Sure! To update an arbitrary weight wji" role="presentation" style="position: relative;">wjiwjiw\_{ji} connecting a neuron in the i-layer to the neuron in the j-layer, we can use a similar approach as for the k-layer weight update. The update rule is:

wjinew=wjiold&#x2212;&#x03B1;&#x2202;E&#x2202;wji" role="presentation" style="text-align: center; position: relative;">wnewji=woldji−α∂E∂wjiwjinew=wjiold−α∂E∂wjiw\_{ji}^{new} = w\_{ji}^{old} - \alpha \frac{\partial E}{\partial w\_{ji}}

where wjiold" role="presentation" style="position: relative;">woldjiwjioldw\_{ji}^{old} is the old value of the weight, wjinew" role="presentation" style="position: relative;">wnewjiwjineww\_{ji}^{new} is the new value of the weight, &#x03B1;" role="presentation" style="position: relative;">αα\alpha is the learning rate, and &#x2202;E&#x2202;wji" role="presentation" style="position: relative;">∂E∂wji∂E∂wji\frac{\partial E}{\partial w\_{ji}} is the partial derivative of the error with respect to the weight.

Expanding the partial derivative of the error with respect to the weight wji" role="presentation" style="position: relative;">wjiwjiw\_{ji} as a product of 3 terms using the multivariate chain rule, we get:

&#x2202;E&#x2202;wji=&#x2211;k=1K&#x2202;E&#x2202;zk&#x2202;zk&#x2202;aj&#x2202;aj&#x2202;zj&#x2202;zj&#x2202;wji" role="presentation" style="text-align: center; position: relative;">∂E∂wji=∑k=1K∂E∂zk∂zk∂aj∂aj∂zj∂zj∂wji∂E∂wji=∑k=1K∂E∂zk∂zk∂aj∂aj∂zj∂zj∂wji\frac{\partial E}{\partial w\_{ji}} = \sum\_{k=1}^K \frac{\partial E}{\partial z\_k} \frac{\partial z\_k}{\partial a\_j} \frac{\partial a\_j}{\partial z\_j} \frac{\partial z\_j}{\partial w\_{ji}}

where K" role="presentation" style="position: relative;">KKK is the number of neurons in the k-layer, and the sum is over all neurons in the k-layer that receive input from the j-layer.

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

&#x2202;E&#x2202;wji=&#x2211;k=1K&#x03B4;k&#x22C5;wkj&#x22C5;&#x03C3;(zj)&#x22C5;(1&#x2212;&#x03C3;(zj))&#x22C5;xi" role="presentation" style="text-align: center; position: relative;">∂E∂wji=∑k=1Kδk⋅wkj⋅σ(zj)⋅(1−σ(zj))⋅xi∂E∂wji=∑k=1Kδk⋅wkj⋅σ(zj)⋅(1−σ(zj))⋅xi\frac{\partial E}{\partial w\_{ji}} = \sum\_{k=1}^K \delta\_k \cdot w\_{kj} \cdot \sigma(z\_j) \cdot (1-\sigma(z\_j)) \cdot x\_i

So the update rule for the weight wji" role="presentation" style="position: relative;">wjiwjiw\_{ji} is:

wjinew=wjiold&#x2212;&#x03B1;&#x2211;k=1K&#x03B4;k&#x22C5;wkj&#x22C5;&#x03C3;(zj)&#x22C5;(1&#x2212;&#x03C3;(zj))&#x22C5;xi" role="presentation" style="text-align: center; position: relative;">wnewji=woldji−α∑k=1Kδk⋅wkj⋅σ(zj)⋅(1−σ(zj))⋅xiwjinew=wjiold−α∑k=1Kδk⋅wkj⋅σ(zj)⋅(1−σ(zj))⋅xiw\_{ji}^{new} = w\_{ji}^{old} - \alpha \sum\_{k=1}^K \delta\_k \cdot w\_{kj} \cdot \sigma(z\_j) \cdot (1-\sigma(z\_j)) \cdot x\_i

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.

Note that the expression for the weight update involves a sum over all the neurons in the k-layer, and for each neuron in the k-layer, we need to compute the corresponding &#x03B4;k" role="presentation" style="position: relative;">δkδk\delta\_k term.
물론이죠! i 레이어의 뉴런과 j 레이어의 뉴런을 연결하는 임의의 가중치 wji"role="presentation" style="위치: 상대적;">wjiwjiw\_{ji}를 업데이트하려면 k 레이어 가중치 업데이트와 유사한 접근 방식을 사용할 수 있습니다. 업데이트 규칙은 다음과 같습니다:wjinew=wjiold&#x2212;&#x03B1;&#x2202;E&#x2202;wji" role="presentation" style="text-align: center; position: 상대적;">wnewji=woldji-α∂E∂wjiwjinew=wjiold-α∂E∂wjiw\_{ji}^{new} = w\_{ji}^{old} - \alpha \frac{\partial E}{\partial w\_{ji}}여기서 wjiold" role="presentation" style="위치: 상대적;">woldjiwjioldw\_{ji}^{old}는 가중치의 이전 값이고, wjinew" role="presentation" style="위치: 상대적;">wnewjiwjineww\_{ji}^{new}는 가중치의 새 값이며, &#x03B1;" role="presentation" style="position: 상대적;">αα\알파는 학습률이고, &#x2202;E&#x2202;wji" role="presentation" style="위치: 상대적;">∂E∂wji∂E∂wji\frac{\partial E}{\partial w\_{ji}}는 가중치에 대한 오류의 부분 도함수입니다.가중치에 대한 오차의 부분 도함수를 다변량 연쇄 규칙을 사용하여 3개 항의 곱으로 확장하면 다음과 같이 구할 수 있습니다:&#x2202;E&#x2202;wji=&#x2211;k=1K&#x2202;E&#x2202;zk&#x2202;zk&#x2202;aj&#x2202;aj&#x2202;zj&#x2202;zj&#x2202;wji" role="presentation" style="text-align: center; position: 상대적;">∂E∂wji=∑k=1K∂E∂zk∂zk∂aj∂aj∂zj∂zj∂wji∂E∂wji=∑k=1K∂E∂zk∂zk∂aj∂aj∂zj∂zj∂wji\frac{\partial E}{\partial w\_{ji}} = \sum\_{k=1}^K \frac{\partial E}{\partial z\_k} \frac{\partial z\_k}{\partial a\_j} \frac{\partial a\_j}{\partial z\_j} \frac{\partial z\_j}{\partial w\_{ji}}여기서 K" role="presentation" style="위치: 상대적;">KKK는 k 계층의 뉴런 수이며, 합은 j 계층으로부터 입력을 받는 k 계층의 모든 뉴런을 합한 값입니다.앞서 도출한 결과를 사용하면 이 식을 다음과 같이 단순화할 수 있습니다:&#x2202;E&#x2202;wji=&#x2211;k=1K&#x03B4;k&#x22C5;wkj&#x22C5;&#x03C3;(zj)&#x22C5;(1&#x2212;&#x03C3;(zj))&#x22C5;xi" role="presentation" style="text-align: center; position: 상대적;">∂E∂wji=∑k=1Kδk⋅wkj⋅σ(zj)⋅(1-σ(zj))⋅xi∂E∂wji=∑k=1Kδk⋅wkj⋅σ(zj)⋅(1-σ(zj))⋅xi\frac{\partial E}{\partial w\_{ji}} = \sum\_{k=1}^K \delta\_k \cdot w\_{kj} \cdot \sigma(z\_j) \cdot (1-\sigma(z\_j)) \x\_i따라서 가중치 wji"role="presentation" style="위치: 상대적;">wjiwjiw\_{ji}에 대한 업데이트 규칙은 다음과 같습니다:wjinew=wjiold&#x2212;&#x03B1;&#x2211;k=1K&#x03B4;k&#x22C5;wkj&#x22C5;&#x03C3;(zj)&#x22C5;(1&#x2212;&#x03C3;(zj))&#x22C5;xi" role="presentation" style="text-align: center; position: 상대적;">wnewji=woldjii-α∑k=1Kδk⋅wkj⋅σ(zj)⋅(1-σ(zj))⋅xiwjinew=wjiold-α∑k=1Kδk⋅wkj⋅σ(zj)⋅(1-σ(zj))⋅xiw\_{ji}^{new} = w\_{ji}^{old} - \alpha \sum\_{k=1}^K \delta\_k \cdot w\_{kj} \cdot \sigma(z\_j) \cdot (1-\sigma(z\_j)) \cdot x\_i이 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.가중치 업데이트에 대한 표현식에는 k 계층의 모든 뉴런에 대한 합이 포함되며, k 계층의 각 뉴런에 대해 해당 &#x03B4;k" role="presentation" style="위치: 상대적;">δkδk\delta\_k 항을 계산해야 한다는 점에 유의하세요.