Fractional Classification Loss (FCL) Overview

• Fractional Classification Loss (FCL) is a robust loss function designed for noisy-label learning by interpolating between cross-entropy and MAE through a learnable fractional derivative order.
• It utilizes an active-passive loss framework to balance fast convergence with robustness, automatically adjusting the parameter μ based on dataset noise levels.
• Empirical results show FCL consistently outperforms traditional losses under high noise conditions, eliminating manual hyperparameter tuning while maintaining competitive accuracy.

Searching arXiv for papers on “Fractional Classification Loss” and closely related formulations. Fractional Classification Loss (FCL) is an overloaded designation that refers to several distinct objective constructions in classification research rather than a single canonical loss. In the contemporary deep-learning literature, the most explicit use of the term denotes a robust loss for learning under noisy labels, built by combining a fractional-derivative form of Cross-Entropy (CE) with Mean Absolute Error (MAE) inside the active-passive loss framework (Kurucu et al., 8 Aug 2025). In earlier work on binary classification metrics, closely related “fractional” terminology refers not to a pointwise classification loss but to a calibrated surrogate utility for directly optimizing linear-fractional performance measures such as the $F_\beta$-measure and Jaccard index (Bao et al., 2019). The acronym FCL is also used for other concepts, including Focal Calibration Loss for probability calibration (Liang et al., 2024) and focal contrastive loss in supervised contrastive learning (Vito et al., 2022). This terminological heterogeneity makes precise contextualization essential.

1. Terminological scope and competing meanings

The most direct modern formulation of Fractional Classification Loss appears in "Introducing Fractional Classification Loss for Robust Learning with Noisy Labels" (Kurucu et al., 8 Aug 2025). There, FCL is presented as a robust classification objective for noisy-label learning, with a learnable fractional order $\mu$ that interpolates between CE-like and MAE-like behavior. Its stated purpose is to balance fast convergence against robustness without requiring dataset-specific manual tuning of robustness hyperparameters.

A distinct use of the phrase appears in "Calibrated Surrogate Maximization of Linear-fractional Utility in Binary Classification" (Bao et al., 2019). In that setting, the “fractional” aspect refers to the target metric family rather than to a fractional derivative. The objective is a surrogate utility for non-decomposable linear-fractional metrics, not a modern additive deep-network loss in the usual per-example sense.

Other uses of the acronym FCL are semantically unrelated. "Calibrating Deep Neural Network using Euclidean Distance" defines FCL as Focal Calibration Loss, a combination of focal loss and a squared Euclidean calibration term (Liang et al., 2024). "An Asymmetric Contrastive Loss for Handling Imbalanced Datasets" uses FCL to denote focal contrastive loss, a focal reweighting of supervised contrastive loss (Vito et al., 2022). By contrast, "PolyLoss: A Polynomial Expansion Perspective of Classification Loss Functions" explicitly notes that its framework is not fractional-order, although it remarks that the expansion “can be generalized to non-integer $j$” while focusing only on integer powers in the paper (Leng et al., 2022).

This multiplicity of meanings implies that “FCL” must be interpreted relative to the problem class: noisy labels, linear-fractional metric optimization, calibration, or contrastive representation learning.

2. Fractional Classification Loss for noisy-label learning

In the noisy-label formulation, FCL is constructed in the active-passive loss (APL) framework, where an active term promotes the true class and a passive term provides a more noise-tolerant regularization signal (Kurucu et al., 8 Aug 2025). The paper defines the generic APL form as

$L_{\mathrm{APL} = \alpha\,L_{\mathrm{Active} + \beta\,L_{\mathrm{Passive},$

and then instantiates FCL without separate hand-tuned coefficients: $$\ell_{FCL}^{\mu}(f(x), y) := \ell_{FCE}^{\mu}(f(x),y) + \ell_{MAE}(f(x),y).$$

For a $K$-class classifier with softmax output

$$p(k|x)=\frac{\exp(f_k(x))}{\sum_{j=1}^K \exp(f_j(x))},$$

and one-hot target distribution $q(k|x)$, CE is given by

$L_{CE}(f(x), y)= -\sum_{k=1}^K q(k|x)\log p(k|x).$

The active term uses a fractional derivative with respect to the negative log-likelihood

$u(p(k|x)) = -\log p(k|x),$

yielding the Fractional Cross-Entropy term

$\mu$0

The full loss is then

$\mu$1

The central parameter is the fractional derivative order $\mu$2. The paper identifies two anchor cases (Kurucu et al., 8 Aug 2025). When $\mu$3,

$\mu$4

which is CE-like and therefore faster but less robust to label noise. When $\mu$5, the fractional CE term becomes constant, so

$\mu$6

which is MAE-like and therefore more robust but slower to converge. Intermediate values produce a continuous interpolation between these regimes.

The paper’s motivating claim is that CE is not robust to label noise, whereas MAE is more robust but often converges slowly, underfits, or may fail to converge well (Kurucu et al., 8 Aug 2025). FCL is designed precisely to occupy the continuum between these behaviors and to learn the appropriate operating point automatically.

3. Learning the fractional order and the robustness–convergence trade-off

A defining feature of the noisy-label FCL formulation is that $\mu$7 is not fixed a priori but optimized jointly with the network parameters $\mu$8 (Kurucu et al., 8 Aug 2025). The empirical risk is written as

$\mu$9

Training proceeds by computing $j$0 and $j$1, updating $j$2 every mini-batch, accumulating gradients for $j$3, and updating $j$4 once per epoch: $j$5

Several implementation details are emphasized (Kurucu et al., 8 Aug 2025). The fractional order is updated once per epoch rather than every mini-batch, it is held fixed for the first 5 epochs, and a larger learning rate is used for $j$6 than for the network weights. Standard optimizers such as Adam or SGD can be used.

The derivative of the fractional CE term with respect to $j$7 is given explicitly: $j$8 This smooth dependence on $j$9 is presented as one reason gradient-based adaptation is feasible (Kurucu et al., 8 Aug 2025).

Conceptually, the paper argues that increasing $L_{\mathrm{APL} = \alpha\,L_{\mathrm{Active} + \beta\,L_{\mathrm{Passive},$0 has two opposing effects. It reduces the log penalty on difficult or mislabeled examples, which improves robustness, but it also raises the penalty on easy, correctly labeled examples, which can reduce model confidence (Kurucu et al., 8 Aug 2025). The loss landscape is therefore reshaped asymmetrically: it becomes flatter for low-probability target predictions and steeper for high-probability target predictions. This trade-off is presented as the mechanism that allows stable learning of $L_{\mathrm{APL} = \alpha\,L_{\mathrm{Active} + \beta\,L_{\mathrm{Passive},$1, because the optimizer can move $L_{\mathrm{APL} = \alpha\,L_{\mathrm{Active} + \beta\,L_{\mathrm{Passive},$2 higher on noisier datasets and lower on cleaner datasets.

A plausible implication is that FCL does not merely down-weight hard examples uniformly; instead, it alters the relative geometry of hard and easy regions in the loss curve. That interpretation is consistent with the paper’s emphasis on a “critical trade-off” between robustness and confidence (Kurucu et al., 8 Aug 2025).

4. Empirical behavior under clean and noisy labels

The noisy-label FCL paper evaluates the method on MNIST, CIFAR-10, and CIFAR-100 under clean data, symmetric label noise at $L_{\mathrm{APL} = \alpha\,L_{\mathrm{Active} + \beta\,L_{\mathrm{Passive},$3, and asymmetric label noise at $L_{\mathrm{APL} = \alpha\,L_{\mathrm{Active} + \beta\,L_{\mathrm{Passive},$4 (Kurucu et al., 8 Aug 2025). Baselines include CE, MAE, GCE, SCE, NCE+MAE, and NCE+RCE.

The paper’s main empirical claims are that FCL is consistently top-performing or near-top across datasets and noise settings, that it is particularly strong under high symmetric noise, and that it eliminates the need for manual hyperparameter tuning while remaining competitive with or better than tuned baselines (Kurucu et al., 8 Aug 2025). Selected reported results include:

Setting	Reported FCL result
MNIST, symmetric $L_{\mathrm{APL} = \alpha\,L_{\mathrm{Active} + \beta\,L_{\mathrm{Passive},$5	99.47%
MNIST, symmetric $L_{\mathrm{APL} = \alpha\,L_{\mathrm{Active} + \beta\,L_{\mathrm{Passive},$6	82.70%
CIFAR-10, symmetric $L_{\mathrm{APL} = \alpha\,L_{\mathrm{Active} + \beta\,L_{\mathrm{Passive},$7	57.32%
CIFAR-100, symmetric $L_{\mathrm{APL} = \alpha\,L_{\mathrm{Active} + \beta\,L_{\mathrm{Passive},$8	45.19%

The ablations further report that mistuned hyperparameters significantly degrade GCE, SCE, and other APL-style baselines, whereas FCL avoids this failure mode because $L_{\mathrm{APL} = \alpha\,L_{\mathrm{Active} + \beta\,L_{\mathrm{Passive},$9 is learned automatically (Kurucu et al., 8 Aug 2025). The learned $$\ell_{FCL}^{\mu}(f(x), y) := \ell_{FCE}^{\mu}(f(x),y) + \ell_{MAE}(f(x),y).$$0 tends toward $$\ell_{FCL}^{\mu}(f(x), y) := \ell_{FCE}^{\mu}(f(x),y) + \ell_{MAE}(f(x),y).$$1 under high noise and toward $$\ell_{FCL}^{\mu}(f(x), y) := \ell_{FCE}^{\mu}(f(x),y) + \ell_{MAE}(f(x),y).$$2 under low or no noise, while $$\ell_{FCL}^{\mu}(f(x), y) := \ell_{FCE}^{\mu}(f(x),y) + \ell_{MAE}(f(x),y).$$3 is recommended as a balanced initialization.

The computational overhead is non-negligible but bounded: the paper reports about $$\ell_{FCL}^{\mu}(f(x), y) := \ell_{FCE}^{\mu}(f(x),y) + \ell_{MAE}(f(x),y).$$4 the training time of CE, MAE, and GCE, and about $$\ell_{FCL}^{\mu}(f(x), y) := \ell_{FCE}^{\mu}(f(x),y) + \ell_{MAE}(f(x),y).$$5 the time of some APL losses, due to the additional optimization of $$\ell_{FCL}^{\mu}(f(x), y) := \ell_{FCE}^{\mu}(f(x),y) + \ell_{MAE}(f(x),y).$$6 (Kurucu et al., 8 Aug 2025).

These results position FCL as an adaptive alternative to fixed robust losses. This suggests that the paper’s principal contribution lies less in the isolated fractional derivative formula than in turning the robustness parameter into an optimization variable.

5. Fractional utility surrogates for non-decomposable metrics

A different research line uses “fractional” in the sense of linear-fractional utilities rather than fractional calculus. In binary classification, the target family is

$$\ell_{FCL}^{\mu}(f(x), y) := \ell_{FCE}^{\mu}(f(x),y) + \ell_{MAE}(f(x),y).$$7

with

$$\ell_{FCL}^{\mu}(f(x), y) := \ell_{FCE}^{\mu}(f(x),y) + \ell_{MAE}(f(x),y).$$8

where $$\ell_{FCL}^{\mu}(f(x), y) := \ell_{FCE}^{\mu}(f(x),y) + \ell_{MAE}(f(x),y).$$9 and $K$0 (Bao et al., 2019).

This family includes the $K$1-measure

$K$2

and the Jaccard index

$K$3

The surrogate construction replaces the 0/1 components with a convex surrogate loss $K$4, defining

$K$5

and hence

$K$6

The key guarantee is the lower bound

$K$7

provided $K$8 for all $K$9 (Bao et al., 2019). The central calibration notion is $$p(k|x)=\frac{\exp(f_k(x))}{\sum_{j=1}^K \exp(f_j(x))},$$0-calibration: if $$p(k|x)=\frac{\exp(f_k(x))}{\sum_{j=1}^K \exp(f_j(x))},$$1, then $$p(k|x)=\frac{\exp(f_k(x))}{\sum_{j=1}^K \exp(f_j(x))},$$2. Sufficient conditions are given via the notion of a $$p(k|x)=\frac{\exp(f_k(x))}{\sum_{j=1}^K \exp(f_j(x))},$$3-discrepant loss: $$p(k|x)=\frac{\exp(f_k(x))}{\sum_{j=1}^K \exp(f_j(x))},$$4

For $$p(k|x)=\frac{\exp(f_k(x))}{\sum_{j=1}^K \exp(f_j(x))},$$5, the paper proves calibration under differentiability almost everywhere, convexity, monotone non-increase, a lower bound on $$p(k|x)=\frac{\exp(f_k(x))}{\sum_{j=1}^K \exp(f_j(x))},$$6, and $$p(k|x)=\frac{\exp(f_k(x))}{\sum_{j=1}^K \exp(f_j(x))},$$7-discrepancy for some $$p(k|x)=\frac{\exp(f_k(x))}{\sum_{j=1}^K \exp(f_j(x))},$$8 (Bao et al., 2019). An analogous result is proved for Jaccard with $$p(k|x)=\frac{\exp(f_k(x))}{\sum_{j=1}^K \exp(f_j(x))},$$9.

Optimization is treated as fractional programming. The surrogate numerator is concave when $q(k|x)$0 is convex, and the full ratio is quasi-concave on the region where the numerator is nonnegative (Bao et al., 2019). The resulting Hybrid Optimization Algorithm first maximizes the numerator until it becomes positive and then performs normalized gradient ascent on the ratio. The paper also proposes a normalized BFGS variant and reports that U-BFGS converges much faster than U-GD, often within about 30 iterations, with particular effectiveness in small-sample settings (Bao et al., 2019).

Although this work does not define a deep-learning FCL in the noisy-label sense, it establishes an important broader meaning of “fractional classification” as direct optimization of ratio-valued evaluation metrics.

6. Related losses and conceptual contrasts

Several nearby loss constructions clarify what Fractional Classification Loss is and is not.

PolyLoss provides a polynomial expansion view of classification loss functions in the basis $q(k|x)$1, defining

$q(k|x)$2

with cross-entropy recovered by $q(k|x)$3 and focal loss expressed as a horizontal shift in the polynomial powers (Leng et al., 2022). The paper explicitly states that this is not a fractional-order formulation, although it remarks that non-integer $q(k|x)$4 is possible in principle. Its practical message is that small perturbations of leading coefficients, especially the Poly-1 variant

$q(k|x)$5

can outperform standard CE and focal loss across tasks (Leng et al., 2022). This provides a coefficient-based alternative to the fractional-order generalization used by noisy-label FCL.

For label-error robustness, "Loss Functions Robust to the Presence of Label Errors" introduces Blurry Loss

$q(k|x)$6

and Piecewise-zero Loss

$q(k|x)$7

to de-weight or ignore hard examples that may be mislabeled (Pellegrino et al., 20 Nov 2025). The paper explicitly notes that it does not use the exact name FCL, though it observes that Blurry Loss may be viewed as a “fractional” or down-weighting alternative to focal loss. Its results show improved label-error detection F1 over CE and often over focal loss on artificially corrupted MNIST and Fashion MNIST, with Blurry Loss generally performing best (Pellegrino et al., 20 Nov 2025). This line shares FCL’s concern with noisy labels but implements robustness through low-$q(k|x)$8 suppression rather than a learnable fractional derivative order.

In multilabel learning, sigmoidF1 defines a smooth fractional surrogate for the F1 score: $q(k|x)$9 The paper explicitly states that any confusion-matrix metric can be formulated with a smooth surrogate and presents sigmoidF1 as a batchwise, differentiable approximation to F1 (Bénédict et al., 2021). It does not use the term FCL, but it is plainly an FCL-type objective in the sense of optimizing a fractional metric directly.

These neighboring approaches show that “fractional” may refer to a fractional derivative, a fractional utility, or a fractional metric expression. The precise mathematical object differs accordingly.

7. Acronym collisions: calibration, contrastive learning, and fractional supervision

The acronym FCL also appears in at least three additional senses.

First, Focal Calibration Loss combines focal loss with a squared Euclidean calibration penalty: $L_{CE}(f(x), y)= -\sum_{k=1}^K q(k|x)\log p(k|x).$0 where

$L_{CE}(f(x), y)= -\sum_{k=1}^K q(k|x)\log p(k|x).$1

(Liang et al., 2024). The paper argues that standard focal loss is not a proper scoring rule and may be miscalibrated, whereas the Euclidean calibration term is strictly proper. The proposed FCL is said to mitigate overconfidence and underconfidence, reduce the post-processing gap relative