Renormalize weight matrix using TensorFlow

Question

I'd like to add a max norm constraint to several of the weight matrices in my TensorFlow graph, ala Torch's renorm method.

If the L2 norm of any neuron's weight matrix exceeds max_norm, I'd like to scale its weights down so that their L2 norm is exactly max_norm.

What's the best way to express this using TensorFlow?

Answer 1

Here is a possible implementation:

import tensorflow as tf

def maxnorm_regularizer(threshold, axes=1, name="maxnorm", collection="maxnorm"):
    def maxnorm(weights):
        clipped = tf.clip_by_norm(weights, clip_norm=threshold, axes=axes)
        clip_weights = tf.assign(weights, clipped, name=name)
        tf.add_to_collection(collection, clip_weights)
        return None # there is no regularization loss term
    return maxnorm

Here's how you would use it:

from tensorflow.contrib.layers import fully_connected
from tensorflow.contrib.framework import arg_scope

with arg_scope(
        [fully_connected],
        weights_regularizer=max_norm_regularizer(1.5)):
    hidden1 = fully_connected(X, 200, scope="hidden1")
    hidden2 = fully_connected(hidden1, 100, scope="hidden2")
    outputs = fully_connected(hidden2, 5, activation_fn=None, scope="outs")

max_norm_ops = tf.get_collection("max_norm")

[...]

with tf.Session() as sess:
    sess.run(init)
    for epoch in range(n_epochs):
        for X_batch, y_batch in load_next_batch():
            sess.run(training_op, feed_dict={X: X_batch, y: y_batch})
            sess.run(max_norm_ops)

This creates a 3 layer neural network and trains it with max norm regularization at every layer (with a threshold of 1.5). I just tried it, seems to work. Hope this helps! Suggestions for improvements are welcome. :)

Notes

This code is based on tf.clip_by_norm():

>>> x = tf.constant([0., 0., 3., 4., 30., 40., 300., 400.], shape=(4, 2))
>>> print(x.eval())
[[   0.    0.]
 [   3.    4.]
 [  30.   40.]
 [ 300.  400.]]
>>> clip_rows = tf.clip_by_norm(x, clip_norm=10, axes=1)
>>> print(clip_rows.eval())
[[ 0.          0.        ]
 [ 3.          4.        ]
 [ 6.          8.        ]  # clipped!
 [ 6.00000048  8.        ]] # clipped!

You can also clip columns if you need to:

>>> clip_cols = tf.clip_by_norm(x, clip_norm=350, axes=0)
>>> print(clip_cols.eval())
[[   0.            0.        ]
 [   3.            3.48245788]
 [  30.           34.82457733]
 [ 300.          348.24578857]]
                # clipped!

Answer 2

Using Rafał's suggestion and TensorFlow's implementation of clip_by_norm, here's what I came up with:

def renorm(x, axis, max_norm):
    '''Renormalizes the sub-tensors along axis such that they do not exceed norm max_norm.'''
    # This elaborate dance avoids empty slices, which TF dislikes.
    rank = tf.rank(x)
    bigrange = tf.range(-1, rank + 1)
    dims = tf.slice(
                tf.concat(0, [tf.slice(bigrange, [0], [1 + axis]),
                              tf.slice(bigrange, [axis + 2], [-1])]),
                [1], rank - [1])

    # Determine which columns need to be renormalized.
    l2norm_inv = tf.rsqrt(tf.reduce_sum(x * x, dims, keep_dims=True))
    scale = max_norm * tf.minimum(l2norm_inv, tf.constant(1.0 / max_norm))

    # Broadcast the scalings
    return tf.mul(scale, x)

It seems to have the desired behavior for 2-dimensional matrices and should generalize to tensors:

> x = tf.constant([0., 0., 3., 4., 30., 40., 300., 400.], shape=(4, 2))
> print x.eval()
[[   0.    0.]  # rows have norms of 0, 5, 50, 500
 [   3.    4.]  # cols have norms of ~302, ~402
 [  30.   40.]
 [ 300.  400.]]
> print renorm(x, 0, 10).eval()
[[ 0.          0.        ]  # unaffected
 [ 3.          4.        ]  # unaffected
 [ 5.99999952  7.99999952]  # rescaled
 [ 6.00000048  8.00000095]] # rescaled
> print renorm(x, 1, 350).eval()
[[   0.            0.        ]  # col 0 is unaffected
 [   3.            3.48245788]  # col 1 is rescaled
 [  30.           34.82457733]
 [ 300.          348.24578857]]