# Paper Explanation: Net2Net – Accelerating Learning via Knowledge Transfer

### Motivation

One of the biggest challenges during designing new neural network architectures is time. During real-world workflows, one often trains many different neural networks during the experimentation and design process. This is a wasteful process in which each new model is trained from scratch.

In a typical workflow, one trains multiple models, with each model designed to improve upon the previous model in some way. As a result, this iterative design process relies on fully training and evaluating the innovations from the previous step (design) from scratch. This makes the whole designing process slow due to the latency of evaluating whether each change to the model caused an improvement or not.

This is where the Net2Net procedure come in. It’s (in my opinion) a really cool yet simple method to address the above problem to some extent.

### Net2Net Procedure

The Net2Net strategy involves initializing the student network (the new model) to represent the same function as the teacher (the previous model), then continue training the student network by normal means. What this means is that, before further training the student network, for every input, it will give the same output (label) as the teacher network even though its architecture may be quite different from the teacher network.

Mathematically, suppose that the teacher network is represented by a function y = f(x; θ) where x is the input to the network, y is the output of the network, and θ is the parameters of the network. The Net2Net strategy is to choose a new set of parameters θ’ for the student network g(x; θ’) such that

∀x, f(x; θ) = g(x; θ’)

There are two ways of using Net2Net: Increase the width or the depth of the network.

##### Net2WiderNet

This allows a layer to be replaced with a wider layer, meaning a layer that has more units. For convolution architectures, this means the layers will have more convolution channels.

Suppose that layer i and layer i + 1 are both fully connected layers, and layer i uses an elementwise non-linearity. To widen layer i, replace $W^{(i)}$ and $W^{(i+1)}$. If layer i has m inputs and n outputs, and layer i + 1 has p outputs, then $W^{(i)}\in \Re^{m\times n}$ and $W^{(i+1)}\in \Re^{n\times p}$. Net2WiderNet allows you to replace layer i with a layer that has t outputs, with t > n.

New weight matrices $U^{(i)}$ and $U^{(i+1)}$, replacing $W^{(i)}$ and $W^{(i+1)}$ respectively, representing the weights for these layers in the new student network are introduced. Then the new weights are given by:

Here, the first n columns of $W^{(i)}$ are copied directly into $U^{(i)}$. Columns n+1 through t of $U^{(i)}$ are created by choosing a random column from $W^{(i)}$ using the random mapping function, g. The random selection is performed with replacement, so each column of $W^{(i)}$ is copied potentially many times. For weights in $U^{(i+1)}$, we must account for the replication by dividing the weight by the replication factor given by $\mathbf{\frac{1}{|\left\{x|g(x)\ =\ g(j)\right\}|}}$. This is done to make sure the output of the student network remains the same as that of the teacher network. (To calculate the activation of a particular neuron in a layer we perform a weighted sum of the neurons in the previous layer. If a neuron is repeated multiple times in the previous layer, it’s contribution to the sum will become higher than what is required. Therefore, to account for the multiple neurons, the weights of the repetitions are appropriately divided, if twice than divide by 2, if thrice divide by 3, and so on, to make the sum same as before.)

The random mapping function g can be defined as, g : {1, 2, · · · , t} → {1, 2, · · · , n}, that satisfies:

The above is just one of the many ways to define the function g. These functions are subject to constraints defined by the computation graph of the model, as care needs to be taken to ensure that the remapping function does, in fact, result in function preservation. One example of a constraint imposed by the batch normalization layer. The layer involves a linear transformation (shifting) and also elementwise multiplication (scaling) by learned parameters. The random remapping for the multiplication parameters must match the random remapping for the weight matrix. Otherwise, a new unit could be generated that uses the weight vector for pre-existing unit i but is scaled by the multiplication parameter for unit j. The new unit would not implement the same function as the old unit i or as the old unit j (in the teacher network), which is not the desired result.

##### Net2DeeperNet

This allows us to transform an existing net into a deeper one. A layer $h^{(i)} = \phi (h^{(i-1)T}W^{(i)})$ is replaced with two layers $h(i) = \phi (U^{(i)T}\phi (W^{(i)T}h^{(i-1)})))$. The new matrix U is initialized to an identity matrix. This operation is only applicable when $\phi$ is chosen such that $\phi (I\phi (v)) = \phi (v)$ for all vectors v. The rectified linear unit (ReLU) satisfy this property but the sigmoid activation function doesn’t. When applying this method to convolution networks, simply set the convolution kernels to be identity filter.

The above method of Net2DeeperNet is a specific case of factorizing a layer in a network. It factorizes a layer $L^{(i)}$ into two layers $I$ and $L^{(i)}$, where I is the identity mapping layer.

### How Effective is Net2Net?

Due to the function preserving method used, the new larger network (student network), performs just as well as the original network (teacher network) instead of having to go through a period of low performance.

Also, Networks trained using Net2Net converge faster to the same accuracy as networks initialized randomly. Though, one thing to note is that the final accuracy is independent of the training procedure and only dependent on the size of the network.

Graphs showing the results of experiments performed by the authors of the paper show how much beneficial it is to train using Net2Net when designing new architectures and experimenting.

Experiment with Net2WiderNet:

“Random pad”: The network is widened by adding new units with random weights, rather than by replicating units to perform function preserving initialization. This operation is implemented by padding the pre-existing weight matrices with additional random values.

Experiment with Net2DeeperNet:

Another merit is that, while training the student network no special changes have to be made in the hyperparameters. Nearly all of the hyperparameters that are typically used to train a network from scratch can be used to train a network with Net2Net. This makes Net2Net very simple to use. However, using an initial learning rate of 0.1 times that of the initial learning rate of the teacher network proves to be more effective. An intuition to why this is advised is because the training of the student network can be considered as just the further training of the original network, and as training progresses the learning rate is usually decreased to allow the network to converge at the optimal global minima.