# Paper: Neural Graph Machines

Semi-Supervised Learning is useful in scenarios where you have some labeled data, and lots of unlabeled data. If there is a way to cluster together training examples by a measure of similarity, and we can assume that examples close to each other in these clusterings are likely to have the same labels, then Semi-Supervised Learning can be pretty useful.

Essentially the premise is that labeling all the data is expensive, and we should learn as much as we can from as small a dataset as possible. For their data-labeling needs, the industry either relies on Mechanical Turks or full-time labelers on contract (Google Search is one example where they have a large team of human raters). Overall, it is costly to build a large labeled dataset. Therefore, if we can minimize our dependence on labeled data, and learn from known / inferred similarity within the dataset, that would be great. That’s where Semi-Supervised Learning helps.

Assume there is a graph-structure to our data, where a node is a datum / row, which needs to be labeled. And an edge exists between two nodes if they are similar, along with a weight. In this case, Label Propagation is a classic technique which has been used commonly.

I read this paper from Google Research which does a good job of generalizing and summarizing similar work done by Weston et. al, 2012, around training Neural Nets augmented by such a graph-based structure.

To summarize the work very quickly, the network tries to do two things:

a. For labeled data, try to predict the correct label (of course),

b. For the entire data set, try to learn a representation of each datum (embedding) in the hidden layers of the neural net.

For nodes which are adjacent to each other, the distance between their respective embeddings should be small, and the importance of keeping this distance small is proportional to the edge weight. That is, if there are two adjacent nodes with a high edge weight, if the neural net doesn’t learn to create embeddings such that these two examples are close to each other, there would be a larger penalty, than if the distance was smaller / the edge weight was lower.

Check the figure above. The blue layer is the hidden layer used for generating the embedding, whose output is represented by $h_{\theta}(X_i)$. $y$ is the final output. If $X_i$ and $X_j$ are close-by, the distance between them is represented by $d(h_{\theta}(X_i), h_{\theta}(X_j))$.

The cost-function is below. Don’t let this scare you. The first term is just the total loss from the predictions of the network for labeled data. The next three terms are for tweaking the importance of distances between the various (labeled / unlabeled) -(labeled / unlabeled) pairs, weighed by their respective edge weights, $w_{uv}$.

I skimmed through the paper to get the gist, but I wonder that the core contribution of such a network is to go from a graph-based structure to an embedding. If we were to construct an embedding directly from the graph structure, and train a NN separately using the embedding as the input, my guess is it should fetch similar results with a less complex objective function.

# A Gentle Intro to PyTorch

PyTorch is a fairly new deep-learning framework released by Facebook, which reminds me of the JS framework frenzy. But having played around with PyTorch a slight bit, it already feels fun.

To keep things short, I liked it because:

1. Unlike TensorFlow it allows me to easily print Tensors on the screen (no seriously, this is a big deal for me since I usually take several iterations to get a DL implementation right).
2. TensorFlow adds a layer between Python and TensorFlow. TensorFlow even has it’s own variable scope. This is way too much abstraction, that I don’t appreciate for my experimental interests.
3. Interop with numpy is easy in PyTorch, with the simple .numpy() suffix to convert a Tensor to a numpy array.
4. Unlike Torch, it is not in Lua (also doesn’t need the LuaRocks package manager).
5. Unlike Caffe2, I don’t have to write C++ code and write build scripts.

PyTorch’s website has a 60 min. blitz tutorial, which is laid out pretty well.

Here is the summary to get you started on PyTorch:

• torch.Tensor is your np.array (the NumPy array). torch.Tensor(3,4) will create a Tensor of shape (3,4).
• All the functions are pretty standard. Such as torch.rand can be used to generate random Tensors.
• Indexing in Tensors is pretty similar to NumPy as well.
• .numpy() allows converting Tensor to a numpy array.
• For the purpose of a compute graph, PyTorch lets you create Variables which are similar to placeholder in TF.
• Creating a compute graph and computing the gradient is pretty easy (and automatic).

This is all it takes to compute the gradient, where x is a variable:

Doing backprop simply with the backward method call on the scalar out, computes gradients all the way to x. This is amazing!

• For NN, there is an nn.Module which wraps around the boring boiler-plate.
• A simple NN implementation is below for solving the MNIST dataset, instead of the CIFAR-10 dataset that the tutorial solves:

As seen, in the __init__ method, we just need to define the various NN layers we are going to be using. Then the forward method just runs through them. The view method is analogous to the NumPy reshape method.

The gradients will be applied after the backward pass, which is auto-computed. The code is self-explanatory and fairly easy to understand.

• Torch also keeps track of how to retrieve standard data-sets such as CIFAR-10, MNIST, etc.
• After getting the data, from the data-loader you can proceed to play with it. Below are rest of the pieces required to complete the implementation (almost all of it is from the tutorial):

The criterion object is used to compute your loss function. optim has a bunch of convex optimization algorithms such as vanilla SGD, Adam, etc. As promised, simply calling the backward method on the loss object allows computing the gradient.

• Assuming you are working on the tutorial. Try to solve the tutorial for MNIST data instead of CIFAR-10.
• Instead of the 3-channel (RGB) image of size 24x24 pixels, the MNIST images are single channel 28x28 pixel images.

Overall, I could get to 96% accuracy, with the current setup. The complete gist is here.

# Back to Basics: Linear Regression

Essentially, in Linear Regression, we try to estimate a dependent variable $y$, using independent variables $x_1$, $x_2$, $x_3$, $…$, using a linear model.
More formally, $y = b + W_1 x_1 + W_2 x_2 + … + W_n x_n + \epsilon$. Where, $W$ is the weight vector, $b$ is the bias term, and $\epsilon$ is the noise in the data.
It can be used when there is a linear relationship between the input $X$ (input vector containing all the $x_i$, and $y$).