Neural Network Compression

Brief survey on neural network compression techniques sampled from existing surveys.

Pruning

Sensitivity Methods

The idea here is to measure how sensitive each weight/connection or neuron is.
I.e. if you remove the neuron, how will it change the output?
Typically, weights are pruned by zeroing them out, freezing them, and fine-tuning the unfrozen weights.

In general, the procedure is

1. Train the network with a lot of parameters.
2. Compute sensitivity for each parameter.
3. Delete low-saliency parameters.
4. Continue training to fine-tune remaining parameters.
5. Repeat pruning until the number of parameters is low enough or the error is too high.

Sometimes, pruning can also increase accuracy and improve generalization.

• Mozer and Smolensky (1988) use a gate for each neuron. Then the sensitivity and be estimated with the derivative w.r.t the gate.
• Karnin estimates the sensitivity by monitoring the change in weight during training.
• LeCun e al. present Optimal Brain Damage which uses the second derivative of each weight.

Redundancy Methods

• Srinivas and Babu propose a pair-wise similarity on each neuron: \(\displaystyle s = \Vert a_j^2 \Vert_1 \Vert W_i - W_j \Vert^2_{2}\) where \(\displaystyle a_j\) is the vector of weights on neuron j at the layer above and \(\displaystyle W\) are neuron weights. This combines a weight metric and a similarity metric into one sensitivity metric. When a neuron is pruned, the matrix for the current and next layers need to be updated.

Structured Pruning

Structured pruning focuses on keeping the dense structure of the network such that the pruned network can benefit using standard dense matrix multiplication operations.
This is in contrast to unstructured pruning which zeros out values in the weight matrix but may not necessarilly run faster.

• Wen et al. (2016) propose Structured Sparsity Learning (SSL) on CNNs. Given filters of size (N, C, M, K), i.e. (out-channels, in-channels, height, width), they use a group lasso loss/regularization to penalize usage of extra input and output channels. They also learn filter shapes using this regularization.

Quantization

There are many codebases which use 8-bit or 16-bit representations instead of the standard 32-bit floats.
Work on quantization typically focus on different representations and mixed-precision training, though quantization can also be used to speed up inference.

• Google uses bfloat16 for training on TPUs.
• Gupta et al. train using a custom 16-bit representation with stochastic rounding. They observe little to no degradation on MNIST MLP and CIFAR10 CNN classification accuracy. Stochastic rounding rounds to the nearest value with probability based on distance to that value.

Factorization

Also known as tensor decomposition.

• Denil et al. (2013) propose a low-rank factorization: \(\displaystyle W=UV\) where \(\displaystyle U\) is \(\displaystyle n_v \times n_\alpha\) and \(\displaystyle V\) is \(\displaystyle n_\alpha \times n_h\). Here, vectors are left-multiplied against \(\displaystyle W\). The compare several scenarios: training both U and V, randomly setting U with identity basis vectors, randomly setting U with iid Gaussian entries, and more.

Libraries

Both Tensorflow and PyTorch have built in libraries for pruning:

• PyTorch pruning tutorial
• TF: Pruning with Keras

These support magnitude-based pruning which zero out small weights.

Resources

Surveys

• Pruning algorithms a survey (1993) by Russel Reed
• A Survey of Model Compression and Acceleration for Deep Neural Networks (2017) by Cheng et al.
• An Overview of Neural Network Compression (2020) by James O' Neill