Binary Classification:
- Logistic Regression: y ^ = σ ( w T x + b ) \hat{y}=\sigma{(w^T x+b)} y^=σ(wTx+b) using sigmoid function σ = 1 1 + e − z \sigma = \frac{1}{1+e^{-z}} σ=1+e−z1.
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torch.sigmoid(x)】 Sigmoid ( x ) = 1 1 + e − x \text{Sigmoid}(x)=\frac{1}{1+e^{-x}} Sigmoid(x)=1+e−x1
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- Logistic Regression loss function:
L ( y ^ , y ) = 1 2 ( y ^ − y ) 2 \mathcal{L}(\hat{y},y) = \frac{1}{2} (\hat{y}-y)^2 L(y^,y)=21(y^−y)2 × non-convex
L ( y ^ , y ) = − ( y log y ^ + ( 1 − y ) log ( 1 − y ^ ) ) \mathcal{L}(\hat{y},y) = -(y \log \hat{y} + (1-y) \log (1-\hat{y} )) L(y^,y)=−(ylogy^+(1−y)log(1−y^)) √ convex - Logistic Regression cost function:
J ( w , b ) = 1 m ∑ i = 1 m L ( y ^ ( i ) , y ( i ) ) = − 1 m ∑ i = 1 m ( y ( i ) log y ^ ( i ) + ( 1 − y ( i ) ) log ( 1 − y ^ ( i ) ) ) J(w, b) = \frac{1}{m} \sum^m_{i=1} \mathcal{L}(\hat{y}^{(i)},y^{(i)}) = - \frac{1}{m} \sum^m_{i=1} (y^{(i)} \log \hat{y}^{(i)} + (1-y^{(i)}) \log (1-\hat{y}^{(i)} )) J(w,b)=m1∑i=1mL(y^(i),y(i))=−m1∑i=1m(y(i)logy^(i)+(1−y(i))log(1−y^(i)))
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