Sure, to update an arbitrary weight wi,n" role="presentation" style="position: relative;">wi,nwi,nw\_{i,n} connecting an n-layer datum of the input vector to an i-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 wi,n" role="presentation" style="position: relative;">wi,nwi,nw\_{i,n} as follows:

&#x2202;E&#x2202;wi,n=&#x2202;E&#x2202;zi&#x2202;zi&#x2202;wi,n" role="presentation" style="text-align: center; position: relative;">∂E∂wi,n=∂E∂zi∂zi∂wi,n∂E∂wi,n=∂E∂zi∂zi∂wi,n\frac{\partial E}{\partial w\_{i,n}} = \frac{\partial E}{\partial z\_i} \frac{\partial z\_i}{\partial w\_{i,n}}

Using the chain rule again, we can expand the partial derivative of the weighted sum zi" role="presentation" style="position: relative;">ziziz\_i with respect to the weight wi,n" role="presentation" style="position: relative;">wi,nwi,nw\_{i,n} as follows:

&#x2202;zi&#x2202;wi,n=xn" role="presentation" style="text-align: center; position: relative;">∂zi∂wi,n=xn∂zi∂wi,n=xn\frac{\partial z\_i}{\partial w\_{i,n}} = x\_n

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

&#x2202;E&#x2202;wi,n=&#x2202;E&#x2202;zixn" role="presentation" style="text-align: center; position: relative;">∂E∂wi,n=∂E∂zixn∂E∂wi,n=∂E∂zixn\frac{\partial E}{\partial w\_{i,n}} = \frac{\partial E}{\partial z\_i} x\_n

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

&#x03B4;i=&#x2202;E&#x2202;zi=&#x03C3;&#x2032;(zi)&#x2211;kwk,i&#x03B4;k" role="presentation" style="text-align: center; position: relative;">δi=∂E∂zi=σ′(zi)∑kwk,iδkδi=∂E∂zi=σ′(zi)∑kwk,iδk\delta\_i = \frac{\partial E}{\partial z\_i} = \sigma'(z\_i) \sum\_k w\_{k,i} \delta\_k

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

Using this definition, we can simplify the expression for updating the weight wi,n" role="presentation" style="position: relative;">wi,nwi,nw\_{i,n} as follows:

&#x0394;wi,n=&#x2212;&#x03B1;&#x2202;E&#x2202;wi,n=&#x03B1;&#x03B4;ixn" role="presentation" style="text-align: center; position: relative;">Δwi,n=−α∂E∂wi,n=αδixnΔwi,n=−α∂E∂wi,n=αδixn\Delta w\_{i,n} = -\alpha \frac{\partial E}{\partial w\_{i,n}} = \alpha \delta\_i x\_n

Therefore, we can update the weights in the i-layer using the simplified expression above, where &#x03B4;i" role="presentation" style="position: relative;">δiδi\delta\_i 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 zi" role="presentation" style="position: relative;">ziziz\_i for an arbitrary i-layer neuron.

Similarly, to update the bias bi" role="presentation" style="position: relative;">bibib\_i for an arbitrary i-layer neuron, we can again use the chain rule to expand the partial derivative of the error function with respect to the bias as follows:

&#x2202;E&#x2202;bi=&#x2202;E&#x2202;zi&#x2202;zi&#x2202;bi" role="presentation" style="text-align: center; position: relative;">∂E∂bi=∂E∂zi∂zi∂bi∂E∂bi=∂E∂zi∂zi∂bi\frac{\partial E}{\partial b\_i} = \frac{\partial E}{\partial z\_i} \frac{\partial z\_i}{\partial b\_i}

Using the chain rule again, we can expand the partial derivative of the weighted sum zi" role="presentation" style="position: relative;">ziziz\_i with respect to the bias bi" role="presentation" style="position: relative;">bibib\_i as follows:

&#x2202;zi&#x2202;bi=1" role="presentation" style="text-align: center; position: relative;">∂zi∂bi=1∂zi∂bi=1\frac{\partial z\_i}{\partial b\_i} = 1

Therefore, we can simplify the expression for the partial derivative of E" role="presentation" style="position: relative;">EEE with respect to bi" role="presentation" style="position: relative;">bibib\_i as:

&#x2202;E&#x2202;bi=&#x2202;E&#x2202;zi" role="presentation" style="text-align: center; position: relative;">∂E∂bi=∂E∂zi∂E∂bi=∂E∂zi\frac{\partial E}{\partial b\_i} = \frac{\partial E}{\partial z\_i}

To simplify this expression, we can again use the definition of &#x03B4;i" role="presentation" style="position: relative;">δiδi\delta\_i that we derived earlier:

&#x03B4;i=&#x2202;E&#x2202;zi=&#x03C3;&#x2032;(zi)&#x2211;kwk,i&#x03B4;k" role="presentation" style="text-align: center; position: relative;">δi=∂E∂zi=σ′(zi)∑kwk,iδkδi=∂E∂zi=σ′(zi)∑kwk,iδk\delta\_i = \frac{\partial E}{\partial z\_i} = \sigma'(z\_i) \sum\_k w\_{k,i} \delta\_k

Therefore, we can simplify the expression for updating the bias bi" role="presentation" style="position: relative;">bibib\_i as follows:

&#x0394;bi=&#x2212;&#x03B1;&#x2202;E&#x2202;bi=&#x03B1;&#x03B4;i" role="presentation" style="text-align: center; position: relative;">Δbi=−α∂E∂bi=αδiΔbi=−α∂E∂bi=αδi\Delta b\_i = -\alpha \frac{\partial E}{\partial b\_i} = \alpha \delta\_i

This expression gives us the amount by which we should update the bias bi" role="presentation" style="position: relative;">bibib\_i in order to decrease the error function, $
물론 입력 벡터의 n층 데이텀과 i층 뉴런을 연결하는 임의의 가중치 wi,n" role="presentation" style="위치: 상대적;">wi,nwi,nw\_{i,n}을 업데이트하려면 가중치에 대한 오차 함수의 부분 미분인 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. wi,n" role="presentation" style="위치: 상대적;">wi,nwi,nw\_{i,n}:&#x2202;E&#x2202;wi,n=&#x2202;E&#x2202;zi&#x2202;zi&#x2202;wi,n" role="presentation" style="text-align: center; position: 상대적;">∂E∂wi,n=∂E∂zi∂zi∂wi,n∂E∂wi,n=∂E∂zi∂zi∂wi,n\frac{\partial E}{\partial w\_{i,n}} = \frac{\partial E}{\partial z\_i} \frac{\partial z\_i}{\partial w\_{i,n}}연쇄 규칙을 다시 사용하여 가중 합계의 부분 도함수 zi" role="presentation" style="위치: 상대적;">ziziz\_i를 가중치 wi,n" role="presentation" style="위치: 상대적;">wi,nwi,nw\_{i,n}에 대해 다음과 같이 확장할 수 있습니다:&#x2202;zi&#x2202;wi,n=xn" role="presentation" style="text-align: 가운데; 위치: 상대적;">∂zi∂wi,n=xn∂zi∂wi,n=xn\frac{\partial z\_i}{\partial w\_{i,n}} = x\_n따라서 E" role="presentation" style="위치: 상대적;">의 부분 도함수에 대한 식을 wi,n" role="presentation" style="위치: 상대적;">wi,nwi,nw\_{i,n}에 대해 다음과 같이 단순화할 수 있습니다:&#x2202;E&#x2202;wi,n=&#x2202;E&#x2202;zixn" role="presentation" style="text-align: 가운데; 위치: 상대적;">∂E∂wi,n=∂E∂zixn∂E∂wi,n=∂E∂zixn\frac{\partial E}{\partial w\_{i,n}} = \frac{\partial E}{\partial z\_i} x\_n이 식을 단순화하기 위해, 임의의 i층 뉴런에 대한 가중 합계에 대한 오류 함수의 부분 미분의 연쇄 규칙 확장에서 처음 두 항을 &#x03B4;i" role="presentation" style="위치: 상대적;">δiδi\delta\_i로 다시 정의할 수 있습니다:&#x03B4;i=&#x2202;E&#x2202;zi=&#x03C3;&#x2032;(zi)&#x2211;kwk,i&#x03B4;k" role="presentation" style="text-align: center; position: 상대;">δi=∂E∂zi=σ′(zi)∑kwk,iδkδi=∂E∂zi=σ′(zi)∑kwk,iδk\delta\_i = \frac{\partial E}{\partial z\_i} = \sigma'(z\_i) \sum\_k w\_{k,i} \델타\_k여기서 합은 i 계층 뉴런으로부터 연결을 받는 모든 k 계층 뉴런에 대해 취합니다.이 정의를 사용하면 가중치를 업데이트하는 표현식을 다음과 같이 단순화할 수 있습니다. wi,n" role="presentation" style="위치: 상대적;">wi,nwi,nw\_{i,n}:&#x0394;wi,n=&#x2212;&#x03B1;&#x2202;E&#x2202;wi,n=&#x03B1;&#x03B4;ixn" role="presentation" style="text-align: center; position: 상대;">Δwi,n=-α∂E∂wi,n=αδixnΔwi,n=-α∂E∂wi,n=αδixn\Delta w\_{i,n} = -\알파 \frac{\partial E}{\partial w\_{i,n}} = \알파 \delta\_i x\_n따라서 위의 단순화된 식을 사용하여 i 계층의 가중치를 업데이트할 수 있는데, 여기서 &#x03B4;i" role="presentation" style="위치: 상대적;">δiδi\delta\_i는 임의의 i 계층 뉴런에 대한 가중 합 zi" role="presentation" style="위치: 상대적;">ziziz\_i의 오류 함수 부분 미분 연쇄 규칙의 처음 두 항으로 정의됩니다.마찬가지로 임의의 i층 뉴런에 대해 바이어스 bi" role="presentation" style="위치: 상대적;">bibib\_i를 업데이트하려면 다시 체인 규칙을 사용하여 다음과 같이 바이어스에 대한 오류 함수의 부분 미분을 확장할 수 있습니다:&#x2202;E&#x2202;bi=&#x2202;E&#x2202;zi&#x2202;zi&#x2202;bi" role="presentation" style="text-align: center; position: 상대적;">∂E∂bi=∂E∂zi∂zi∂bi∂E∂bi=∂E∂zi∂zi∂bi\frac{\partial E}{\partial b\_i} = \frac{\partial E}{\partial z\_i} \frac{\partial z\_i}{\partial b\_i}연쇄 규칙을 다시 사용하여 가중 합 zi" role="presentation" style="위치: 상대적;">ziziz\_i의 편미분을 바이어스 bi" role="presentation" style="위치: 상대적;">bibib\_i에 대해 다음과 같이 확장할 수 있습니다:&#x2202;zi&#x2202;bi=1" role="presentation" style="text-align: 가운데; 위치: 상대적;">∂zi∂bi=1∂zi∂bi=1\frac{\partial z\_i}{\partial b\_i} = 1입니다.따라서, E의 부분 도함수에 대한 식을 다음과 같이 단순화할 수 있습니다." role="presentation" style="위치: 상대적;">E를 bi에 대한" role="presentation" style="위치: 상대적;">bibib\_i로:&#x2202;E&#x2202;bi=&#x2202;E&#x2202;zi" role="presentation" style="text-align: 가운데; 위치: 상대적;">∂E∂bi=∂E∂zi∂E∂bi=∂E∂zi\frac{\partial E}{\partial b\_i} = \frac{\partial E}{\partial z\_i}이 식을 단순화하기 위해 앞서 도출한 &#x03B4;i" role="presentation" style="위치: 상대적;">δiδi\delta\_i의 정의를 다시 사용할 수 있습니다:&#x03B4;i=&#x2202;E&#x2202;zi=&#x03C3;&#x2032;(zi)&#x2211;kwk,i&#x03B4;k" role="presentation" style="text-align: center; position: 상대;">δi=∂E∂zi=σ′(zi)∑kwk,iδkδi=∂E∂zi=σ′(zi)∑kwk,iδk\delta\_i = \frac{\partial E}{\partial z\.