文章目錄
1- layers.py
from builtins import range
import numpy as np
def affine_forward(x, w, b):
"""
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
###########################################################################
# TODO: Implement the affine forward pass. Store the result in out. You #
# will need to reshape the input into rows. #
###########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
# 查看一下x的形狀
#print(x.shape)
# x的第一維度是數據的數量，直接根據第一維度reshape成二維矩陣，則x每一行都是將x的維度展開成了一個行向量
out = (x.reshape(x.shape[0], -1)).dot(w) + b
x = x.reshape(x.shape)
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
###########################################################################
# END OF YOUR CODE #
###########################################################################
cache = (x, w, b)
return out, cache
def affine_backward(dout, cache):
"""
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)
- b: Biases, of shape (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
###########################################################################
# TODO: Implement the affine backward pass. #
###########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
dx = dout.dot(w.T)
dw = (x.reshape(x.shape[0], -1).T).dot(dout)
db = (np.ones((1, dout.shape[0]))).dot(dout)
dx = dx.reshape(x.shape)
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
###########################################################################
# END OF YOUR CODE #
###########################################################################
return dx, dw, db
def relu_forward(x):
"""
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
###########################################################################
# TODO: Implement the ReLU forward pass. #
###########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
out = np.maximum(x, 0)
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
###########################################################################
# END OF YOUR CODE #
###########################################################################
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
###########################################################################
# TODO: Implement the ReLU backward pass. #
###########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
# ReLU對於x大於零的部分偏導爲1，其餘爲0
dx = dout
dx[x<=0] = 0
# 這裏一開始想複雜了。。還以爲是要求出整個ReLU加前一層網絡輸入和權重的偏導，導致結果一直不對
# 後來才發現這裏只是僅僅求出單純的ReLU函數的偏導形式
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
###########################################################################
# END OF YOUR CODE #
###########################################################################
return dx
def batchnorm_forward(x, gamma, beta, bn_param):
"""
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
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)
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
if mode == 'train':
#######################################################################
# TODO: Implement the training-time forward pass for batch norm. #
# Use minibatch statistics to compute the mean and variance, use #
# these statistics to normalize the incoming data, and scale and #
# shift the normalized data using gamma and beta. #
# #
# You should store the output in the variable out. Any intermediates #
# that you need for the backward pass should be stored in the cache #
# variable. #
# #
# You should also use your computed sample mean and variance together #
# with the momentum variable to update the running mean and running #
# variance, storing your result in the running_mean and running_var #
# variables. #
# #
# Note that though you should be keeping track of the running #
# variance, you should normalize the data based on the standard #
# deviation (square root of variance) instead! #
# Referencing the original paper (https://arxiv.org/abs/1502.03167) #
# might prove to be helpful. #
#######################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
sample_mean = x.mean(axis = 0) # 求行方向（豎直方向）的mean，豎直方向壓縮矩陣
sample_var = x.var(axis = 0)
x_norm = (x-sample_mean)/np.sqrt(sample_var+eps)
out = gamma * x_norm + beta
cache = (x, x_norm, sample_var, sample_mean, gamma, beta, eps)
running_mean = momentum * running_mean + (1 - momentum) * sample_mean
running_var = momentum * running_var + (1 - momentum) * sample_var
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
#######################################################################
# END OF YOUR CODE #
#######################################################################
elif mode == 'test':
#######################################################################
# TODO: Implement the test-time forward pass for batch normalization. #
# Use the running mean and variance to normalize the incoming data, #
# then scale and shift the normalized data using gamma and beta. #
# Store the result in the out variable. #
#######################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
x_norm = (x-running_mean)/np.sqrt(running_var)
out = gamma * x_norm + beta
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
#######################################################################
# END OF YOUR CODE #
#######################################################################
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):
"""
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
###########################################################################
# TODO: Implement the backward pass for batch normalization. Store the #
# results in the dx, dgamma, and dbeta variables. #
# Referencing the original paper (https://arxiv.org/abs/1502.03167) #
# might prove to be helpful. #
###########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
x, x_norm, sample_var, sample_mean, gamma, beta, eps = cache
dgamma = np.sum(dout * x_norm, axis=0) # 因爲前向傳播時候out=x_norm*gamma，即gamma(D,)只有一行，但對x_norm(N*D)的N個數據都產生了影響，所以反向傳播時候N個數據都要對gamma的導數有影響
dbeta = np.sum(dout, axis=0) # 同gamma
# dx的求導，先求out對x_norm再求x_norm對x，x_norm對x是難點
delta_x_norm = gamma * dout # out對x_norm的導數
N = dout.shape[0]
dvar = np.sum(delta_x_norm * (x-sample_mean)*(-0.5)*(sample_var+eps)**(-1.5), axis=0)
dmean = np.sum(delta_x_norm * (-1)/np.sqrt(sample_var+eps), axis=0) + dvar * np.sum((-2) * (x-sample_mean), axis=0)/N
dx = delta_x_norm/np.sqrt(sample_var+eps) + dvar*2*(x-sample_mean)/N + dmean/N
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
###########################################################################
# END OF YOUR CODE #
###########################################################################
return dx, dgamma, dbeta
def batchnorm_backward_alt(dout, cache):
"""
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.
See the jupyter notebook for more hints.
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
###########################################################################
# TODO: Implement the backward pass for batch normalization. Store the #
# results in the dx, dgamma, and dbeta variables. #
# #
# After computing the gradient with respect to the centered inputs, you #
# should be able to compute gradients with respect to the inputs in a #
# single statement; our implementation fits on a single 80-character line.#
###########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
x, x_norm, sample_var, sample_mean, gamma, beta, eps = cache
dgamma = np.sum(dout * x_norm, axis=0) # 因爲前向傳播時候out=x_norm*gamma，即gamma(D,)只有一行，但對x_norm(N*D)的N個數據都產生了影響，所以反向傳播時候N個數據都要對gamma的導數有影響
dbeta = np.sum(dout, axis=0) # 同gamma
# dx的求導，先求out對x_norm再求x_norm對x，x_norm對x是難點
delta_x_norm = gamma * dout # out對x_norm的導數
N = dout.shape[0]
dvar = np.sum(delta_x_norm * (x-sample_mean)*(-0.5)*(sample_var+eps)**(-1.5), axis=0)
dmean = np.sum(delta_x_norm * (-1)/np.sqrt(sample_var+eps), axis=0) + dvar * np.sum((-2) * (x-sample_mean), axis=0)/N
dx = delta_x_norm/np.sqrt(sample_var+eps) + dvar*2*(x-sample_mean)/N + dmean/N
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
###########################################################################
# END OF YOUR CODE #
###########################################################################
return dx, dgamma, dbeta
def layernorm_forward(x, gamma, beta, ln_param):
"""
Forward pass for layer normalization.
During both training and test-time, the incoming data is normalized per data-point,
before being scaled by gamma and beta parameters identical to that of batch normalization.
Note that in contrast to batch normalization, the behavior during train and test-time for
layer normalization are identical, and we do not need to keep track of running averages
of any sort.
Input:
- x: Data of shape (N, D)
- gamma: Scale parameter of shape (D,)
- beta: Shift paremeter of shape (D,)
- ln_param: Dictionary with the following keys:
- eps: Constant for numeric stability
Returns a tuple of:
- out: of shape (N, D)
- cache: A tuple of values needed in the backward pass
"""
out, cache = None, None
eps = ln_param.get('eps', 1e-5)
###########################################################################
# TODO: Implement the training-time forward pass for layer norm. #
# Normalize the incoming data, and scale and shift the normalized data #
# using gamma and beta. #
# HINT: this can be done by slightly modifying your training-time #
# implementation of batch normalization, and inserting a line or two of #
# well-placed code. In particular, can you think of any matrix #
# transformations you could perform, that would enable you to copy over #
# the batch norm code and leave it almost unchanged? #
###########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
###########################################################################
# END OF YOUR CODE #
###########################################################################
return out, cache
def layernorm_backward(dout, cache):
"""
Backward pass for layer normalization.
For this implementation, you can heavily rely on the work you've done already
for batch normalization.
Inputs:
- dout: Upstream derivatives, of shape (N, D)
- cache: Variable of intermediates from layernorm_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
###########################################################################
# TODO: Implement the backward pass for layer norm. #
# #
# HINT: this can be done by slightly modifying your training-time #
# implementation of batch normalization. The hints to the forward pass #
# still apply! #
###########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
###########################################################################
# END OF YOUR CODE #
###########################################################################
return dx, dgamma, dbeta
def dropout_forward(x, dropout_param):
"""
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 keep 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: 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.
NOTE: Please implement **inverted** dropout, not the vanilla version of dropout.
See http://cs231n.github.io/neural-networks-2/#reg for more details.
NOTE 2: Keep in mind that p is the probability of **keep** a neuron
output; this might be contrary to some sources, where it is referred to
as the probability of dropping a neuron output.
"""
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':
#######################################################################
# TODO: Implement training phase forward pass for inverted dropout. #
# Store the dropout mask in the mask variable. #
#######################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
#######################################################################
# END OF YOUR CODE #
#######################################################################
elif mode == 'test':
#######################################################################
# TODO: Implement the test phase forward pass for inverted dropout. #
#######################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
#######################################################################
# END OF YOUR CODE #
#######################################################################
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':
#######################################################################
# TODO: Implement training phase backward pass for inverted dropout #
#######################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
#######################################################################
# END OF YOUR CODE #
#######################################################################
elif mode == 'test':
dx = dout
return dx
def conv_forward_naive(x, w, b, conv_param):
"""
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 WW.
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.
During padding, 'pad' zeros should be placed symmetrically (i.e equally on both sides)
along the height and width axes of the input. Be careful not to modfiy the original
input x directly.
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)
"""
out = None
###########################################################################
# TODO: Implement the convolutional forward pass. #
# Hint: you can use the function np.pad for padding. #
###########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
###########################################################################
# END OF YOUR CODE #
###########################################################################
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 with respect to w
- db: Gradient with respect to b
"""
dx, dw, db = None, None, None
###########################################################################
# TODO: Implement the convolutional backward pass. #
###########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
###########################################################################
# END OF YOUR CODE #
###########################################################################
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
No padding is necessary here. Output size is given by
Returns a tuple of:
- out: Output data, of shape (N, C, H', W') where H' and W' are given by
H' = 1 + (H - pool_height) / stride
W' = 1 + (W - pool_width) / stride
- cache: (x, pool_param)
"""
out = None
###########################################################################
# TODO: Implement the max-pooling forward pass #
###########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
###########################################################################
# END OF YOUR CODE #
###########################################################################
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
###########################################################################
# TODO: Implement the max-pooling backward pass #
###########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
###########################################################################
# END OF YOUR CODE #
###########################################################################
return dx
def spatial_batchnorm_forward(x, gamma, beta, bn_param):
"""
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
###########################################################################
# TODO: Implement the forward pass for spatial batch normalization. #
# #
# HINT: You can implement spatial batch normalization by calling the #
# vanilla version of batch normalization you implemented above. #
# Your implementation should be very short; ours is less than five lines. #
###########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
###########################################################################
# END OF YOUR CODE #
###########################################################################
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
###########################################################################
# TODO: Implement the backward pass for spatial batch normalization. #
# #
# HINT: You can implement spatial batch normalization by calling the #
# vanilla version of batch normalization you implemented above. #
# Your implementation should be very short; ours is less than five lines. #
###########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
###########################################################################
# END OF YOUR CODE #
###########################################################################
return dx, dgamma, dbeta
def spatial_groupnorm_forward(x, gamma, beta, G, gn_param):
"""
Computes the forward pass for spatial group normalization.
In contrast to layer normalization, group normalization splits each entry
in the data into G contiguous pieces, which it then normalizes independently.
Per feature shifting and scaling are then applied to the data, in a manner identical to that of batch normalization and layer normalization.
Inputs:
- x: Input data of shape (N, C, H, W)
- gamma: Scale parameter, of shape (C,)
- beta: Shift parameter, of shape (C,)
- G: Integer mumber of groups to split into, should be a divisor of C
- gn_param: Dictionary with the following keys:
- eps: Constant for numeric stability
Returns a tuple of:
- out: Output data, of shape (N, C, H, W)
- cache: Values needed for the backward pass
"""
out, cache = None, None
eps = gn_param.get('eps',1e-5)
###########################################################################
# TODO: Implement the forward pass for spatial group normalization. #
# This will be extremely similar to the layer norm implementation. #
# In particular, think about how you could transform the matrix so that #
# the bulk of the code is similar to both train-time batch normalization #
# and layer normalization! #
###########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
###########################################################################
# END OF YOUR CODE #
###########################################################################
return out, cache
def spatial_groupnorm_backward(dout, cache):
"""
Computes the backward pass for spatial group 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
###########################################################################
# TODO: Implement the backward pass for spatial group normalization. #
# This will be extremely similar to the layer norm implementation. #
###########################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
###########################################################################
# END OF YOUR CODE #
###########################################################################
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
"""
shifted_logits = x - np.max(x, axis=1, keepdims=True)
Z = np.sum(np.exp(shifted_logits), axis=1, keepdims=True)
log_probs = shifted_logits - np.log(Z)
probs = np.exp(log_probs)
N = x.shape[0]
loss = -np.sum(log_probs[np.arange(N), y]) / N
dx = probs.copy()
dx[np.arange(N), y] -= 1
dx /= N
return loss, dx
2- layer_utils.py加入四個求解batch/layer norm的函數
def affine_bn_relu_forward(x, w, b, gamma, beta, bn_param):
"""
affine-bn-relu forward layer
"""
a, fc_cache = affine_forward(x, w, b)
bn, bn_cache = batchnorm_forward(a, gamma, beta, bn_param)
out, relu_cache = relu_forward(bn)
cache = (fc_cache, bn_cache, relu_cache)
return out, cache
def affine_bn_relu_backward(dout, cache):
"""
affine-bn-relu backward layer
"""
fc_cache, bn_cache, relu_cache = cache
drelu = relu_backward(dout, relu_cache)
dbn, dgamma, dbeta = batchnorm_backward(drelu, bn_cache)
dx, dw, db = affine_backward(dbn, fc_cache)
return dx, dw, db, dgamma, dbeta
def affine_ln_relu_forward(x, w, b, gamma, beta, ln_param):
"""
affine-ln-relu forward layer
"""
a, fc_cache = affine_forward(x, w, b)
ln, ln_cache = layernorm_forward(a, gamma, beta, ln_param)
out, relu_cache = relu_forward(ln)
cache = (fc_cache, ln_cache, relu_cache)
return out, cache
def affine_ln_relu_backward(dout, cache):
"""
affine-ln-relu backward layer
"""
fc_cache, ln_cache, relu_cache = cache
drelu = relu_backward(dout, relu_cache)
dln, dgamma, dbeta = layernorm_backward(drelu, ln_cache)
dx, dw, db = affine_backward(dln, fc_cache)
return dx, dw, db, dgamma, dbeta
3- fc_net.py的完善
from builtins import range
from builtins import object
import numpy as np
from cs231n.layers import *
from cs231n.layer_utils import *
class TwoLayerNet(object):
"""
A two-layer fully-connected neural network with ReLU nonlinearity and
softmax loss that uses a modular layer design. We assume an input dimension
of D, a hidden dimension of H, and perform classification over C classes.
The architecure should be affine - relu - affine - softmax.
Note that this class does not implement gradient descent; instead, it
will interact with a separate Solver object that is responsible for running
optimization.
The learnable parameters of the model are stored in the dictionary
self.params that maps parameter names to numpy arrays.
"""
def __init__(self, input_dim=3*32*32, hidden_dim=100, num_classes=10,
weight_scale=1e-3, reg=0.0):
"""
Initialize a new network.
Inputs:
- input_dim: An integer giving the size of the input
- hidden_dim: An integer giving the size of the hidden layer
- num_classes: An integer giving the number of classes to classify
- weight_scale: Scalar giving the standard deviation for random
initialization of the weights.
- reg: Scalar giving L2 regularization strength.
"""
self.params = {}
self.reg = reg
############################################################################
# TODO: Initialize the weights and biases of the two-layer net. Weights #
# should be initialized from a Gaussian centered at 0.0 with #
# standard deviation equal to weight_scale, and biases should be #
# initialized to zero. All weights and biases should be stored in the #
# dictionary self.params, with first layer weights #
# and biases using the keys 'W1' and 'b1' and second layer #
# weights and biases using the keys 'W2' and 'b2'. #
############################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
self.params['W1'] = np.random.randn(input_dim, hidden_dim) * weight_scale
self.params['b1'] = np.zeros(hidden_dim)
self.params['W2'] = np.random.randn(hidden_dim, num_classes) * weight_scale
self.params['b2'] = np.zeros(num_classes)
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
############################################################################
# END OF YOUR CODE #
############################################################################
def loss(self, X, y=None):
"""
Compute loss and gradient for a minibatch of data.
Inputs:
- X: Array of input data of shape (N, d_1, ..., d_k)
- y: Array of labels, of shape (N,). y[i] gives the label for X[i].
Returns:
If y is None, then run a test-time forward pass of the model and return:
- scores: Array of shape (N, C) giving classification scores, where
scores[i, c] is the classification score for X[i] and class c.
If y is not None, then run a training-time forward and backward pass and
return a tuple of:
- loss: Scalar value giving the loss
- grads: Dictionary with the same keys as self.params, mapping parameter
names to gradients of the loss with respect to those parameters.
"""
scores = None
############################################################################
# TODO: Implement the forward pass for the two-layer net, computing the #
# class scores for X and storing them in the scores variable. #
############################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
#z1 = (X.reshape(X.shape[0], -1)).dot(self.params['W1']) + self.params['b1']
z1, cache_fc1 = affine_forward(X, self.params['W1'], self.params['b1'])
a1, cache_relu = relu_forward(z1)
scores, cache_fc2 = affine_forward(a1, self.params['W2'], self.params['b2'])
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
############################################################################
# END OF YOUR CODE #
############################################################################
# If y is None then we are in test mode so just return scores
if y is None:
return scores
loss, grads = 0, {}
############################################################################
# TODO: Implement the backward pass for the two-layer net. Store the loss #
# in the loss variable and gradients in the grads dictionary. Compute data #
# loss using softmax, and make sure that grads[k] holds the gradients for #
# self.params[k]. Don't forget to add L2 regularization! #
# #
# NOTE: To ensure that your implementation matches ours and you pass the #
# automated tests, make sure that your L2 regularization includes a factor #
# of 0.5 to simplify the expression for the gradient. #
############################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
N = X.shape[0]
# 自帶的softmax可以求出loss以及其對scores的偏導dsoftmax
loss, dsoftmax = softmax_loss(scores, y)
loss += 0.5 * self.reg * (np.sum(self.params['W2'] * self.params['W2']) + np.sum(self.params['W1'] * self.params['W1']))
relu_loss = dsoftmax.dot((self.params['W2']).T)
dx2, grads['W2'], grads['b2'] = affine_backward(dsoftmax, cache_fc2)
# 因爲在自帶的softmax_loss函數中，第一次求偏導時已經除以了數據總個數N，所以在求W1/W2時候就不用再除以N了
grads['W2'] = grads['W2'] + self.reg * self.params['W2']
relu_back = relu_backward(relu_loss, z1)
dx1, grads['W1'], grads['b1'] = affine_backward(relu_back, cache_fc1)
grads['W1'] = grads['W1'] + self.reg * self.params['W1']
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
############################################################################
# END OF YOUR CODE #
############################################################################
return loss, grads
class FullyConnectedNet(object):
"""
A fully-connected neural network with an arbitrary number of hidden layers,
ReLU nonlinearities, and a softmax loss function. This will also implement
dropout and batch/layer normalization as options. For a network with L layers,
the architecture will be
{affine - [batch/layer norm] - relu - [dropout]} x (L - 1) - affine - softmax
where batch/layer normalization and dropout are optional, and the {...} block is
repeated L - 1 times.
Similar to the TwoLayerNet above, learnable parameters are stored in the
self.params dictionary and will be learned using the Solver class.
"""
def __init__(self, hidden_dims, input_dim=3*32*32, num_classes=10,
dropout=1, normalization=None, reg=0.0,
weight_scale=1e-2, dtype=np.float32, seed=None):
"""
Initialize a new FullyConnectedNet.
Inputs:
- hidden_dims: A list of integers giving the size of each hidden layer.
- input_dim: An integer giving the size of the input.
- num_classes: An integer giving the number of classes to classify.
- dropout: Scalar between 0 and 1 giving dropout strength. If dropout=1 then
the network should not use dropout at all.
- normalization: What type of normalization the network should use. Valid values
are "batchnorm", "layernorm", or None for no normalization (the default).
- reg: Scalar giving L2 regularization strength.
- weight_scale: Scalar giving the standard deviation for random
initialization of the weights.
- dtype: A numpy datatype object; all computations will be performed using
this datatype. float32 is faster but less accurate, so you should use
float64 for numeric gradient checking.
- seed: If not None, then pass this random seed to the dropout layers. This
will make the dropout layers deteriminstic so we can gradient check the
model.
"""
self.normalization = normalization
self.use_dropout = dropout != 1
self.reg = reg
self.num_layers = 1 + len(hidden_dims)
self.dtype = dtype
self.params = {}
############################################################################
# TODO: Initialize the parameters of the network, storing all values in #
# the self.params dictionary. Store weights and biases for the first layer #
# in W1 and b1; for the second layer use W2 and b2, etc. Weights should be #
# initialized from a normal distribution centered at 0 with standard #
# deviation equal to weight_scale. Biases should be initialized to zero. #
# #
# When using batch normalization, store scale and shift parameters for the #
# first layer in gamma1 and beta1; for the second layer use gamma2 and #
# beta2, etc. Scale parameters should be initialized to ones and shift #
# parameters should be initialized to zeros. #
############################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
L = len(hidden_dims) # 一共有L個隱含層
np.random.seed(seed)
shape_one = input_dim
for i in range(L):
self.params['W'+str(i+1)] = np.random.randn(shape_one, hidden_dims[i]) * weight_scale
self.params['b'+str(i+1)] = np.zeros(hidden_dims[i])
shape_one = hidden_dims[i] # 把第一個參數的第二維度賦給後一個參數的第一維度，是連續的
if self.normalization != None:
self.params['gamma'+str(i+1)] = np.ones(hidden_dims[i])
self.params['beta'+str(i+1)] = np.zeros(hidden_dims[i])
self.params['W'+str(L+1)] = np.random.randn(shape_one, num_classes) * weight_scale # 最後一個全連接層，用於輸出
self.params['b'+str(L+1)] = np.zeros(num_classes)
pass
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
############################################################################
# END OF YOUR CODE #
############################################################################
# When using dropout we need to pass a dropout_param dictionary to each
# dropout layer so that the layer knows the dropout probability and the mode
# (train / test). You can pass the same dropout_param to each dropout layer.
self.dropout_param = {}
if self.use_dropout:
self.dropout_param = {'mode': 'train', 'p': dropout}
if seed is not None:
self.dropout_param['seed'] = seed
# With batch normalization we need to keep track of running means and
# variances, so we need to pass a special bn_param object to each batch
# normalization layer. You should pass self.bn_params[0] to the forward pass
# of the first batch normalization layer, self.bn_params[1] to the forward
# pass of the second batch normalization layer, etc.
self.bn_params = []
if self.normalization=='batchnorm':
self.bn_params = [{'mode': 'train'} for i in range(self.num_layers - 1)] # 列表元素是字典，對應每一個mode
if self.normalization=='layernorm':
self.bn_params = [{} for i in range(self.num_layers - 1)]
# Cast all parameters to the correct datatype
for k, v in self.params.items():
self.params[k] = v.astype(dtype)
def loss(self, X, y=None):
"""
Compute loss and gradient for the fully-connected net.
Input / output: Same as TwoLayerNet above.
"""
X = X.astype(self.dtype) # 轉換數據類型
mode = 'test' if y is None else 'train'
# Set train/test mode for batchnorm params and dropout param since they
# behave differently during training and testing.
if self.use_dropout:
self.dropout_param['mode'] = mode
if self.normalization=='batchnorm':
for bn_param in self.bn_params:
bn_param['mode'] = mode
scores = None
############################################################################
# TODO: Implement the forward pass for the fully-connected net, computing #
# the class scores for X and storing them in the scores variable. #
# #
# When using dropout, you'll need to pass self.dropout_param to each #
# dropout forward pass. #
# #
# When using batch normalization, you'll need to pass self.bn_params[0] to #
# the forward pass for the first batch normalization layer, pass #
# self.bn_params[1] to the forward pass for the second batch normalization #
# layer, etc. #
############################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
if self.normalization == 'batchnorm':
bn_relu_cache = {}
input_layer = X
for i in range(self.num_layers-1):
bn_relu_out, bn_relu_cache[i] = affine_bn_relu_forward(input_layer, self.params['W'+str(i+1)], self.params['b'+str(i+1)], self.params['gamma'+str(i+1)], self.params['beta'+str(i+1)], self.bn_params[i])
input_layer = bn_relu_out
# 最後一層是沒有relu的，只有全連接，這裏把cache放在relu_cache字典了只是爲了方便，其實這一層不算relu_cache
final_layer_out, bn_relu_cache[self.num_layers-1] = affine_forward(bn_relu_out, self.params['W'+str(self.num_layers)], self.params['b'+str(self.num_layers)])
scores = final_layer_out
elif self.normalization == 'layernorm':
ln_relu_cache = {}
input_layer = X
for i in range(self.num_layers-1):
ln_relu_out, ln_relu_cache[i] = affine_ln_relu_forward(input_layer, self.params['W'+str(i+1)], self.params['b'+str(i+1)], self.params['gamma'+str(i+1)], self.params['beta'+str(i+1)], self.bn_params[i])
input_layer = ln_relu_out
# 最後一層是沒有relu的，只有全連接，這裏把cache放在relu_cache字典了只是爲了方便，其實這一層不算relu_cache
final_layer_out, ln_relu_cache[self.num_layers-1] = affine_forward(ln_relu_out, self.params['W'+str(self.num_layers)], self.params['b'+str(self.num_layers)])
scores = final_layer_out
else:
relu_cache = {}
input_layer = X
for i in range(self.num_layers-1):
relu_out, relu_cache[i] = affine_relu_forward(input_layer, self.params['W'+str(i+1)], self.params['b'+str(i+1)])
input_layer = relu_out
# 最後一層是沒有relu的，只有全連接，這裏把cache放在relu_cache字典了只是爲了方便，其實這一層不算relu_cache
final_layer_out, relu_cache[self.num_layers-1] = affine_forward(relu_out, self.params['W'+str(self.num_layers)], self.params['b'+str(self.num_layers)])
scores = final_layer_out
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
############################################################################
# END OF YOUR CODE #
############################################################################
# If test mode return early
if mode == 'test':
return scores
loss, grads = 0.0, {}
############################################################################
# TODO: Implement the backward pass for the fully-connected net. Store the #
# loss in the loss variable and gradients in the grads dictionary. Compute #
# data loss using softmax, and make sure that grads[k] holds the gradients #
# for self.params[k]. Don't forget to add L2 regularization! #
# #
# When using batch/layer normalization, you don't need to regularize the scale #
# and shift parameters. #
# #
# NOTE: To ensure that your implementation matches ours and you pass the #
# automated tests, make sure that your L2 regularization includes a factor #
# of 0.5 to simplify the expression for the gradient. #
############################################################################
# *****START OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
loss, dout = softmax_loss(scores, y)
for i in range(self.num_layers): # 正則化求loss的最終值
loss += 0.5 * self.reg * np.sum(self.params['W'+str(i+1)] * self.params['W'+str(i+1)])
if self.normalization == 'batchnorm':
# 最後一層的第一次反向傳播，因爲變量要重新賦值，這一次單獨拿出來
dx, final_dw, final_db = affine_backward(dout, bn_relu_cache[self.num_layers-1])
grads['W'+str(self.num_layers)] = final_dw + self.reg * self.params['W'+str(self.num_layers)]
grads['b'+str(self.num_layers)] = final_db
# 對剩餘的層可以用循環來求解
for i in range(self.num_layers-1, 0, -1):
dx, dw, db, dgamma, dbeta = affine_bn_relu_backward(dx, bn_relu_cache[i-1])
grads['W'+str(i)] = dw + self.reg * self.params['W'+str(i)]
grads['b'+str(i)] = db
grads['gamma'+str(i)] = dgamma
grads['beta'+str(i)] = dbeta
elif self.normalization == 'layernorm':
# 最後一層的第一次反向傳播，因爲變量要重新賦值，這一次單獨拿出來
dx, final_dw, final_db = affine_backward(dout, ln_relu_cache[self.num_layers-1])
grads['W'+str(self.num_layers)] = final_dw + self.reg * self.params['W'+str(self.num_layers)]
grads['b'+str(self.num_layers)] = final_db
# 對剩餘的層可以用循環來求解
for i in range(self.num_layers-1, 0, -1):
dx, dw, db, dgamma, dbeta = affine_ln_relu_backward(dx, ln_relu_cache[i-1])
grads['W'+str(i)] = dw + self.reg * self.params['W'+str(i)]
grads['b'+str(i)] = db
grads['gamma'+str(i)] = dgamma
grads['beta'+str(i)] = dbeta
else:
# 最後一層的第一次反向傳播，因爲變量要重新賦值，這一次單獨拿出來
dx, final_dw, final_db = affine_backward(dout, relu_cache[self.num_layers-1])
grads['W'+str(self.num_layers)] = final_dw + self.reg * self.params['W'+str(self.num_layers)]
grads['b'+str(self.num_layers)] = final_db
# 對剩餘的層可以用循環來求解
for i in range(self.num_layers-1, 0, -1):
dx, dw, db = affine_relu_backward(dx, relu_cache[i-1])
grads['W'+str(i)] = dw + self.reg * self.params['W'+str(i)]
grads['b'+str(i)] = db
# *****END OF YOUR CODE (DO NOT DELETE/MODIFY THIS LINE)*****
############################################################################
# END OF YOUR CODE #
############################################################################
return loss, grads
4- Batchnorm for deep networks訓練結果
訓練結果如下圖，從顯示的結果來看，使用了Batchnorm的網絡的train acc比沒有使用的更高，但是test acc卻低了一些，覺得主要原因還是因爲網絡不夠成熟。
但是從可視化的結果來看的話，batchnorm確實對訓練有一定的增強
4.1- batch norm的inline question
Describe the results of this experiment. What does this imply about the relationship between batch normalization and batch size? Why is this relationship observed?
根據batch norm的計算方法來看，當batch size較小時，計算樣本的均值和方差結果對總體的估計偏差較大，這樣BN效果就不好了，也就是batch size越大BN的效果應該越好。