---
kind: concept
slug: bounding-box-regression-loss
title: Bounding-box regression loss
summary: >-
  The loss that trains a detector's predicted box against its target box, built on IoU with optional penalty terms. Three wiki cards: Inner-IoU swaps the IoU term for the IoU of scaled auxiliary boxes; Edge YOLO uses plain CIoU; a loss survey gives the GIoU formula. Use a penalised IoU loss by default. The Inner-IoU ratio needs tuning, and its gains are single runs.
tags: [object-detection, loss-function]
papers: ["2205.14942", "2307.02694", "2311.02877"]
related: [object-detection, focal-loss, oriented-object-detection, label-assignment]
status: draft
updated_at: 2026-11-01
---

## Overview

A box regression loss scores how far a predicted box is from its target box,
and its gradient moves the prediction during training. The common base is one
minus the intersection over union (IoU) of the two boxes [arxiv:2307.02694
§5.1.3]. Later losses keep that term and add penalties for things IoU ignores:
boxes that do not overlap, centre distance, aspect ratio, and shape or angle
[arxiv:2311.02877 §1] [arxiv:2311.02877 §2.2].

The invariant all variants share is that the loss is zero only when the
predicted and target boxes coincide, and that it decreases as the overlap grows
[arxiv:2311.02877 eq. 1-2].

## Variants

### Penalised IoU losses (GIoU, DIoU, CIoU, EIoU, SIoU)

These keep the IoU term and add a penalty computed on the full boxes. The
penalties cover non-overlapping boxes, centre distance and aspect ratio, width
and height, or angle, distance and shape [arxiv:2311.02877 §1] [arxiv:2311.02877
§2.2]. The Inner-IoU card uses CIoU and SIoU as its baselines [arxiv:2311.02877
§4.2]. Neither the Inner-IoU card nor the survey card gives a DIoU or EIoU
penalty formula, only the names [arxiv:2311.02877 eq. 23-27] [arxiv:2307.02694
§5.1.3].

GIoU's penalty is given in the survey: the part of the smallest box enclosing
both boxes that the union does not cover, divided by that enclosing area
[arxiv:2307.02694 §5.1.4]. The survey lists DIoU and CIoU as detection losses
but gives no formula for them [arxiv:2307.02694 §5.1.4] [arxiv:2307.02694
tab. 2].

### Inner-IoU: IoU of scaled auxiliary boxes

Inner-IoU replaces the IoU term of a penalised loss with the IoU of two
auxiliary boxes. Each auxiliary box has the same centre as its original box,
with width and height multiplied by one shared ratio [arxiv:2311.02877 §3.2]
[arxiv:2311.02877 eq. 15-18]. The penalty terms stay on the full boxes
[arxiv:2311.02877 eq. 23-27]. The method adds no new loss term
[arxiv:2311.02877 §1]. Card: [Inner-IoU](/papers/2311.02877).

The reason given for the ratio is a gradient analysis on one box of size 10 and
auxiliary boxes of sizes 8 and 12 [arxiv:2311.02877 §3.1]. For high-IoU
samples the smaller auxiliary box gives a larger gradient than the real box,
so a ratio below 1 is argued to speed up their regression. For low-IoU samples
the larger auxiliary box gives the larger gradient, so a ratio above 1 is
argued to help them [arxiv:2311.02877 §3.1]. The paper's general range for the
ratio is about 0.5 to 1.5 [arxiv:2311.02877 §3.2].

### Plain CIoU in an edge YOLO detector

Edge YOLO uses CIoU as its box term, in a single-stage YOLOv4-style detector
for in-vehicle hardware [arxiv:2205.14942 §III-B] [arxiv:2205.14942 eq. 5].
The total loss is a plain sum of a class term, a confidence term and the CIoU
term, with no weight on any of them [arxiv:2205.14942 eq. 2]. Card:
[Edge YOLO](/papers/2205.14942).

The paper reports no ablation of its loss. Its speed and accuracy results
compare whole detectors, so they cannot show what the box term contributes
[arxiv:2205.14942 §IV]. Use it as an example of CIoU in a YOLO-style pipeline,
not as evidence that CIoU beats another box loss.

## When to use which

| Situation | Use | Why |
|---|---|---|
| Default, no time to tune a loss | A penalised IoU loss such as CIoU or SIoU | These are the baselines the Inner-IoU paper compares against [arxiv:2311.02877 tab. 1] [arxiv:2311.02877 tab. 2]. No ratio to choose. |
| Single-stage YOLO-style detector on edge hardware | Plain CIoU, as Edge YOLO does | The paper uses CIoU in an unweighted loss sum [arxiv:2205.14942 eq. 2] [arxiv:2205.14942 eq. 5]. It gives no ablation, so there is no evidence here that CIoU beats another box loss [arxiv:2205.14942 §IV]. |
| Need a reference formula for GIoU | The survey's GIoU definition | The survey gives it with the IoU loss [arxiv:2307.02694 §5.1.3] [arxiv:2307.02694 §5.1.4]. It is a survey with no experiments of its own [arxiv:2307.02694 §1], so it gives no measured gains. |
| Most training boxes already overlap well, and you can afford a sweep | Inner variant with ratio below 1 | The paper's gradient argument covers high-IoU samples [arxiv:2311.02877 §3.1]. Its VOC runs used 0.70 to 0.80 [arxiv:2311.02877 tab. 1]. |
| Many boxes overlap poorly, or the objects are very small and the ratio is free to tune | Inner variant with ratio above 1 | The paper's argument covers low-IoU samples [arxiv:2311.02877 §3.1]. Its AI-TOD runs used 1.10 to 1.15 on objects averaging 12.8 px [arxiv:2311.02877 §4.2] [arxiv:2311.02877 tab. 2]. The paper does not tie the ratio to object size, so the size link is the paper's data, not its argument. |
| Loss other than CIoU or SIoU, such as GIoU, DIoU or EIoU | Test before relying on it | The method is written for these losses [arxiv:2311.02877 eq. 23-27], but the paper's experiments use only CIoU and SIoU [arxiv:2311.02877 §4.2]. |
| Tight training budget | Penalised IoU loss first | Each ratio is a separate training run. The paper's VOC runs used 150 epochs [arxiv:2311.02877 §4.2]. The paper reports no extra runtime cost for the inner boxes, and does not measure it [arxiv:2311.02877 §4.2]. |

Two cautions apply to the inner variant. First, the paper gives no rule for
choosing the ratio, and its tables show several values on the test set
[arxiv:2311.02877 §4.2]. Second, the reported gains come from one run per row,
with no variance reported [arxiv:2311.02877 §4.2]. A 0.6-point gain cannot be
judged from the paper alone. Use a held-out split to pick the ratio, and run
several seeds before claiming a gain.

## Implementation sketch

The pseudocode below follows the description in the paper. It does not
reproduce any released code. Its inputs are (x_centre, y_centre, width,
height) boxes, which the paper does not state explicitly [arxiv:2311.02877
eq. 15-18].

```pseudocode
# Inner IoU of a predicted and a target box  [arxiv:2311.02877 eq. 15-21]
# pred, target: (x_centre, y_centre, width, height); ratio > 0

def inner_iou(pred, target, ratio):
    # Auxiliary boxes: same centre, sides multiplied by ratio  [eq. 15-18]
    aux_pred = scale_about_centre(pred, ratio)
    aux_target = scale_about_centre(target, ratio)

    # Overlap area of the two auxiliary boxes  [eq. 19]
    inter = overlap_area(aux_pred, aux_target)

    # Union from the scaled areas: ratio^2 times the sum, minus overlap
    # [eq. 20]
    union = ratio**2 * (area(pred) + area(target)) - inter

    # Inner IoU  [eq. 21]
    return inter / union
```

```pseudocode
# Penalised loss with the inner IoU term  [arxiv:2311.02877 eq. 22-27]
# base_penalty: a penalty computed on the full boxes, e.g. CIoU or SIoU

def inner_penalised_loss(pred, target, ratio, base_penalty):
    # IoU term is replaced by the inner IoU  [eq. 22]
    iou_term = 1 - inner_iou(pred, target, ratio)

    # Penalty stays on the full, unscaled boxes  [eq. 23-27]
    return iou_term + base_penalty(pred, target)
```

```pseudocode
# GIoU loss for two boxes  [arxiv:2307.02694 §5.1.3-§5.1.4]
# pred, target: boxes; the survey does not fix the box format

def giou_loss(pred, target):
    # Intersection and union areas  [§5.1.3]
    inter = overlap_area(pred, target)
    union = area(pred) + area(target) - inter
    iou = inter / union

    # Smallest box enclosing both boxes  [§5.1.4]
    enclosing = enclosing_area(pred, target)

    # 1 - IoU + (enclosing area not covered by the union) / enclosing area
    # [§5.1.4]
    return 1 - iou + (enclosing - union) / enclosing
```

Where the variants differ: the penalty function is the part that changes
between CIoU, SIoU, GIoU and the other penalised losses. The paper's SIoU
penalty has an angle term, a distance term and a shape term, and it has a
shape exponent theta set to 4 in its experiments [arxiv:2311.02877 eq. 9-14]
[arxiv:2311.02877 eq. 13]. The paper gives the range for theta as 2 to 6
[arxiv:2311.02877 §1]. Edge YOLO's CIoU penalty is a centre-distance term over
an enclosing-box size c, plus alpha times an aspect-ratio term v, with alpha
set from v and the IoU [arxiv:2205.14942 eq. 5-7].

Edge cases the paper does not discuss:

- With a ratio below 1, two boxes that overlap a little can have auxiliary
  boxes that do not overlap. The inner IoU is then zero, and the IoU term gives
  no gradient [arxiv:2311.02877 eq. 15-19]. The penalty term still gives a
  gradient for CIoU and SIoU.
- The union must stay positive, which requires a ratio above zero and boxes
  with positive width and height [arxiv:2311.02877 eq. 20-21].
- Edge YOLO does not define the enclosing-box size c in its text, so an
  implementation has to choose it [arxiv:2205.14942 eq. 5].

## Contradictions

Within the Edge YOLO paper, the box loss is named CIoU in its method section
[arxiv:2205.14942 eq. 5], but its experiments section says DIoU
[arxiv:2205.14942 §IV-B]. Its loss equation is an unweighted sum
[arxiv:2205.14942 eq. 2], while the card records that the released config sets
a box-term normalizer of 0.07 [arxiv:2205.14942 eq. 2]. The repository has no
loss source to check this against, so the weight behind the paper's results
is not settled.

Within the Inner-IoU paper, the text says the VOC gains are above 0.5 points
in AP50 and mAP50:95 [arxiv:2311.02877 §4.2]. The AI-TOD table shows one ratio
(1.10) lowering mAP50:95 by 0.17 points [arxiv:2311.02877 tab. 2]. The paper's
own numbers are therefore mixed across datasets.

The survey gives no numeric results, so it does not conflict with the two
experimental papers [arxiv:2307.02694 §1].

## Papers

- [Edge YOLO](/papers/2205.14942), arXiv:2205.14942: a YOLOv4-style edge detector whose box term is CIoU in an unweighted loss sum.
- [Loss Functions and Metrics in Deep Learning](/papers/2307.02694), arXiv:2307.02694: a survey of losses and metrics; gives the IoU and GIoU formulas for detection.
- [Inner-IoU](/papers/2311.02877), arXiv:2311.02877: auxiliary-box IoU term with a ratio in place of the plain IoU term, tested with CIoU and SIoU on VOC and AI-TOD.