L2-norms of gradients increasing during training of deep neural network

Question

I'm training a convolutional neural network (CNN) with 5 conv-layers and 2 fully-connected layers for binary classification using stochastic gradient descent (SGD) with momentum. The accuracy of the network is fine, but I am curious about the behavior of the gradients. I would expect that the 2-norm of the gradients would get lowered during training due to continuous lowering of the learning rate. However, for all of my layers the 2-norm increases slightly during training. How can that be?

More importantly, what should I do to fix it?

Below is an image of the 2-norm of the gradients for my last fully-connected layer (it has 450 neurons). The development for the remaining layers look identical, but the 2-norm is shifted as the gradients are lower for earlier layers (as expected).

Update: This is the code I used to find the gradient of the CNN implemented in Caffe. It should be working correctly based on the answers in this StackOverflow post. The batch-size of my network is 32 and I use leaky ReLU after each layer and dropout after the fully-connected layers.

def find_loss_gradients(self, solver):
    net = solver.net
    diffs = net.backward(diffs=['hzvt1', 'hzvt2', 'hzvt3', 'hzvt4', 'hzvt5', 'hzvt6', 'fc1'])
    l2norms = {}

    for key in diffs.keys():
       grad = diffs[key].flatten()
       l2norms.update({key:np.linalg.norm(grad)})
    return l2norms

def graph_gradients(save_dir, gradient_norm):
    graph_gradient_norm(save_dir + "gradient_all.png", gradient_norm)    
    values_dict = gradient_norm[:,1]
    layers = values_dict[0].keys()
    for key in layers:
        graph_gradient_norm(save_dir + "gradient_{}.png".format(key), gradient_norm, key)

def graph_gradient_norm(save_name, gradient_norm, key=None):
    plt.style.use('ggplot')
    fig = plt.figure()
    _, ax1 = P.subplots()

    x = gradient_norm[:,0]
    values_dict = gradient_norm[:,1]
    handles = []

    if key is None:
        layers = values_dict[0].keys()
    else:
        layers = [key]

    for key in layers:
        val = cm.utils.extract_array_from_dict(values_dict, key, index_into=False)
        h, = ax1.semilogy(x,val)
        handles.append(h)

    ax1.set_xlabel('Iterations')
    ax1.set_ylabel('2-Norm of Gradient')
    plt.legend(handles, layers)
    P.savefig(save_name, bbox_inches='tight')
    plt.close(fig)

Below is a simplified version of my training code

grad_norms = []
for i in range(n_steps):
    solver.step(1)
    grad_norms.append(find_loss_gradients(solver))
grad_norms = np.array(grad_norms)
graph_gradients(run_dir + "visualization/", grad_norms)

Update 2

Note that the loss stops decreasing quite quickly. The same is true for the validation accuracy. I'm currently trying to increase the batch size and see if it has an influence.

Answer 1

I guess I got what is a problem with a gradient norm value. Basically negative gradient shows a direction to a local minimum value, but it doesn't say how far it is. For this reason you are able to configure you step proportion. When your weight combination is closer to the minimum value your constant step could be bigger than is necessary and some times it hits in wrong direction and in next epooch network try to solve this problem. Momentum algorithm use modified approach. After each iteration it increases weight update if sign for the gradient the same (by an additional parameter that is added to the $\Delta w$ value). In terms of vectors this addition operation can increase magnitude of the vector and change it direction as well, so you are able to miss perfect step even more. To fix this problem network sometimes needs a bigger vector, because minimum value a little further than in the previous epoch.

To prove that theory I build small experiment. First of all I reproduce the same behaviour but for simpler network architecture with less number of iterations.

import numpy as np
from numpy.linalg import norm
import matplotlib.pyplot as plt
from sklearn.datasets import make_regression
from sklearn import preprocessing
from sklearn.pipeline import Pipeline
from neupy import algorithms

plt.style.use('ggplot')

grad_norm = []
def train_epoch_end_signal(network):
    global grad_norm
    # Get gradient for the last layer
    grad_norm.append(norm(network.gradients[-1]))

data, target = make_regression(n_samples=10000, n_features=50, n_targets=1)

target_scaler = preprocessing.MinMaxScaler()
target = target_scaler.fit_transform(target)

mnet = Pipeline([
    ('scaler', preprocessing.MinMaxScaler()),
    ('momentum', algorithms.Momentum(
        (50, 30, 1),
        step=1e-10,
        show_epoch=1,
        shuffle_data=True,
        verbose=False,
        train_epoch_end_signal=train_epoch_end_signal,
    )),
])

mnet.fit(data, target, momentum__epochs=100)

After training I checked all gradients on plot. Below you can see similar behaviour as yours.

plt.figure(figsize=(12, 8))
plt.plot(grad_norm)
plt.title("Momentum algorithm final layer gradient 2-Norm")
plt.ylabel("Gradient 2-Norm")
plt.xlabel("Epoch")
plt.show()

Also if look closer into the training procedure results after each epoch you will find that errors are vary as well.

plt.figure(figsize=(12, 8))
network = mnet.steps[-1][1]
network.plot_errors()
plt.show()

Next I using almost the same settings create another network, but for this time I select Golden search algorithm for step selection on each epoch.

grad_norm = []
def train_epoch_end_signal(network):
    global grad_norm
    # Get gradient for the last layer
    grad_norm.append(norm(network.gradients[-1]))
    if network.epoch % 20 == 0:
        print("Epoch #{}: step = {}".format(network.epoch, network.step))

mnet = Pipeline([
    ('scaler', preprocessing.MinMaxScaler()),
    ('momentum', algorithms.Momentum(
        (50, 30, 1),
        step=1e-10,
        show_epoch=1,
        shuffle_data=True,
        verbose=False,
        train_epoch_end_signal=train_epoch_end_signal,
        optimizations=[algorithms.LinearSearch]
    )),
])

mnet.fit(data, target, momentum__epochs=100)

Output below shows step variation at each 20 epoch.

Epoch #0: step = 0.5278640466583575
Epoch #20: step = 1.103484809236065e-13
Epoch #40: step = 0.01315561773591515
Epoch #60: step = 0.018180616551587894
Epoch #80: step = 0.00547810271094794

And if you after that training look closer into the results you will find that variation in 2-norm is much smaller

plt.figure(figsize=(12, 8))
plt.plot(grad_norm)
plt.title("Momentum algorithm final layer gradient 2-Norm")
plt.ylabel("Gradient 2-Norm")
plt.xlabel("Epoch")
plt.show()

And also this optimization reduce variation of errors as well

plt.figure(figsize=(12, 8))
network = mnet.steps[-1][1]
network.plot_errors()
plt.show()

As you can see the main problem with gradient is in the step length.

It's important to note that even with a high variation your network can give you improve in your prediction accuracy after each iteration.

Answer 2

This is normal situation in training of convolutional NNN. Your loss function decrease, but loss gradients go up. This is true both for Vanilla SGD, and for SGD with momentum. The reason for this is that initial learning rate is too high for some layers. There is very good Ian Goodfellow' tutorial on Optimization of NN, where he explains why this happens: http://videolectures.net/deeplearning2015_goodfellow_network_optimization/ , minute 29