【吴恩达深度学习笔记系列】Logistic Regression 【理论】

原创已于 2024-06-29 21:32:50 修改 · 1.1k 阅读

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于 2024-06-29 21:23:46 首次发布

                    
                    Binary Classification:
Logistic Regression: y^=σ(wTx+b)\hat{y}=\sigma{(w^T x+b)}y^​=σ(wTx+b) using sigmoid function σ=11+e−z\sigma = \frac{1}{1+e^{-z}}σ=1+e−z1​.
【torch.sigmoid(x)】Sigmoid(x)=11+e−x\text{Sigmoid}(x)=\frac{1}{1+e^{-x}}Sigmoid(x)=1+e−x1​

Logistic Regression loss function:

L(y^,y)=12(y^−y)2\mathcal{L}(\hat{y},y) = \frac{1}{2} (\hat{y}-y)^2L(y^​,y)=21​(y^​−y)2 × non-convex

L(y^,y)=−(ylog⁡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)=1m∑i=1mL(y^(i),y(i))=−1m∑i=1m(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=1m​L(y^​(i),y(i))=−m1​∑i=1m​(y(i)logy^​(i)+(1−y(i))log(1−y^​(i)))