Sure! To update an arbitrary weight wkj" role="presentation" style="position: relative;">wkjwkjw\_{kj} connecting a neuron in the j-layer to the neuron in the k-layer, we can use gradient descent to find the direction of steepest descent for the error function. The update rule is:

wkjnew=wkjold&#x2212;&#x03B1;&#x2202;E&#x2202;wkj" role="presentation" style="text-align: center; position: relative;">wnewkj=woldkj−α∂E∂wkjwkjnew=wkjold−α∂E∂wkjw\_{kj}^{new} = w\_{kj}^{old} - \alpha \frac{\partial E}{\partial w\_{kj}}

where wkjold" role="presentation" style="position: relative;">woldkjwkjoldw\_{kj}^{old} is the old value of the weight, wkjnew" role="presentation" style="position: relative;">wnewkjwkjneww\_{kj}^{new} is the new value of the weight, &#x03B1;" role="presentation" style="position: relative;">αα\alpha is the learning rate, and &#x2202;E&#x2202;wkj" role="presentation" style="position: relative;">∂E∂wkj∂E∂wkj\frac{\partial E}{\partial w\_{kj}} is the partial derivative of the error with respect to the weight.

Expanding the partial derivative of the error with respect to the weight wkj" role="presentation" style="position: relative;">wkjwkjw\_{kj} as a product of 3 terms, we get:

&#x2202;E&#x2202;wkj=&#x2202;E&#x2202;ak&#x2202;ak&#x2202;zk&#x2202;zk&#x2202;wkj" role="presentation" style="text-align: center; position: relative;">∂E∂wkj=∂E∂ak∂ak∂zk∂zk∂wkj∂E∂wkj=∂E∂ak∂ak∂zk∂zk∂wkj\frac{\partial E}{\partial w\_{kj}} = \frac{\partial E}{\partial a\_k} \frac{\partial a\_k}{\partial z\_k} \frac{\partial z\_k}{\partial w\_{kj}}

Using the results we derived earlier, we can simplify this expression as follows:

&#x2202;E&#x2202;wkj=(ak&#x2212;tk)&#x22C5;&#x03C3;(zk)&#x22C5;(1&#x2212;&#x03C3;(zk))&#x22C5;aj" role="presentation" style="text-align: center; position: relative;">∂E∂wkj=(ak−tk)⋅σ(zk)⋅(1−σ(zk))⋅aj∂E∂wkj=(ak−tk)⋅σ(zk)⋅(1−σ(zk))⋅aj\frac{\partial E}{\partial w\_{kj}} = (a\_k - t\_k) \cdot \sigma(z\_k) \cdot (1-\sigma(z\_k)) \cdot a\_j

So the update rule for the weight wkj" role="presentation" style="position: relative;">wkjwkjw\_{kj} is:

wkjnew=wkjold&#x2212;&#x03B1;(ak&#x2212;tk)&#x22C5;&#x03C3;(zk)&#x22C5;(1&#x2212;&#x03C3;(zk))&#x22C5;aj" role="presentation" style="text-align: center; position: relative;">wnewkj=woldkj−α(ak−tk)⋅σ(zk)⋅(1−σ(zk))⋅ajwkjnew=wkjold−α(ak−tk)⋅σ(zk)⋅(1−σ(zk))⋅ajw\_{kj}^{new} = w\_{kj}^{old} - \alpha (a\_k - t\_k) \cdot \sigma(z\_k) \cdot (1-\sigma(z\_k)) \cdot a\_j

This rule tells us to adjust the weight in the direction of steepest descent of the error function with respect to that weight, which is proportional to the negative of the partial derivative of the error with respect to that weight. The learning rate &#x03B1;" role="presentation" style="position: relative;">αα\alpha controls the step size of the weight update.
물론이죠! j 레이어의 뉴런과 k 레이어의 뉴런을 연결하는 임의의 가중치 wkj"role="presentation" style="위치: 상대적;">wkjwkjw\_{kj}를 업데이트하려면 경사 하강을 사용하여 오류 함수에 대해 가장 가파른 하강 방향을 찾을 수 있습니다. 업데이트 규칙은 다음과 같습니다:wkjnew=wkjold&#x2212;&#x03B1;&#x2202;E&#x2202;wkj" role="presentation" style="text-align: center; position: 상대적;">wnewkj=woldkj-α∂E∂wkjwkjnew=wkjold-α∂E∂wkjw\_{kj}^{new} = w\_{kj}^{old} - \alpha \frac{\partial E}{\partial w\_{kj}}여기서 wkjold" role="presentation" style="위치: 상대적;">woldkjwkjoldw\_{kj}^{old}는 가중치의 이전 값, wkjnew" role="presentation" style="위치: 상대적;">wnewkjwkjneww\_{kj}^{new}는 가중치의 새 값, &#x03B1;" role="presentation" style="position: 상대적;">αα\알파는 학습률이고, &#x2202;E&#x2202;wkj" role="presentation" style="위치: 상대적;">∂E∂wkj∂E∂wkj\frac{\partial E}{\partial w\_{kj}}는 가중치에 대한 오류의 부분 도함수입니다.가중치에 대한 오차의 편미분을 세 항의 곱으로 확장하면 다음과 같은 식을 얻을 수 있습니다:&#x2202;E&#x2202;wkj=&#x2202;E&#x2202;ak&#x2202;ak&#x2202;zk&#x2202;zk&#x2202;wkj" role="presentation" style="text-align: center; position: 상대적;">∂E∂wkj=∂E∂ak∂ak∂zk∂zk∂wkj∂E∂wkj=∂E∂ak∂ak∂zk∂zk∂zk∂wkj\frac{\partial E}{\partial w\_{kj}} = \frac{\partial E}{\partial a\_k} \frac{\partial a\_k}{\partial z\_k} \frac{\partial z\_k}{\partial w\_{kj}}앞서 도출한 결과를 사용하면 이 식을 다음과 같이 단순화할 수 있습니다:&#x2202;E&#x2202;wkj=(ak&#x2212;tk)&#x22C5;&#x03C3;(zk)&#x22C5;(1&#x2212;&#x03C3;(zk))&#x22C5;aj" role="presentation" style="text-align: center; position: 상대적;">∂E∂wkj=(ak-tk)⋅σ(zk)⋅(1-σ(zk))⋅aj∂E∂wkj=(ak-tk)⋅σ(zk)⋅(1-σ(zk))⋅aj\frac{\partial E}{\partial w\_{kj}} = (a\_k - t\_k) \cdot \sigma(z\_k) \cdot (1-\sigma(z\_k)) \cdot a\_j따라서 가중치에 대한 업데이트 규칙은 wkj" role="presentation" style="위치: 상대적;">wkjwkjw\_{kj}입니다:wkjnew=wkjold&#x2212;&#x03B1;(ak&#x2212;tk)&#x22C5;&#x03C3;(zk)&#x22C5;(1&#x2212;&#x03C3;(zk))&#x22C5;aj" role="presentation" style="text-align: center; position: 상대적;">wnewkj=woldkj-α(ak-tk)⋅σ(zk)⋅(1-σ(zk))⋅ajwkjnew=wkjold-α(ak-tk)⋅σ(zk)⋅(1-σ(zk))⋅ajw\_{kj}^{new} = w\_{kj}^{old} - \알파 (a\_k - t\_k) \cdot \sigma(z\_k) \cdot (1-\sigma(z\_k)) \cdot a\_j이 규칙은 해당 가중치에 대한 오차 함수의 가장 가파른 하강 방향으로 가중치를 조정하도록 지시하며, 이는 해당 가중치에 대한 오차의 편미분의 음에 비례합니다. 학습률 &#x03B1;" role="presentation" style="위치: 상대적;">αα\알파는 가중치 업데이트의 단계 크기를 제어합니다.