机器学习单层感知机梯度推导

**函数：

图示：

梯度求解：
$f(x) = {1 \above{0.5px} 1 + {e^{-x}}}$f(x)=1+e−x1​
$f'(x) = {e^{-x} \above{0.5px} ({1 + e^{-x}})^2}$f′(x)=(1+e−x)2e−x​
$f'(x) = {(1 + e^{-x}) - 1 \above{0.5px}({1 + e^{-x}})^2}$f′(x)=(1+e−x)2(1+e−x)−1​
$f'(x) = {\frac 1 {1 + e^{-x}} - \frac 1 {{(1 + e^{-x}})^2}}$f′(x)=1+e−x1​−(1+e−x)21​
$\sigma'(x) = \sigma(x)(1 - \sigma(x))$σ′(x)=σ(x)(1−σ(x))

$E = {\frac 1 2} (O_0^1 - t)^2 \\ (t是标签值，O_0^1是经过sigmoid**函数后的输出值, 1/2是为了求导时消掉常数项w_{j_0}表示第j个权值）$E=21​(O01​−t)2(t是标签值，O01​是经过sigmoid激活函数后的输出值,1/2是为了求导时消掉常数项wj0​​表示第j个权值）
$\frac {\varphi_E} {\varphi_{w_{j_0}}} = (O_0^1 - t){\frac {\varphi_O} {\varphi_{w_{j_0}}}} \\ (这里为了便于书写，用O来代替O_0^1)$φwj0​​​φE​​=(O01​−t)φwj0​​​φO​​(这里为了便于书写，用O来代替O01​)
$O = \sigma(x) \\ \frac {\varphi_E} {\varphi_{w_{j_0}}} = (O - t){\frac {\varphi_{\sigma(x_0^1)}} {\varphi_{(w_{j_0})}}} \\$O=σ(x)φwj0​​​φE​​=(O−t)φ(wj0​​)​φσ(x01​)​​
$\frac {\varphi_E} {\varphi_{w_{j_0}}} = (O - t){\frac {\varphi_{\sigma(x_0^1)}} {\varphi_{(x_0^1)}}} {\frac {\varphi_{(x_0^1)}} {\varphi_{\sigma(w_{j_0})}}}$φwj0​​​φE​​=(O−t)φ(x01​)​φσ(x01​)​​φσ(wj0​​)​φ(x01​)​​
$\frac {\varphi_E} {\varphi_{w_{j_0}}} = (O - t) \sigma(x_0^1))(1 - \sigma(x_0^1))){\frac {\varphi_{(x_0^1)}} {\varphi_{\sigma(w_{j_0})}}}$φwj0​​​φE​​=(O−t)σ(x01​))(1−σ(x01​)))φσ(wj0​​)​φ(x01​)​​
$(根据线性关系 )\\ x_0^1 = \Sigma w_{j_0}x_j^0 \\ \frac {\varphi_E} {\varphi_{w_{j_0}}} = (O - t) \sigma(x_0^1))(1 - \sigma(x_0^1))) {x_j^0} \\ (所有变量都为已知，求出梯度)$(根据线性关系)x01​=Σwj0​​xj0​φwj0​​​φE​​=(O−t)σ(x01​))(1−σ(x01​)))xj0​(所有变量都为已知，求出梯度)