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| | Machine Learning |
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| Machine Learning | | Machine Learning |
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| so the hessian is positive semi-definite | | so the hessian is positive semi-definite |
| }} | | }} |
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| ===Cross Entropy===
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| The cross entropy loss is
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| * <math>J(\theta) = \sum [(y^{(i)})\log(h_\theta(x)) + (1-y^{(i)})\log(1-h_\theta(x))]</math>
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| ;Notes
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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
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| {{hidden | Proof |
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| <math>
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| \begin{aligned}
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| \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)}))]\\
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| &= -\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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| &= -\sum [(y^{(i)})(1-g(\theta^t x^{(i)}))x^{(i)} - (1-y^{(i)})g(\theta^t x^{(i)})x^{(i)}]\\
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| &= -\sum [(y^{(i)})x^{(i)} -(y^{(i)}) g(\theta^t x^{(i)}))x^{(i)} - g(\theta^t x^{(i)})x^{(i)} + y^{(i)}g(\theta^t x^{(i)})x^{(i)}]\\
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| &= -\sum [(y^{(i)})x^{(i)} - 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)}]\\
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| &= \sum [(g(\theta^t x^{(i)}))(1-g(\theta^t x^{(i)})) x^{(i)} (x^{(i)})^T]
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| \end{aligned}
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| </math><br>
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| which is a PSD matrix
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| }}
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| ===Hinge Loss===
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| ==Optimization== | | ==Optimization== |