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backprop_nn.py
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backprop_nn.py
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import numpy as np
from activations import Softmax, ActivationFunction
from losses import CrossEntropy, Unity, QuadraticLoss, LossFunction
from typing import List, Tuple
from copy import deepcopy
class NeuralNetwork(object):
def __init__(
self,
layers: list,
activations: List[ActivationFunction],
loss: LossFunction,
rng: np.random._generator.Generator = None
):
assert (len(layers) == len(activations) + 1)
self.layers = layers
self.activations = activations
self.loss = loss
self.weights = []
self.biases = []
self.rng = np.random.RandomState(123) if not rng else rng
for l in range(len(layers) - 1):
# xavier initialization of weights
self.weights.append(np.sqrt(2 / layers[l]) * self.rng.randn(layers[l + 1], layers[l]))
self.biases.append(np.zeros((layers[l + 1], 1)))
def feedforward(self, x: np.ndarray) -> Tuple[list, list]:
"""
Function to feed forward a signal x through the network
:param x: signal / feature matrix
:return: list of weighted inputs and activations for each layer
"""
# return the feedforward value for x
a = np.copy(x)
z_s = []
a_s = [a]
for l in range(len(self.weights)):
z_s.append(self.weights[l].dot(a) + self.biases[l])
a = self.activations[l](z_s[-1])
a_s.append(a)
return z_s, a_s
def backpropagation(self, y: np.ndarray, z_s: list, a_s: list) -> Tuple[list, list]:
"""
Function that implements the backpropagation algorithm
:param y: 1xobservation numpy array containing the labels
:param z_s: list containing array of weighted inputs calculated at each neuron for each layer
:param a_s: list containing array of activations calculated at each neuron for each layer
:return: list containing gradients of cost function w.r.t weights and biases for each layer
"""
# delta = dC/dZ known as error for each layer
deltas = [np.zeros_like(self.weights[i]) for i in range(len(self.weights))]
# insert the last layer error
# first derivative of cross entropy has an easy form if last layers activation is softmax. Call it here
if isinstance(self.activations[-1], Softmax) and isinstance(self.loss, CrossEntropy):
deltas[-1] = np.atleast_2d(
self.loss.first_derivative_softmax(a_s[-1], y)
)
# otherwise, the first derivative is the hadamard product of gradient of loss and gradient of output
else:
deltas[-1] = np.atleast_2d(
self.loss.first_derivative(a_s[-1], y) * (self.activations[-1].first_derivative(z_s[-1]))
)
# perform backpropagation
for l in reversed(range(len(deltas) - 1)):
deltas[l] = self.weights[l + 1].T.dot(deltas[l + 1]) * (
self.activations[l].first_derivative(z_s[l])
)
batch_size = y.shape[1] # observations in columns, features/classes in rows
# calculate the derivatives of the loss w.r.t weights using the deltas (error terms)
db = [d.dot(np.ones((batch_size, 1))) / float(batch_size) for d in deltas]
dw = [d.dot(a_s[i].T) / float(batch_size) for i, d in enumerate(deltas)]
return dw, db
def train(
self, x_train, y_train, batch_size=10, epochs=100, lr=0.01, val: tuple = None, momentum=0
) -> Tuple[dict, dict]:
"""
Function that implements stochastic gradient descent with standard backpropagation for training the
neural network. If a validation set is provided, the best parameters are saved and overwrite the
current weights and biases at the end of the training
:param x_train: input features of training data. Required shape is umpy 2D array with D rows and N columns
:param y_train: labels of training data. Required shape is numpy 2D array with 1 row and N columns
:param batch_size: size of the mini batch
:param epochs: number of training iterations
:param lr: learning rate alpha
:param val: validation set (features, labels)
:param momentum: momentum parameter, if momentum should be used
:return: dictionary of train set and validation set loss for the different epochs
"""
# update weights and biases based on the output
losses = {}
val_losses = {}
# define the function to be used to evaluate the loss
eval_func = lambda x, y: self.loss(self.feedforward(x)[1][-1], y)
# save the best parameters in terms of validation set loss
best_weights = self.weights.copy()
best_biases = self.biases.copy()
# initialize velocity vectors for sgd with momentum
weight_velocity = [
np.zeros_like(weights) for weights in self.weights
]
bias_velocity = [
np.zeros_like(biases) for biases in self.biases
]
for e in range(epochs):
x, y = self.shuffle_training_data(x_train, y_train)
i = 0
while i < y.shape[1]:
x_batch = x[:, i:i + batch_size]
y_batch = y[:, i:i + batch_size]
i = i + batch_size
z_s, a_s = self.feedforward(x_batch)
dw, db = self.backpropagation(y_batch, z_s, a_s)
if momentum:
weight_velocity = [
momentum * velocity - lr * dweight for velocity, dweight in zip(weight_velocity, dw)
]
bias_velocity = [
momentum * velocity - lr * dbias for velocity, dbias in zip(bias_velocity, db)
]
self.weights = [
w + velocity for w, velocity in zip(self.weights, weight_velocity)
]
self.biases = [
w + velocity for w, velocity in zip(self.biases, bias_velocity)
]
else:
self.weights = [
w - lr * weight_gradient for w, weight_gradient in zip(self.weights, dw)
]
self.biases = [
w - lr * bias_gradient for w, bias_gradient in zip(self.biases, db)
]
losses[e] = eval_func(x, y)
if val:
val_losses[e] = eval_func(*val)
if val_losses[e] <= min(val_losses.values()):
# save the best parameters in terms of validation set loss
best_weights = deepcopy(self.weights)
best_biases = deepcopy(self.biases)
else:
val_losses[e] = np.nan
print("Training loss = {}. Validation loss = {}".format(losses[e], val_losses[e]))
self.weights = best_weights
self.biases = best_biases
return losses, val_losses
def train_ekf(
self, x, y, P=None, Q=None, R=None, epochs=10, val: tuple = None, eta=0.3
) -> Tuple[dict, dict]:
"""
this function executes the EKF-algorithm iteratively for training the neural network. It iterates
through the specified number of training iterations and within each training iteration through every
training pattern provided in arrays x and y. If a validation set is provided, the best parameters are
saved and overwrite the current weights and biases at the end of the training.
:param x: input features of training data. Required shape is numpy 2D array with D rows and N columns
:param y: labels of training data. Required shape is numpy 2D array with 1 row and N columns
:param P: square matrix of size equal to the number of parameters or scalar value
:param R: square matrix of size equal to the number of output neurons or scalar value
:param Q: square matrix of size equal to the number of parameters, scalar value or None
:param epochs: number of training iterations
:param val: tuple containing a validation set: (features, labels)
:param eta: forgetting factor for updating Q and R
:return: dictionary of train set and validation set loss for the different epochs
"""
losses = {}
val_losses = {}
self._init_kalman(P, R, Q)
best_weights = self.weights.copy()
best_biases = self.biases.copy()
# Require that loss function used in backpropagation is the identity, i.e. make Jacobian of
# output w.r.t. weights
if not isinstance(self.loss, Unity):
eval_func = lambda x, y: self.loss(self.feedforward(x)[1][-1], y)
self.loss = Unity()
else:
eval_func = lambda x, y: QuadraticLoss()(self.feedforward(x)[1][-1], y)
# iterate over epochs and training instances
for e in range(epochs):
x_shuffled, y_shuffled = self.shuffle_training_data(x, y)
# Train
for i in range(y.shape[1]):
x_batch = x_shuffled[:, [i]]
y_batch = y_shuffled[:, [i]]
# Forward propagation
z_s, a_s = self.feedforward(x_batch)
# Do the learning
self._ekf(x_batch, y_batch, z_s, a_s, eta)
losses[e] = eval_func(x, y)
if val:
val_losses[e] = eval_func(*val)
if val_losses[e] <= min(val_losses.values()):
best_weights = deepcopy(self.weights)
best_biases = deepcopy(self.biases)
else:
val_losses[e] = np.nan
print("Training loss = {}. Validation loss = {}".format(losses[e], val_losses[e]))
print("New R: {}, or learning rate: {}".format(self.R, 1/self.R))
self.weights = best_weights
self.biases = best_biases
return losses, val_losses
def shuffle_training_data(self, x_train, y_train) -> Tuple[np.ndarray, np.ndarray]:
"""
In order to randomize the training process, the training data is randomly permuted.
:param x_train: design matrix of training instances
:param y_train: labels of training instances
:param rng: Random number generator
:return: shuffled dataset
"""
x_train = np.copy(x_train)
y_train = np.copy(y_train)
permutation = self.rng.permutation(y_train.shape[1])
y_train = np.take(y_train, permutation, axis=1)
x_train = np.take(x_train, permutation, axis=1)
return x_train, y_train
def _init_kalman(self, P, R, Q):
"""
Initialize P, Q and R matrices. One can only pass scalar values for the respective matrices to the ekf train
method of the neural network. In this case, the matrices P, Q and R are constructed by multiplying the
scalar with the identity matrix.
"""
num_params = sum(map(np.size, self.weights)) + sum(map(np.size, self.biases))
if np.isscalar(P):
self.P = P*np.eye(num_params)
else:
assert P.shape == (num_params, num_params)
self.P = P
if Q is None:
self.Q = np.zeros((num_params, num_params))
elif np.isscalar(Q):
self.Q = Q*np.eye(num_params)
else:
assert Q.shape == (num_params, num_params)
self.Q = Q
if np.isscalar(R):
self.R = R*np.eye(self.layers[-1])
def _ekf(self, x, y, z_s, a_s, eta):
"""
This function implements the Extended Kalman Filter Recursion that determines the weight
update for the neural network.
"""
# Compute NN jacobian
# perform backpropagation
Hw, Hb = self.backpropagation(y, z_s, a_s)
# NN Jacobian H is a N x W matrix, with N the number of instances and W the number of
# parameters. Stack the weights and biases horizontally in this step
Hw = np.hstack([dweight.flatten() for dweight in Hw])
Hb = np.hstack([dbias.flatten() for dbias in Hb])
H = np.hstack((Hw, Hb)).reshape(1, -1)
# compute Kalman gain
A = H.dot(self.P).dot(H.T) + self.R
K = self.P.dot(H.T).dot(np.linalg.inv(A))
# compute the weight delta
innovation = self.loss(a_s[-1], y)
dtheta = K.dot(innovation)
# split dthetha into weight and bias gradients
dw_flat = dtheta[:Hw.size]
db_flat = dtheta[Hw.size:]
# bring the gradients in the same shape as the weight and bias matrices
idx_weight, idx_bias = 0, 0
# update the weights and biases
for i, (layer_weights, layer_biases) in enumerate(zip(self.weights, self.biases)):
self.weights[i] += dw_flat[idx_weight:idx_weight+layer_weights.size].reshape(
layer_weights.shape
)
self.biases[i] += db_flat[idx_bias:idx_bias+layer_biases.size].reshape(
layer_biases.shape
)
idx_weight += layer_weights.size
idx_bias += layer_biases.size
if eta < 1:
# update measurement noise
a_posteriori_y = self.feedforward(x)[-1]
residual = self.loss(a_posteriori_y[-1], y)
self.R = eta * self.R + (1-eta) * (np.dot(residual.T, residual) + H.dot(self.P).dot(H.T))
# update process noise
self.Q = eta * self.Q + (1-eta) * np.dot(K, np.dot(innovation.T, innovation)).dot(K.T)
# update error covariance
self.P = self.P - np.dot(K, H.dot(self.P)) + self.Q