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| Original file line number | Diff line number | Diff line change |
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| @@ -0,0 +1,134 @@ | ||
| --- | ||
| title: "Vectorising the Digit Classifier" | ||
| date: 2024-07-30 | ||
| draft: false | ||
| tags: ["machine learning"] | ||
| --- | ||
|
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| ## Overview | ||
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| In the previous post, I trained a neural network to classify hand-written digits. I got an accuracy of ~85% on the test data. It took more than an hour to train the model. I will now vectorise the implementation, to speed up the training process. | ||
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| ## Implementation | ||
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| Instead of iterating through the training examples one by one, I will stack them up in a matrix `X` and one-hot encode the labels in `Y`. Then I can just multiply `X` with `W` and add `B` to get the output `y`. | ||
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| Before vectorising, I multiplied the weights with the training examples and added the bias values, in the following way | ||
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| {{< katex >}} | ||
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| $$ | ||
| \{ | ||
| \begin{bmatrix} | ||
| w_{11} & w_{12} & w_{13} & w_{14}\newline | ||
| w_{21} & w_{22} & w_{23} & w_{24}\newline | ||
| w_{31} & w_{32} & w_{33} & w_{34} | ||
| \end{bmatrix} | ||
| \begin{bmatrix} | ||
| x_{11}\newline | ||
| x_{12}\newline | ||
| x_{13}\newline | ||
| x_{14} | ||
| \end{bmatrix} | ||
| + | ||
| \begin{bmatrix} | ||
| b_1\newline | ||
| b_2\newline | ||
| b_3 | ||
| \end{bmatrix} | ||
| \} | ||
| $$ | ||
|
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| $$ | ||
| \{ | ||
| = | ||
| \begin{bmatrix} | ||
| w_{11}x_{11}+w_{12}x_{12}+w_{13}x_{13}+w_{14}x_{14}+b_1\newline | ||
| w_{21}x_{11}+w_{22}x_{12}+w_{23}x_{13}+w_{24}x_{14}+b_2\newline | ||
| w_{31}x_{11}+w_{32}x_{12}+w_{33}x_{13}+w_{34}x_{14}+b_3 | ||
| \end{bmatrix} | ||
| = | ||
| \begin{bmatrix} | ||
| a_1\newline | ||
| a_2\newline | ||
| a_3 | ||
| \end{bmatrix} | ||
| \} | ||
| $$ | ||
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| Now, I can multiply all the training examples with the weights and add the bias values, in one step | ||
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| $$ | ||
| \{ | ||
| \begin{bmatrix} | ||
| w_{11} & w_{12} & w_{13} & w_{14}\newline | ||
| w_{21} & w_{22} & w_{23} & w_{24}\newline | ||
| w_{31} & w_{32} & w_{33} & w_{34} | ||
| \end{bmatrix} | ||
| \begin{bmatrix} | ||
| x_{11} & x_{21} & x_{31} & x_{41}\newline | ||
| x_{12} & x_{22} & x_{32} & x_{42}\newline | ||
| x_{13} & x_{23} & x_{33} & x_{43}\newline | ||
| x_{14} & x_{24} & x_{34} & x_{44} | ||
| \end{bmatrix} | ||
| + | ||
| \begin{bmatrix} | ||
| b_1\newline | ||
| b_2\newline | ||
| b_3 | ||
| \end{bmatrix} | ||
| \} | ||
| $$ | ||
|
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||
| $$ | ||
| \{ | ||
| = | ||
| \begin{bmatrix} | ||
| a_{11} & a_{21} & a_{31} & a_{41}\newline | ||
| a_{12} & a_{22} & a_{32} & a_{42}\newline | ||
| a_{13} & a_{23} & a_{33} & a_{43} | ||
| \end{bmatrix} | ||
| \} | ||
| $$ | ||
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| In the code, I got rid of one `for` loop. | ||
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| ```py | ||
| for run in range(epoch): | ||
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| # forward prop | ||
| Z1 = np.dot(W1, X) + B1 | ||
| A1 = 1/(1+np.exp(-Z1+1e-5)) | ||
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| Z2 = np.dot(W2, A1) + B2 | ||
| A2 = 1/(1+np.exp(-Z2+1e-5)) | ||
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| Z3 = np.dot(W3, A2) + B3 | ||
| y = np.exp(Z3+1e-5) | ||
| y /= sum(y) | ||
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| # back prop | ||
| dz3 = y - Y_one | ||
| dw3 = np.dot(dz3, A2.T)/m | ||
| db3 = sum(dz3.T)/m | ||
| da2 = np.matmul(W3.T, dz3) | ||
| dz2 = A2*(1-A2)*da2 | ||
| dw2 = np.dot(dz2, A1.T)/m | ||
| db2 = sum(dz2.T)/m | ||
| da1 = np.matmul(W2.T, dz2) | ||
| dz1 = A1*(1-A1)*da1 | ||
| dw1 = np.dot(dz1, X.T)/m | ||
| db1 = sum(dz1.T)/m | ||
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| # update params | ||
| W3 -= alpha*dw3 | ||
| B3 -= alpha*db3.reshape(-1, 1) | ||
| W2 -= alpha*dw2 | ||
| B2 -= alpha*db2.reshape(-1, 1) | ||
| W1 -= alpha*dw1 | ||
| B1 -= alpha*db1.reshape(-1, 1) | ||
| ``` | ||
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| ## Conclusion | ||
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| In the vectorised form, the training process became 15x faster, producing similar results to the non-vectorised form. |
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