Huber loss / np.where

Huber loss:

\[L_{\delta }(y,f(x)) = {\begin{cases}{ \frac{1}{2}}(y-f(x))^{2} & {\textrm {for}}|y-f(x)| \leq \delta, \newline \delta \,|y-f(x)|-{\frac{1}{2}} \delta^{2} & {\textrm {otherwise.}} \end{cases}}\]

• Usually set $\delta = 1$
• Huber loss is less sensitive to outliers (those $y$ such that $\vert y-f(x) \vert > \delta$) than the MSE
• And converges faster than the mean absolute error (because of larger gradients).

A python implementation:

def huber_fn(y_true, y_pred):
    # Suppose delta == 1
    error = y_true - y_pred
    
    is_small_error = tf.abs(error) < 1
    
    squared_loss = tf.square(error) / 2
    linear_loss = tf.abs(error) - 0.5
    
    return tf.where(is_small_error, squared_loss, linear_loss)

If all the arrays are 1-D, np.where(condition, X, Y) is equivalent to:

for (c, x, y) in zip(condition, X, Y):
    yield c ? x : y