Lovasz Hinge Loss

Overview

Standard losses (BCE, Dice) are proxies for IoU — they correlate with it but don't optimize it directly. Lovasz hinge loss computes the exact subgradient of the Jaccard index by sorting prediction errors, computing cumulative intersection/union, and using the Lovasz extension of submodular functions. This makes it a tight convex surrogate for 1-IoU. In practice, it consistently outperforms BCE and Dice on IoU-based metrics by 1-3%, especially when mask shapes are irregular or class distributions are skewed.

Quick Start

import torch
import torch.nn.functional as F

def lovasz_grad(gt_sorted):
    p = len(gt_sorted)
    gts = gt_sorted.sum()
    intersection = gts - gt_sorted.float().cumsum(0)
    union = gts + (1 - gt_sorted).float().cumsum(0)
    jaccard = 1.0 - intersection / union
    if p > 1:
        jaccard[1:p] = jaccard[1:p] - jaccard[0:-1]
    return jaccard

def lovasz_hinge_flat(logits, labels):
    signs = 2.0 * labels.float() - 1.0
    errors = 1.0 - logits * signs
    errors_sorted, perm = torch.sort(errors, dim=0, descending=True)
    gt_sorted = labels[perm.data]
    grad = lovasz_grad(gt_sorted)
    return torch.dot(F.relu(errors_sorted), grad)

# Usage: pass raw logits (before sigmoid)
loss = lovasz_hinge_flat(logits.view(-1), masks.view(-1))

Workflow

1. Compute signed errors: 1 - logit * sign(label)
2. Sort errors in descending order
3. Compute cumulative Jaccard gradients via lovasz_grad
4. Dot product of ReLU'd errors with gradients gives the loss
5. Operates on raw logits — do NOT apply sigmoid first

Key Decisions

• Input: Raw logits, not probabilities — the hinge formulation needs unbounded values
• Per-image vs batch: Compute per-image then average for stable gradients
• Warm-up: Train with BCE for first few epochs, then switch to Lovasz for fine-tuning
• Multi-class: Use lovasz_softmax variant for multi-class segmentation

References

• Loss Function Library - Keras & PyTorch