K-FAC: Kronecker-Factored Approximate Curvature

K-FAC in TensorFlow is an implementation of K-FAC, an approximate second-order optimization method, in TensorFlow. When applied to feedforward and convolutional neural networks, K-FAC can converge >3.5x faster in >14x fewer iterations than SGD with Momentum.

What is K-FAC?

K-FAC, short for “Kronecker-factored Approximate Curvature”, is an approximation to the Natural Gradient algorithm designed specifically for neural networks. It maintains a block-diagonal approximation to the Fisher Information matrix, whose inverse preconditions the gradient.

K-FAC can be used in place of SGD, Adam, and other Optimizer implementations. Experimentally, K-FAC converges >3.5x faster than well-tuned SGD.

Unlike most optimizers, K-FAC exploits structure in the model itself (e.g. “What are the weights for layer i?”). As such, you must add some additional code while constructing your model to use K-FAC.

Why should I use K-FAC?

K-FAC can take advantage of the curvature of the optimization problem, resulting in faster training. For an 8-layer Autoencoder, K-FAC converges to the same loss as SGD with Momentum in 3.8x fewer seconds and 14.7x fewer updates. See how training loss changes as a function of number of epochs, steps, and seconds:

Is K-FAC for me?

If you have a feedforward or convolutional model for classification that is converging too slowly, K-FAC is for you. K-FAC can be used in your model if:

• Your model defines a posterior distribution.
• Your model uses only fully-connected or convolutional layers (residual connections OK).
• You are training on CPU or GPU.
• You can modify model code to register layers with K-FAC.

How do I use K-FAC?

Using K-FAC requires three steps:

1. Registering layer inputs, weights, and pre-activations with a LayerCollection.
2. Minimizing the loss with a KfacOptimizer.
3. Keeping K-FAC's preconditioner updated.

# Build model.
w = tf.get_variable("w", ...)
b = tf.get_variable("b", ...)
logits = tf.matmul(x, w) + b
loss = tf.reduce_mean(
  tf.nn.softmax_cross_entropy_with_logits(labels=y, logits=logits))

# Register layers.
layer_collection = LayerCollection()
layer_collection.register_fully_connected((w, b), x, logits)
layer_collection.register_categorical_predictive_distribution(logits)

# Construct training ops.
optimizer = KfacOptimizer(..., layer_collection=layer_collection)
train_op = optimizer.minimize(loss)

# Minimize loss.
with tf.Session() as sess:
  ...
  sess.run([train_op, optimizer.cov_update_op, optimizer.inv_update_op])

See examples/ for runnable, end-to-end illustrations.

Authors

• Alok Aggarwal
• Daniel Duckworth
• James Martens
• Matthew Johnson
• Olga Wichrowska
• Roger Grosse