Entropy Minimization for Semi-supervised Learning

Minimum entropy regularization can be applied to any model of posterior distribution.

The learning set is denoted ${\mathcal {L}}_{n}=\{X^{(i)},Z^{(i)}\}_{i=1}^{n}$, where $Z^{(i)}\in \{0,1\}^{K}$: If $X^{(i)}$ is labeled as $w_{i}$, then $Z_{k}^{(i)}=1$ and $Z_{l}^{(i)}=0$ for $l\not =k$; if $X^{(i)}$ is unlabeled, then $Z_{l}^{(i)}=1$ for $l=1\dots K$.

The conditional entropy of class labels conditioned on the observed variables:

$H(Y|X,Z;{\mathcal {L}}_{n})=-{\frac {1}{n}}\sum _{i=1}^{n}\sum _{k=1}^{K}P(Y^{(i)}=w_{k}|X^{(i)},Z^{(i)}){\text{log}}P(Y^{(i)}=w_{k}|X^{(i)},Z^{(i)})$

Assuming that labels are missing at random, we have that

$P(Y^{(i)}=w_{k}|X^{(i)},Z^{(i)})={\frac {Z_{k}^{(i)}P(Y^{(i)}=w_{k}|X^{(i)})}{\sum _{k=1}^{K}Z_{l}^{(i)}P(Y^{(i)}=w_{k}|X^{(i)})}}$

The posterior distribution is defined as

${\begin{alignedat}{2}C({\boldsymbol {\theta }},\lambda ;{\mathcal {L}}_{n})&=L({\boldsymbol {\theta }};{\mathcal {L}}_{n})-\lambda H(Y|X,Z;{\mathcal {L}}_{n})\\&=\sum _{i=1}^{n}{\text{log}}(\sum _{k=1}^{K}Z_{ik}P(Y^{i}=w_{k}|X^{i}))+\lambda \sum _{i=1}^{n}\sum _{k=1}^{K}P(Y^{i}=w_{k}|X^{i},Z^{i}){\text{log}}P(Y^{i}=w_{k}|X^{i},Z^{i})\end{alignedat}}$