[Bug] Incorrect Parenthesis Location in Cross Term of AUCMLoss(version='v2') #74

Environment: libauc 2.0.1, torch 2.2.2

Location: libauc/losses/auc.py $\rightarrow$ AUCMLoss.forward branch for version == 'v2'

Issue:
The cross term is currently written as:

self.mean((y_pred*neg_mask) - self.mean(y_pred*pos_mask))

The inner self.mean(y_pred*pos_mask) computes a scalar $\mu_+$, which is broadcasted and subtracted across the entire vector of length $N$. This causes the positive sample positions—originally zeroed out by neg_mask—to become $-\mu_+$ (non-zero). Consequently, the outer self.mean's count_nonzero includes these positions in its denominator, making the denominator $N$ instead of $N_-$:

$$\text{cross}_{\text{actual}} = \frac{1}{N} \sum_{i \in \text{neg}} s_i - \mu_+ = (1-p) \cdot \mu_- - \mu_+$$

$$\text{cross}_{\text{expected}} = \mu_- - \mu_+$$

This means the negative class mean is unexpectedly scaled by $(1-p)$. It should be fixed by adjusting the parenthesis:

self.mean(y_pred*neg_mask) - self.mean(y_pred*pos_mask)  # Fix parenthesis location

Minimal Reproducible Example:

import torch
from libauc.losses.auc import AUCMLoss

s = torch.tensor([0.1, 0.4, 0.35, 0.8]).view(-1, 1)
y = torch.tensor([0., 0., 1., 1.]).view(-1, 1)

f = AUCMLoss(margin=1.0, version='v2')
with torch.no_grad():
    f.a.fill_(0.3); f.b.fill_(0.6); f.alpha.fill_(0.5)
out = f(s, y); out.backward()
print(float(out), float(f.alpha.grad))    # 0.57125002  0.10000002

# Manual calculation based on class-wise means:
mu_pos, mu_neg = (0.35+0.8)/2, (0.1+0.4)/2           # 0.575, 0.25
expect = ((0.35-0.3)**2 + (0.8-0.3)**2)/2 \
        + ((0.1-0.6)**2 + (0.4-0.6)**2)/2 \
        + 2*0.5*(1.0 + mu_neg - mu_pos) - 0.5**2
print(expect)                               # 0.69625000

Cross-Validation (Key Evidence):
The version='v1' implementation in your library is correct. Since v1 and the class-wise mean formulation differ by a constant scaling factor $p(1-p)$, the identity $\text{v2} = \text{v1} / (p(1-p))$ should strictly hold. Using the same data:

f1 = AUCMLoss(margin=1.0, version='v1')
with torch.no_grad():
    f1.a.fill_(0.3); f1.b.fill_(0.6); f1.alpha.fill_(0.5)
print(float(f1(s, y)) / (0.5*0.5))        # 0.69625002  ← Matches manual calculation, contradicts v2

This identity holds for the current v2 implementation when $\alpha=0$ (cross term inactive) but breaks when $\alpha \neq 0$, further isolating the bug to the cross term.

Impact:

1. Loss value deviation: Closed-form deviation is $-2 \cdot \alpha \cdot p \cdot \mu_-$;
2. Underestimated dual variable pressure: $\partial L / \partial \alpha$ is significantly smaller than expected (e.g., $0.10$ vs. expected $0.35$ in the example above), underestimating the margin constraint penalty;
3. Asymmetric gradients: The gradient on the network output for negative samples is scaled down by $(1-p)$, introducing asymmetry against positive samples. Under recommended training setups like DualSampler(sampling_rate=0.5), negative sample gradients are reduced to only half of their intended magnitude.

Recommendation:
Fix the parenthesis placement. If backward compatibility is a concern, consider providing a feature toggle or documenting numerical changes in the release notes.