Sure! To update the bias bk" role="presentation" style="position: relative;">bkbkb\_k for a neuron in the k-layer, we can use a similar update rule to the weight update rule. The update rule is:

bknew=bkold&#x2212;&#x03B1;&#x2202;E&#x2202;bk" role="presentation" style="text-align: center; position: relative;">bnewk=boldk−α∂E∂bkbknew=bkold−α∂E∂bkb\_k^{new} = b\_k^{old} - \alpha \frac{\partial E}{\partial b\_k}

where bkold" role="presentation" style="position: relative;">boldkbkoldb\_k^{old} is the old value of the bias, bknew" role="presentation" style="position: relative;">bnewkbknewb\_k^{new} is the new value of the bias, &#x03B1;" role="presentation" style="position: relative;">αα\alpha is the learning rate, and &#x2202;E&#x2202;bk" role="presentation" style="position: relative;">∂E∂bk∂E∂bk\frac{\partial E}{\partial b\_k} is the partial derivative of the error with respect to the bias.

Expanding the partial derivative of the error with respect to the bias bk" role="presentation" style="position: relative;">bkbkb\_k as a product of 3 terms, we get:

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

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

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

So the update rule for the bias bk" role="presentation" style="position: relative;">bkbkb\_k is:

bknew=bkold&#x2212;&#x03B1;(ak&#x2212;tk)&#x22C5;&#x03C3;(zk)&#x22C5;(1&#x2212;&#x03C3;(zk))" role="presentation" style="text-align: center; position: relative;">bnewk=boldk−α(ak−tk)⋅σ(zk)⋅(1−σ(zk))bknew=bkold−α(ak−tk)⋅σ(zk)⋅(1−σ(zk))b\_k^{new} = b\_k^{old} - \alpha (a\_k - t\_k) \cdot \sigma(z\_k) \cdot (1-\sigma(z\_k))

This rule tells us to adjust the bias in the direction of steepest descent of the error function with respect to the bias, which is proportional to the negative of the partial derivative of the error with respect to the bias. The learning rate &#x03B1;" role="presentation" style="position: relative;">αα\alpha controls the step size of the bias update.
물론이죠! k 계층의 뉴런에 대한 바이어스 bk" role="presentation" style="위치: 상대적;">bkbkb\_k를 업데이트하려면 가중치 업데이트 규칙과 유사한 업데이트 규칙을 사용할 수 있습니다. 업데이트 규칙은 다음과 같습니다:bknew=bkold&#x2212;&#x03B1;&#x2202;E&#x2202;bk" role="presentation" style="text-align: center; position: 상대적;">bnewk=boldk-α∂E∂bkbknew=bkold-α∂E∂bkb\_k^{new} = b\_k^{old} - \alpha \frac{\partial E}{\partial b\_k}여기서 bkold" role="presentation" style="위치: 상대적;">boldkbkoldb\_k^{old}는 바이어스의 이전 값이고, bknew" role="presentation" style="위치: 상대적;">bnewkbknewb\_k^{new}는 바이어스의 새 값이며, &#x03B1;" role="presentation" style="position: 상대적;">αα\알파는 학습률이고, &#x2202;E&#x2202;bk" role="presentation" style="위치: 상대적;">∂E∂bk∂E∂bk\frac{\partial E}{\partial b\_k}는 바이어스에 대한 오류의 부분 도함수입니다.편향에 대한 오차의 편미분을 bk" role="presentation" style="위치: 상대적;">bkbkb\_k를 세 항의 곱으로 확장하면 다음과 같이 구할 수 있습니다:&#x2202;E&#x2202;bk=&#x2202;E&#x2202;ak&#x2202;ak&#x2202;zk&#x2202;zk&#x2202;bk" role="presentation" style="text-align: center; position: 상대적;">∂E∂bk=∂E∂ak∂ak∂zk∂zk∂zk∂bk∂E∂bk=∂E∂ak∂ak∂zk∂zk∂zk∂bk\frac{\partial E}{\partial b\_k} = \frac{\partial E}{\partial a\_k} \frac{\partial a\_k}{\partial z\_k} \frac{\partial z\_k}{\partial b\_k}앞서 도출한 결과를 사용하면 이 식을 다음과 같이 단순화할 수 있습니다:&#x2202;E&#x2202;bk=(ak&#x2212;tk)&#x22C5;&#x03C3;(zk)&#x22C5;(1&#x2212;&#x03C3;(zk))&#x22C5;1" role="presentation" style="text-align: center; position: 상대;">∂E∂bk=(ak-tk)⋅σ(zk)⋅(1-σ(zk))⋅1∂E∂bk=(ak-tk)⋅σ(zk)⋅(1-σ(zk))⋅1\frac{\partial E}{\partial b\_k} = (a\_k - t\_k) \cdot \sigma(z\_k) \cdot (1-\sigma(z\_k)) \cdot 1따라서 바이어스 bk"role="presentation" style="위치: 상대적;">bkbkb\_k에 대한 업데이트 규칙은 다음과 같습니다:bknew=bkold&#x2212;&#x03B1;(ak&#x2212;tk)&#x22C5;&#x03C3;(zk)&#x22C5;(1&#x2212;&#x03C3;(zk))" role="presentation" style="text-align: center; position: 상대적;">bnewk=boldk-α(ak-tk)⋅σ(zk)⋅(1-σ(zk))bknew=bkold-α(ak-tk)⋅σ(zk)⋅(1-σ(zk))b\_k^{new} = b\_k^{old} - \알파 (a\_k - t\_k) \cdot \sigma(z\_k) \cdot (1-\sigma(z\_k))이 규칙은 바이어스에 대한 오차 함수의 가장 가파른 하강 방향으로 바이어스를 조정하도록 지시하며, 이는 바이어스에 대한 오차의 부분 도함수의 음수에 비례합니다. 학습률 &#x03B1;" role="presentation" style="위치: 상대적;">αα\알파는 바이어스 업데이트의 단계 크기를 제어합니다.