Focal Loss Calculator

Focal loss was introduced by Lin, Goyal, Girshick, He and Dollár in “Focal Loss for Dense Object Detection” (ICCV 2017) to fix a specific problem: in dense detectors like RetinaNet there are tens of thousands of easy background boxes for every real object, and under ordinary cross-entropy their many small losses add up and drown out the handful of hard, informative examples. Focal loss reshapes cross-entropy so easy examples contribute far less.

Start from the probability the model assigned to the true class, p_t. In the binary case with label y ∈ {0, 1} and predicted positive-class probability p, Eq. 2 of the paper defines p_t = p when y = 1 and p_t = 1 − p otherwise. Plain cross-entropy (Eq. 1) is then:

CE(p_t) = −ln(p_t)

Focal loss multiplies that by a modulating factor (Eq. 3) and an optional α-balancing weight (Eq. 4), giving the final form (Eq. 5):

FL(p_t) = −α_t · (1 − p_t)^γ · ln(p_t) = α_t · (1 − p_t)^γ · CE

1. Modulating factor (1 − p_t)^γ. When an example is well-classified (p_t near 1) this is near 0, so its loss almost vanishes; when it is misclassified (p_t small) the factor is near 1 and the loss is barely touched. γ (gamma) sets how aggressively this happens. With γ = 0 the factor is 1 and focal loss is exactly cross-entropy.
2. α-balancing α_t. When enabled, α_t = α for the positive class and 1 − α for the negative class; when disabled, α_t = 1. It nudges the balance between the two classes on top of the γ down-weighting.
3. Clamp for safety. p_t is clamped to [1e-7, 1 − 1e-7] before the log so a predicted 0 gives a large finite loss instead of ln 0 = −∞.
4. Batch mean. The trainable objective is the average (1/N)·Σ FL_i over all samples, which the batch section computes.

The recommended defaults are γ = 2 and α = 0.25 — the best-performing settings in the paper (§5). Multiclass mode uses the same equations with p_t taken as the softmax probability of the correct class and α as a single per-sample class weight. To keep every result trustworthy the tool independently recomputes each binary focal loss the way torchvision.ops.sigmoid_focal_loss does — building cross-entropy from (p, y) first, then applying the modulating factor and α_t — and confirms the two agree to floating-point precision.

• Lin, Goyal, Girshick, He & Dollár — Focal Loss for Dense Object Detection (ICCV 2017), Eq. 1–5, defaults γ = 2, α = 0.25
• PyTorch — torchvision.ops.sigmoid_focal_loss (reference implementation cross-checked here)
• scikit-learn — sklearn.metrics.log_loss (cross-entropy baseline; γ = 0 reduces focal loss to this)

The equations on this page were last cross-checked against these sources on 2026-07-12. Focal loss is a fixed mathematical definition, so this tool needs no rate or schedule updates — only the worked examples are periodically re-reconciled.