Ratio-Based Loss Functions

• Ratio-based loss functions are supervised objectives that measure prediction quality via the ratio (e.g., (u(t)+c)/(y+c)), making them inherently scale-aware and ideal for multiplicative error models.
• They encompass a variety of families—such as logarithm-based, relative, and inverse-relative forms—and are engineered to handle differences in over- and under-estimation across applications like pricing, dosing, and signal processing.
• Analytical properties including continuity, differentiability, and convexity are thoroughly examined, with extensions to density ratio estimation and invariance-aware designs enhancing model calibration and robustness.

Ratio-based loss functions are supervised objectives in which prediction quality is evaluated through a ratio rather than a difference. In the general formulation surveyed by Helgerth and Christmann, a loss is ratio-based if there exist a representing function $\ell:(0,\infty)\to[0,\infty)$ with $\ell(1)=0$, a constant $c\ge 0$, and a monotonic, measurable, surjective link $u:\mathbb R\to Y$ such that

$L(x,y,t)=\ell\!\biggl(\frac{u(t)+c}{y+c}\biggr).$

If $c=0$, the loss is strictly ratio-based. This class is motivated by multiplicative error structure and relative error objectives, in contrast to distance-based losses $L(x,y,t)=\psi(y-t)$, which are aligned with additive models such as $y=t+\epsilon$ (Helgerth et al., 7 May 2026).

1. Formal scope and modeling rationale

Ratio-based losses encode the deviation of a prediction from a target through the ratio $(u(t)+c)/(y+c)$, or equivalently through the inverse ratio $(y+c)/(u(t)+c)$ after replacing $\ell(1)=0$0 by $\ell(1)=0$1. In this sense, they are naturally scale-aware: they penalize relative discrepancies and therefore remain meaningful when the operative notion of error is multiplicative rather than additive (Helgerth et al., 7 May 2026).

This modeling distinction is substantive. Distance-based losses measure absolute deviations and are invariant under simultaneous translation $\ell(1)=0$2, $\ell(1)=0$3. Ratio-based losses instead target relative or multiplicative errors and are scale-invariant in the strict form $\ell(1)=0$4. The survey identifies applications involving multiplicative noise and relative-effect scales, including prices, biological measures, inflation, medical dosing, count-models, and dose–response settings. It also notes that one can pass from a ratio-loss to a distance-loss in the log-domain via $\ell(1)=0$5, but that this can obscure the interpretation in the original scale (Helgerth et al., 7 May 2026).

A closely related but distinct use of the term “ratio-based” appears in share-based losses. There, the target and realization are normalized by totals, $\ell(1)=0$6 and $\ell(1)=0$7, and the loss takes the form

$\ell(1)=0$8

Coleman studies these losses alongside level-based losses $\ell(1)=0$9 and proves asymptotic proportionality under decomposable product-form weights and regularity conditions (Coleman, 17 Nov 2025).

2. Canonical constructions and representative families

The survey “Ratio-based Loss Functions” organizes the univariate representing functions $c\ge 0$0 into a broad catalog. Under the common choice $c\ge 0$1, $c\ge 0$2, and $c\ge 0$3, the ratio is $c\ge 0$4, and representative families include logarithmic, relative, inverse-relative, product-relative, max-relative, quantile-style, and robust clipped forms (Helgerth et al., 7 May 2026).

Family	Representative $c\ge 0$5	Note
Logarithm-based	$c\ge 0$6	Symmetric in log-scale
Relative	$c\ge 0$7	Direct relative deviation
Inverse-relative	$c\ge 0$8	Focus on underestimation
LARE / LPRE	$c\ge 0$9, $u:\mathbb R\to Y$0	Joint relative and inverse-relative structure
GRE	$u:\mathbb R\to Y$1	General measurable bivariate construction
Max / pinball-type	$u:\mathbb R\to Y$2	Quantile-style asymmetric variants exist

The same survey also lists Huber-type logarithmic, Huber-type relative, Huber-type inverse-relative, $u:\mathbb R\to Y$3-insensitive, Hampel-type clipping, and smooth robust flattening constructions. These families are not merely stylistic variants; they instantiate different trade-offs among smoothness, robustness, asymmetry, and under- versus over-estimation sensitivity (Helgerth et al., 7 May 2026).

A different ratio construction is developed by Tyralis and Papacharalampous from the index of agreement. For predictions $u:\mathbb R\to Y$4, observations $u:\mathbb R\to Y$5, and sample mean $u:\mathbb R\to Y$6, they study Willmott’s negatively oriented loss

$u:\mathbb R\to Y$7

and propose the Euclidean norm-ratio variant

$u:\mathbb R\to Y$8

The latter is presented as a theoretical improvement because it replaces the denominator of $u:\mathbb R\to Y$9 with the sum of Euclidean distances, better aligning with the underlying geometric intuition (Tyralis et al., 16 Oct 2025).

3. Analytical properties and optimization trade-offs

The basic analytical questions for ratio-based losses are continuity, differentiability, convexity, Lipschitz continuity, and boundedness of the associated risk. Helgerth and Christmann show that if $L(x,y,t)=\ell\!\biggl(\frac{u(t)+c}{y+c}\biggr).$0 and $L(x,y,t)=\ell\!\biggl(\frac{u(t)+c}{y+c}\biggr).$1 are continuous, then $L(x,y,t)=\ell\!\biggl(\frac{u(t)+c}{y+c}\biggr).$2 is continuous uniformly in $L(x,y,t)=\ell\!\biggl(\frac{u(t)+c}{y+c}\biggr).$3. If $L(x,y,t)=\ell\!\biggl(\frac{u(t)+c}{y+c}\biggr).$4 and $L(x,y,t)=\ell\!\biggl(\frac{u(t)+c}{y+c}\biggr).$5 are $L(x,y,t)=\ell\!\biggl(\frac{u(t)+c}{y+c}\biggr).$6, then

$L(x,y,t)=\ell\!\biggl(\frac{u(t)+c}{y+c}\biggr).$7

They also emphasize a frequent misconception: $L(x,y,t)=\ell\!\biggl(\frac{u(t)+c}{y+c}\biggr).$8 convex and $L(x,y,t)=\ell\!\biggl(\frac{u(t)+c}{y+c}\biggr).$9 convex do not in general imply that $c=0$0 is convex. Indeed, for $c=0$1 bounded or $c=0$2 logistic, any nonconstant $c=0$3 cannot be convex. By contrast, in the strictly-ratio case $c=0$4, $c=0$5, $c=0$6, specific constructions do yield convex losses in $c=0$7; explicit examples include

$c=0$8

and

$c=0$9

The same survey derives global Lipschitz conditions when $L(x,y,t)=\psi(y-t)$0 and $L(x,y,t)=\psi(y-t)$1 are globally Lipschitz and $L(x,y,t)=\psi(y-t)$2 with $L(x,y,t)=\psi(y-t)$3, and develops bounded-risk and Nemitski-type conditions ensuring continuity of the risk functional in $L(x,y,t)=\psi(y-t)$4 or $L(x,y,t)=\psi(y-t)$5 norms (Helgerth et al., 7 May 2026).

The index-of-agreement losses $L(x,y,t)=\psi(y-t)$6 and $L(x,y,t)=\psi(y-t)$7 furnish a concrete finite-sample example of these properties. Both are bounded within $L(x,y,t)=\psi(y-t)$8, translation invariant, and scale invariant. For $L(x,y,t)=\psi(y-t)$9, boundedness follows from the Euclidean triangle inequality, while for $y=t+\epsilon$0 it follows from the elementwise triangle inequality followed by the Euclidean norm. The paper also derives a closed-form calibration for the linear model $y=t+\epsilon$1: the unique minimizer of $y=t+\epsilon$2 is

$y=t+\epsilon$3

with minimum loss

$y=t+\epsilon$4

As $y=t+\epsilon$5, parameter estimates from squared error, $y=t+\epsilon$6, and $y=t+\epsilon$7 converge. In hydrologic model calibration with the GR4J rainfall–runoff model on ten gauged French catchments, calibration and validation performance under MSE, $y=t+\epsilon$8, and $y=t+\epsilon$9 differed only marginally, reflecting the high-correlation regime analyzed theoretically (Tyralis et al., 16 Oct 2025).

4. Density-ratio estimation and classification-derived ratio losses

Density-ratio estimation (DRE) is one of the most technically developed settings for ratio-based losses, but conventions vary across papers. One formulation uses $(u(t)+c)/(y+c)$0, while another uses $(u(t)+c)/(y+c)$1; both are treated as the core object of interest and optimized through variational or classification surrogates (Kitazawa, 2024, Kitazawa, 2024).

In the $(u(t)+c)/(y+c)$2-divergence formulation, for a convex generator $(u(t)+c)/(y+c)$3 with $(u(t)+c)/(y+c)$4, the divergence is

$(u(t)+c)/(y+c)$5

and a variational representation based on the convex conjugate $(u(t)+c)/(y+c)$6 yields a standard empirical objective

$(u(t)+c)/(y+c)$7

Sakura and Sugiyama derive upper and lower bounds on $(u(t)+c)/(y+c)$8 for Lipschitz estimators trained by minimizing an $(u(t)+c)/(y+c)$9-divergence loss. Both bounds scale like $(y+c)/(u(t)+c)$0, and the lower bound contains an explicit exponential term depending on $(y+c)/(u(t)+c)$1 when $(y+c)/(u(t)+c)$2, showing that the $(y+c)/(u(t)+c)$3 error can deteriorate exponentially in the KL divergence. Their numerical experiments report that different $(y+c)/(u(t)+c)$4-divergence losses yield almost identical error curves, supporting a “generator-free” view of the bounds (Kitazawa, 2024).

A separate line of work targets the optimization pathologies of specific DRE losses. The $(y+c)/(u(t)+c)$5-divergence loss

$(y+c)/(u(t)+c)$6

is derived from the variational form of $(y+c)/(u(t)+c)$7. For $(y+c)/(u(t)+c)$8, $(y+c)/(u(t)+c)$9 is bounded by $\ell(1)=0$00, mini-batch gradients are unbiased, and the gradient does not vanish at extreme local minima; the paper reports stable convergence for $\ell(1)=0$01, while $\ell(1)=0$02 caused divergence or collapse. At the same time, the experiments show no significant advantage over the KL-divergence loss function in terms of RMSE, suggesting that optimization stability and final ratio accuracy need not coincide (Kitazawa, 2024).

Classifier-based DRE exposes another design axis: the binary loss itself. Zellinger characterizes all composite binary losses whose excess risk equals one half of a prescribed Bregman divergence $\ell(1)=0$03, thereby inverting the usual classifier-to-ratio pipeline. This produces a constructive recipe for loss design, including polynomial and exponential weight families that place increasing emphasis on accurate estimation of large density-ratio values. In average performance across 484 real-world tasks including sensor signals, texts, and images, the resulting novel loss functions outperform related approaches for parameter choice in 11 deep domain adaptation algorithms (Zellinger, 2024).

The frontier extension is quasiprobabilistic DRE. “Quasiprobabilistic Density Ratio Estimation with a Reverse Engineered Classification Loss Function” generalizes classifier-based density-ratio estimation to settings where probability densities can be negative. Its abstract identifies a key defect of most existing loss functions: they implicitly define a relationship between the optimal classifier and the target quasiprobabilistic density ratio which is discontinuous or not surjective. The paper introduces a convex loss function adapted to both probabilistic and quasiprobabilistic density-ratio estimation, defines an extended Sliced-Wasserstein distance compatible with quasiprobability distributions, and reports state-of-the-art results on di-Higgs production in association with jets via gluon-gluon fusion (Drnevich et al., 22 Dec 2025).

5. Invariance-aware and perceptual ratio objectives

A major modern use of ratio-based losses is the removal of nuisance degrees of freedom. In speech enhancement, the scale-invariant signal-to-distortion ratio (SI-SDR) projects the estimate $\ell(1)=0$04 onto the target $\ell(1)=0$05 by

$\ell(1)=0$06

and then measures

$\ell(1)=0$07

Kolbæk et al. compare SI-SDR to time-domain MSE, STFT-amplitude MSE, STOI, ESTOI, and PMSQE. Systems trained with a given loss perform best in their own metric, but SI-SDR yields strong general performance across STOI, PESQ, and SI-SDR itself. The same study also warns that waveform matching performance metrics must be used with caution because they can fail completely, and that the learning rate is a crucial design parameter even for adaptive gradient-based optimizers (Kolbæk et al., 2019).

An analogous best-scale projection appears in gridless direction-of-arrival estimation. For a ground-truth covariance matrix $\ell(1)=0$08 and a predicted Gram matrix $\ell(1)=0$09, the optimal real scaling is

$\ell(1)=0$10

and the scale-invariant loss is

$\ell(1)=0$11

This loss is invariant under $\ell(1)=0$12, flattens the solution set along scale directions, and in practice leads to more stable training with larger learning rates and fewer oscillations. Numerical results show that the scale-invariant loss outperforms its non-invariant Frobenius counterpart, but remains inferior to the subspace loss that is invariant to change of basis. The reported ordering is

$\ell(1)=0$13

which the authors interpret as evidence that greater degrees of invariance are advantageous in deep learning-based gridless DoA estimation (Chen et al., 16 Mar 2025).

Ratio-based design can also be explicitly perceptual. In deep neural network audio watermarking, the Noise-to-Mask Ratio (NMR) loss uses the psychoacoustic masking threshold $\ell(1)=0$14 and the induced noise pattern $\ell(1)=0$15 to define

$\ell(1)=0$16

Training combines this perceptual term with a binary cross-entropy term for message extraction,

$\ell(1)=0$17

The paper reports that models trained with NMR loss generate more transparent watermarks than models trained with conventionally used MSE loss, with both objective quality assessed by PEAQ and subjective quality assessed by a MUSHRA test supporting the same conclusion (Moritz et al., 2024).

6. Asymptotic equivalence, ranking, and contemporary extensions

Share-based ratio losses and level-based losses can become asymptotically interchangeable, but only under explicit structural conditions. Coleman proves that for weighted exponentiated losses with decomposable product-form weights, and under assumptions including finite $\ell(1)=0$18-th moments, bounded and uniformly Cesàro-integrable weights, stable totals, Cesàro limits of weights, and sparse deviations, the averaged level-based and share-based losses satisfy

$\ell(1)=0$19

A direct implication is that numerical and distributive accuracy converge almost surely, and rankings based on level or share losses coincide for large cross-sections. The same paper explicitly cautions that for small $\ell(1)=0$20, differences can be material; finite-sample equivalence should not be assumed (Coleman, 17 Nov 2025).

A modern large-scale extension appears in recommender systems, where Softmax Loss (SL) and Cosine Contrastive Loss (CCL) are analyzed as ratio-based objectives under a common distributionally robust optimization template. The proposed Rényi-divergence-based loss DrRL replaces the divergence constraint with a Cressie–Read or $\ell(1)=0$21-Rényi divergence, recovers SL as $\ell(1)=0$22, and recovers CCL as $\ell(1)=0$23. The worst-case negative distribution acquires polynomial rather than exponential weighting, which is argued to mitigate false-negative sensitivity while preserving a truncation mechanism. Across the datasets Gowalla, AmazonKitchen, AmazonBeauty, and AmazonElectronics, and the backbones MF, LightGCN, and XSimGCL, DrRL is reported to consistently outperform the best SL and CCL variants in Recall@20 and NDCG@20, and to degrade more gracefully under injected false negatives and temporal distribution shift (Zhang et al., 18 Jun 2025).

The current research frontier therefore treats ratio-based losses less as a narrow family of relative-error penalties and more as a design principle: encode the geometry, nuisance invariances, and evaluation asymmetries of the target task directly into the loss. At the same time, the open questions remain those identified by the survey literature: characterize all convex and Lipschitz ratio-based losses under natural conditions; derive representer theorems, learning rates, and statistical robustness for kernel methods using ratio-losses; extend the framework to deep-network settings, bounded and zero-inflated output spaces, and nonconvex or structured penalties; and explore more flexible GRE families and data-dependent tuning functions (Helgerth et al., 7 May 2026).