# The Norm Must Go On: Dynamic Unsupervised Domain Adaptation by Normalization

M. Jehanzeb Mirza1,2 Jakub Micorek1 Horst Possegger1 Horst Bischof1,2
1Institute for Computer Graphics and Vision, Graz University of Technology.
2Christian Doppler Laboratory for Embedded Machine Learning.
{muhammad.mirza, jakub.micorek, possegger<EMAIL_ADDRESS>
###### Abstract

Domain adaptation is crucial to adapt a learned model to new scenarios, such
as domain shifts or changing data distributions. Current approaches usually
require a large amount of labeled or unlabeled data from the shifted domain.
This can be a hurdle in fields which require continuous dynamic adaptation or
suffer from scarcity of data, _e.g_. autonomous driving in challenging weather
conditions. To address this problem of continuous adaptation to distribution
shifts, we propose Dynamic Unsupervised Adaptation (DUA). By continuously
adapting the statistics of the batch normalization layers we modify the
feature representations of the model. We show that by sequentially adapting a
model with only a fraction of unlabeled data, a strong performance gain can be
achieved. With even less than 1% of _unlabeled data_ from the target domain,
DUA already achieves competitive results to strong baselines. In addition, the
computational overhead is minimal in contrast to previous approaches. Our
approach is simple, yet effective and can be applied to any architecture which
uses batch normalization. We show the utility of DUA by evaluating it on a
variety of domain adaptation datasets and tasks including object recognition,
digit recognition and object detection.

##  1 Introduction

Present day Deep Neural Networks (DNNs) show promising results when both
training and testing data belong to the same distribution [26, 16, 69].
However, if there is a domain shift, _i.e_. when the testing data comes from a
different domain, neural networks struggle to generalize [38, 4, 10]. In fact,
even if there is only a slight distribution shift, the performance of neural
networks is reported to already degrade significantly [18, 48].

(a) CIFAR-10C results.

(b) Detections in foggy weather without (top) and with (bottom) DUA. We
overlay the detection results (blue) and the ground truth (orange).

Figure 1: Exemplary DUA results. a) Mean classification error over 15
different corruption types (at the most severe level 5) on CIFAR-10C [18]. We
outperform the state-of-the-art NORM [55, 42] and TTT [61], while using less
than 1% of unlabeled data from the corrupted test set only. Our proposed
adaptive momentum scheme leads to both fast and stable improvements in
contrast to fixing the momentum parameter $\rho$. b) Qualitative results for
object detection in degrading weather conditions: our DUA (bottom)
significantly improves the performance of a KITTI [11] pre-trained YOLOv3 [49]
on KITTI-Fog [14]. Best viewed in color.

One way to overcome the performance drop during domain shifts is to obtain
labeled data from the shifted domain and re-train the network. However, manual
labeling of large amounts of data imposes significant human and monetary
costs. These issues are addressed by Unsupervised Domain Adaptation (UDA)
approaches, _e.g_. [4, 12, 10, 33, 68, 77, 17, 53, 24, 34]. For UDA, the goal
is to modify the network parameters in such a way that it can adapt in an
unsupervised manner to out-of-distribution testing data. Traditionally, these
approaches require labeled training data along with a large amount of
unlabeled testing data.

In many practical scenarios, the traditional requirements, _i.e_. access to
both labeled training and large amounts of unlabeled testing data can often
not be fulfilled. For example, in the medical domain, pre-trained models are
often provided without access to the training data (which is kept private due
to privacy regulations). Likewise, some application domains benefit from
dynamic adaptation to a changing environment. For example, consider object
detectors for autonomous vehicles, which are usually trained on mostly clear
weather images, _e.g_. [11, 15, 59]. In real-world scenarios, however, weather
can suddenly deteriorate, resulting in significant performance degradation
[41, 40]. In such cases, it is not feasible to obtain labeled training data
captured in degrading weather and re-train the detector from scratch. A better
solution is to dynamically adapt the detector, given only a few (unlabeled)
bad weather examples.

In this work, we show that one hindrance in domain generalization is the
statistical difference in mean and variance between train and (shifted) test
data. Thus, during inference, we adapt the running mean and variance which are
calculated during training by the batch normalization layer [21]. Moreover, we
adapt the statistics dynamically in an online manner on a tiny fraction of
test data. For adaptation, we form a small batch by augmenting each incoming
sample. In order to ensure stable adaptation and fast convergence we propose
an adaptive update schema.

Related approaches [28, 55, 42, 64] typically ignore the training statistics
and recalculate the batch statistics from scratch for the test data. This,
however, requires large batches of test data. We argue that a large batch of
test data might not always be available in real-world applications, _e.g_.
autonomous cars adapting to challenging weather (see Figure 1). We show that a
strong performance gain can be achieved by adapting the running mean and
variance in an online manner (one sample at a time). In particular, we require
only a small number of sequential samples from the out-of-distribution data.

Our contributions can be summarized as follows:

  * •

We show that online adaptation of batch normalization parameters on a tiny
fraction of unlabeled out-of-distribution test data can provide a strong
performance gain. With even less than 1% of unlabeled test data, DUA already
performs competitively to strong baselines which use the entire test set for
adaptation.

  * •

DUA is simple, unsupervised, dynamic and requires no back propagation [51] to
work. Since the computational overhead is also negligible, it is perfectly
suited for real-time applications.

  * •

We evaluate DUA on a variety of domain shift benchmarks, demonstrating its
strong performance. We achieve state-of-the-art results on most benchmarks
while being competitive on the remaining.

  * •

We show that our dynamic adaptation method works on a variety of different
tasks and different architectures. To the best of our knowledge, we are the
first to show dynamic adaptation for object detection.

##  2 Related Work

Unsupervised Domain Adaptation (UDA) has received a significant amount of
interest recently. We summarize these approaches in four categories:
minimizing discrepancy between domains, adversarial approaches, self-
supervised approaches and correcting domain statistics.

Discrepancy reduction between source and target domains is usually performed
at specific network layers or in a contrastive manner. Long et al. [37] match
the mean embeddings from task specific layers. Sun et al. [57, 58] minimize
the second order statistics to align the source and target domains. They apply
a linear transformation on the source domain to align it with the target
domain. Zellinger et al. [75] propose to match higher order moments by
introducing a Central Moment Discrepancy (CMD) metric to learn domain
invariant features. Chen et al. [2] propose to match third and fourth order
statistics of the source and target domains for unsupervised domain
adaptation. On the other hand, [65, 23] use contrastive learning [6] to reduce
discrepancy between domains.

Adversarial discriminative approaches align the features from the source and
target domains mostly by using the domain confusion loss. Ganin et al. [10]
propose a method which is based on the philosophy that predictions must be
made on features which are non-discriminative during training. A novel
gradient reversal layer is proposed which brings the features from the source
and target domains closer by maximizing the domain confusion loss. Tzeng et
al. [63] also rely on maximizing the domain confusion loss for unsupervised
domain adaptation. Hong et al. [19] use a fully convolutional network and use
generative adversarial networks [13] to address the problem of synthetic-to-
real feature alignment. Chen et al. [5] align the global and class wise
features by using a generative adversarial network. Similar approaches for UDA
have also been followed for a variety of tasks, including object detection
[17, 4, 24, 66, 68, 70, 71, 77, 8], object classification [33, 34, 36, 47, 30]
and semantic segmentation [22, 3, 78, 1, 72, 29] for both 2D and 3D data.

Self Supervision has also been used for the purpose of unsupervised domain
adaptation. Sun et al. [60] combine different self supervised auxiliary tasks
for domain adaptation. Sun et al. [61] also propose Test Time Training (TTT)
with self supervision. They put forward the idea of removing the self-imposed
condition of a fixed decision boundary at test time. In their work they use
the rotation prediction task [12] as a self supervised task in order to adapt
the network to out-of-distribution test data.

Correcting the domain statistics calculated by the batch normalization layer
[21] has also been used for UDA. Li et al. [28] propose Adaptive Batch
Normalization where they show that recalculating the batch normalization
parameters from scratch for the test set can improve generalization of DNNs.
Carlucci et al. [39] propose domain adaptation layers which can learn a
hyperparameter during training to find the optimal mixing of statistics from
the source and target domain. Singh et al. [56] study the effect of lower
batch sizes during training and show that DNNs using batch normalization
layers are effected by lower batch sizes. They propose an auxiliary loss for
remedy. Similarly, [42, 55, 76] also show that recalculation of batch
normalization statistics from scratch for test data can be helpful to address
the problem of distribution shift between source and target domains. Wang et
al. [64] also recalculate the batch normalization statistics for the test
data. Further, they calculate the loss from the entropy of predictions and
adapt the scale and shift parameters of batch normalization layers. It is
important to point out that [28, 42, 55, 64] share the same philosophy of a
variable decision boundary at test time as TTT [61].

Our work closely resonates with [28, 42, 55, 64] and is also similar in
philosophy to TTT [61], aiming for a variable decision boundary at test time.
However, we differ from them in several fundamental ways: In [28, 42, 55, 64],
the training statistics are ignored and the batch statistics are recalculated
from the test set. For this reason they require large batches from the test
set. However, in general, large batches of test data might not be available.
Instead, we adapt the statistics calculated from the training data in an
online manner (on each incoming sample). We show competitive results by using
less than 1% of unlabeled test data in contrast to all previous approaches
which use the complete test set. Further, contrary to previous approaches,
such as [64, 61], our method does not require back propagation. Our scenario
is more realistic for dynamic adaptation where we can obtain only a single
test frame at one time.

(a) Source Model

(b) Model Adapted with DUA

Figure 2: Density plots of output distribution for the 64 channels of the last
batch normalization layer of ResNet-26 trained on CIFAR-10. a) Yellow is the
output distribution of the training data. Red is the output distribution of
the shifted test data, _i.e_. corrupted with Contrast Level-5 [18]. The
misalignment of the feature responses is one reason for the performance drop.
b) Yellow is the output distribution of the training data. Blue is the output
distribution of the shifted test data, _i.e_. corrupted with Contrast Level-5,
after adaptation with DUA. DUA aligns the output distribution from corrupted
data closely with the clean (training) distribution. Best viewed in color.

##  3 Approach

We first summarize batch normalization [21] in Section 3.1 as it lies at the
center of our approach. Section 3.2 then details our DUA approach.

###  3.1 Batch Normalization

Ioffe and Szegedy [21] proposed a batch normalization layer which has become
an important component of modern day DNNs. Each batch normalization layer in
the network calculates the mean and variance for each activation coming from
the training data $X$, and normalizes each incoming sample $x$ as

$\hat{x}=\frac{x-{\operatorname{\mathbb{E}}\\!\left[{X}\right]}}{\sqrt{{\operatorname{Var}\\!\left[{X}\right]}+\epsilon}}\cdot\gamma+\beta,$
(1)

where $\gamma$ and $\beta$ are the scale and shift parameters, and $\epsilon$
is used for numerical stability. The expected value
$\operatorname{\mathbb{E}}\\!\left[{X}\right]$ of the training statistics is
estimated through the running mean,

${\hat{\mu}_{k}}=(1-\rho)\cdot{\hat{\mu}_{k-1}}+\rho\cdot{\mu_{k}},$ (2)

and the variance of the training statistics
$\operatorname{Var}\\!\left[{X}\right]$ is estimated through running variance,

${\hat{\sigma}_{k}^{2}}=(1-\rho)\cdot{\hat{\sigma}_{k-1}^{2}}+\rho\cdot\sigma_{k}^{2}.$
(3)

Here, $\hat{\mu}$ and $\hat{\sigma}^{2}$ are the estimated mean and variance
from the training data, whereas $\mu$ and $\sigma^{2}$ represent the mean and
variance of the incoming batch. The hyperparameter $\rho$ is the momentum term
(default $\rho=0.1$) and $k$ denotes each training step. Intuitively, $\rho$
can be thought of as the factor which controls how much the existing estimate
of statistics is affected by the statistics of the incoming batch. A larger
momentum value would essentially give more weight to the calculated statistics
of the incoming batch. Empirically, it has been shown that batch normalization
helps to train faster and also stabilize the training process[54].

The behavior of batch normalization differs during training and testing as
follows:

#### Training:

During training, the batch normalization layer calculates the running mean and
variance over the complete training set. The scale parameter $\gamma$ and
shift parameter $\beta$ from Eq. (1) are learned by back propagation. Running
mean and variance is updated during each forward pass with the new batch
statistics.

#### Testing:

During inference, the running mean and variance of the batch normalization
layer is fixed. Each new sample encountered during testing is normalized by
using the population statistics calculated during training.

###  3.2 Dynamic Unsupervised Adaptation

Let $\Phi_{\text{src}}$ be the network trained solely with source data
$X_{\text{src}}$. Our goal is to adapt the trained model to out-of-
distribution target data $X_{\text{tar}}$ in an unsupervised manner. The batch
normalization layer performs consistently well when train and test data belong
to a similar distribution [16, 69]. However, in many practical scenarios this
is not the case. It has been shown that when out-of-distribution test data is
encountered, batch normalization can hamper the performance significantly [28,
42, 55, 64, 67, 9]. One reason for the performance degradation is the
misalignment of the activation distribution between training and out-of-
distribution test data as shown in Figure 2(a). Thus, our adaptation process
aligns the activation distribution between training and shifted test data as
depicted in Figure 2(b).

In our proposed adaptation schema, all the parameters of the network
$\Phi_{\text{src}}$, other than the running mean and running variance are
fixed. We only adapt the $\operatorname{\mathbb{E}}\\!\left[{X}\right]$ and
$\operatorname{Var}\\!\left[{X}\right]$ from Eq. (1) to the new statistics of
$X_{\text{tar}}$, by using the training statistics obtained from
$X_{\text{src}}$ as a prior. The training statistics are updated by using one
image after the other, _i.e_. processing new examples of the (shifted) test
data in a sequential manner as they arrive. The naïve approach would be to
update the statistics by using Eqs. (2) and (3) with a fixed momentum
parameter $\rho$. However, as shown in Figure 1(a), such fixed momentum leads
to a couple of problems: The adaptation performance is either unstable or
converges rather slowly. This is because with the default parameters the
adaptation of the running mean and variance is highly unstable, as shown in
Figure 3(a). Thus, for stable and fast convergence we adapt the momentum with
each incoming sample. More formally, we update the mean and variance
consecutively:

${\hat{\mu}_{k}}=(1-(\rho_{k}+\zeta))\cdot{\hat{\mu}_{k-1}}+(\rho_{k}+\zeta)\cdot\mu_{k},$
(4)

with

${\hat{\mu}_{0}}={\hat{\mu}_{s}},\quad\rho_{k}=\rho_{k-1}\cdot\omega,\quad\rho_{0}=0.1,$
(5)

and

${\hat{\sigma}^{2}_{k}}=(1-(\rho_{k}+\zeta))\cdot{\hat{\sigma}^{2}_{k-1}}+(\rho_{k}+\zeta)\cdot\sigma^{2}_{k},$
(6)

with

${\hat{\sigma}^{2}_{0}}={\hat{\sigma}^{2}_{s}},\quad\rho_{k}=\rho_{k-1}\cdot\omega,\quad\rho_{0}=0.1.$
(7)

Here, $\omega$ $\in$ $(0,1)$, is the momentum decay parameter, whereas
$\zeta$, with $0\,$<$\,\zeta\,$<$\,\rho_{0}$, is a constant and defines the
lower bound of the momentum. As the momentum $\rho_{k}$ decays, the later
samples will have a smaller impact. Our adaptive momentum scheme has a direct
impact on how the running mean and variance are adapted. The adaptation
becomes stable as compared to default parameters, which is visible in Figure
3(b).

(a) Default Parameters

(b) DUA

Figure 3: Running mean and variance of a single channel from the last batch
normalization layer for different corruptions in CIFAR-10C. a) Running mean
and variance values at each adaptation iteration when default momentum
parameters are used. The values are highly unstable which leads to unstable
adaptation. b) DUA proposes to use an adaptive momentum schema which leads to
fast and stable convergence. This is because of the stability of the running
mean and variance values. Initially, the distributions are far apart and thus,
we want larger update steps (faster assimilation), whereas later on smaller
update steps are beneficial.

Whenever we obtain a new sample from the (shifted) test distribution, we make
a small batch by augmenting the incoming sample. In particular, we use random
horizontal flipping, random cropping and rotation. We take care that the
augmentations we use do not correlate with the shifted test data in our
experiments (_e.g_. we do not augment with any of the corruptions in
CIFAR-10/100C [18]). An example of a batch we use for adaptation is provided
in the supplemental material. Throughout our evaluations, we found that making
a small batch from a single image stabilizes the adaptation process and
improves the results, although this is not strictly necessary for our
adaptation scheme to work. The effect of batches and augmentations is analyzed
in our ablation study in Sec. 5.

| gaus | shot | impul | defcs | gls | mtn | zm | snw | frst | fg | brt | cnt | els | px | jpg | mean
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Source | 67.7 | 63.1 | 69.9 | 55.3 | 56.6 | 42.2 | 50.1 | 31.6 | 46.3 | 39.1 | 17.1 | 74.6 | 34.2 | 57.9 | 31.7 | 49.2
TTT | 45.6 | 41.8 | 50.0 | 21.8 | 46.1 | 23.0 | 23.9 | 29.9 | 30.0 | 25.1 | 12.2 | 23.9 | 22.6 | 47.2 | 27.2 | 31.4
NORM | 44.6 | 43.7 | 49.1 | 29.4 | 45.2 | 26.2 | 26.9 | 25.8 | 27.9 | 23.8 | 18.3 | 34.3 | 29.3 | 37.0 | 32.5 | 32.9
DUA | 34.9 | 32.6 | 42.2 | 18.7 | 40.2 | 24.0 | 18.4 | 23.9 | 24.0 | 20.9 | 12.3 | 27.1 | 27.2 | 26.2 | 28.7 | 26.8
Source | 28.8 | 22.9 | 26.2 |  09.5 | 20.6 | 10.6 |  09.3 | 14.2 | 15.3 | 17.5 |  07.6 | 20.9 | 14.7 | 41.3 | 14.7 | 18.3
TENT | 15.8 | 13.5 | 18.7 |  08.1 | 18.7 |  09.1 |  08.0 | 10.3 | 10.8 | 11.7 |  06.7 | 11.6 | 14.1 | 11.7 | 15.2 | 12.3
DUA | 15.4 | 13.4 | 17.3 |  08.0 | 18.0 |  09.1 |  07.7 | 10.8 | 10.8 | 12.1 |  06.6 | 10.9 | 13.6 | 13.0 | 14.3 | 12.1
Table 1: Top-1 Classification Error (%) for each corruption in CIFAR-10C at the highest severity (Level 5). _Source_ shows the results from the same model trained on the clean train set and tested on the corrupted test set. For a fair comparison with TTT and NORM, we use ResNet-26 (top), while for TENT, we use the WRN-40-2 (bottom) from their official implementation. Smallest error is shown in bold. | gaus | shot | impul | defcs | gls | mtn | zm | snw | frst | fg | brt | cnt | els | px | jpg | mean
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Source | 89.5 | 88.8 | 95.5 | 68.4 | 83.3 | 65.0 | 63.5 | 62.4 | 74.9 | 70.3 | 42.9 | 83.0 | 61.1 | 84.4 | 65.5 | 73.2
TTT | 83.8 | 83.0 | 86.8 | 59.9 | 77.7 | 57.9 | 59.2 | 61.5 | 70.6 | 70.5 | 44.5 | 69.8 | 56.5 | 80.2 | 60.3 | 68.1
NORM | 72.5 | 72.7 | 77.1 | 48.6 | 69.3 | 49.7 | 47.9 | 59.5 | 59.7 | 58.4 | 41.8 | 53.1 | 58.8 | 57.3 | 67.7 | 59.6
DUA | 67.9 | 67.3 | 72.6 | 47.9 | 66.1 | 51.6 | 46.6 | 58.1 | 57.6 | 54.4 | 41.3 | 58.6 | 55.3 | 53.3 | 60.7 | 57.3
Source | 65.7 | 60.1 | 59.1 | 32.0 | 51.0 | 33.6 | 32.4 | 41.4 | 45.2 | 51.4 | 31.6 | 55.5 | 40.3 | 59.7 | 42.4 | 46.7
TENT | 40.3 | 39.9 | 41.8 | 29.8 | 42.3 | 31.0 | 30.0 | 34.5 | 35.2 | 39.5 | 28.0 | 33.9 | 38.4 | 33.4 | 41.4 | 36.0
DUA | 42.2 | 40.9 | 41.0 | 30.5 | 44.8 | 32.2 | 29.9 | 38.9 | 37.2 | 43.6 | 29.5 | 39.2 | 39.0 | 35.3 | 41.2 | 37.6
Table 2: Top-1 Classification Error (%) for each corruption in CIFAR-100C at the highest severity (Level 5). | gaus | shot | impul | defcs | gls | mtn | zm | snw | frst | fg | brt | cnt | els | px | jpg | mean
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Source | 98.4 | 97.7 | 98.4 | 90.6 | 93.4 | 89.8 | 81.8 | 89.5 | 85.0 | 86.3 | 51.1 | 97.2 | 85.3 | 76.9 | 71.7 | 86.2
TTT | 96.9 | 95.5 | 96.5 | 89.9 | 93.2 | 86.5 | 81.5 | 82.9 | 82.1 | 80.0 | 53.0 | 85.6 | 79.1 | 77.2 | 74.7 | 83.6
NORM | 87.1 | 89.6 | 90.5 | 87.6 | 89.4 | 80.0 | 71.9 | 70.6 | 81.5 | 66.9 | 47.8 | 89.8 | 73.5 | 64.2 | 68.5 | 77.3
DUA | 89.4 | 87.6 | 88.1 | 88.0 | 88.6 | 84.7 | 74.3 | 77.8 | 78.4 | 68.6 | 45.6 | 95.9 | 72.2 | 66.5 | 67.4 | 78.2
Table 3: Top-1 Classification Error (%) for each corruption in ImageNet-C at
the highest severity (Level 5). _Source_ refers to results obtained from a
model pre-trained on the original ImageNet and tested on the corrupted test
sets. All results are obtained using a ResNet-18 backbone. Smallest error is
shown in bold.

##  4 Results

In the following, we evaluate DUA on a variety of tasks and benchmarks. First,
we summarize the datasets. Next, we introduce the approaches to which we
compare. Lastly, we present our detailed results.

###  4.1 Benchmarks and Tasks

#### CIFAR-10/100C:

CIFAR-10C and CIFAR-100C [18] are image classification benchmarks to test a
model’s robustness w.r.t. covariate shifts. These benchmarks add different
corruptions to the original test set of CIFAR-10/100 [25] at 5 severity
levels. Following the common protocol [64, 42, 61], we evaluate on 15 types of
corruptions.

#### ImageNet-C:

Similar to the CIFAR-10/100C benchmarks, ImageNet-C [18] is also an image
classification dataset introducing different corruptions at several severity
levels to the original test set of ImageNet [7].

#### KITTI:

To test DUA’s adaptation capabilities on the task of object detection for
autonomous vehicles we use the well-known KITTI [11] dataset. Further, we also
use the KITTI-Rain and KITTI-Fog datasets [14] to test the adaptation
performance of a KITTI pre-trained model in degrading weather.

###  4.2 Baselines

We compare our DUA against the following approaches:

  * •

Source: denotes the results of the corresponding baseline model trained only
on source data, _i.e_. without any adaptation to the test data.

  * •

TTT: Test Time Training (TTT) [61] adapts the network parameters by using an
auxiliary task on each (out-of-distribution) data sample before testing it.

  * •

NORM [55, 42]: ignores the train statistics completely and recalculates the
batch normalization statistics on the entire test set, leveraging larger batch
sizes.

  * •

TENT: Test time entropy minimization (TENT) [64] recalculates the batch
normalization statistics and additionally modifies the scale and shift
parameters ($\gamma$ and $\beta$) of the batch normalization layers through
back propagation. They obtain the gradients by calculating the prediction
entropy on large batches from the out-of-distribution test data.

###  4.3 Experiments

In this section, we provide a description of all the results obtained on
different datasets and benchmarks. We test our DUA during slight distribution
shifts and severe domain shifts as well. For our results we always use less
than 1% of unlabeled test data and adapt on each incoming sample in a
sequential manner (the exact numbers of test samples used for adaptation in
our experiments are listed in the supplemental material). Results for all
other baselines have been obtained by adapting on the complete test set as
stated in their original papers. Note, we also do not need to control for
shuffling of the test set in contrast to other approaches [64, 55, 42]. For
adaptation, unless stated otherwise, we fix the momentum decay parameter
$\omega=0.94$, and the lower bound $\zeta=0.005$. Further details for all the
experiments are provided in the supplemental material. For reproducibility,
code for DUA is available at this repository: https://github.com/jmiemirza/DUA

### CIFAR-10/100C

| Car | Pedestrian | Cyclist
---|---|---|---
Source only | 30.9 | 34.1 | 16.2
DUA | 51.4 | 48.5 | 33.1
Fully Supervised | 71.3 | 64.5 | 63.2
(a)

| Car | Pedestrian | Cyclist
---|---|---|---
Source only | 80.7 | 66.7 | 54.6
DUA | 86.3 | 70.3 | 66.7
Fully Supervised | 92.3 | 76.1 | 78.2
(b)

Table 4: Results for KITTI pre-trained YOLOv3 tested on rain and fog datasets.
We report the Mean Average Precision (mAP@50). a) Results for the most severe
fog level, _i.e_. $30$m visibility. b) Results for the most severe rain level,
_i.e_. $200$mm/hr rain intensity.

For a fair comparison to TTT and NORM we use ResNet-26 [16] and follow the
parametrization in their official implementations111TTT:
https://github.com/yueatsprograms/ttt_cifar_release${}^{\text{,}}$222NORM:
https://github.com/bethgelab/robustness. Similarly, for TENT we use the Wide-
ResNet-40-2 [73] from their official implementation333TENT:
https://github.com/DequanWang/tent.

Table 1 shows the results for the highest severity level on CIFAR-10C. Note
that we achieve a new state-of-the-art. All other approaches use the complete
test set and most also use larger batch sizes and control shuffling of test
data. Results on CIFAR-100C are listed in Table 2. Here, DUA outperforms TTT
and NORM while being competitive with TENT. Results for lower severity levels
are provided in the supplemental material, demonstrating that DUA provides
strong results for less severe corruptions as well.

### ImageNet-C

For evaluations on ImageNet-C we take an off-the-shelf pre-trained ResNet-18
from PyTorch [44]. Table 3 shows the top-1 error for the highest severity
level. DUA performs on-par with all the baselines for ImageNet-C. Results for
lower severity levels are provided in the supplemental.

### Object Detection

We also test our approach for object detection and show considerable
improvement. We conduct these experiments with YOLOv3 [49]. However, our
approach could also be applied to other base architectures such as [50, 31,
35, 62], which use batch normalization.

For evaluating our approach on object detection we consider the two following
scenarios:

  * •

Evaluation during covariate shifts; These evaluations are performed to adapt
to rain and fog conditions.

  * •

Evaluation during domain shifts; These evaluations test for domain adaptation
between datasets.

In degrading weather, a sharp drop in performance of present day object
detectors has been noted [41, 40]. Our goal is to dynamically adapt a detector
trained on clear weather data to degrading weather conditions.

Adaptation results for the most severe forms of fog and rain augmented on
KITTI are shown in Table LABEL:tab:kitti_fog_rain:fog and
LABEL:tab:kitti_fog_rain:rain, respectively. For fog, the mean improvement
across all commonly evaluated classes (_i.e_. car, pedestrian and cyclist)
over the source model is 17.7% mAP. Similarly, we also achieve notable
improvements while adapting to rain. Here, the mean improvement over the
source model is 7.1% mAP. Additional results for varying severity of fog and
rain are provided in the supplemental material.

### Additional Results

We also demonstrate the benefits of DUA on several other datasets and
adaptation tasks in the supplemental material. In particular, we evaluate DUA
on:

#### Digit Recognition:

DUA can successfully be used for domain adaptation across datasets which we
demonstrate for the task of digit recognition. In particular, we use MNIST
[27] and USPS [20], which are datasets consisting of handwritten digits.
Additionally, we use SVHN [43], a dataset containing house numbers obtained
from Google street view images.

#### Office-31

[52]: is a visual domain adaptation dataset for object classification,
containing 31 categories of common objects found in an office environment,
captured in three different settings. These include images captured by Webcam,
DSLR and gathered from Amazon. We test for domain adaptation across all three
settings.

#### VIS-DA:

The Visual Domain Adaptation [45] dataset (VIS-DA) is a large scale image
recognition dataset which contains 12 classes. The training set consists of
synthetically rendered images. The test set consists of real images cropped
from the MS-COCO dataset [32].

#### SODA10M:

The large scale object detection dataset for autonomous vehicles SODA10M [15]
provides data captured during day and night. We test for adaptation from day
to night. Further, we test for domain adaptation between KITTI and SODA10M.

##  5 Ablation Studies

In this section we present detailed ablation studies in order to examine our
approach more closely.

###  5.1 Sample Order Does Not Matter

To understand if the ordering of the incoming samples for adaptation holds any
significance, we run DUA for 300 independent runs (on CIFAR-10C) and randomly
shuffle the test set in each run. The initial, source-only, mean error is
$49.2\,\%$. The largest standard deviation over the 300 runs occurs after
adaptation on 5 samples, where we achieve $36.4\pm 0.4$. Already after 25
samples, we achieve $28.3\pm 0.19$. The performance saturates after 100
samples and we achieve $27.2\pm 0.09$ (a detailed plot is provided in
supplemental material). Thus, the performance of DUA is stable across all
independent runs with very little deviation. These results are important to
understand that DUA can be used with any arrangement of incoming data.

###  5.2 Continuous Dynamic Adaptation

In order to understand how good can DUA handle the real-life scenarios where
different weather conditions can occur interchangeably, we test DUA for such a
scenario on KITTI-Fog: day$\to$fog$\to$day$\to$_etc._ Results are shown in
Fig. 4. Note the only minor drop on the source domain (_e.g_. 5.8% worse than
the _day baseline_ at iteration 100) and that we can quickly recover the same
performance again. This shows that despite not being an incremental learning
approach, DUA still remembers information from the previous domain. We
conjecture that this is because learned weights are not changed during
adaptation. DUA only shifts the activation distributions at test time.

Figure 4: Dynamic adaptation scenario for DUA. We let a KITTI-pretrained model
adapt to fog and then back to the original KITTI dataset for two cycles in
order to show how well DUA can dynamically adapt to changing weather
conditions.

###  5.3 Ablating Batch Normalization Layers

We investigate the effect of adapting only selected batch normalization layers
in Figure 5. For this, we adapt batch normalization layers of specific
ResNet-26 blocks while keeping all others fixed. As can be seen from the plot,
the best performance is obtained by adapting all batch normalization layers in
the architecture. Individual improvements are slightly larger at later batch
normalization layers.

Figure 5: Results on CIFAR-10C after adapting the batch normalization layers
of specific ResNet-26 blocks. ‘All’ refers to adapting all batch normalization
layers. This includes the last batch normalization layer after the three
ResNet blocks.

###  5.4 Effect of Augmentation

As explained in Section 3.2, we form a small batch of augmented versions from
each incoming sample. In Figure 6, we ablate different augmentations and batch
sizes to study their effects. Apart from providing stability to our adaptation
procedure, making a small batch and augmenting it randomly also provides
further improvements. For our experiments, we form a batch of size 64 from
each incoming image by augmenting it. However, even a batch size of only 8
suffices to benefit from DUA (with only minor sacrifice in performance).

Figure 6: Effects of different batch sizes and augmentations on CIFAR-10C with
ResNet-26. DUA makes a batch out of each sequential image by using different
random augmentations. Note that the source-only error (without adaptation) is
49.2%.

##  6 Conclusion

We have shown that even slight distribution shifts between train and test data
can greatly hamper the performance of present day neural networks. We address
this limitation by our DUA, which adapts the statistics of a trained model in
a sequential manner on each unlabeled sample coming from out-of-distribution
test data. To ensure fast and stable adaptation, we introduce an adaptive
momentum scheme. DUA does not require access to training data but only needs a
fraction of test data to achieve competitive results to strong baselines.
Extensive experimentation on a variety of challenging benchmarks and tasks
demonstrate the utility of our method on a broad range of batch normalization-
based architectures. Since we can dynamically adapt to shifting distributions
at a minimal computational overhead, DUA is also well-suited for both real-
time systems and embedded devices.

#### Acknowledgments

We gratefully acknowledge the financial support by the Austrian Federal
Ministry for Digital and Economic Affairs, the National Foundation for
Research, Technology and Development and the Christian Doppler Research
Association. This work was also partially funded by the Austrian Research
Promotion Agency (FFG) under the project High-Scene (884306).

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In the following, we summarize our evaluation details to support
reproducibility (Section A) and provide additional detailed results (Section
B).

##  Appendix A Evaluation Details

#### Number of Samples:

DUA requires only a tiny fraction of the (unlabeled) test set to achieve
competitively strong results. The exact numbers are listed in Table 5. Note
that after adaptation on these specific number of samples, the adaptation
performance always saturates. We conducted all experiments on a single NVIDIA®
GeForce® RTX 3090.

Dataset | Total no. of samples in | No. of samples used | % of samples used
---|---|---|---
test set | for adaptation | for adaptation
CIFAR-10C [18] | 10000 |  080 | 0.8%
CIFAR-100C [18] | 10000 |  080 | 0.8%
ImageNet-C [18] | 50000 | 100 | 0.2%
MNIST [27] | 10000 |  030 | 0.3%
SVHN [43] | 26032 |  030 | 0.1%
USPS [20] |  02007 |  020 | 0.9%
Office-31 [52] |  04110 |  040 | 0.9%
VIS-DA [45] |  05534 |  040 | 0.7%
KITTI [11] |  03741 |  025 | 0.6%
KITTI-Rain [14] |  03741 |  025 | 0.6%
KITTI-Fog [14] |  03741 |  025 | 0.6%
SODA-10M [15] |  04991 |  030 | 0.6%
SODA10M-Day [15] |  03347 |  030 | 0.8%
SODA10M-Night [15] |  01644 |  015 | 0.9%
Table 5: Available number of samples in the test sets _vs._ number of samples
used for adaptation. Note that all datasets except for KITTI have dedicated
test sets. On KITTI, we divide the training set of 7481 images into 3740 train
and 3741 test images.

#### Batch Augmentations:

As evaluated in our ablation study (main manuscript Section 5), augmentations
help to further improve the adaptation performance. In our experiments we use
random cropping, random horizontal flipping and rotating by specific angles
($0,90,180,270$ degrees). Figure 11 shows an exemplary batch of size 64.

#### Corruption Benchmarks:

To train the ResNet-26 [16] backbone on CIFAR-10/100 [25], we use a batch size
of 128. We train the model for 150 epochs and use Stochastic Gradient Descent
(SGD) as optimizer with learning rate 0.1, momentum 0.9 and weight decay
$5\cdot 10^{-4}$. We use the multi-step learning rate scheduler from PyTorch
with milestones at $75$ and $125$ and set its $\gamma=0.1$. For training we
use random cropping and horizontal flipping as augmentations.

#### Domain Adaptation for Classification:

We use an ImageNet-pretrained ResNet-18 from PyTorch and finetune it on the
Office-31 [52] train split. For this finetuning, we train the model for 100
epochs with SGD, initial learning rate $0.1$, momentum $0.9$ and weight decay
$5\cdot 10^{-4}$. We use the PyTorch multi-step learning rate scheduler with
milestones at $60$ and $75$ and set its $\gamma=0.1$. For VIS-DA [45] we use a
ResNet-50, while all other settings are kept the same as for Office-31. We
again use random cropping and horizontal flipping as augmentations during
training.

#### Digit Recognition:

For these experiments we use a simple architecture, consisting of 2
convolution layers, 2 linear layers and 2 batch normalization layers. We also
use 2 dropout layers in the classification head for regularization with
dropout probabilities $p=0.25$ and $p=0.5$, respectively. For each of the
datasets, we train this model using ADADELTA [74] for 15 epochs and use a
batch size of 64.

#### Object Detection:

For all our object detection experiments we use a YOLOv3 [49] pretrained on
MS-COCO [32]. Then, we retrain the model for 100 epochs on each of the
training splits of the respective datasets. We use a batchsize of 20, while
all other optimization settings and augmentation routines for training are
taken from the PyTorch implementation444https://github.com/ultralytics/yolov3
of YOLOv3.

##  Appendix B Additional Ablation Results

###  B.1 Sample Order Does Not Matter

As described in the main manuscript (Section 5), the sample order does not
matter. Here, we include the detailed plot for all the 300 runs in Figure 7,
which shows that DUA is consistently stable across all runs.

Figure 7: Adaptation results on CIFAR-10C for 300 runs with $\omega=0.94$,
$\rho_{0}=0.1$. For each run we randomly shuffle the corrupted test sets. We
plot the mean error (on 15 corruptions) and the standard deviation after
adaptation on each incoming sample. Figure 8: Adaptation results for CIFAR-10C
with different values of Momentum Decay Parameter $\omega$. At $\omega=0.94$,
we achieve a good balance between stable adaptation and fast convergence. For
all these experiments $\rho_{0}=0.1$ is used.

###  B.2 Choice of Momentum Decay Parameter

The momentum decay parameter $\omega$ helps to provide stable and fast
adaptation on the incoming samples. The effect of varying $\omega$ values is
analyzed in Figure 8. From this experiment we understand that the adaptation
is unstable for a higher value of $\omega$, while it is slower for a lower
value. Thus, we use $\omega=0.94$ for all reported experiments, which
empirically provides both stable and fast adaptation.

Corruption Type | Abbreviation
---|---
Gaussian Noise | gaus
Shot Noise | shot
Impulse Noise | impul
Defocus Blur | defcs
Glass Blur | gls
Motion Blur | mtn
Zoom Blur | zm
Snow | snw
Frost | frst
Fog | fg
Brightness | brt
Contrast | cnt
Elastic | els
Pixelate | px
JPEG Compression | jpg
Table 6: Abbreviations of corruption types in CIFAR-10/100C and ImageNet-C.

##  Appendix C DUA for Additional Adaptation Tasks

###  C.1 Domain Adaptation for Digit Recognition

Table 7 summarizes results for cross-dataset domain adaptation in digit
recognition. For all these experiments we use a simple architecture consisting
of 2 convolution layers, 2 batch normalization layers and 2 fully connected
layers. Our method improves the results for all the popular domain adaptation
benchmarks. Note that we only use random cropping for digit recognition
experiments.

###  C.2 Domain Adaptation for Visual Recognition

In Table 8, we list the results obtained by testing on two popular domain
adaptation benchmarks, Office-31 and VIS-DA. For VIS-DA we take a ResNet-50,
pre-trained on ImageNet and then fine tune it on the VIS-DA train split. For
the Office-31 dataset, we use an ImageNet pre-trained ResNet-18 which we
finetune on the Office-31 train split.

We show that our method achieves improvements on both Office-31 and VIS-DA
datasets. It should be noted that the purpose of these evaluations is not to
outperform the state-of-the-art methods which specialize on these tasks.
Instead, we show that DUA is applicable to a variety of tasks and
architectures. It can be used as an initial adaptation method before applying
other established domain adaptation methods such as [61, 57, 10, 46].

###  C.3 Corruption Benchmarks

We provide detailed results for the lower severity corruption levels 1–4
(highest severity level 5 is included in the main manuscript) for CIFAR-10C
(Table 9), CIFAR-100C (Table 10) and ImageNet-C (Table 11). Note that the
abbreviations of the different corruption types are summarized in Table 6. As
can be seen from these results, our DUA achieves consistent improvements over
all corruption types and severity levels.

###  C.4 Object Detection

#### Natural Domain Shifts:

Table LABEL:tab:results-soda10m_day_to_night shows the results for adapting a
detector pre-trained on day images only and tested on night images. Table
LABEL:tab:results-kitti_to_soda10m and Table LABEL:tab:results-
soda10m_to_kitti report the domain adaptation results between KITTI and
SODA10M. DUA provides notable gains for all the different scenarios.

#### Degrading Weather:

We provide results for the lower severity levels of KITTI-Fog and KITTI-Rain
[14] in Table 13 (highest severity/lowest visibility is included in the main
manuscript). We see that by adapting a model with DUA, the detection
performance increases even in lower severities of rain and fog.

Source data: |  SVHN [43] | SVHN |  MNIST [27] | MNIST |  USPS [20] | USPS | Exemplary Images
---|---|---|---|---|---|---|---
Target data: | USPS | MNIST | USPS | SVHN | MNIST | SVHN | MNIST | SVHN | USPS
Source only | 66 | 58 | 78 | 16 | 42 |  09 |  |  |
DUA | 71 | 68 | 86 | 33 | 55 | 28
Fully Supervised | 94 | 96 | 95 | 85 | 97 | 91
Table 7: Domain adaptation results (measured by accuracy in %) for digit
recognition. All results are obtained by using a simple architecture
consisting of 2 convolution layers, 2 fully connected layers and batch
normalization layers.

Source data: | Amazon | Amazon | DSLR | DSLR | Webcam | Webcam
---|---|---|---|---|---|---
Target data: | Webcam | DSLR | Webcam | Amazon | DSLR | Amazon
Source only | 38.4 | 34.3 | 12.9 | 61.4 | 4.2 | 66.1
DUA | 33.2 | 29.6 |  05.7 | 51.9 | 2.0 | 59.7
Fully Supervised | 27.1 | 11.3 |  03.9 | 34.1 | 1.2 | 22.3
(a)

| Classification
---|---
| Error (%)
Source only | 57.4
DUA | 47.2
Fully Supervised | 23.7
(b)

Table 8: Domain adaptation results (measured by classification error in %) for a) Office-31 using a ResNet-18 backbone, and for b) VIS-DA using a ResNet-50 backbone. | gaus | shot | impul | defcs | gls | mtn | zm | snw | frst | fg | brt | cnt | els | px | jpg | mean
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Level 4 |
Source | 63.9 | 53.7 | 57.0 | 28.9 | 58.9 | 32.4 | 38.1 | 25.9 | 33.9 | 17.5 | 10.4 | 33.7 | 26.7 | 40.7 | 27.2 | 36.6
TTT | 41.5 | 35.4 | 39.8 | 15.0 | 47.8 | 19.1 | 18.4 | 20.1 | 24.0 | 13.5 | 10.0 | 14.1 | 17.7 | 29.4 | 24.5 | 24.7
NORM | 40.7 | 37.4 | 43.2 | 16.7 | 47.4 | 21.8 | 20.2 | 29.9 | 30.3 | 19.0 | 16.1 | 20.5 | 26.5 | 26.6 | 35.7 | 28.8
DUA | 31.0 | 27.6 | 35.8 | 13.2 | 40.7 | 20.3 | 15.4 | 22.2 | 20.6 | 12.7 | 10.1 | 14.8 | 20.5 | 18.6 | 24.6 | 21.9
Source | 24.1 | 17.1 | 16.4 |  06.6 | 23.5 |  08.4 |  07.4 | 12.2 | 11.5 |  08.3 |  06.2 |  09.2 | 10.6 | 19.4 | 13.1 | 12.9
TENT | 13.8 | 11.7 | 14.3 |  06.7 | 18.6 |  08.2 |  07.1 | 10.6 |  09.7 |  07.5 |  06.1 |  08.4 | 10.9 |  08.5 | 13.2 | 10.3
DUA | 13.7 | 11.8 | 13.5 |  05.9 | 18.3 |  07.6 |  06.6 | 10.3 |  09.0 |  07.4 |  05.8 |  07.2 |  09.9 |  09.3 | 13.0 | 10.0
Level 3 |
Source | 58.0 | 47.5 | 38.5 | 17.7 | 46.2 | 32.8 | 30.6 | 22.7 | 31.8 | 12.6 |  09.5 | 19.3 | 20.7 | 23.7 | 24.7 | 29.1
TTT | 37.2 | 31.6 | 28.6 | 11.5 | 35.8 | 19.1 | 15.8 | 17.8 | 23.3 | 11.0 |  09.1 | 11.6 | 14.3 | 18.9 | 22.3 | 20.5
NORM | 37.8 | 35.1 | 34.7 | 14.1 | 38.2 | 21.7 | 18.2 | 27.5 | 29.0 | 16.6 | 15.2 | 18.6 | 19.6 | 21.1 | 33.3 | 25.4
DUA | 28.3 | 24.6 | 27.0 | 10.4 | 30.7 | 20.2 | 14.4 | 20.4 | 19.3 | 11.0 |  09.2 | 12.3 | 14.6 | 15.1 | 23.1 | 18.7
Source | 20.4 | 14.6 |  09.7 |  05.4 | 12.9 |  08.6 |  06.5 |  09.9 | 11.4 |  06.3 |  05.5 |  07.2 |  07.4 |  09.6 | 12.1 |  09.8
TENT | 12.6 | 10.4 | 10.4 |  06.0 | 12.7 |  08.1 |  06.7 |  09.5 |  09.1 |  06.7 |  05.9 |  07.5 |  08.4 |  07.5 | 12.7 |  08.9
DUA | 12.2 | 10.5 |  09.3 |  05.5 | 11.9 |  07.8 |  06.1 |  09.1 |  09.1 |  06.1 |  05.4 |  06.3 |  07.0 |  07.4 | 11.7 |  08.4
Level 2 |
Source | 43.1 | 27.8 | 29.3 | 10.2 | 49.5 | 23.4 | 22.4 | 26.4 | 21.3 | 10.3 |  08.7 | 13.4 | 14.7 | 17.9 | 22.3 | 22.7
TTT | 28.8 | 20.7 | 23.0 |  09.0 | 36.6 | 15.4 | 13.1 | 20.2 | 16.9 |  09.2 |  08.3 | 10.2 | 12.5 | 14.8 | 19.7 | 17.2
NORM | 31.0 | 25.3 | 28.7 | 13.5 | 38.8 | 18.8 | 16.3 | 27.8 | 23.9 | 15.4 | 14.6 | 17.1 | 18.7 | 19.6 | 30.6 | 22.7
DUA | 22.3 | 16.8 | 22.9 |  09.2 | 30.3 | 16.0 | 12.7 | 21.5 | 15.7 |  09.6 |  08.7 | 11.1 | 12.7 | 13.3 | 20.8 | 16.2
Source | 13.4 |  08.8 |  08.0 |  05.1 | 14.2 |  06.5 |  05.8 |  09.2 |  08.5 |  05.3 |  05.3 |  06.1 |  06.5 |  07.8 | 10.9 |  08.1
TENT | 10.2 |  07.6 |  08.6 |  05.9 | 13.0 |  07.2 |  06.2 |  08.1 |  07.8 |  06.3 |  05.8 |  06.9 |  07.5 |  07.0 | 11.8 |  08.0
DUA | 10.0 |  07.5 |  07.6 |  05.1 | 12.4 |  06.4 |  05.7 |  08.3 |  07.3 |  05.2 |  05.2 |  05.7 |  06.4 |  06.8 | 10.9 |  07.4
Level 1 |
Source | 25.8 | 18.4 | 19.0 |  08.5 | 51.1 | 14.7 | 18.2 | 15.0 | 13.8 |  08.3 |  08.3 |  08.7 | 14.4 | 11.3 | 16.5 | 16.8
TTT | 19.1 | 15.8 | 16.5 |  08.0 | 37.9 | 11.7 | 12.2 | 12.8 | 11.9 |  08.2 |  08.0 |  08.3 | 12.6 | 11.1 | 15.5 | 14.0
NORM | 24.0 | 20.9 | 22.5 | 13.4 | 38.1 | 16.5 | 15.5 | 20.5 | 18.8 | 14.9 | 14.0 | 15.3 | 19.1 | 16.9 | 24.7 | 19.7
DUA | 16.5 | 13.8 | 16.6 |  08.3 | 30.4 | 12.4 | 12.6 | 14.5 | 12.2 |  08.4 |  08.4 |  08.8 | 13.4 | 11.0 | 15.9 | 13.6
Source |  08.7 |  06.5 |  06.2 |  04.9 | 14.1 |  05.5 |  05.9 |  06.4 |  06.5 |  04.9 |  05.0 |  05.0 |  06.9 |  05.8 |  08.7 |  06.7
TENT |  07.5 |  06.9 |  07.2 |  05.7 | 12.4 |  06.2 |  06.3 |  06.8 |  06.5 |  05.9 |  05.7 |  06.0 |  07.9 |  06.5 |  09.1 |  07.1
DUA |  07.3 |  06.2 |  06.2 |  05.1 | 11.9 |  05.5 |  05.8 |  06.2 |  06.1 |  05.1 |  05.1 |  05.1 |  07.0 |  05.8 |  08.5 |  06.5
Table 9: Error (%) for each corruption in CIFAR-10C severity (Level 1–4) is reported. Source refers to results obtained from a model trained on clean train set and tested on corrupted test sets. For a fair comparison with TTT and NORM, ResNet-26 is used. For comparison with TENT we take the Wide-ResNet-40-2 model from their official Github Repository. Lowest error is highlighted for each corruption. | gaus | shot | impul | defcs | gls | mtn | zm | snw | frst | fg | brt | cnt | els | px | jpg | mean
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Level 4 |
Source | 88.3 | 85.3 | 92.8 | 54.3 | 84.8 | 57.3 | 57.2 | 54.9 | 65.9 | 48.6 | 35.8 | 61.1 | 52.9 | 73.0 | 61.0 | 64.9
TTT | 81.6 | 78.3 | 81.1 | 48.6 | 78.7 | 52.5 | 53.4 | 53.8 | 62.8 | 49.5 | 38.5 | 50.3 | 50.2 | 66.2 | 57.2 | 60.2
NORM | 70.5 | 67.6 | 72.1 | 41.1 | 69.9 | 46.1 | 44.5 | 54.7 | 55.2 | 46.7 | 38.8 | 44.1 | 50.8 | 49.9 | 64.2 | 54.4
DUA | 66.0 | 62.9 | 66.4 | 41.3 | 66.5 | 48.2 | 43.0 | 53.5 | 52.8 | 43.2 | 37.6 | 45.7 | 48.2 | 46.6 | 57.6 | 52.0
Source | 60.7 | 51.6 | 47.9 | 27.1 | 54.4 | 30.3 | 28.9 | 37.4 | 39.0 | 35.4 | 27.2 | 35.9 | 34.4 | 39.0 | 40.1 | 39.3
TENT | 38.9 | 36.3 | 36.6 | 27.3 | 42.0 | 28.9 | 28.4 | 34.8 | 32.8 | 32.1 | 26.3 | 29.8 | 33.3 | 29.9 | 38.4 | 33.1
DUA | 43.0 | 39.0 | 37.9 | 26.9 | 44.7 | 29.5 | 28.3 | 36.5 | 34.3 | 33.8 | 26.6 | 31.0 | 33.8 | 31.0 | 38.9 | 34.3
Level 3 |
Source | 86.0 | 81.4 | 83.8 | 42.7 | 78.3 | 57.7 | 52.4 | 53.0 | 64.1 | 40.1 | 33.3 | 49.3 | 46.6 | 54.4 | 58.2 | 58.8
TTT | 79.6 | 74.6 | 69.3 | 42.5 | 73.0 | 53.2 | 49.8 | 51.2 | 61.4 | 42.1 | 36.8 | 43.5 | 45.8 | 52.9 | 55.2 | 55.4
NORM | 67.7 | 64.6 | 62.3 | 60.9 | 63.4 | 42.6 | 52.6 | 54.5 | 41.5 | 37.8 | 37.8 | 41.1 | 44.3 | 45.0 | 61.7 | 51.9
DUA | 63.8 | 60.0 | 57.9 | 37.3 | 58.4 | 48.2 | 41.2 | 50.8 | 52.4 | 39.2 | 35.5 | 41.4 | 41.7 | 42.1 | 55.3 | 48.4
Source | 55.2 | 45.9 | 36.9 | 25.7 | 39.9 | 30.5 | 27.4 | 33.3 | 38.1 | 29.5 | 25.5 | 30.5 | 28.6 | 30.3 | 38.0 | 34.4
TENT | 37.1 | 34.3 | 32.0 | 26.1 | 34.6 | 29.2 | 27.7 | 32.3 | 32.3 | 29.0 | 25.6 | 28.4 | 29.2 | 28.3 | 37.4 | 30.9
DUA | 40.8 | 36.5 | 32.2 | 25.3 | 37.0 | 29.6 | 27.1 | 32.7 | 34.4 | 29.0 | 25.2 | 28.6 | 28.4 | 28.7 | 37.3 | 31.5
Level 2 |
Source | 79.9 | 66.1 | 73.7 | 33.7 | 79.6 | 48.9 | 46.4 | 56.8 | 52.4 | 35.2 | 31.3 | 42.1 | 40.5 | 48.0 | 55.2 | 52.7
TTT | 71.5 | 61.8 | 59.8 | 36.5 | 73.1 | 47.2 | 46.0 | 55.7 | 52.8 | 38.0 | 35.2 | 39.7 | 42.0 | 47.9 | 52.6 | 50.7
NORM | 61.1 | 54.9 | 55.9 | 36.3 | 61.1 | 43.0 | 40.7 | 53.0 | 49.0 | 39.6 | 37.0 | 39.5 | 42.8 | 43.3 | 59.0 | 47.7
DUA | 57.1 | 50.7 | 52.0 | 35.2 | 58.4 | 43.8 | 39.1 | 52.8 | 47.6 | 35.9 | 34.2 | 38.9 | 39.9 | 39.7 | 52.9 | 45.2
Source | 44.6 | 34.5 | 30.7 | 24.3 | 41.5 | 27.7 | 26.2 | 32.7 | 31.8 | 26.8 | 24.4 | 27.5 | 27.9 | 28.0 | 36.5 | 31.0
TENT | 33.3 | 29.5 | 28.8 | 25.7 | 34.9 | 27.4 | 27.0 | 30.6 | 29.6 | 27.0 | 25.4 | 27.3 | 29.1 | 27.6 | 36.3 | 29.3
DUA | 35.9 | 31.2 | 28.8 | 24.1 | 36.9 | 27.2 | 26.0 | 32.2 | 30.9 | 26.1 | 24.0 | 26.6 | 27.9 | 27.6 | 35.9 | 29.4
Level 1 |
Source | 65.0 | 53.4 | 54.2 | 30.3 | 80.1 | 40.2 | 42.9 | 39.6 | 43.0 | 31.0 | 30.2 | 31.8 | 40.3 | 37.0 | 48.1 | 44.5
TTT | 60.4 | 53.0 | 48.0 | 34.7 | 74.0 | 41.3 | 41.3 | 41.5 | 44.2 | 34.6 | 34.4 | 34.8 | 41.8 | 39.4 | 47.0 | 44.7
NORM | 52.9 | 48.8 | 47.7 | 36.2 | 60.6 | 40.1 | 39.5 | 43.9 | 44.0 | 36.8 | 36.3 | 36.7 | 43.2 | 40.6 | 52.0 | 44.0
DUA | 49.6 | 45.5 | 44.4 | 32.5 | 58.0 | 39.4 | 38.3 | 41.7 | 41.9 | 33.2 | 33.1 | 33.5 | 40.8 | 36.0 | 47.0 | 41.0
Source | 34.4 | 29.6 | 26.9 | 23.8 | 42.9 | 25.6 | 26.1 | 26.1 | 27.4 | 24.0 | 23.8 | 24.3 | 28.4 | 25.2 | 32.4 | 28.1
TENT | 29.3 | 27.6 | 26.9 | 25.5 | 34.7 | 26.6 | 26.7 | 27.0 | 27.3 | 25.6 | 25.2 | 26.0 | 29.8 | 26.7 | 32.9 | 27.9
DUA | 31.2 | 28.5 | 26.4 | 23.8 | 36.9 | 25.3 | 25.9 | 26.2 | 27.1 | 24.1 | 23.9 | 23.9 | 28.6 | 25.4 | 32.0 | 27.3
Table 10: Error (%) for each corruption in CIFAR-100C severity (Level 1–4) is reported. Source refers to results obtained from a model trained on clean train set and tested on corrupted test sets. For a fair comparison with TTT and NORM, ResNet-26 is used. For comparison with TENT we take the Wide-ResNet-40-2 model from their official Github Repository. Lowest error is highlighted for each corruption. | gaus | shot | impul | defcs | gls | mtn | zm | snw | frst | fg | brt | cnt | els | px | jpg | mean
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Level 4 |
Source | 93.2 | 94.7 | 94.3 | 84.5 | 89.4 | 85.3 | 77.2 | 83.4 | 79.4 | 72.8 | 44.5 | 88.1 | 63.4 | 71.2 | 58.8 | 78.7
NORM | 84.4 | 77.6 | 87.3 | 86.4 | 88.0 | 76.8 | 70.7 | 76.9 | 70.9 | 56.0 | 37.8 | 64.4 | 53.5 | 58.6 | 57.8 | 69.8
DUA | 78.1 | 82.8 | 80.3 | 82.8 | 83.3 | 78.7 | 69.8 | 76.1 | 74.2 | 59.7 | 40.4 | 87.3 | 55.8 | 61.8 | 54.9 | 71.1
Level 3 |
Source | 80.9 | 82.7 | 82.9 | 74.1 | 85.4 | 73.9 | 71.9 | 73.6 | 77.8 | 66.2 | 39.6 | 65.3 | 51.1 | 56.7 | 49.3 | 68.8
NORM | 65.5 | 69.8 | 63.2 | 71.1 | 79.5 | 63.6 | 58.2 | 68.1 | 65.4 | 55.3 | 39.3 | 55.0 | 41.0 | 51.8 | 50.9 | 59.8
DUA | 67.6 | 68.5 | 69.0 | 70.7 | 78.6 | 66.4 | 65.3 | 66.9 | 70.6 | 54.3 | 37.4 | 61.8 | 46.5 | 47.6 | 48.2 | 61.3
Level 2 |
Source | 63.6 | 67.8 | 74.6 | 59.4 | 65.4 | 57.7 | 65.2 | 77.5 | 67.0 | 56.3 | 36.4 | 51.5 | 60.9 | 43.3 | 46.3 | 59.5
NORM | 50.7 | 61.2 | 65.2 | 61.3 | 54.4 | 46.4 | 55.8 | 69.1 | 58.0 | 45.6 | 36.1 | 40.3 | 64.1 | 41.3 | 40.6 | 52.7
DUA | 56.4 | 58.0 | 62.0 | 56.5 | 62.0 | 53.1 | 58.0 | 69.9 | 63.0 | 49.3 | 35.8 | 50.5 | 56.6 | 40.6 | 45.3 | 54.5
Level 1 |
Source | 50.5 | 53.1 | 61.9 | 51.3 | 52.4 | 45.2 | 55.9 | 55.5 | 49.7 | 49.0 | 34.2 | 44.0 | 40.4 | 41.3 | 42.6 | 48.5
NORM | 47.0 | 48.4 | 54.1 | 50.2 | 47.8 | 39.5 | 50.3 | 47.7 | 44.1 | 41.1 | 38.4 | 36.6 | 42.0 | 37.0 | 43.2 | 44.5
DUA | 45.7 | 49.0 | 54.7 | 50.6 | 49.8 | 43.4 | 50.1 | 51.7 | 46.9 | 45.8 | 31.8 | 43.3 | 38.9 | 39.1 | 41.5 | 45.5
Table 11: Error (%) for each corruption in ImageNet-C severity (Level 1–4) is
reported. Source refers to results obtained from a model pre-trained on
ImageNet and tested on corrupted test sets. All results are obtained by using
ResNet-18 as the backbone architecture. Lowest error is highlighted for each
corruption.

| Car | Ped | Cyclist
---|---|---|---
SO | 75.3 | 48.3 | 50.1
DUA | 77.2 | 49.9 | 51.6
FS | 86.9 | 55.7 | 63.2
(a)

| Car | Ped | Cyclist
---|---|---|---
SO | 43.6 | 13.9 | 18.9
DUA | 49.4 | 17.2 | 23.1
FS | 84.8 | 46.1 | 61.2
(b)

| Car | Ped | Cyclist
---|---|---|---
SO | 80.1 | 55.5 | 33.9
DUA | 81.3 | 56.3 | 35.1
FS | 91.8 | 71.3 | 76.5
(c)

Table 12: Domain adaptation results (mAP@50) for object detection with YOLOv3.
a) We train on the SODA10M day split and test on the SODA10M night split. b)
We train on the KITTI train split and test on the SODA10M test split. c) We
train on the SODA10M day split and test on the KITTI validation split (details
in Table 5). SO: source-only, FS: fully supervised.

| Car | Pedestrian | Cyclist
---|---|---|---
40m fog visibility
Source only | 39.6 | 39.3 | 20.2
DUA | 61.3 | 53.5 | 39.9
Fully Supervised | 77.8 | 68.1 | 70.5
50m fog visibility
Source only | 48.2 | 45.9 | 27.3
DUA | 69.6 | 59.7 | 48.4
Fully Supervised | 82.2 | 71.3 | 74.2
(a)

| Car | Pedestrian | Cyclist
---|---|---|---
100 mm/hr rain
Source only | 93.2 | 80.5 | 81.1
DUA | 94.9 | 81.7 | 83.2
Fully Supervised | 95.4 | 83.2 | 84.1
75 mm/hr rain
Source only | 95.5 | 82.6 | 83.6
DUA | 96.1 | 83.2 | 86.0
Fully Supervised | 96.6 | 84.1 | 87.2
(b)

Table 13: Results for KITTI pretrained YOLOv3 tested on rain and fog datasets.
Mean Average Precision (mAP@50) is reported for the three common classes in
KITTI dataset. a) Results for $40$m and $50$m visibility in fog. b) Results
for $100$mm/hr and $75$mm/hr rain intensity. Figure 11: Example of a batch
which we create from a single image by augmenting it randomly. The input
sample is taken from ImageNet-C [18] (Level 5) Gaussian Noise Corruption.


# The stability of a biadditive functional equation via a new direct method

Gang Lu∗ Gang Lu
School of General Education, Guangzhou College of Technology and Business,
Guangzhou 510850, P.R. China<EMAIL_ADDRESS>,  Yang Liu  Yang Liu
Department of Mathematics, Yanbian University, Yanji 133001, P.R. China
<EMAIL_ADDRESS>,  Yuanfeng Jin∗ Yuanfeng Jin
Department of Mathematics, Yanbian University, Yanji 133001, P.R. China
<EMAIL_ADDRESS>and  Qi Liu  Qi Liu
School of Mathematics and Physics, Anqing Normal University, Anqing
246001,P.R. China<EMAIL_ADDRESS>
###### Abstract.

Addressing stability in functional equations is a critical task with broad
implications across mathematics and its applications. In this paper, we
present a novel direct method for proving the stability of the following
equation,

$\displaystyle f(x,y)=\alpha f(f_{1}(x,y))+\beta f(f_{2}(x,y))$

subjecting to certain constraints on the constants $\alpha$ and $\beta$, as
well as the functions $f_{1}(x,y)$ and $f_{2}(x,y)$. We introduce the direct
method to prove the stability of the following functional equation and
inequality,

$\displaystyle\|f(x+y,z-w)+f(x-y,z+w)-2f(x,z)+2f(y,w)\|\leq\|x\|^{p}\|y\|^{p}\|z\|^{p}\|w\|^{p};$
$\displaystyle\begin{split}\;&\|f\left(x+y,z-w\right)+af\left(\frac{x-y}{a},z+w\right)-2f(x,z)+2f(y,w)\|\\\
\;&\leq\left\|\rho\left(f(x+y,z-w)+f(x-y,z+w)-2f(x,z)+2f(y,w)\right)\right\|.\end{split}$

This technique efficiently confirms stability, outperforming traditional proof
methods in simplicity and efficacy. The application of our direct approach to
various equations solidifies its validity, ensuring stable behavior across
different mathematical contexts.

###### Key words and phrases:

Hyers-Ulam stability; Functional Equations, Stability, Direct Method,
Inequality.
∗Corresponding authors<EMAIL_ADDRESS>(G. Lu<EMAIL_ADDRESS>(Y. Jin).

###### 2010 Mathematics Subject Classification:

Primary 39B82, 39B52, 47H10, 46B25

##  1\. Introduction and preliminaries

The concept of investigating the stability of functional equations originated
from a challenge presented by S.M. Ulam regarding the approximations of
homomorphisms between groups (refer to [31]). Hyers [17] provided a partial
resolution to this problem

Subsequently, numerous papers on Cauchy equations emerged, utilizing the
direct approach to demonstrate stability.(see [4, 16, 18, 24, 28]). Indeed,
the importance tool is to discuss the stability of a suitable functional
equation. For some information concerning various functional equations (see
[2, 3, 6, 5, 10, 7, 8, 9, 11, 21, 22, 23, 25, 26, 32]). In[14], Forti provided
a broad methodology that can yield numerous stability outcomes without the
need for repetitive procedures. However, his result cannot be used for each
functional equation. Sikorska [29] built the a direct method (see, e.g., [34,
35]) to improve the approximating constant for the following mappings

$\displaystyle\begin{split}\;&\|f(x)-af(h(x))-bf(-h(x))\|\leq\epsilon(x);\\\
\;&\|f(x)-af(h^{n}(x))-bf(h^{n+1}(x))\|\leq\epsilon(x).\end{split}$

Recently, several mathematicians investigated the problem of Hyers-Ulam
stability of various functional equations (see [1, 12, 15, 19, 20, 27, 29, 30,
33]). The present approach enables us to study several equations not covered
by earlier results.

We introduce the a direct method to obtain the Hyers-Ulam stability result for
a two-variable function equations in Banach spaces.

Throughout the paper, let $\mathbb{N}$ be the set of all positive integers,
$\mathbb{R}_{+}$ denote the set of all positive reals, $Y^{X\times X}$ be the
family of all mappings from $X\times X$ to $Y$. The main theme of the present
work is to analyze the stability of the solution of the following functional
equation

$\displaystyle\|f(x,y)-\alpha f(f_{1}(x,y))-\beta
f(f_{2}(x,y))\|\leq\mu(x,y),$ (1.1)

where $\alpha$ and $\beta$ are real numbers, and mappings $f_{1},f_{2}:X\times
X\rightarrow X,\mu:X\times X\rightarrow[0,\infty)$ are arbitrarily given. In
fact, the above inequality is still more general form. In Section 2, we give
some improvements to the approximations known so far. In Section 3, we find
several applications of the stability results. For the sake of simplicity we
provide our results for functions with values in Banach spaces, but with some
small additional assumptions they can be formulated in complete metric spaces
or other spaces.

##  2\. Main results

######  Theorem 2.1.

Suppose that $X$ is a linear normed space and $Y$ is a Banach space. Let
$f:X\times X\rightarrow Y$ be a mapping satisfying

$\displaystyle\|f(x,y)-\alpha f(f_{1}(x,y))-\beta f(f_{2}(x,y))\|\leq\mu(x,y)$
(2.1)

for all $(x,y)\in X\times X$. If $\alpha$ and $\beta$ are real numbers, and
the mappings $f_{1},f_{2}:X\times X\rightarrow X\times X,\mu:X\times
X\rightarrow[0,\infty)$ are arbitrarily given,

$\displaystyle\sum_{n=0}^{\infty}(\Lambda^{n}\mu)(x,y)=:\mu^{*}(x,y)<\infty$

holds and $\Lambda$ is a linear operator defined by

$\displaystyle(\Lambda\delta)(x,y):=|A|\delta(f_{1}(x,y))+|B|\delta(f_{2}(x,y))$

for $\delta:X\times X\rightarrow[0,\infty)$ and $(x,y)\in X\times X$, then
there exists a unique determined mapping $K:X\times X\rightarrow Y$

$\displaystyle K(x,y)=\alpha K(f_{1}(x,y))+\beta K(f_{2}(x,y)),(x,y)\in
X\times X,$

such that

$\displaystyle\|f(x,y)-K(x,y)\|\leq\mu^{*}(x,y)$ (2.2)

for all $(x,y)\in X\times X$.

###### Proof.

Let $T:Y^{X\times X}\rightarrow Y^{X\times X}$ be an operator satisfying
$(Tf)(x,y))=\alpha f(f_{1}(x,y))+\beta f(f_{2}(x,y))$ in (2.1). Then we get

$\displaystyle\|f(x,y)-(Tf)(x,y))\|\leq\mu(x,y).$ (2.3)

At same time, we can see

$\displaystyle\begin{split}\;&\|(T\xi)(x,y)-(T\zeta)(x,y)\|\\\
\;&\leq|\alpha|\|\xi(f_{1}(x,y))-\zeta(f_{1}(x,y))\|+|\beta|\|\xi(f_{2}(x,y))-\zeta(f_{2}(x,y))\|\end{split}$
(2.4)

for all $\xi,\zeta\in Y^{X\times X},(x,y)\in X\times X$. First, we get by
induction that, for every $n\in\mathbb{N}$,

$\displaystyle\|(T^{n}f)(x,y))-(T^{n+1}f)(x,y)\|\leq(\Lambda^{n}\mu)(x,y),(x,y)\in
X\times X.$ (2.5)

Obviously, from (2.3), the case $n=0$ holds. Now fix $n\in\mathbb{N}$ and
suppose that the inequality (2.5) is valid. Then, using (2.4), for all
$(x,y)\in X\times X$, we have

$\displaystyle\begin{split}\;&\|(T^{n+1}f)(x,y)-(T^{n+2}f)(x,y)\|\\\
\;&\leq|\alpha|\|(T^{n}f)(f_{1}(x,y))-(T^{n+1}f)(f_{1}(x,y))\|+|\beta|\|(T^{n}f)(f_{2}(x,y))-(T^{n+1}f)(f_{2}(x,y))\|\\\
\;&\leq|\alpha|(\Lambda^{n}\mu)(f_{1}(x,y))+|\beta|(\Lambda^{n}\mu)(f_{2}(x,y))=(\Lambda^{n+1}\mu)(x,y).\end{split}$
(2.6)

Thus, we complete the proof of (2.5). For $n,k\in\mathcal{N},k>0$,

$\displaystyle\begin{split}\;&\|(T^{n}f)(x,y)-(T^{n+k}f)(x,y)\|\leq\sum_{i=0}^{k-1}\|(T^{n}f)(x,y)-(T^{n+i+1}f)(x,y)\|\\\
\;&\leq\sum_{i=n}^{n+k-1}(\Lambda^{i}\mu)(x,y)\leq\mu^{*}(x,y),(x,y)\in
X\times X.\end{split}$ (2.7)

From the convergence of the series $\sum(\Lambda^{n}\mu)(x,y)$, for every
$(x,y)\in X\times X$, $\\{(T^{n}f)(x,y)\\}_{n\in\mathcal{N}}$ is a Cauchy
sequence. Since $Y$ is a Banach space, we can define
$\lim_{n\rightarrow\infty}(T^{n}f)(x,y):=\psi(x,y)$. Taking $n=0$ and
$k\rightarrow\infty$ in (2.7), we know that (2.2) holds, and

$\displaystyle\|(T\psi)(x,y)-(T^{n+1}f)(x,y)\|\leq(\Lambda^{n+1})\mu(x,y),n\in\mathcal{N},(x,y)\in
X\times X,$

and thus

$\displaystyle(T\psi)(x,y)=lim_{n\rightarrow\infty}(T^{n+1}\psi)(x,y)=\psi(x,y),(x,y)\in
X\times X.$

In order to prove the uniqueness of $\psi$, suppose that $\psi_{1},\psi_{2}\in
Y^{X\times X}$ are two fixed points of $T$ with
$\|\psi_{i}(x,y)-f(x,y)\|\leq\mu^{*}(x,y)$ for every $(x,y)\in X\times
X,i=1,2.$ We can easily show that

$\displaystyle\|\psi_{1}(x,y)-\psi_{2}(x,y)\|=\|(T^{m}\psi_{1})(x,y)-(T^{m}\psi_{2})(x,y)\|\leq
2\sum_{i=m}^{\infty}(\Lambda^{i}\mu)(x,y),x\in X.$

Thus $\psi_{1}(x,y)=\psi_{2}(x,y)$. ∎

######  Remark 2.2.

We can easily prove

$\displaystyle\left\|f(x,y)-\sum_{i=0}^{n}a_{i}(f(f_{i}(x,y)))\right\|\leq\mu(x)$

for all $(x,y)\in X\times X$. Thus there exists a unique determined mapping
$K:X\rightarrow Y$

$\displaystyle K(x,y)=\sum_{i=0}^{n}a_{i}K(f_{i}(x,y)),(x,y)\in X\times X,$

such that

$\displaystyle\|f(x,y)-K(x,y)\|\leq\mu^{*}(x,y)$

for all $(x,y)\in X\times X$.

##  3\. Application

In this section, we provide an example of how the various functional equations
can be applied in practice.

###  3.1. General two-variable functional mapping

Let us recall that a mapping $f:X\times X\rightarrow Y$ is biadditive provided

$f(x+y,w)=f(x,w)+f(y,w),\quad f(x,z+w)=f(x,z)+f(x,w)$

for all $x,y,z,w\in X$. El-Fassi [13] investigated the stability of the
following functional equation

$\displaystyle f(x+y,z-w)+f(x-y,z+w)=2f(x,z)-2f(y,w).$

The problem is of profound significance for the stability of functional
equations.

Next, the type of the following function equation can be demonstrated by using
a different method.

$\displaystyle f(x+y,z-w)+f(x-y,z+w)=2f(x,z)-2f(y,w).$

######  Theorem 3.1.

Suppose that $X$ is a linear normed space and $Y$ is a Banach space. Let
$f:X\times X\rightarrow Y$ be a mapping satisfying

$\displaystyle\|f(x+y,z-w)+f(x-y,z+w)-2f(x,z)+2f(y,w)\|\leq\|x\|^{p}\|y\|^{p}\|z\|^{p}\|w\|^{p},$
(3.1)

with $p>3$, for all $x,y,z,w\in X$. If $f:X\times X\rightarrow Y$ is symmetric
(i.e., $f(x,y)=f(y,x)$ for $x,y\in X$), then there is a biadditive mapping $K$
such that

$\displaystyle\|K(x,y)-f(x,y)\|\leq\frac{1}{1-\left(3\left(\frac{3}{5}\right)^{2p}+2\left(\frac{4}{5}\right)^{2p}\right)}{\left(\frac{12}{25}\right)^{p}\|x\|^{2p}\|y\|^{2p}}$

for all $(x,y)\in X\times X$.

###### Proof.

From the inequality (3.1), we get

$\displaystyle\left\|f(x,y)+f\left(\frac{-x}{5},3y\right)+2f\left(\frac{3x}{5},y\right)-2f\left(\frac{2x}{5},2y\right)\right\|\leq\left(\frac{12}{25}\right)^{p}\|x\|^{2p}\|y\|^{2p}$
(3.2)

for all $x,y\in X$.

Consider $T:Y^{X\times X}\rightarrow Y^{X\times X}$ and $\mu:X\times
X\rightarrow\mathbb{R}_{+}$ given as

$\displaystyle
T\xi(x,y)=-\xi\left(\frac{-x}{5},3y\right)+2\xi\left(\frac{3x}{5},y\right)-2\xi\left(\frac{2x}{5},2y\right)$

for all $x,y\in X,\xi\in Y^{X\times X}$ and

$\mu(x,y)={\left(\frac{12}{25}\right)^{p}\|x\|^{2p}\|y\|^{2p}},\quad x\in X.$

The inequality (3.2) becomes

$\displaystyle\|Tf(x,y)-f(x,y)\|\leq\mu(x,y),\forall x,y\in X.$

For every $g,h\in Y^{X\times X}$, $x,y\in X$,

$\displaystyle\begin{split}\;&\|Tg(x,y)-Th(x,y)\|\\\
\;&=\left\|2g\left(\frac{2x}{5},2y\right)-g\left(\frac{-x}{5},3y\right)-2g\left(\frac{3x}{5},y\right)\right.\\\
\;&\left.-2h\left(\frac{2x}{5},2y\right)+h\left(\frac{-x}{5},3y\right)+2h\left(\frac{3x}{5},y\right)\right\|\\\
\;&\leq
2\left\|g\left(\frac{3x}{5},y\right)-h\left(\frac{3x}{5},y\right)\right\|\\\
\;&+\left\|g\left(\frac{-x}{5},3y\right)-h\left(\frac{-x}{5},3y\right)\right\|+2\left\|g\left(\frac{2x}{5},2y\right)-h\left(\frac{2x}{5},2y\right)\right\|.\end{split}$

So $T$ satisfies the inequality (2.4)
with$f_{1}(x,y)=\left(\frac{3x}{5},y\right);f_{2}(x,y)=\left(-\frac{x}{5},3y\right);f_{3}(x,y)=\left(-\frac{2x}{5},2y\right)$
and $\alpha_{1}=\alpha_{3}=2,\alpha_{2}=1$. Next, we define the operator
$\Lambda:\mathds{R}_{+}^{X}\rightarrow\mathds{R}_{+}^{X}$, which is given by

$\displaystyle\Lambda\eta(x,y)=2\eta\left(\frac{3x}{5},y\right)+\eta\left(-\frac{x}{5},3y\right)+2\eta\left(\frac{2x}{5},2y\right)$

for all $x,y\in X$. In particular,

$\displaystyle\begin{split}\;\Lambda\mu(x,y)&=2\mu\left(\frac{3x}{5},y\right)+\mu\left(-\frac{x}{5},3y\right)+2\mu\left(\frac{2x}{5},2y\right)\\\
\;&=\left(3\left(\frac{3}{5}\right)^{2p}+2\left(\frac{4}{5}\right)^{2p}\right)\mu(x).\end{split}$

Since $\Lambda$ is linear, we can get

$\displaystyle\Lambda^{n}\mu(x)=\left(3\left(\frac{3}{5}\right)^{2p}+2\left(\frac{4}{5}\right)^{2p}\right)^{n}\mu(x),x\in
X,n\in\mathds{N}_{0}.$

Since
$\left(3\left(\frac{3}{5}\right)^{2p}+2\left(\frac{4}{5}\right)^{2p}\right)<1$,
the series $\sum_{n=0}^{\infty}\Lambda^{n}\mu(x)$ is convergent for every
$x\in X$ and

$\displaystyle\begin{split}\;&\mu^{*}(x)=\sum_{n=0}^{\infty}\Lambda^{n}\mu(x)=\sum_{n=0}^{\infty}\left(3\left(\frac{3}{5}\right)^{2p}+2\left(\frac{4}{5}\right)^{2p}\right)^{n}\mu(x)\\\
\;&=\frac{1}{1-\left(3\left(\frac{3}{5}\right)^{2p}+2\left(\frac{4}{5}\right)^{2p}\right)}\mu(x)\\\
\;&=\frac{1}{1-\left(3\left(\frac{3}{5}\right)^{2p}+2\left(\frac{4}{5}\right)^{2p}\right)}{\left(\frac{12}{25}\right)^{p}\|x\|^{2p}\|y\|^{2p}},x\in
X.\end{split}$

By Theorem 2.1, there exists a mapping $K:X\rightarrow Y$ such that

$\displaystyle\begin{split}\;&K(x,y)=\lim_{n\rightarrow\infty}T^{n}f(x,y)\\\
\;&K(x,y)=-K\left(\frac{-x}{5},3y\right)+2K\left(\frac{3x}{5},y\right)-2K\left(\frac{2x}{5},2y\right).\end{split}$

By [13, Property 2.2], $K$ satisfies the biadditive functional equation. ∎

###  3.2. The functional inequality

In this subsection, we solve and investigate the biadditive $\rho$-functional
inequality in normed spaces, which as the following form

$\displaystyle\begin{split}\;&\|f\left(x+y,z-w\right)+af\left(\frac{x-y}{a},z+w\right)-2f(x,z)+2f(y,w)\|\\\
\;&\leq\left\|\rho\left(f(x+y,z-w)+f(x-y,z+w)-2f(x,z)+2f(y,w)\right)\right\|.\end{split}$

######  Proposition 3.2.

Suppose that $X$ is a linear normed space, $Y$ is a Banach space, and
$a\in\mathds{R}$, $a\neq 1$. Let $f:X\times X\rightarrow Y$ be a mapping
satisfying

$\displaystyle\begin{split}\;&\|f\left(x+y,z-w\right)+af\left(\frac{x-y}{a},z+w\right)-2f(x,z)+2f(y,w)\|\\\
\;&\leq\left\|\rho\left(f(x+y,z-w)+f(x-y,z+w)-2f(x,z)+2f(y,w)\right)\right\|.\end{split}$
(3.3)

for all $x,y,z,w\in X$. If $\rho<\frac{2}{5}$, then $f:X\times X\rightarrow Y$
is biadditive.

###### Proof.

Suppose that $f:X\times X\rightarrow Y$ satisfies (3.3). Letting $y=x=w=z=0$
in (3.3), we can get $f(0,0)=0$. Similarly, assume that $x=y=w=0$ in (3.3),
$f(0,x)=0$ can be obtained. By the same way, we can have$f(x,0)=0$.

Next, if $y=x,w=0$ in (3.3), then

$\|f(2x,z)-2f(x,z)\|\leq|\rho|\|f(2x,z)-2f(x,z)\|$

for all $x,z\in X$. Thus $f(2x,z)=2f(x,z)$. The same technique is applied to
obtain $af\left(\frac{x}{a},z\right)=f(x,z)$.

Letting $w=0$ in (3.3), we get

$\displaystyle\begin{split}\;&\|f(x+y,z)+f(x-y,z)-2f(x,z)\|\\\
\;&=\left\|f\left(x+y,z\right)+af\left(\frac{x-y}{a},z\right)-2f\left(x,z\right)\right\|\\\
\;&\leq|\rho|\|f(x+y,z)+f(x-y,z)-2f(x,z)\|,\end{split}$

ans so $f(x+y,z)+f(x-y,z)=2f(x,z)$ for all $x,y,z\in X$. Next, we can obtain
that $f:X\times X\rightarrow Y$ is additive in the second variable. Therefore
$f(x,y)$ is biadditive. ∎

######  Theorem 3.3.

Suppose that $X$ is a linear normed space, $Y$ is a Banach space, and
$a,\rho\in\mathds{R}$ with $2\rho\leq|1+a|$. Let $f:X\times X\rightarrow Y$ be
a mapping satisfying

$\displaystyle\begin{split}\;&\left\|f\left(x+y,z-w\right)+af\left(\frac{x-y}{a},z+w\right)-2f(x,z)+2f(y,w)\right\|\\\
\;&\leq\left\|\rho\left(f(x+y,z-w)+f(x-y,z+w)-2f(x,z)+2f(y,w)\right)\right\|\\\
\;&+\left(\|x\|^{r}+\|y\|^{r}+\|z\|^{r}+\|w\|^{r}\right),\end{split}$ (3.4)

for all $x,y,z,w\in X$. If $r>2$, then there exists a unique biadditive
mapping $K:X\rightarrow Y$ such that

$\displaystyle\|f(x,z)-K(x,z)\|\leq\frac{2^{3+r}}{2^{r}-1}\frac{\|x/2\|^{r}+\|z/2\|^{r}}{1-|\rho|},x,z\in
X.$

###### Proof.

Letting $x=y=z=w=0$ in (3.4), we get $f(0,0)=0$. Letting $y=x,w=-z$ in (3.4),
we get

$\displaystyle\begin{split}\;&\|f(2x,2z)+af(0,0)-2f(x,z)+2f(x,-z)\|\\\
\;&\leq\|\rho(f(2x,2z)+f(0,0)-2f(x,z)+2f(x,-z))\|+2\|x\|^{r}+2\|z\|^{r},\end{split}$
(3.5)

for all $x,z\in X$.

It follows from (3.5) that

$\displaystyle\|f(x,z)-2f(x/2,z/2)+2f(x/2,-z/2)\|\leq\frac{2\|x/2\|^{r}+2\|z/2\|^{r}}{1-|\rho|}.$
(3.6)

Consider $T:Y^{X}\rightarrow Y^{X}$ and $\mu:X\times
X\rightarrow\mathbb{R}_{+}$ given as

$\displaystyle Tf(x,z)=2f\left(x/2,z/2\right)-2f\left(x/2,-z/2\right)$ (3.7)

for all $x,z\in X,f\in Y^{X\times X}$ and

$\mu(x,z)=\frac{2\|x/2\|^{r}+2\|z/2\|^{r}}{1-|\rho|},\quad x,z\in X.$

The inequality (3.6) becomes

$\displaystyle\|(Tf)(x,z)-f(x,z)\|\leq\mu(x,z),\forall x,z\in X.$ (3.8)

For every $g,h\in Y^{X\times X}$ and $x,z\in X$,

$\displaystyle\begin{split}\;&\|Tg(x,z)-Th(x,z)\|\\\
\;&=\left\|2g\left(x/2,z/2\right)-2g\left(x/2,-z/2\right)-2h\left(x/2,z/2\right)+2h\left(x/2,-z/2\right)\right\|\\\
\;&\leq
2\left\|g\left(x/2,z/2\right)-h\left(x/2,z/2\right)\right\|+2\left\|g\left(x/2,-z/2\right)-h\left(x/2,-z/2\right)\right\|.\end{split}$
(3.9)

Let$f_{1}(x,z)=(x/2,z/2),f_{2}(x,z)=(x/2,-z/2)$ and $a_{1}(x)=a_{2}(x)=2$ in
Remak 2.2. The operator $\Lambda:\mathds{R}_{+}^{X\times
X}\rightarrow\mathds{R}_{+}^{X\times X}$ is given by

$\displaystyle\Lambda\eta(x,z)=2\eta\left(x/2,z/2\right)+2\eta\left(x/2,-z/2\right)$
(3.10)

for all $x,z\in X$. In particular,

$\displaystyle\begin{split}\;&\Lambda\mu(x,z)=2\mu\left(x/2,z/2\right)+2\mu\left(x/2,-z/2\right)\\\
\;&=2\frac{2\|x/4\|^{r}+2\|z/4\|^{r}}{1-|\rho|}+2\frac{2\|x/4\|^{r}+2\|z/4|^{r}}{1-|\rho|}\\\
\;&=\left(\frac{4}{2^{r}}\right)\mu(x).\end{split}$ (3.11)

Since $\Lambda$ is linear, we can get

$\displaystyle\Lambda^{n}\mu(x,z)=\left(\frac{4}{2^{r}}\right)^{n}\mu(x,z),x,z\in
X,n\in\mathds{N}_{0}.$ (3.12)

Since $\left(\frac{4}{2^{r}}\right)<1$, the series
$\sum_{n=0}^{\infty}\Lambda^{n}\mu(x)$ is convergent for every $x\in X$ and

$\displaystyle\begin{split}\;&\mu^{*}(x)=\sum_{n=0}^{\infty}\Lambda^{n}\mu(x)=\sum_{n=0}^{\infty}\left(\frac{4}{2^{r}}\right)^{n}\mu(x)\\\
\;&=\frac{4}{1-\frac{1}{2^{r}}}\mu(x,z)\\\
\;&=\frac{2^{3+r}}{2^{r}-1}\frac{\|x/2\|^{r}+\|z/2\|^{r}}{1-|\rho|},x,z\in
X.\end{split}$

By Theorem 2.1, there exists a mapping $K:X\rightarrow Y$ such that

$\displaystyle\begin{split}\;&K(x,z)=\lim_{n\rightarrow\infty}T^{n}f(x,z)\\\
\;&K(x,z)=2K\left(\frac{x}{2},\frac{z}{2}\right)-2K\left(\frac{x}{2},-\frac{z}{2}\right)\\\
\;&\|f(x,z)-K(x,z)\|\leq\frac{4}{1-\frac{1}{2^{r}}}\mu(x,z)\\\
\;&=\frac{2^{3+r}}{2^{r}-1}\frac{\|x/2\|^{r}+\|z/2\|^{r}}{1-|\rho|},x,z\in
X.\end{split}$

Next, we prove that $K$ satisfies the biadditive functional equation.
Considering (3.4), we get

$\displaystyle\left\|f(x+y)+f(x-y)-Af\left(ax\right)-Bf\left(by\right)-Df\left(-y\right)\right\|\leq\theta\left(\left\|x\right\|^{p}+\left\|y\right\|^{p}\right)$
(3.13)

for all $x,y\in X$. Then, from the above inequality, we get

$\displaystyle\begin{split}\;&\|f(x+y,z-w)+f(x-y,z+w)-2f(x,z)+2f(y,w)\|\\\
\;&\leq
2\left(\left\|x\right\|^{r}+\left\|z\right\|^{r}\right)+\|y\|^{r}+\|w\|^{r},x,y,z,w\in
X.\end{split}$ (3.14)

Hence

$\displaystyle\begin{split}\;&\|(Tf)(x+y,z-w)+(Tf)(x-y,z+w)-2(Tf)(x,z)+2(Tf)(y,w)\|\\\
\;&\leq\frac{4}{2^{r}}\frac{2\|x\|^{r}+\|y\|^{r}+2\|z\|^{r}+\|w\|^{r}}{1-|\rho|}\end{split}$
(3.15)

and so

$\displaystyle\begin{split}\;&\|J^{n}f(x+y,z-w)+J^{n}f(x-y,z+w)-2J^{n}f(x,z)+2J^{n}f(y,w)\|\\\
\;&\leq(\frac{4}{2^{r}})^{n}\frac{2\|x\|^{r}+\|y\|^{r}+2\|z\|^{r}+\|w\|^{r}}{1-|\rho|}\end{split}$
(3.16)

for all $n\in\mathds{N}_{0}$ and $x,y\in X$. Letting $n\rightarrow\infty$, we
obtain

$\displaystyle K(x+y,z-w)+K(x-y,z+w)-2K(x,z)+2K(y,w)=0,x,y\in X.$ (3.17)

As a result, $K(x,y)$ is a biadditive mapping. ∎

##  4\. Conclusion

In the paper, we examined the stability of the biadditive function equation of
the form

$\displaystyle f(x,y)=\alpha f(f_{1}(x,y))+\beta f(f_{2}(x,y)).$

The employed approach is utilized to examine several functional equations that
involve multiple variables, and stability theorems have been derived. Notably,
this method does not impose any constraints on the parity, domain, or range of
the function under scrutiny, unlike other techniques. We provided two examples
to demonstrate the effectiveness of our approach.

## Declarations

Availablity of data and materials
Not applicable.

Conflict of interest
The authors declare that they have no competing interests.

Fundings
This work was supported by the National Natural Science Foundation of China
(no. 11761074), Project of Jilin Science and Technology Development for
Leading Talent of Science and Technology Innovation in Middle and Young and
Team Project (no. 20200301053RQ), Natural Science Foundation of Jilin Province
(no. YDZJ202101ZYTS136), and Scientifc Research Project of Guangzhou College
of Technology and Business in 2023 (no. KYPY2023012).

Authors’ contributions
The authors equally conceived of the study, participated in its design and
coordination, drafted the manuscript, participated in the sequence alignment,
and read and approved the final manuscript.

Acknowledgements
Not applicable.

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