The Discriminative Approach to Classification for MNIST

In the last post we described the simplest possible generative approach to classifying digits in the MNIST database. In this post, we are going to explore the discriminative approach to classification.

The Model

Let \(\mathcal{D} = \left\{ y^n \in S,x^n \in \mathbb{R}^f \right\}_{n=1,\dots,N}\) be a data set. For notational convenience, we embed \(S\) into Euclidean space: \[S = \{ s_1,\dots,s_c \} \hookrightarrow \mathbb{R}^c \quad s_i \mapsto e_i.\] Given a parameter space \(\Theta\) and functions \(f_1,\dots,f_c : \Theta \times \mathbb{R}^f \to (0,1)\), we define the following probability distribution: \[p(y | \theta,x) = \prod_i f_i(\theta,x)^{y_i}\] Given a prior distribution \(p(\theta)\) over \(\Theta\), Bayes rule implies that \[\log p(\theta | \mathcal{D}) = \log p(\theta) + \sum_{n,i} y^n_i \log
f_i(\theta,x^n).\] We can choose parameters given data using maximum likelihood estimation. More precisely, we want to minimize the following expression: \[-\sum_{n,i} y^n_i \log f_i(\theta,x^n)\] Intuitively, we are choosing parameters which minimize the cross entropy between the predicted distribution on \(y\) and the observed distribution on \(y\). Minimizing the cross entropy is only half the story. We also need to choose the parameter space \(\Theta\) and functions \(f_1,\dots,f_c\). To keep things simple, we shall take \[(f_1,\dots,f_c)(W,b,x) = {\rm softmax}(Wx + b)\] where \(W\) is a \(c \times f\) matrix, \(b \in \mathbb{R}^c\) and softmax transforms the output into a probability distribution. These ideas are demonstrated in this notebook.