Focal loss · AI Engineering

Focal loss is cross-entropy with a weight that makes easy examples count less. A detector sees many easy background regions and few objects. With plain cross-entropy, the easy examples give most of the loss. Focal loss multiplies each term by (1 - p_t)^\gamma, so the model learns more from the hard examples.

Task

Write three functions.

binary_focal_loss(logits, targets, gamma, alpha) returns the mean binary focal loss.
binary_focal_grad(logits, targets, gamma, alpha) returns the gradient of that mean loss with respect to the logits.
focal_loss(logits, targets, gamma, alpha) returns the mean multi-class focal loss.

For one example, p_t is the predicted probability of the true class and

\mathrm{FL}(p_t) = -\alpha_t \, (1 - p_t)^\gamma \log p_t.

In the binary case, the logit is x and the target is y \in \{0, 1\}. Then p_t = \sigma(x) when y = 1 and p_t = 1 - \sigma(x) when y = 0, with \sigma(x) = 1 / (1 + e^{-x}). With a float alpha, \alpha_t = \alpha when y = 1 and \alpha_t = 1 - \alpha when y = 0.

In the multi-class case, p_t is the softmax probability of the target class over the row of logits. With an array alpha of size C, \alpha_t is alpha[target].

When alpha is None, \alpha_t = 1. Each loss is the mean of \mathrm{FL} over the batch. With \gamma = 0 and no alpha, both losses are the ordinary cross-entropy.

Input and output

Name	Type	Shape	Meaning
`logits` (binary)	float array	`(N,)`	one logit for each example
`targets` (binary)	float array	`(N,)`	0 or 1 for each example
`logits` (multi-class)	float array	`(N, C)`	one row of logits for each example
`targets` (multi-class)	int array	`(N,)`	the class index of each example, from 0 to `C - 1`
`gamma`	float		the focusing exponent, 0 or more
`alpha`	float, float array or `None`	`()` or `(C,)`	the class weight: a float in the binary case, an array of size `C` in the multi-class case

binary_focal_loss and focal_loss return a Python float.
binary_focal_grad returns a float array of shape (N,).

Constraints

1 <= N <= 1000, 2 <= C <= 100, 0 <= gamma <= 5 and 0 < alpha < 1 in the binary case.
Logits can be as large as 1000 in absolute value. The loss and the gradient must stay finite.
Do not change the input arrays.
Use numpy only.

Examples

python

logits = np.array([2.0, -1.0, 0.5, -3.0])
targets = np.array([1.0, 1.0, 0.0, 0.0])
binary_focal_loss(logits, targets, 0.0)
binary_focal_loss(logits, targets, 2.0)
binary_focal_loss(logits, targets, 2.0, 0.25)
binary_focal_grad(logits, targets, 2.0, 0.25)

text

0.6157135085787611
0.270298187607633
0.1147646574248966
[-3.04433762e-04 -4.80146702e-02  9.86534666e-02  5.90384712e-05]

python

logits = np.array([[2.0, 1.0, 0.1], [0.5, 2.5, 0.3]])
targets = np.array([0, 1])
focal_loss(logits, targets, 0.0)
focal_loss(logits, targets, 2.0)
focal_loss(logits, targets, 2.0, np.array([0.5, 1.0, 2.0]))

text

0.31853976964918573
0.02853873533647399
0.016415649484550623