Ramp loss Usually, the margin-based bound for
The γ \gamma γ
Φ γ ( t ) = { 0 if t ≥ γ 1 − t γ if 0 < t < γ 1 if t ≤ 0 \Phi_\gamma(t) = \begin{cases}
0 & \text{if } t \geq \gamma \\
1 - \frac{t}{\gamma} & \text{if } 0 < t < \gamma \\
1 & \text{if } t \leq 0
\end{cases} Φ γ ( t ) = ⎩ ⎨ ⎧ 0 1 − γ t 1 if t ≥ γ if 0 < t < γ if t ≤ 0
In relation with Hinge loss:
l ramp ( w , ( x , y ) ) = min { 1 , l hinge ( w , ( x , y ) ) } = min { 1 , max { 0 , 1 − y ⟨ w , x ⟩ } } \mathcal{l}^{\text{ramp}}(\textbf{w}, (\textbf{x},y)) = \min \{1, \mathcal{l}^{\text{hinge}}(\textbf{w}, (\textbf{x},y))\} = \min \{1, \max\{0, 1 - y \langle w, x \rangle\}\} l ramp ( w , ( x , y )) = min { 1 , l hinge ( w , ( x , y ))} = min { 1 , max { 0 , 1 − y ⟨ w , x ⟩}} Note that we use Hinge loss for SVM is due to the fact that ramp-loss is a non-convex functions, meaning it is more computationally efficient to minimise Hinge loss in comparison to ramp loss
