weights and biases in each layer are as follows:

For k-layer:

&#x394;wkj=&#x2212;&#x3B7;&#x3B4;kajand&#x394;bk=&#x2212;&#x3B7;&#x3B4;k \Delta w\_{kj} = -\eta \delta\_k a\_j \quad \text{and} \quad \Delta b\_k = -\eta \delta\_k" role="presentation" style="text-align: center; position: relative;">Δwkj=−ηδkajandΔbk=−ηδkΔwkj=−ηδkajandΔbk=−ηδk \Delta w\_{kj} = -\eta \delta\_k a\_j \quad \text{and} \quad \Delta b\_k = -\eta \delta\_k<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>w</mi><mrow><mi>k</mi><mi>j</mi></mrow></msub><mo>=</mo><mo>−</mo><mi>η</mi><msub><mi>δ</mi><mi>k</mi></msub><msub><mi>a</mi><mi>j</mi></msub><mspace width="1em"></mspace><mtext>and</mtext><mspace width="1em"></mspace><mi mathvariant="normal">Δ</mi><msub><mi>b</mi><mi>k</mi></msub><mo>=</mo><mo>−</mo><mi>η</mi><msub><mi>δ</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Delta w\_{kj} = -\eta \delta\_k a\_j \quad \text{and} \quad \Delta b\_k = -\eta \delta\_k</annotation></semantics></math>Δwkj​=−ηδk​aj​andΔbk​=−ηδk​For j-layer:

&#x394;wji=&#x2212;&#x3B7;&#x3B4;jaiand&#x394;bj=&#x2212;&#x3B7;&#x3B4;j \Delta w\_{ji} = -\eta \delta\_j a\_i \quad \text{and} \quad \Delta b\_j = -\eta \delta\_j" role="presentation" style="text-align: center; position: relative;">Δwji=−ηδjaiandΔbj=−ηδjΔwji=−ηδjaiandΔbj=−ηδj \Delta w\_{ji} = -\eta \delta\_j a\_i \quad \text{and} \quad \Delta b\_j = -\eta \delta\_j<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>w</mi><mrow><mi>j</mi><mi>i</mi></mrow></msub><mo>=</mo><mo>−</mo><mi>η</mi><msub><mi>δ</mi><mi>j</mi></msub><msub><mi>a</mi><mi>i</mi></msub><mspace width="1em"></mspace><mtext>and</mtext><mspace width="1em"></mspace><mi mathvariant="normal">Δ</mi><msub><mi>b</mi><mi>j</mi></msub><mo>=</mo><mo>−</mo><mi>η</mi><msub><mi>δ</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">\Delta w\_{ji} = -\eta \delta\_j a\_i \quad \text{and} \quad \Delta b\_j = -\eta \delta\_j</annotation></semantics></math>Δwji​=−ηδj​ai​andΔbj​=−ηδj​For i-layer:

&#x394;win=&#x2212;&#x3B7;&#x3B4;ixnand&#x394;bi=&#x2212;&#x3B7;&#x3B4;i \Delta w\_{in} = -\eta \delta\_i x\_n \quad \text{and} \quad \Delta b\_i = -\eta \delta\_i" role="presentation" style="text-align: center; position: relative;">Δwin=−ηδixnandΔbi=−ηδiΔwin=−ηδixnandΔbi=−ηδi \Delta w\_{in} = -\eta \delta\_i x\_n \quad \text{and} \quad \Delta b\_i = -\eta \delta\_i<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">Δ</mi><msub><mi>w</mi><mrow><mi>i</mi><mi>n</mi></mrow></msub><mo>=</mo><mo>−</mo><mi>η</mi><msub><mi>δ</mi><mi>i</mi></msub><msub><mi>x</mi><mi>n</mi></msub><mspace width="1em"></mspace><mtext>and</mtext><mspace width="1em"></mspace><mi mathvariant="normal">Δ</mi><msub><mi>b</mi><mi>i</mi></msub><mo>=</mo><mo>−</mo><mi>η</mi><msub><mi>δ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\Delta w\_{in} = -\eta \delta\_i x\_n \quad \text{and} \quad \Delta b\_i = -\eta \delta\_i</annotation></semantics></math>Δwin​=−ηδi​xn​andΔbi​=−ηδi​These expressions tell us how to adjust the weights and biases in each layer to minimize the error using gradient descent.

Note that these expressions assume a batch size of 1. To extend these expressions to mini-batch or full-batch training, you would need to accumulate the weight and bias updates over the entire batch before applying them to the model parameters.

You can use Mathematica or MATLAB Symbolic Math Toolbox to symbolically derive these expressions based on your setup.
각 레이어의 가중치와 편향은 다음과 같습니다:K- 레이어의 경우&#x394;wkj=&#x2212;&#x3B7;&#x3B4;kajand&#x394;bk=&#x2212;&#x3B7;&#x3B4;k \Delta w\_{kj} = -\eta \delta\_k a\_j \quad \text{and} \quad \Delta b\_k = -\eta \delta\_k" role="presentation" style="text-align: 가운데; 위치: 상대적;">Δwkj=-ηδkajandΔbk=-ηδkΔwkj=-ηδkajandΔbk=-ηδk \Delta w\_{kj} = -\eta \delta\_k a\_j \quad \text{and} \quad \Delta b\_k = -\eta \delta\_k<math xmlns="http://www.w3. org/1998/Math/MathML" display="block"><시맨틱><mrow><mi mathvariant="normal">Δ</mi><msub><mi>w</mi><mrow><mi>k</mi><mi>j</mi></mrow></msub><mo>=</mo><mo>-</mo><mi>η</mi><msub><mi>δ</mi><mi>k</mi></msub><msub><mi>a</mi><mi>j</mi></msub><mspace width="1em"></mspace><mtext>및</mtext><mspace width="1em"></mspace><mi mathvariant="normal">Δ</mi><msub><mi>b</mi><mi>k</mi></msub><mo>=</mo><mo>-</mo><mi>η</mi><msub><mi>δ</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-. tex">\Delta w\_{kj} = -\eta \delta\_k a\_j \quad \text{and} \쿼드 \델타 b\_k = -\eta \delta\_k</주석></semantics></math>Δwkj=-ηδkajandΔbk=-ηδk제이 레이어의 경우:&#x394;wji=&#x2212;&#x3B7;&#x3B4;jai및&#x394;bj=&#x2212;&#x3B7;&#x3B4;j \Delta w\_{ji} = -\eta \delta\_j a\_i \quad \text{and} \quad \Delta b\_j = -\eta \delta\_j" role="presentation" style="text-align: 가운데; 위치: 상대적;">Δwji=-ηδjaiandΔbj=-ηδjΔwji=-ηδjaiandΔbj=-ηδj \Delta w\_{ji} = -\eta \delta\_j a\_i \quad \text{and} \쿼드 \델타 b\_j = -\eta \델타\_j<math xmlns="http://www.w3. org/1998/Math/MathML" display="block"><시맨틱><mrow><mi mathvariant="normal">Δ</mi><msub><mi>w</mi><mrow><mi>j</mi><mi>i</mi></mrow></msub><mo>=</mo><mo>-</mo><mi>η</mi><msub><mi>δ</mi><mi>j</mi></msub><msub><mi>a</mi><mi>i</mi></msub></msub><mspace width="1em"></mspace><mtext>및</mtext><mspace width="1em"></mspace><mi mathvariant="normal">Δ</mi><msub><mi>b</mi><mi>j</mi></msub><mo>=</mo><mo>-</mo><mi>η</mi><msub><mi>δ</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-. tex">\Delta w\_{ji} = -\eta \delta\_j a\_i \quad \text{and} \quad \Delta b\_j = -\eta \delta\_j</주석></semantics></math>Δwji=-ηδjaiandΔbj=-ηδj아이 레이어의 경우:&#x394;win=&#x2212;&#x3B7;&#x3B4;ixnand&#x394;bi=&#x2212;&#x3B7;&#x3B4;i \Delta w\_{in} = -\eta \delta\_i x\_n \쿼드 \text{and} \quad \Delta b\_i = -\eta \delta\_i" role="presentation" style="text-align: 가운데; 위치: 상대적;">Δwin=-ηδixnandΔbi=-ηδiΔwin=-ηδixnandΔbi=-ηδi \Delta w\_{in} = -\eta \delta\_i x\_n \quad \text{and} \쿼드 \델타 b\_i = -\eta \delta\_i<math xmlns="http://www.w3. org/1998/Math/MathML" display="block"><시맨틱><mrow><mi mathvariant="normal">Δ</mi><msub><mi>w</mi><mrow><mi>i</mi><mi>n</mi></mrow></msub><mo>=</mo><mo>-</mo><mi>η</mi><msub><mi>δ</mi><mi>i</mi></msub><msub><mi>x</mi><mi>n</mi></msub><mspace width="1em"></mspace><mtext>및</mtext><mspace width="1em"></mspace><mi mathvariant="normal">Δ</mi><msub><mi>b</mi><mi>i</mi></msub><mo>=</mo><mo>-</mo><mi>η</mi><msub><mi>δ</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-. tex">\Delta w\_{in} = -\eta \delta\_i x\_n \quad \text{and} \quad \Delta b\_i = -\eta \delta\_i</주석></semantics></math>Δwin=-ηδixnandΔbi=-ηδi이 표현식은 경사 하강을 사용하여 오차를 최소화하기 위해 각 레이어의 가중치와 바이어스를 조정하는 방법을 알려줍니다.이 표현식은 배치 크기가 1이라고 가정합니다. 이 표현식을 미니 배치 또는 전체 배치 훈련으로 확장하려면 모델 파라미터에 적용하기 전에 전체 배치에 걸쳐 가중치 및 바이어스 업데이트를 누적해야 합니다.설정에 따라 Mathematica 또는 MATLAB 기호 수학 도구상자를 사용하여 이러한 표현식을 기호적으로 도출할 수 있습니다.