Sure, to update an arbitrary weight wj,i" role="presentation" style="position: relative;">wj,iwj,iw\_{j,i} connecting an i-layer neuron to a j-layer neuron, we need to find an expression that is proportional to the negative of the partial derivative of the error function, E" role="presentation" style="position: relative;">EEE, with respect to the weight.

By using the chain rule, we can expand the partial derivative of the error function with respect to the weight wj,i" role="presentation" style="position: relative;">wj,iwj,iw\_{j,i} as follows:

&#x2202;E&#x2202;wj,i=&#x2202;E&#x2202;zj&#x2202;zj&#x2202;wj,i" role="presentation" style="text-align: center; position: relative;">∂E∂wj,i=∂E∂zj∂zj∂wj,i∂E∂wj,i=∂E∂zj∂zj∂wj,i\frac{\partial E}{\partial w\_{j,i}} = \frac{\partial E}{\partial z\_j} \frac{\partial z\_j}{\partial w\_{j,i}}

Using the chain rule again, we can expand the partial derivative of the weighted sum zj" role="presentation" style="position: relative;">zjzjz\_j with respect to the weight wj,i" role="presentation" style="position: relative;">wj,iwj,iw\_{j,i} as follows:

&#x2202;zj&#x2202;wj,i=ai" role="presentation" style="text-align: center; position: relative;">∂zj∂wj,i=ai∂zj∂wj,i=ai\frac{\partial z\_j}{\partial w\_{j,i}} = a\_i

Therefore, we can simplify the expression for the partial derivative of E" role="presentation" style="position: relative;">EEE with respect to wj,i" role="presentation" style="position: relative;">wj,iwj,iw\_{j,i} as:

&#x2202;E&#x2202;wj,i=&#x2202;E&#x2202;zjai" role="presentation" style="text-align: center; position: relative;">∂E∂wj,i=∂E∂zjai∂E∂wj,i=∂E∂zjai\frac{\partial E}{\partial w\_{j,i}} = \frac{\partial E}{\partial z\_j} a\_i

To simplify this expression, we can again define &#x03B4;j" role="presentation" style="position: relative;">δjδj\delta\_j as the first two terms in the chain rule expansion of the partial derivative of the error function with respect to the weighted sum zj" role="presentation" style="position: relative;">zjzjz\_j for an arbitrary j-layer neuron:

&#x03B4;j=&#x2202;E&#x2202;zj=&#x2202;E&#x2202;aj&#x2202;aj&#x2202;zj=&#x03C3;&#x2032;(zj)&#x2211;kwk,j&#x03B4;k" role="presentation" style="text-align: center; position: relative;">δj=∂E∂zj=∂E∂aj∂aj∂zj=σ′(zj)∑kwk,jδkδj=∂E∂zj=∂E∂aj∂aj∂zj=σ′(zj)∑kwk,jδk\delta\_j = \frac{\partial E}{\partial z\_j} = \frac{\partial E}{\partial a\_j} \frac{\partial a\_j}{\partial z\_j} = \sigma'(z\_j) \sum\_k w\_{k,j} \delta\_k

where the sum is taken over all k-layer neurons that receive connections from the j-layer neuron.

Using this definition, we can simplify the expression for updating the weight wj,i" role="presentation" style="position: relative;">wj,iwj,iw\_{j,i} as follows:

&#x0394;wj,i=&#x2212;&#x03B1;&#x2202;E&#x2202;wj,i=&#x03B1;&#x03B4;jai" role="presentation" style="text-align: center; position: relative;">Δwj,i=−α∂E∂wj,i=αδjaiΔwj,i=−α∂E∂wj,i=αδjai\Delta w\_{j,i} = -\alpha \frac{\partial E}{\partial w\_{j,i}} = \alpha \delta\_j a\_i

Therefore, we can update the weights in the j-layer using the simplified expression above, where &#x03B4;j" role="presentation" style="position: relative;">δjδj\delta\_j is defined as the first two terms in the chain rule expansion of the partial derivative of the error function with respect to the weighted sum zj" role="presentation" style="position: relative;">zjzjz\_j for an arbitrary j-layer neuron.
물론, i 계층 뉴런과 j 계층 뉴런을 연결하는 임의의 가중치 wj,i" role="presentation" style="위치: 상대적;">wj,iwj,iw\_{j,i}를 업데이트하려면 가중치에 대한 오류 함수의 편미분인 E" role="presentation" style="위치: 상대적;">EEE의 음에 비례하는 식을 찾아야 합니다.연쇄 규칙을 사용하면 가중치에 대한 오차 함수의 편미분을 다음과 같이 확장할 수 있습니다. wj,i" role="presentation" style="위치: 상대적;">wj,iwj,iw\_{j,i}:&#x2202;E&#x2202;wj,i=&#x2202;E&#x2202;zj&#x2202;zj&#x2202;wj,i" role="presentation" style="text-align: center; position: 상대적;">∂E∂wj,i=∂E∂zj∂zj∂wj,i∂E∂wj,i=∂E∂zj∂zj∂wj,i\frac{\partial E}{\partial w\_{j,i}} = \frac{\partial E}{\partial z\_j} \frac{\partial z\_j}{\partial w\_{j,i}}연쇄 규칙을 다시 사용하면 가중 합계의 부분 도함수 zj" role="presentation" style="위치: 상대적;">zjzz\_j를 가중치 wj,i" role="presentation" style="위치: 상대적;">wj,iwj,iw\_{j,i}에 대하여 다음과 같이 확장할 수 있습니다:&#x2202;zj&#x2202;wj,i=ai" role="presentation" style="text-align: 가운데; 위치: 상대적;">∂zj∂wj,i=ai∂zj∂wj,i=ai\frac{\partial z\_j}{\partial w\_{j,i}} = a\_i&#x2202;E&#x2202;wj,i=&#x2202;E&#x2202;zjai" role="presentation" style="text-align: center; position: 상대적;">∂E∂wj,i=∂E∂zjai∂E∂wj,i=∂E∂zjai\frac{\partial E}{\partial w\_{j,i}} = \frac{\partial E}{\partial z\_j} a\_i이 식을 단순화하기 위해, 임의의 j층 뉴런에 대한 가중 합 zj에 대한 오류 함수의 부분 도함수의 연쇄 규칙 확장에서 처음 두 항으로 &#x03B4;j" role="presentation" style="위치: 상대적;">δjδj\delta\_j를 다시 정의할 수 있습니다:&#x03B4;j=&#x2202;E&#x2202;zj=&#x2202;E&#x2202;aj&#x2202;aj&#x2202;zj=&#x03C3;&#x2032;(zj)&#x2211;kwk,j&#x03B4;k" role="presentation" style="text-align: center; position: 상대적;">δj=∂E∂zj=∂E∂aj∂aj∂zj=σ′(zj)∑kwk,jδkδj=∂E∂zj=∂E∂aj∂aj∂zj=σ′(zj)∑kwk,jδk\delta\_j = \frac{\partial E}{\partial z\_j} = \frac{\partial E}{\partial a\_j} \frac{\partial a\_j}{\partial z\_j} = \sigma'(z\_j) \sum\_k w\_{k,j} \델타\_k여기서 합은 j 계층 뉴런으로부터 연결을 받는 모든 k 계층 뉴런에 대해 취합니다.이 정의를 사용하면 가중치를 업데이트하는 표현식을 다음과 같이 단순화할 수 있습니다. wj,i" role="presentation" style="위치: 상대적;">wj,iwj,iw\_{j,i}:&#x0394;wj,i=&#x2212;&#x03B1;&#x2202;E&#x2202;wj,i=&#x03B1;&#x03B4;jai" role="presentation" style="text-align: center; position: 상대적;">Δwj,i=-α∂E∂wj,i=αδjaiΔwj,i=-α∂E∂wj,i=αδjai\Delta w\_{j,i} = -\알파 \frac{\partial E}{\partial w\_{j,i}} = \알파 \delta\_j a\_i따라서 위의 단순화된 식을 사용하여 j 계층의 가중치를 업데이트할 수 있는데, 여기서 &#x03B4;j" role="presentation" style="위치: 상대적;">δjδj\delta\_j는 임의의 j 계층 뉴런에 대한 가중 합 zj" role="presentation" style="위치: 상대적;">zjzz\_j의 오류 함수 부분 미분의 체인 규칙 확장에서 처음 두 항으로 정의됩니다.