Operating-Condition-Dependent Trainable a-DCF Loss

• The paper introduces an operating-condition-dependent trainable a-DCF loss that embeds smooth surrogates to enable gradient-based optimization for integrated ASV and countermeasure systems.
• It leverages dynamic threshold optimization and a convex BCE combination to balance trade-offs between user convenience and spoofing robustness.
• Empirical results on SASV benchmarks demonstrate up to 47% relative improvement, confirming its effectiveness in adapting to specific operating conditions.

Operating-condition-dependent trainable a-DCF loss is a supervised learning loss function specifically designed for integrated automatic speaker verification (ASV) and countermeasure (CM) systems, particularly with the goal of optimizing performance under explicit trade-offs between user-convenience and spoofing-robustness. This approach directly embeds the architecture-agnostic detection cost function (a-DCF) into the learning objective, parameterizing the loss with respect to user-defined operating conditions (such as miss/false alarm costs and class priors). The method replaces non-differentiable decision statistics with smooth surrogates, making the a-DCF loss fully differentiable and amenable to gradient-based optimization. It is integrated alongside standard binary cross-entropy (BCE), and can include dynamic threshold optimization, ensuring alignment between the training objective and the final evaluation metric used in SASV benchmarks (Kurnaz et al., 2024, Kurnaz et al., 2 Feb 2026).

1. Formal Definition and Mathematical Formulation

The a-DCF generalizes the detection cost function for scenarios involving speaker verification, zero-effort impostors, and spoofing attacks. For a system emitting a real-valued score $g(x)$ and threshold $\tau$, the hard error rates are defined as: $$P_{\rm miss}^{\rm tar}(\tau) = \frac{1}{N_{\rm tar}} \sum_{x \in {\rm tar}} \mathbf{1}(g(x) \leq \tau)$$

$$P_{\rm fa}^{\rm non}(\tau) = \frac{1}{N_{\rm non}} \sum_{x \in {\rm non}} \mathbf{1}(g(x) > \tau)$$

$$P_{\rm fa}^{\rm spf}(\tau) = \frac{1}{N_{\rm spf}} \sum_{x \in {\rm spf}} \mathbf{1}(g(x) > \tau)$$

where $\mathbf{1}(\cdot)$ denotes the indicator function. The architecture-agnostic DCF combines these as: $${\rm a\!-\!DCF}(\tau) = C_{\rm miss}\,\pi_{\rm tar}\,P_{\rm miss}^{\rm tar}(\tau) + C_{\rm fa}^{\rm non}\,\pi_{\rm non}\,P_{\rm fa}^{\rm non}(\tau) + C_{\rm fa}^{\rm spf}\,\pi_{\rm spf}\,P_{\rm fa}^{\rm spf}(\tau)$$ with explicit operating-condition parameters $$\{C_{\rm miss}, C_{\rm fa}^{\rm non}, C_{\rm fa}^{\rm spf}, \pi_{\rm tar}, \pi_{\rm non}, \pi_{\rm spf}\}$$ (Kurnaz et al., 2024, Kurnaz et al., 2 Feb 2026).

2. Differentiability via Soft Surrogates and Threshold Optimization

Hard counts in a-DCF loss are non-differentiable due to step-wise indicator functions and the threshold selection $\arg\min_\tau$. For gradient-based training, these are replaced by sigmoid-based soft surrogates: $$\hat{P}_{\rm miss}^{\rm tar}(\tau) = \frac{1}{N_{\rm tar}} \sum_{x \in {\rm tar}} \sigma(\tau - g(x))$$

$$\hat{P}_{\rm fa}^{\rm non}(\tau) = \frac{1}{N_{\rm non}} \sum_{x \in {\rm non}} \sigma(g(x) - \tau)$$

$$\hat{P}_{\rm fa}^{\rm spf}(\tau) = \frac{1}{N_{\rm spf}} \sum_{x \in {\rm spf}} \sigma(g(x) - \tau)$$

with $\sigma(z) = 1/(1+e^{-z})$. The soft a-DCF loss is

$$\mathcal{L}_{\rm a\!-\!DCF}^{\rm soft}(\tau) = C_{\rm miss}\,\pi_{\rm tar}\,\hat{P}_{\rm miss}^{\rm tar}(\tau) + C_{\rm fa}^{\rm non}\,\pi_{\rm non}\,\hat{P}_{\rm fa}^{\rm non}(\tau) + C_{\rm fa}^{\rm spf}\,\pi_{\rm spf}\,\hat{P}_{\rm fa}^{\rm spf}(\tau)$$

A threshold search is performed at each epoch to minimize this loss over $\tau$, yielding a differentiable surrogate that supports backpropagation (Kurnaz et al., 2024).

3. Joint Loss and Training Regimen

To maintain class separation beyond what a-DCF or BCE alone can achieve, a convex combination of soft a-DCF and binary cross-entropy (BCE) is used: $$J(\theta, \tau) = \alpha\,\mathcal{L}_{\rm BCE}(\theta) + (1-\alpha)\,\mathcal{L}_{\rm a\!-\!DCF}^{\rm soft}(\tau;\theta)$$ where

$$\mathcal{L}_{\rm BCE} = -\frac1N \sum_{i=1}^N \left[y_i\log \hat y_i + (1-y_i)\log(1-\hat y_i)\right]$$

Hyperparameters such as $\alpha$ (trade-off parameter, typically 0.5), batch size, learning rate, and training epochs are selected to maximize empirical generalization. Small-scale grid search for $\tau$ is conducted within each epoch to obtain the loss-minimizing threshold (Kurnaz et al., 2024).

Example setup (ASVspoof2019 LA):

Operating Parameter	Value
$C_{\rm miss}$	1
$C_{\rm fa}^{\rm non}$	10
$C_{\rm fa}^{\rm spf}$	20
$\pi_{\rm tar}$	0.9
$\pi_{\rm non}$	0.05
$\pi_{\rm spf}$	0.05

The system utilizes an embedding fusion back-end (concatenation of ECAPA-TDNN ASV and AASIST CM outputs) and a DNN classifier (Kurnaz et al., 2024).

4. Operating-Condition Parameterization

Operating condition parameters control the cost trade-offs and class priors that the network is explicitly optimized for. Adjusting these parameters:

• Increasing $\pi_{\rm spf}$ or $C_{\rm fa}^{\rm spf}$ shifts the optimal threshold $\tau^*$ upward, decreasing the false acceptance rate for spoofs at the cost of higher miss rates.
• Raising $\pi_{\rm non}$ or $C_{\rm fa}^{\rm non}$ similarly suppresses non-target false alarms. Guidelines recommend setting $\pi_{\rm tar}$ to match the bona-fide rate in the target deployment context and $C_{\rm fa}^{\rm spf} \gg C_{\rm miss}$ when spoof prevention is prioritized (Kurnaz et al., 2024, Kurnaz et al., 2 Feb 2026).

A key property is the “operating-condition-dependence”: the same network architecture can be trained for any operating point simply by instantiating different $\{C, \pi\}$ in the loss. This enables practitioners to directly target either user convenience (lower misses) or spoofing robustness (lower false accepts) in accordance with their deployment risk profile.

5. Integration with SASV Architectures

The trainable a-DCF loss is integrated after all back-end fusion and calibration operations, acting on the final fused decision score. In typical SASV systems:

• The feature encoders (ASV and CM) remain frozen, and only the fusion/calibration back-end is updated.
• All back-end parameters, including re-weighting, non-linear fusion, and threshold $\tau$, are amenable to optimization.
• The approach supports a variety of fusion mechanisms, including non-linear score fusion schemes (Kurnaz et al., 2 Feb 2026).

For example, in the WildSpoof challenge setting, the loss is back-propagated through the entire differentiable graph—including through sigmoid surrogates in $\mathcal{L}_{\rm a\!-\!DCF}^{\rm soft}$ and through the fusion transform. The WildSpoof paper reports best performance with this method when pretrained feature encoders are frozen (Kurnaz et al., 2 Feb 2026).

6. Empirical Performance and Comparative Results

Direct optimization for operating-condition-dependent a-DCF, with joint BCE regularization and dynamic thresholding, yields measurable gains over BCE-only or static a-DCF baselines. In the ASVspoof2019 LA scenario:

System	Dev	Eval
BCE Only (S1)	0.1234	0.1445
Soft a-DCF (S2, $\tau$ fixed)	0.1355	0.2352
Soft a-DCF + BCE (S3, $\tau$ fixed)	0.1182	0.1398
Full (S4, BCE + threshold search)	0.1109	0.1254

Relative improvements range from 13% (over BCE-only) to 47% (over soft a-DCF-only with $\tau$ fixed) in evaluation a-DCF (Kurnaz et al., 2024). In challenge scenarios, final competitive a-DCF values of 0.0515 (progress) and 0.2163 (final) are achieved (Kurnaz et al., 2 Feb 2026).

7. Implementation Considerations and Practical Guidelines

• Soft-count relaxations are essential for enabling gradient-based training.
• Threshold search per epoch ensures that the loss is minimized at the operationally most relevant boundary.
• Regularization via BCE assists in stabilizing convergence and supporting sufficient class separation.
• Practitioners are advised to match training and evaluation operating points; discrepancy between these can degrade final performance.
• Best results are reported with feature encoders frozen and back-end fusion/calibration components trained on the a-DCF+BCE objective.
• Hyperparameters such as learning rate, batch size, and loss weighting influence convergence speed and stability.

• “Optimizing a-DCF for Spoofing-Robust Speaker Verification” (Kurnaz et al., 2024)
• “Joint Optimization of ASV and CM tasks: BTUEF Team's Submission for WildSpoof Challenge” (Kurnaz et al., 2 Feb 2026)