SVM Subgradient Step

Perform one linear-SVM subgradient step on the weights w. The objective is $\frac{1}{2}\mathrm{\parallel }w{\mathrm{\parallel }}^2+\text{reg}{\sum }_i\max (0,1-y_{i}w\text{\hspace{0.17em}⁣}\cdot \text{\hspace{0.17em}⁣}{x}_i)\backslash tfrac12\backslash |w\backslash |^2+\backslash text\{reg\}\backslash sum\_i\backslash max(0,\backslash ,1-y\_i\backslash ,w\backslash !\backslash cdot\backslash !x\_i)$ with labels in $\{-1,+1\}\backslash \{-1,+1\backslash \}$ and no bias. Examples whose margin $y_{i}w\text{\hspace{0.17em}⁣}\cdot \text{\hspace{0.17em}⁣}{x}_{i}y\_i\backslash ,w\backslash !\backslash cdot\backslash !x\_i$ is at least 1 contribute nothing to the hinge part.

Implement solve(X, y, w, lr, reg).

Returns. Return the updated weights $w-\text{lr}\cdot gw-\backslash text\{lr\}\backslash cdot\; g$ as a NumPy array, where . The input w is not modified.

Examples

Example 1

Input:

solve([[1.0, 0.0], [0.0, 1.0]], [1, -1], [0.0, 0.0], 0.1, 1.0)

Output:

[0.1, -0.1]

Example 2

Input:

solve([[2.0, 0.0]], [1], [1.0, 0.0], 0.1, 1.0)

Output:

[0.9, 0.0]