Layers Python Implementation
Affine (Fully connected layer):
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)
"""
# Number of images in the batch.
NN = x.shape[0]
# Reshape each input in our batch to a vector.
reshaped_input = np.reshape(x,[NN, -1])
# FC layer forward pass.
out = np.dot(reshaped_input, w) + b
cache = (x, w, b)
return out, cachedef 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)
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
# Number of images in the batch.
NN = x.shape[0]
# Reshape each input in our batch to a vector.
reshaped_x = np.reshape(x,[NN, -1]) # input is an image, need to resize first.
# Calculate dx = w*dout - remember to reshape back to shape of x.
dx = np.dot(dout, w.T)
dx = np.reshape(dx, x.shape)
# Calculate dw = x*dout
dw = np.dot(reshaped_x.T,dout)
# Calculate db = dout
db = np.sum(dout, axis=0)
return dx, dw, dbRelu
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
"""
# Forward Relu.
out = x.copy() # Must use copy in numpy to avoid pass by reference.
out[out < 0] = 0
cache = x
return out, cachedef 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
"""
x = cache
# For Relu we only backprop to non-negative elements of x
relu_mask = (x >= 0)
dx = dout * relu_mask
return dxBatch Norm
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. #
#######################################################################
# Take sample mean & var of our minibatch across each dimension.
sample_mean = np.mean(x, axis=0)
sample_var = np.var(x, axis=0)
# Normalize our batch then shift and scale with gamma/beta.
normalized_data = (x - sample_mean) / np.sqrt(sample_var + eps)
out = gamma * normalized_data + beta
# Update our running mean and variance then store.
running_mean = momentum * running_mean + (1 - momentum) * sample_mean
running_var = momentum * running_var + (1 - momentum) * sample_var
bn_param['running_mean'] = running_mean
bn_param['running_var'] = running_var
# Store intermediate results needed for backward pass.
cache = {
'x_minus_mean': (x - sample_mean),
'normalized_data': normalized_data,
'gamma': gamma,
'ivar': 1./np.sqrt(sample_var + eps),
'sqrtvar': np.sqrt(sample_var + eps),
}
elif mode == 'test':
# Test time batch norm using learned gamma/beta and calculated running mean/var.
out = (gamma / (np.sqrt(running_var + eps)) * x) + (beta - (gamma*running_mean)/np.sqrt(running_var + 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, cachedef 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,)
"""
# Get cached results from the forward pass.
N, D = dout.shape
normalized_data = cache.get('normalized_data')
gamma = cache.get('gamma')
ivar = cache.get('ivar')
x_minus_mean = cache.get('x_minus_mean')
sqrtvar = cache.get('sqrtvar')
# Backprop dout to calculate dbeta and dgamma.
dbeta = np.sum(dout, axis=0)
dgamma = np.sum(dout * normalized_data, axis=0)
# Carry on the backprop in steps to calculate dx.
# Step1
dxhat = dout*gamma
# Step2
dxmu1 = dxhat*ivar
# Step3
divar = np.sum(dxhat*x_minus_mean, axis=0)
# Step4
dsqrtvar = divar * (-1/sqrtvar**2)
# Step5
dvar = dsqrtvar * 0.5 * (1/sqrtvar)
# Step6
dsq = (1/N)*dvar*np.ones_like(dout)
# Step7
dxmu2 = dsq * 2 * x_minus_mean
# Step8
dx1 = dxmu1 + dxmu2
dmu = -1*np.sum(dxmu1 + dxmu2, axis=0)
# Step9
dx2 = (1/N)*dmu*np.ones_like(dout)
# Step10
dx = dx2 + dx1
return dx, dgamma, dbetaLast updated
