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Diffstat (limited to 'README.md')
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@@ -6,11 +6,17 @@ A simple linear regression implementation using gradient descent to predict car - Python 3 - matplotlib (for visualization only) +- pandoc (to generate HTML documentation) ``` pip install matplotlib ``` +To generate the HTML version of this README (to see the equations): +``` +pandoc README.md --mathml -s -o README.html +``` + ## Usage ### Train the model @@ -51,4 +57,22 @@ The model fits a linear function: estimatePrice(mileage) = θ0 + θ1 * mileage ``` -Parameters are found via gradient descent with min-max normalization on the input data. After training, thetas are denormalized so they work directly on raw mileage values. +Parameters are found via gradient descent. The input data is normalized before training using min-max normalization, and the resulting thetas are denormalized afterward so they work directly on raw mileage values. + +### Why normalization? + +The two variables have very different scales: mileage ranges from ~22,000 to ~240,000 while prices range from ~3,600 to ~8,300. This causes the gradient for $\theta_1$ (which is multiplied by mileage) to be orders of magnitude larger than the gradient for $\theta_0$. No single learning rate can work well for both parameters simultaneously. + +Min-max normalization scales each variable to $[0, 1]$: + +$$x_{\text{norm}} = \frac{x - x_{\min}}{x_{\max} - x_{\min}}$$ + +Without normalization, if you pick a learning rate small enough to prevent $\theta_1$ from overshooting, $\theta_0$ barely moves and needs millions of iterations. If you pick a larger learning rate so $\theta_0$ converges in a reasonable time, $\theta_1$ overshoots, oscillates, and diverges to infinity (NaN). + +Normalization brings both gradients to the same scale, allowing gradient descent to converge efficiently with a single learning rate. + +After training on normalized data, the thetas are converted back to work on raw values: + +$$\theta_1' = \theta_1 \cdot \frac{p_{\max} - p_{\min}}{km_{\max} - km_{\min}}$$ + +$$\theta_0' = \theta_0 \cdot (p_{\max} - p_{\min}) + p_{\min} - \theta_1' \cdot km_{\min}$$ |