Precision-Focused BCE Loss (PFBCE)

• PFBCE is a loss function designed to optimize the precision–recall trade-off in safety-critical vehicular platooning by decoupling false-positive penalties from attack sensitivity.
• It employs adaptive weighting to penalize high-confidence false positives while preserving detection recall, addressing the limitations of standard BCE loss.
• Experimental results show that PFBCE halves the false positive rate and improves F₁-score by 3–5 points with minimal recall loss in Transformer-based misbehavior detection systems.

Precision-Focused Binary Cross-Entropy (PFBCE) is a loss function designed to optimize the precision–recall trade-off for misbehavior detection in safety-critical vehicular platooning scenarios. Introduced in the context of Transformer-based misbehavior detection systems for vehicle platoons, such as AIMformer, PFBCE penalizes false positives (FPs) more heavily than symmetric alternatives, thereby directly addressing the operational risks of unnecessary alarms while sustaining adequate detection recall (Kalogiannis et al., 17 Dec 2025).

1. Motivation and Problem Context

In vehicular platooning, coordinated using Vehicle-to-Everything (V2X) communications, the misbehavior detection system (MDS) must maintain extremely low FP rates. High FPs can trigger unwarranted braking or de-platooning, undermining traffic efficiency and eroding trust in MDS—potentially leading to operator disregard of legitimate warnings. Standard Binary Cross-Entropy (BCE) loss treats false positives and false negatives equivalently, lacking mechanisms to explicitly control the FP rate without detrimental effects on attack recall. PFBCE introduces decoupled weighting to selectively suppress erroneous attack predictions on benign samples. This enables direct management of the precision–recall trade-off, which standard BCE and focal losses do not provide in such an interpretable manner (Kalogiannis et al., 17 Dec 2025).

2. Mathematical Formulation

Let $y_{i,j} \in \{0,1\}$ denote the ground-truth (0: benign, 1: attack) for vehicle $i$ at time $j$, $\hat{y}_{i,j}$ the model's raw logit, $\sigma(\cdot)$ the sigmoid activation, $m_{i,j}$ a validity mask, and $\tau \in (0,1)$ an FP threshold. Hyperparameters include $\lambda_{FP} > 1$ (FP-penalty), $\lambda_{pos} \geq 1$ (positive-class weight), and $\epsilon \ll 1$ (stability constant). For the set of valid indices $\mathcal{M}$, the per-sample BCE loss is:

$$\ell_{i,j} = -y_{i,j}\log(\sigma(\hat{y}_{i,j})) - (1 - y_{i,j})\log(1 - \sigma(\hat{y}_{i,j}))$$

Adaptive weights are:

• False-positive penalty:

$$w_{FP}^{(i,j)} = \begin{cases} \lambda_{FP} & \text{if } y_{i,j}=0 \ \text{and} \ \sigma(\hat{y}_{i,j}) > \tau \ 1 & \text{otherwise} \end{cases}$$

• Positive-class weight:

$$w_{pos}^{(i,j)} = \begin{cases} \lambda_{pos} & \text{if } y_{i,j}=1 \ 1 & \text{if } y_{i,j}=0 \end{cases}$$

The total loss:

$$\mathcal{L}_{\mathrm{PFBCE}}(Y, \hat{Y}) = \frac{1}{|\mathcal{M}|} \sum_{(i,j)\in\mathcal{M}} \ell_{i,j} \cdot w_{FP}^{(i,j)} \cdot w_{pos}^{(i,j)} \cdot m_{i,j}$$

All notation is consistent with (Kalogiannis et al., 17 Dec 2025).

3. Theoretical Rationale and Asymmetry

PFBCE is designed for highly asymmetric operational risk, where FPs are more costly in safety-critical settings than FNs. Unlike focal loss, which scales loss by $(1-p_t)^\gamma$ regardless of ground-truth, PFBCE’s selective up-weighting ensures only confident FP predictions ($\sigma(\hat{y}_{i,j}) > \tau$ on benign samples) incur high penalties. Simultaneously, an independent positive-class weight preserves detection sensitivity by up-weighting attack samples. Tuning $\lambda_{FP}$ and $\lambda_{pos}$ independently forms a continuous precision–recall frontier, enabling practitioners to target, for example, precision $\geq 0.95$ and recall $\geq 0.90$ as required in platooning (Kalogiannis et al., 17 Dec 2025).

4. Integration in Transformer-Based MDS Training

Within AIMformer’s supervised training, PFBCE replaces the standard BCE objective at the output of the Transformer encoder. Logits $\hat{y}_{i,j}$ for each vehicle and time step are transformed via the procedure above. Forward passes compute the required masks and adaptive weights; Adam or other optimizers minimize the average PFBCE. All auxiliary training routines—dropout, learning rate schedules, normalization—remain unchanged. This enables seamless adoption in any binary classification framework for safety-critical domains (Kalogiannis et al., 17 Dec 2025).

5. Hyperparameter Tuning Strategies

Hyperparameters are critical for effective deployment. In (Kalogiannis et al., 17 Dec 2025), Hyperband via Keras Tuner yielded optimal values: $\lambda_{FP}=1.7$, $\lambda_{pos}=0.6$, $\tau=0.6$. General guidelines are:

• $\lambda_{FP}$ dictates FP suppression—raising it increases precision but may reduce recall.
• $\lambda_{pos}$ corrects for class imbalance; increase if recall is unsatisfactory.
• $\tau$ defines what constitutes a “high-confidence” FP; values of $0.5$–$0.7$ are empirically robust.

Grid or bandit-style hyperparameter searches over a validation set are recommended, prioritizing operational precision constraints before recall maximization.

6. Experimental Comparison and Performance Impact

Experiments presented in (Kalogiannis et al., 17 Dec 2025) compare PFBCE against standard BCE and an F1-based BCE surrogate (F1BCE) across four platoon controllers and six vehicle positions. Results:

Loss	Precision	Recall (drop)	FP Rate	F₁ Improvement	AUC (ROC)
BCE	~0.90	Baseline	Baseline	Baseline	~0.94
F1BCE	Lower	—	—	—	—
PFBCE	≥0.95	≤1–2%	Halved	+3–5 points	0.96–0.99

PFBCE consistently halved the FP rate in maneuver scenarios (join/exit), raised F₁-score by 3–5 points, and increased AUC to 0.96–0.99. Recall remained within 1–2% of baseline BCE. These outcomes directly reduced spurious de-platooning and improved trust in MDS alerts in all test regimes, as detailed in Sections 5.2 and 5.3 of (Kalogiannis et al., 17 Dec 2025).

7. Implementation Guidelines

The loss computation is straightforward in PyTorch and TensorFlow, as illustrated by the following pseudo-code:

import torch def pfbce_loss(logits, targets, mask, lambda_fp=1.7, lambda_pos=0.6, tau=0.6, eps=1e-6): probs = torch.sigmoid(logits) bce = - (targets * torch.log(probs + eps) + (1-targets) * torch.log(1-probs + eps)) fp_penalty = torch.where((targets==0) & (probs > tau), lambda_fp, 1.0) pos_weight = torch.where(targets==1, lambda_pos, 1.0) weighted_loss = bce * fp_penalty * pos_weight * mask normalizer = mask.sum() + eps return weighted_loss.sum() / normalizer

A direct adaptation for TensorFlow/Keras is equally feasible using analogous tensor operations and Keras custom loss wrappers. Hyperparameters should be optimized via a systematic search using validation data, ensuring that precision targets are prioritized in line with safety-critical requirements (Kalogiannis et al., 17 Dec 2025).