn_param.get(‘momentum‘, 0.9) N, D = x.shape running_mean = bn_param.get(‘running_mean‘, np.zeros(D, dtype=x.dtype)) running_var = bn_param.get(‘running_var‘, np.zeros(D, dtype=x.dtype)) out, cache = None, None sample_mean=np.mean(x,axis=0) sample_var=np.var(x,axis=0) if mode == ‘train‘: running_mean = momentum * running_mean + (1 - momentum) * sample_mean running_var = momentum * running_var + (1 - momentum) * sample_var std=np.sqrt(sample_var+eps) out1=(x-sample_mean)/(std) std=np.sqrt(sample_var+eps) out2=(out1+beta)*gamma
out=out2
cache=(x,sample_mean,sample_var,std,out1,beta,gamma,eps)
elif mode == ‘test‘: x=(x-running_mean)/np.sqrt(running_var+eps) out1=(x+beta)*gamma
out=out1
#cache=(x,sample_mean,sample_var,std,out1,beta,gamma,eps) else: raise ValueError(‘Invalid forward batchnorm mode "%s"‘ % mode) # Store the updated running means back into bn_param bn_param[‘running_mean‘] = running_mean bn_param[‘running_var‘] = running_var return out, cache def batchnorm_backward(dout, cache): 这个很经典 可以试着画出图表来计算 曾经因为卡eps卡了一段时间 """ Backward pass for batch normalization. For this implementation, you should write out a computation graph for batch normalization on paper and propagate gradients backward through intermediate nodes. Inputs: - dout: Upstream derivatives, of shape (N, D) - cache: Variable of intermediates from batchnorm_forward. Returns a tuple of: - dx: Gradient with respect to inputs x, of shape (N, D) - dgamma: Gradient with respect to scale parameter gamma, of shape (D,) - dbeta: Gradient with respect to shift parameter beta, of shape (D,) """ dx, dgamma, dbeta = None, None, None
x,sample_mean,sample_var,std,out1,beta,gamma,eps=cache x_n=x.shape[0] dgamma=np.sum((out1+beta)*dout,axis=0) dbeta=np.sum(gamma*dout,axis=0) ‘‘‘ tdout_media = dout * gamma tdgamma = np.sum(dout * (out1 + beta),axis = 0) tdbeta = np.sum(dout * gamma,axis = 0) tdx = tdout_media / np.sqrt(sample_var + eps) tdmean = -np.sum(tdout_media / np.sqrt(sample_var+eps),axis = 0) tdstd = np.sum(-tdout_media * (x - sample_mean) / (sample_var + eps),axis = 0) tdvar = 1./2./np.sqrt(sample_var+eps) * tdstd tdx_minus_mean_square = tdvar / x.shape[0] tdx_minus_mean = 2 * (x-sample_mean) * tdx_minus_mean_square print tdx_minus_mean tdx += tdx_minus_mean tdmean += np.sum(-tdx_minus_mean,axis = 0) tdx += tdmean / x.shape[0] ‘‘‘ dout1=gamma*dout dx_minus_xmean=dout1/np.sqrt(sample_var+eps) dx1=dx_minus_xmean dxmean=np.sum(-dx_minus_xmean,axis=0) dx2=np.ones((x_n,1)).dot((dxmean/x_n).reshape(1,-1)) dsqrtvar=np.sum( ((sample_mean-x)/(sample_var+eps))*dout1 ,axis=0 ) dvar=1./(2.*(np.sqrt(sample_var+eps))) *dsqrtvar #stupid dx3=2*(x-sample_mean)*dvar/x_n dtest=np.sum(-dx3,axis=0) dx=dx1+dx2+dx3 return dx, dgamma, dbeta def batchnorm_backward_alt(dout, cache): 耐心推一下还是可以推出来的,利用示性函数和中间变量 """ Alternative backward pass for batch normalization. For this implementation you should work out the derivatives for the batch normalizaton backward pass on paper and simplify as much as possible. You should be able to derive a simple expression for the backward pass. Note: This implementation should expect to receive the same cache variable as batchnorm_backward, but might not use all of the values in the cache. Inputs / outputs: Same as batchnorm_backward """ dx, dgamma, dbeta = None, None, None x,sample_mean,sample_var,std,out1,beta,gamma,eps=cache n_x=x.shape[0] dgamma=np.sum((out1+beta)*dout,axis=0) dbeta=np.sum(gamma*dout,axis=0) #dx=(((1-1./n_x)*gamma*(sample_var+eps)-((x-sample_mean)**2)/n_x )/((sample_var+eps)**1.5)) *dout1 dx = (1. / n_x) * gamma * (sample_var + eps)**(-1. / 2.) * (n_x * dout - np.sum(dout, axis=0) - (x - sample_mean) * (sample_var + eps)**(-1.0) * np.sum(dout * (x - sample_mean), axis=0)) return dx, dgamma, dbeta def dropout_forward(x, dropout_param): 也是防止过拟合 """ Performs the forward pass for (inverted) dropout. Inputs: - x: Input data, of any shape - dropout_param: A dictionary with the following keys: - p: Dropout parameter. We drop each neuron output with probability p. - mode: ‘test‘ or ‘train‘. If the mode is train, then perform dropout; if the mode is test, then just return the input. - seed: Seed for the random number generator. Passing seed makes this function deterministic, which is needed for gradient checking but not in real networks. Outputs: - out: Array of the same shape as x. - cache: A tuple (dropout_param, mask). In training mode, mask is the dropout mask that was used to multiply the input; in test mode, mask is None. """ p, mode = dropout_param[‘p‘], dropout_param[‘mode‘] if ‘seed‘ in dropout_param: np.random.seed(dropout_param[‘seed‘]) mask = None out = None if mode == ‘train‘: mask=np.random.rand(1,x.shape[1]) mask=(mask>p)/p out=x*mask elif mode == ‘test‘: out=x
cache = (dropout_param, mask) out = out.astype(x.dtype, copy=False) return out, cache def dropout_backward(dout, cache): """ Perform the backward pass for (inverted) dropout. Inputs: - dout: Upstream derivatives, of any shape - cache: (dropout_param, mask) from dropout_forward. """ dropout_param, mask = cache mode = dropout_param[‘mode‘] dx = None if mode == ‘train‘: dx=dout*mask elif mode == ‘test‘: dx = dout return dx def conv_forward_naive(x, w, b, conv_param): 四重循环,这个单元没让写快速的版本(估计也不太会写/:( """ A naive implementation of the forward pass for a convolutional layer. The input consists of N data points, each with C channels, height H and width W. We convolve each input with F different filters, where each filter spans all C channels and has height HH and width HH. Input: - x: Input data of shape (N, C, H, W) - w: Filter weights of shape (F, C, HH, WW) - b: Biases, of shape (F,) - conv_param: A dictionary with the following keys: - ‘stride‘: The number of pixels between adjacent receptive fields in the horizontal and vertical directions. - ‘pad‘: The number of pixels that will be used to zero-pad the input. Returns a tuple of: - out: Output data, of shape (N, F, H‘, W‘) where H‘ and W‘ are given by H‘ = 1 + (H + 2 * pad - HH) / stride W‘ = 1 + (W + 2 * pad - WW) / stride - cache: (x, w, b, conv_param) """ N,C,H,W=x.shape F,C,HH,WW=w.shape pad=conv_param[‘pad‘] stride=conv_param[‘stride‘] H2=1+(H+2*pad-HH)/stride W2=1+(W+2*pad-WW)/stride out = None out=np.zeros((N,F,H2,W2)) x_pad=np.pad(x,((0,0),(0,0),(pad,pad),(pad,pad)),mode=‘constant‘,constant_values=0) for i in xrange(N): for j in xrange(F): for k in xrange(H2): for q in xrange(W2): out[i,j,k,q]=np.sum(w[j]*x_pad[i,:,k*stride:(k)*stride+HH,q*stride:(q)*stride+WW])+b[j] cache = (x, w, b, conv_param) return out, cache def conv_backward_naive(dout, cache): """ A naive implementation of the backward pass for a convolutional layer. Inputs: - dout: Upstream derivatives. - cache: A tuple of (x, w, b, conv_param) as in conv_forward_naive Returns a tuple of: - dx: Gradient with respect to x - dw: Gradient wikth respect to w - db: Gradient with respect to b """ dx, dw, db = None, None, None x,w,b,conv_param=cache pad=conv_param[‘pad‘] stride=conv_param[‘stride‘] N,C,H,W=x.shape N,C,H2,W2=dout.shape F,C,HH,WW=w.shape F=b.shape[0] x_pad=np.pad(x,((0,0),(0,0),(pad,pad),(pad,pad)),mode=‘constant‘,constant_values=0) dx=np.zeros_like(x) dx_pad=np.zeros_like(x_pad) dw=np.zeros_like(w) db=np.zeros_like(b) for i in xrange(N): for j in xrange(H2): for k in xrange(W2): for q in xrange(F): dw[q]+=dout[i][q][j][k]*x_pad[i,:,j*stride:j*stride+HH,k*stride:k*stride+WW] db[q]+=dout[i][q][j][k] dx_pad[i,:,j*stride:j*stride+HH,k*stride:k*stride+WW]+=dout[i][q][j][k]*w[q] dx=dx_pad[:,:,pad:H+pad,pad:W+pad] return dx, dw, db def max_pool_forward_naive(x, pool_param): """ A naive implementation of the forward pass for a max pooling layer. Inputs: - x: Input data, of shape (N, C, H, W) - pool_param: dictionary with the following keys: - ‘pool_height‘: The height of each pooling region - ‘pool_width‘: The width of each pooling region - ‘stride‘: The distance between adjacent pooling regions Returns a tuple of: - out: Output data - cache: (x, pool_param) """ out = None stride=pool_param[‘stride‘] pool_height=pool_param[‘pool_height‘] pool_width=pool_param[‘pool_width‘] N,C,H,W=x.shape H2=(H-pool_height)/stride + 1 W2=(W-pool_width)/stride + 1 out=np.zeros((N,C,H2,W2)) for n in xrange(N): for c in xrange(C): for h in xrange(H2): for w in xrange(W2): out[n,c,h,w]=np.max(x[n,c,h*stride:h*stride+pool_width,w*stride:w*stride+pool_width]) cache = (x, pool_param) return out, cache def max_pool_backward_naive(dout, cache): """ A naive implementation of the backward pass for a max pooling layer. Inputs: - dout: Upstream derivatives - cache: A tuple of (x, pool_param) as in the forward pass. Returns: - dx: Gradient with respect to x """ dx = None x,pool_param=cache stride=pool_param[‘stride‘] pool_height=pool_param[‘pool_height‘] pool_width=pool_param[‘pool_width‘] N,C,H,W=x.shape N,C,H2,W2=dout.shape dx=np.zeros_like(x) for n in xrange(N): for c in xrange(C): for h in xrange(H2): for w in xrange(W2): window=x[n,c,h*stride:h*stride+pool_height,w*stride:w*stride+pool_width] t=np.max(window) #pass by yinyong ----sf dx[n,c,h*stride:h*stride+pool_height,w*stride:w*stride+pool_width] = (window==t)*dout[n][c][h][w] return dx def spatial_batchnorm_forward(x, gamma, beta, bn_param): 套用以前的代码,让生活变的简单 """ Computes the forward pass for spatial batch normalization. Inputs: - x: Input data of shape (N, C, H, W) - gamma: Scale parameter, of shape (C,) - beta: Shift parameter, of shape (C,) - bn_param: Dictionary with the following keys: - mode: ‘train‘ or ‘test‘; required - eps: Constant for numeric stability - momentum: Constant for running mean / variance. momentum=0 means that old information is discarded completely at every time step, while momentum=1 means that new information is never incorporated. The default of momentum=0.9 should work well in most situations. - running_mean: Array of shape (D,) giving running mean of features - running_var Array of shape (D,) giving running variance of features Returns a tuple of: - out: Output data, of shape (N, C, H, W) - cache: Values needed for the backward pass """ out, cache = None, None N,C,H,W=x.shape x=x.transpose(0,2,3,1).reshape(N*W*H,C) out,cache=batchnorm_forward(x,gamma,beta,bn_param) out=out.reshape(N,H,W,C).transpose(0,3,1,2) return out, cache def spatial_batchnorm_backward(dout, cache): """ Computes the backward pass for spatial batch normalization. Inputs: - dout: Upstream derivatives, of shape (N, C, H, W) - cache: Values from the forward pass Returns a tuple of: - dx: Gradient with respect to inputs, of shape (N, C, H, W) - dgamma: Gradient with respect to scale parameter, of shape (C,) - dbeta: Gradient with respect to shift parameter, of shape (C,) """ dx, dgamma, dbeta = None, None, None N,C,H,W=dout.shape dout=dout.transpose(0,2,3,1).reshape(N*H*W,C) dx,dgamma,dbeta=batchnorm_backward(dout,cache) dx=dx.reshape(N,H,W,C).transpose(0,3,1,2) return dx, dgamma, dbeta def svm_loss(x, y): """ Computes the loss and gradient using for multiclass SVM classification. Inputs: - x: Input data, of shape (N, C) where x[i, j] is the score for the jth class for the ith input. - y: Vector of labels, of shape (N,) where y[i] is the label for x[i] and 0 <= y[i] < C Returns a tuple of: - loss: Scalar giving the loss - dx: Gradient of the loss with respect to x """ N = x.shape[0] correct_class_scores = x[np.arange(N), y] margins = np.maximum(0, x - correct_class_scores[:, np.newaxis] + 1.0) margins[np.arange(N), y] = 0 loss = np.sum(margins) / N num_pos = np.sum(margins > 0, axis=1) dx = np.zeros_like(x) dx[margins > 0] = 1 dx[np.arange(N), y] -= num_pos dx /= N return loss, dx def softmax_loss(x, y): """ Computes the loss and gradient for softmax classification. Inputs: - x: Input data, of shape (N, C) where x[i, j] is the score for the jth class for the ith input. - y: Vector of labels, of shape (N,) where y[i] is the label for x[i] and 0 <= y[i] < C Returns a tuple of: - loss: Scalar giving the loss - dx: Gradient of the loss with respect to x """ probs = np.exp(x - np.max(x, axis=1, keepdims=True)) probs /= np.sum(probs, axis=1, keepdims=True) N = x.shape[0] loss = -np.sum(np.log(probs[np.arange(N), y])) / N dx = probs.copy() dx[np.arange(N), y] -= 1 dx /= N return loss, dx

  

 

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