Machine Learning: Difference between revisions
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* If our model is <math>g(\theta^Tx^{(i)})</math> where <math>g(x)</math> is the sigmoid function <math>\frac{e^x}{1+e^x}</math> then this is convex | * If our model is <math>g(\theta^Tx^{(i)})</math> where <math>g(x)</math> is the sigmoid function <math>\frac{e^x}{1+e^x}</math> then this is convex | ||
{{hidden | Proof | | |||
We show that the Hessian is positive semi definite.<br> | We show that the Hessian is positive semi definite.<br> | ||
<math> | <math> | ||
\begin{ | \begin{alignat}{2} | ||
\nabla_\theta J(\theta) &= -\nabla_\theta \sum [(y^{(i)})\log(g(\theta^t x^{(i)})) + (1-y^{(i)})\log(1-g(\theta^t x^{(i)}))]\\ | \nabla_\theta J(\theta) &= -\nabla_\theta \sum [(y^{(i)})\log(g(\theta^t x^{(i)})) + (1-y^{(i)})\log(1-g(\theta^t x^{(i)}))]\\ | ||
&= -\sum [(y^{(i)})\frac{g(\theta^t x^{(i)})(1-g(\theta^t x^{(i)}))}{g(\theta^t x^{(i)})}x^{(i)} + (1-y^{(i)})\frac{-g(\theta^t x^{(i)})(1-g(\theta^t x^{(i)}))}{1-g(\theta^t x^{(i)})}x^{(i)}]\\ | &= -\sum [(y^{(i)})\frac{g(\theta^t x^{(i)})(1-g(\theta^t x^{(i)}))}{g(\theta^t x^{(i)})}x^{(i)} + (1-y^{(i)})\frac{-g(\theta^t x^{(i)})(1-g(\theta^t x^{(i)}))}{1-g(\theta^t x^{(i)})}x^{(i)}]\\ | ||
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\implies \nabla^2_\theta J(\theta) &= \nabla_\theta -\sum [(y^{(i)})x^{(i)} - g(\theta^t x^{(i)})x^{(i)}]\\ | \implies \nabla^2_\theta J(\theta) &= \nabla_\theta -\sum [(y^{(i)})x^{(i)} - g(\theta^t x^{(i)})x^{(i)}]\\ | ||
&= \sum_i g(\theta^t x^{(i)})(1-g(\theta^t x^{(i)})) x^{(i)} (x^{(i)})^T\\ | &= \sum_i g(\theta^t x^{(i)})(1-g(\theta^t x^{(i)})) x^{(i)} (x^{(i)})^T\\ | ||
\end{ | \end{alignat} | ||
</math><br> | </math><br> | ||
which is a PSD matrix | which is a PSD matrix | ||
}} | }} | ||
===Hinge Loss=== | ===Hinge Loss=== | ||