# Can a rudderless species survive? 111Keywords: branching Markov chain;
return probability

Rinaldo B. Schinazi 222Department of Mathematics, University of Colorado,
Colorado Springs, CO 80933-7150, USA<EMAIL_ADDRESS>
###### Abstract

Some species of salmon and sea turtle are famously good at finding their birth
place to reproduce after having travelled vast expanses of ocean. In contrast,
imagine now a species (maybe ancestral to the salmon or turtle) which has to
find its birth place to reproduce but has no navigation skills and relies on
chance alone. Would such an imaginary species survive? According to our (very
simple) model it would survive if and only if the probability that a given
individual find its birth place is strictly larger than 1/2.

##  1 The model

Let $S$ be a countable set. For $x$ and $y$ in $S$, let $p(x,y)$ be the
transition probability from $x$ to $y$ for an irreducible discrete time Markov
chain X on $S$. Let $O$ be a fixed site in $S$. We define a branching Markov
chain ${\bf Y}$ as follows. At time $0$, ${\bf Y}$ starts with a single
individual at $O$. At every discrete time, if the individual is at $x$ it
jumps to $y$ with probability $p(x,y)$ (the transition probabilities of X).
Before each step the individual has a probability $1-\alpha$ of dying where
$\alpha$ is a fixed parameter in $(0,1]$. Whenever the individual returns to
$O$ it gives birth to another individual which performs the same dynamics. All
individuals behave independently of each other. The process Y is said to
survive if it has at least one individual somewhere in $S$ at all times. Let
$\beta$ be the probability that the Markov chain X starting at $O$ eventually
returns to $O$. The next result shows that $\beta$ determines whether Y may
survive.

######  Theorem 1.

If $\beta\leq 1/2$ the branching Markov chain Y dies out for all $\alpha$ in
$(0,1)$. If $\beta>1/2$ there exists $\alpha_{c}\in(0,1)$ such that Y has a
positive probability of surviving for $\alpha>\alpha_{c}$ but dies out for
$\alpha\leq\alpha_{c}$.

Our branching Markov chain Y is a generalization of a model recently
introduced by Lebensztayn and Pereira (2023). There, $S=\mathbb{Z}$,
$p(x,x+1)=p$ and $p(x,x-1)=1-p$ where $p$ is a parameter in $[0,1]$. In this
setting the probability of return is known to be $\beta=1-|1-2p|$, see for
instance Grimmett and Stirzaker (2001). Note that $\beta>1/2$ if and only if
$1/4<p<3/4$. By direct computation Lebensztayn and Pereira (2023) proved that
survival is possible if and only if $p$ is in that range. This note was
motivated by the desire to understand their nice result.

As a consequence of our result we see that if the Markov chain X is recurrent
(i.e. $\beta=1$) then survival is always possible for some $\alpha$. On the
other hand if the Markov chain is too transient (i.e. $\beta\leq 1/2$) then
survival is possible for no $\alpha$. For instance, survival is possible for
the simple symmetric random walk on $S=\mathbb{Z}^{d}$ for $d=2$ since this is
a recurrent chain but not possible for $d\geq 3$, McCrea and Whipple (1940)
estimated $\beta$ to be about 0.34 in $d=3$.

Going back to our biological application we can think of $(p(x,y))$ as the
probabilities that an individual uses to pick a direction and as $\alpha$ as a
measure of the leniency of the environment. Whether the species will survive
depends on how likely an individual is to find its birth place in a perfectly
lenient environment (i.e. $\alpha=1$). This in turn depends on $S$ and
$(p(x,y))$. This model may have some biological relevance in that species with
great navigation skills may have evolved from species with poor skills.
However, primitive species had to survive in order to evolve into something
else.

##  2 Proof of Theorem 1

Following Lebensztayn and Pereira (2023) we define a Bienaymé-Galton-Watson
process (BGW in short) Z that keeps track of the genealogy of the process Y.
Let $Z_{0}=1$ and let $Z_{1}$ be the number of returns of the initial
individual to $O$. Since at each return a new individual is born $Z_{1}$ also
counts the number of children of the initial individual. We can think of
$Z_{1}$ as the number of individuals in the first generation. We define
$Z_{2}$ as the number of children born from the first generation (i.e. the
grandchildren of the initial individual) and so on. Since all the individuals
are independent of each other and follow the same dynamics ${\bf Z}$ is indeed
a BGW process. Moreover, the process ${\bf Z}$ survives if and only if the
process Y survives.

Note that the total offspring of one individual is the number of times this
individual returns to $O$ without being killed. Hence, the mean offspring per
individual for the process Z is for $0<\alpha<1$,

$\mu(\alpha)=\sum_{n\geq 1}\alpha^{n}p_{n}(O,O),$ (1)

where $p_{n}(O,O)$ denotes the probability that the Markov chain X starting at
time $0$ at $O$ returns to $O$ at time $n$.

We will need the following well known recurrence criterion, see for instance
Theorem 1.1 in Chapter 5 in Schinazi (2010). An irreducible Markov chain ${\bf
X}$ is recurrent if and only if

$\sum_{n\geq 1}p_{n}(O,O)=+\infty,$ (2)

for some state $O$. We also will need the following result for power series,
see Proposition A 1.9 in Port (1994).

######  Lemma 2.

Assume that $(b_{n})$ is a sequence of positive real numbers such that the
series $\sum_{n\geq 1}b_{n}s^{n}$ converges for all $s$ in $[0,1)$. Then,

$\lim_{s\to 1^{-}}\sum_{n\geq 1}b_{n}s^{n}=\sum_{n\geq 1}b_{n},$

where both sides of the equality may be infinite.

There are two cases to consider. Assume first that the Markov chain ${\bf X}$
is recurrent (i.e. $\beta=1$). Then, by Lemma 2 and (2),

$\lim_{\alpha\to 1^{-}}\mu(\alpha)=\sum_{n\geq 1}p_{n}(O,O)=+\infty.$

Since $\mu$ is continuous on $(0,1)$ and $\lim_{\alpha\to 0}\mu(\alpha)=0$,
there exists $\alpha_{c}$ in $(0,1)$ such that $\mu(\alpha_{c})=1$. Since
$\mu$ is strictly increasing, $\mu(\alpha)>1$ if and only if
$\alpha>\alpha_{c}$. Hence, the process ${\bf Z}$ ( and therefore ${\bf Y}$)
survives with positive probability if and only if $\alpha>\alpha_{c}$. This
proves Theorem 1 in the case $\beta=1$.

Consider now the case when the Markov chain ${\bf X}$ is transient. That is,
the probability $\beta$ to return to $O$ is strictly less than 1. By the
Markov property, the offspring distribution for the branching process Z is for
$\alpha=1$,

$P(Z_{1}=j|Z_{0}=1)=(1-\beta)\beta^{j},$

for $j=0,1,2\dots$. Observe that since $0<\beta<1$ this is a proper
probability distribution (it is not when $\beta=1$). Using this offspring
distribution we get that the mean offspring $\mu(\alpha)$ for $\alpha=1$ is,

$\mu(1)=\frac{\beta}{1-\beta}.$

Note that $\mu(1)>1$ if and only if $\beta>1/2$. Moreover, $\mu(\alpha)$ can
also be expressed using equation (1) for all $\alpha\leq 1$ (including
$\alpha=1$).

If $\beta>1/2$ then $\mu(1)>1$. By Lemma 2 the function $\mu$ is continuous on
$(0,1]$. It is also strictly increasing. Hence, there exists $\alpha_{c}<1$
such that $\mu(\alpha_{c})=1$ and $\mu(\alpha)>1$ if and only if
$\alpha>\alpha_{c}$. That is, the process ${\bf Y}$ survives with positive
probability if and only if $\alpha>\alpha_{c}$.

On the other hand if $\beta\leq 1/2$ then $\mu(1)\leq 1$. Since $\mu$ is an
increasing function, $\mu(\alpha)\leq 1$ for all $\alpha\leq 1$. The process
${\bf Y}$ survives for no value of $\alpha$. This concludes the proof of
Theorem 1.

## References

  * [1] Grimmett, G. R., Stirzaker, D. R. (2001). Probability and Random Processes, 3rd ed. Oxford, NY: Oxford Univ. Press.
  * [2] Lebensztayn, E., Pereira, V. (2023). On Random Walks with Geometric Lifetimes, The American Mathematical Monthly, DOI: 10.1080/00029890.2023.2274783
  * [3] McCrea, W. H., Whipple, F. J. W. (1940). Random Paths in Two and Three Dimensions. Proc. Roy. Soc. Edinburgh 60, 281-298.
  * [4] Port, S.C. (1994) Theoretical probability for applications. Wiley.
  * [5] Schinazi, R.B. (2010). Classical and spatial stochastic processes, second ed. Birkhauser.


# Towards Open-Set Test-Time Adaptation
Utilizing the Wisdom of Crowds in Entropy Minimization

Jungsoo Lee1,2${}^{\text{*}}$ Debasmit Das1 Jaegul Choo2 Sungha Choi1†
1Qualcomm AI Research‡ 2KAIST
1{jungsool, debadas<EMAIL_ADDRESS>2{bebeto<EMAIL_ADDRESS>

###### Abstract

Test-time adaptation (TTA) methods, which generally rely on the model’s
predictions (e.g., entropy minimization) to adapt the source pretrained model
to the unlabeled target domain, suffer from noisy signals originating from 1)
incorrect or 2) open-set predictions. Long-term stable adaptation is hampered
by such noisy signals, so training models without such error accumulation is
crucial for practical TTA. To address these issues, including open-set TTA, we
propose a simple yet effective sample selection method inspired by the
following crucial empirical finding. While entropy minimization compels the
model to increase the probability of its predicted label (i.e., confidence
values), we found that noisy samples rather show decreased confidence values.
To be more specific, entropy minimization attempts to raise the confidence
values of an individual sample’s prediction, but individual confidence values
may rise or fall due to the influence of signals from numerous other
predictions (i.e., wisdom of crowds). Due to this fact, noisy signals
misaligned with such ‘wisdom of crowds’, generally found in the correct
signals, fail to raise the individual confidence values of wrong samples,
despite attempts to increase them. Based on such findings, we filter out the
samples whose confidence values are lower in the adapted model than in the
original model, as they are likely to be noisy. Our method is widely
applicable to existing TTA methods and improves their long-term adaptation
performance in both image classification (e.g., 49.4% reduced error rates with
TENT) and semantic segmentation (e.g., 11.7% gain in mIoU with TENT).

††∗Work done during an internship at Qualcomm AI Research.††† Corresponding
author. ‡ Qualcomm AI Research is an initiative of Qualcomm Technologies, Inc.

##  1 Introduction

Figure 1: Utilizing confidence difference for selecting correct samples.
Pseudo-labeling samples (i.e., selecting correct samples) by using a fixed
threshold does not guarantee a reasonable level of pseudo-labeling
performance, which is demonstrated by the significantly low precision values.
On the other hand, we maintain a reasonable level of both precision and recall
by using the confidence difference between $\theta_{o}$ and $\theta_{a}$,
improving the test-time adaptation performance overall.

Despite the recent advancements of deep learning, models still show a
significant performance degradation when confronted with large domain shifts
(e.g., changes of cities with different landscapes during autonomous driving)
[8, 33, 39, 27, 11]. Among various studies, test-time adaptation (TTA) is at
the center of attention due to its practicality in not requiring 1) the source
data during the adaptation stage and 2) ground truth labels of the target
domain [57].

Figure 2: Description of open-set TTA. While previous studies assume covariate
shifts (i.e., Cityscapes to BDD-100K), they fail to address the semantic
shifts (i.e., guardrails only shown in BDD-100K). This paper addresses both
closed-set and open-set test-time adaptation.

TTA models widely utilize a self-training strategy (e.g., entropy
minimization), which uses the model’s prediction as the target of the loss
function [57, 9, 58, 46, 14, 13, 62, 47, 65, 24]. Since TTA models rely on
their own predictions during the adaptation, they are inevitably prone to
utilizing noisy signals. In this paper, noisy signals indicate supervisions
that originated from 1) incorrect or 2) open-set predictions. Fig.
LABEL:fig:teaser shows that performing adaptation with such noisy signals
significantly degrades the TTA performance. Specifically, the pink pixels
indicate the mispredicted pixels (e.g., predicting sidewalks as roads in the
second row), and the red ones are the predictions on open-set classes that
were not included in the train set (e.g., predicting guardrails and the
garbage truck as roads in the first and second rows, respectively). Such an
example clearly demonstrates that TTA in real-world applications needs to
address such open-set classes since mispredicting guardrails as roads may
cause serious accidents during autonomous driving. However, as shown in Fig.
2, previous studies focused on TTA with covariate shifts (i.e., domain shifts)
only and did not address TTA that also includes semantic shifts (i.e.,
including open-set classes). Regarding its significance and practicality,
adaptation with unknown classes included (i.e., open-set TTA) should be also
addressed.

Fig. 1 shows our empirical analysis that discloses an important finding to
address such an issue. While entropy minimization enforces the model to
increase the probability value of its predicted label (i.e., confidence
values), we found that it often fails to increase them on the wrong samples.
While previous studies [58, 46] resorted to finding an adequate confidence
value or loss value to prevent error accumulation, the process of determining
it is cumbersome, and utilizing such a static threshold shows limited
performance. We train TENT [57] with different thresholds for the analysis:
(a) without thresholding, (b) selecting samples with confidence value higher
or equal to 0.9111We used the best confidence value $p$ after grid search of
$p\in\\{0.5,0.8,0.9,0.95,0.99.\\}$, (c) selecting samples with loss values
smaller than the entropy threshold proposed in EATA [46], and (d) selecting
samples that achieve higher confidence value with the adaptation model
$\theta_{a}$ compared to that with the original model $\theta_{o}$. As shown,
using the confidence difference between $\theta_{o}$ and $\theta_{a}$ for
selecting correct samples outperforms utilizing the static thresholds. While
b) and c) show significantly high recall values (i.e., selecting actual
correct samples well), it rather indicates that they simply select most of the
samples and fail to filter out noisy samples considering the substantially low
precision values (i.e., low ratio of correct samples among the selected ones).

The intuition behind using the confidence difference is as follows. Although
entropy minimization enforces the model to increase the confidence value on
the predicted label of an individual sample, the individual confidence value
may rise or fall, influenced by the signals that originated from numerous
other predictions (i.e., wisdom of crowds). To be more specific, the noisy
signals that do not align with such ‘wisdom of crowds’, commonly found in the
correct signals, fail to raise the individual confidence scores of wrong
samples, even with the supervision from entropy minimization to increase them.
By using such an observation, we select samples that achieve higher confidence
value using $\theta_{a}$ compared to that using $\theta_{o}$. Since we reflect
the knowledge state of the model on each individual sample, our selection is
implicitly a dynamic thresholding strategy, which outperforms the previously-
used static strategies. Our simple yet effective sample selection method is
widely applicable to existing TTA methods and improves their performances on
both image classification and semantic segmentation.

Figure 3: As adaptation proceeds, the number of samples with decreased
confidence values increases (purple graph). Additionally, among those samples,
the ratio of wrongly predicted samples also increases (green graph). $t_{i}$
indicates the $i^{th}$ round during the long-term adaptation.

Our contributions are summarized as follows:

  * •

We propose a novel sample selection method that filters out noisy samples
using the confidence difference between $\theta_{a}$ and $\theta_{o}$ based on
the finding that noisy samples, both closed-set wrong samples, and open-set
samples, generally show decreased confidence values on the originally
predicted label.

  * •

This is the first paper to address open-set test-time adaptation, adapting to
a target domain including test samples of unknown classes, which has not been
explored in existing TTA studies despite its importance and practicality.

  * •

Our proposed method can be applied to various test-time adaptation methods and
improves their performances on both image classification using CIFAR-10/100-C
and TinyImageNet-C (e.g., 49.38% reduced error rates with TENT in open-set
TTA), and semantic segmentation (e.g., 11.69% gain in mIoU with TENT) using
real-world datasets including BDD-100K and Mapillary.

##  2 Wisdom of Crowds in Entropy Minimization

###  2.1 Problem Setup

During the test-time adaptation, models adapt to a target domain with $N$
number of test samples in the test set $D_{T}$, $\\{x_{i},\\}^{N}_{i=1}\in
D_{T}$, without target labels provided. Given a pretrained model $\theta_{o}$,
we update $\theta_{o}$ to adapt to a novel target domain, where the adapted
model is then defined as $\theta_{a}$. For a test sample $x$, we define
$\tilde{y}=f(x;\theta_{o})\in\mathbb{R}^{C}$ and
$\hat{y}=f(x;\theta_{a})\in\mathbb{R}^{C}$ as the softmax outputs of the
original model $\theta_{o}$ and the adapted model $\theta_{a}$, respectively,
where $C$ denotes the number of classes. With the predicted class
$c_{o}=\operatorname*{argmax}_{c}f(x;\theta_{o})$ of the original model, we
define the probability value on the predicted label as confidence value
$\tilde{y}^{c_{o}}$. Similarly, the confidence value of the adapted model
$\theta_{a}$ on the label $c_{o}$, predicted by the original model, is defined
as $\hat{y}^{c_{o}}$. The main objective of test-time adaptation is to
correctly predict $c_{a}=\operatorname*{argmax}_{c}f(x;\theta_{a})$ using the
adapted model, especially under large data distribution shifts.

Figure 4: Utilizing the confidence difference distinguishes between the
correct samples (blue) and the wrong samples (red) better (AUROC of 89.42)
than using the confidence values (AUROC of 58.29). We used the same model
(i.e., TENT [57] adapted for 50 rounds) for the visualization.

###  2.2 Motivation

Figure 5: Overall procedure of our sample selection. We forward the mini-batch
of $n$ test images, $\\{x_{i}\\}^{n}_{i=1}$, to the original model
$\theta_{o}$ and the adaptation model $\theta_{a}$. Then, we compare the
probability values $\hat{y}^{c_{o}}$ and $\tilde{y}^{c_{o}}$ and select the
samples achieving ${\hat{y}^{c_{o}}\geq\tilde{y}^{c_{o}}}$. Finally, we apply
the entropy minimization only to the selected samples.

Figure 6:  Cosine similarity of gradients between samples with the same
predicted label. We observe that wrong signals (i.e., off-diagonal elements)
misalign with the correct signals (i.e., diagonal elements) that dominate the
wisdom of crowds.

#### Decreased confidence values

While entropy minimization enforces the model to increase the confidence value
of its originally predicted label, we empirically found that wrong samples
mostly show decreased values (i.e., $\hat{y}^{c_{o}}<\tilde{y}^{c_{o}}$). For
the experiment, we perform test-time adaptation using TENT [57] for 50 rounds
using CIFAR-10-C to simulate a long-term adaptation. One round includes
continuously changing 15 corruption types, so we repeat it 50 times without
resetting the model. With $t_{i}$ indicating the $i^{th}$ round, Fig. 3
(purple graph) shows that the number of samples achieving
$\hat{y}^{c_{o}}<\tilde{y}^{c_{o}}$, showing decreased confidence values,
among $N$ number of test samples increases as adaptation proceeds even with
the entropy minimization that enforces the model to increase its confidence
value on the originally predicted label. In fact, the green graph in Fig. 3
shows that the ratio of wrong samples among the samples with decreased
confidence values also increases as adaptation proceeds. The main reason for
such an observation is due to the ‘wisdom of crowds’, the signals learned from
numerous other samples influencing the confidence level of individual samples.
Specifically, although the individual signal from each sample compels the
model to increase the confidence value of its own predicted label, this effect
may be canceled out if other dominant signals show different patterns.

#### Wisdom of crowds from correct samples

We empirically found that models generally learn the wisdom of crowds from the
correct samples. Fig. 4 demonstrates such a point with the histogram of 1)
confidence values and 2) confidence difference,
$\hat{y}^{c_{o}}-\tilde{y}^{c_{o}}$, using TENT [57] adapted for 50 rounds. We
observe that a substantial number of the samples achieving
$\hat{y}^{c_{o}}-\tilde{y}^{c_{o}}\geq 0$ are correct samples (blue). To be
more specific, utilizing the confidence difference for distinguishing correct
samples from wrong samples (red) achieves an AUROC of 89.42, which outperforms
utilizing the confidence value of the adaptation model, achieving an AUROC of
58.29.

Such an observation discloses two findings. First, since samples achieving
$\hat{y}^{c_{o}}\geq\tilde{y}^{c_{o}}$ are mostly correct ones, the dominant
signals necessary for increasing the confidence values (i.e., wisdom of
crowds) are originated from the correct samples. Second,
$\hat{y}^{c_{o}}-\tilde{y}^{c_{o}}$ is an adequate metric to distinguish
between correct and wrong samples.

#### Misaligned wrong signals

We further empirically analyze why signals from wrong samples fail to increase
the confidence values of the original model. The main reason is that signals
originated from wrong samples misalign with the ‘wisdom of crowds’ obtained
from the correct samples. For the analysis, we compute the cosine similarity
of gradients between two samples with the same predicted label in Fig. 2.2.
For a given predicted label $i$ (column), we compute $s^{j,i}$, the cosine
similarity of gradients obtained between samples of ground truth label $j$
(row) and those of predicted label $i$ as,

$s^{j,i}=\frac{1}{M_{1}M_{2}}\sum^{M_{1}}_{k=1}\sum^{M_{2}}_{l=1}\frac{g^{j,i}_{k}\cdot
g^{i,i}_{l}}{\|g^{j,i}_{k}\|\|g^{i,i}_{l}\|},l\neq k\hskip
2.84544pt\text{if}\hskip 2.84544ptj=i,$ (1)

where $g^{j,i}_{k}$ indicates the gradient vector of $k^{th}$ sample among
$M_{1}$ number of samples with the ground truth label $j$ and the predicted
label $i$, $g^{i,i}_{l}$ indicates the gradient vector of $l^{th}$ sample
among $M_{2}$ number of samples with the ground truth label $i$ and the
predicted label $i$ (i.e., correct samples), and $i,j\in C$. In other words,
given a certain predicted label, we compare the gradients of the correct
samples and those of the samples with the same predicted label either correct
or wrong. Thus, the diagonal elements are the results obtained by comparing
the gradients between correct samples and the off-diagonal elements are
obtained by comparing the gradients between correct samples and the wrong
samples with the same predicted label. We add the description of how the
cosine similarity of each pair is computed on the right side of Fig. 2.2.

Given a certain column in Fig. 2.2, all entropy minimization losses enforce
the model to increase the probability value of the same predicted label.
However, we found that the signals (i.e., gradients) may differ depending on
the actual ground truth labels. Specifically, the correct samples show high
cosine similarity of gradients (diagonal elements, e.g., $s_{2,2}$) compared
to the ones with wrong samples (off-diagonal elements, e.g., $s_{0,2}$). Since
Fig. 4 shows that the correct signals dominate the wisdom of crowds required
for increasing the confidence value of the originally predicted label, signals
that are different from these dominant signals can be suppressed and do not
raise confidence values.

We want to clarify that the wisdom of crowds does not guarantee a model to
utilize the correct signals only. Even with the wisdom of crowds, the model
supervises itself with wrong predictions if the noisy losses are not filtered
out. Such self-training with wrong knowledge significantly deteriorates the
TTA performance of models, especially during the long-term adaptation [25]. In
fact, such an issue has been widely studied in fields beyond TTA, known as the
confirmation bias [63, 2, 56, 35, 34]. To address such an issue in TTA, we
propose a sample selection method to filter out noisy samples by using the
wisdom of crowds.

Method | CIFAR-10-C | CIFAR-100-C | TinyImageNet-C | Average
---|---|---|---|---
Closed | Open | Closed | Open | Closed | Open | Closed | Open
Source [61] | 18.27 | 18.27 | 46.75 | 46.75 | 76.71 | 76.71 | 47.24 | 47.24
BN Adapt [43] | 14.49 | 15.73 | 39.26 | 42.67 | 61.90 | 63.00 | 38.55 | 40.47
GCE [64] | 43.76 | 87.94 | 44.45 | 88.69 | 97.25 | 99.00 | 61.82 | 91.88
Conjugate [14] | 49.57 | 92.25 | 98.97 | 98.79 | 99.38 | 99.46 | 82.64 | 96.83
ENT | 87.06 | 89.26 | 56.35 | 98.76 | 99.43 | 99.50 | 80.95 | 95.84
\+ Ours | 17.33 (-69.73) | 23.98 (-65.28) | 37.69 (-18.66) | 40.48 (-58.28) | 58.93 (-40.50) | 64.01 (-35.49) | 37.98 (-42.97) | 42.82 (-53.02)
TENT [57] | 45.84 | 85.22 | 42.34 | 85.22 | 98.10 | 99.16 | 62.09 | 89.87
\+ Ours | 14.10 (-31.74) | 15.77 (-69.45) | 38.62 (-3.72) | 42.57 (-42.65) | 60.87 (-37.23) | 63.13 (-36.03) | 37.86 (-24.23) | 40.49 (-49.38)
EATA [46] | 29.78 | 82.05 | 49.31 | 98.75 | 59.82 | 63.47 | 46.30 | 81.42
\+ Ours | 14.07 (-15.71) | 15.65 (-66.40) | 38.44 (-10.87) | 42.47 (-56.28) | 59.80 (-0.02) | 62.08 (-1.39) | 37.44 (-8.86) | 40.07 (-41.35)
SWR [9] | 10.21 | 90.55 | 35.78 | 73.05 | 62.39 | 76.13 | 36.13 | 79.91
\+ Ours | 10.12 (-0.09) | 72.58 (-17.97) | 35.64 (-0.14) | 45.68 (-27.37) | 55.15 (-7.24) | 61.91 (-14.22) | 33.64 (-2.49) | 60.06 (-19.85)

Table 1: Error rates of image classification after 50 rounds of adaptation
(i.e., long-term test-time adaptation). We note the performance gain by
reduced error rates.

Method | CIFAR-10-C | CIFAR-100-C | TinyImageNet-C | Average
---|---|---|---|---
Closed | Open | Closed | Open | Closed | Open | Closed | Open
Source [61] | 18.27 | 18.27 | 46.75 | 46.75 | 76.71 | 76.71 | 47.24 | 47.24
BN Adapt [43] | 14.49 | 15.73 | 39.26 | 42.67 | 61.90 | 63.00 | 38.55 | 40.47
GCE [64] | 12.81 | 25.70 | 35.83 | 45.78 | 62.84 | 71.41 | 37.16 | 47.63
Conjugate [14] | 12.84 | 24.96 | 36.67 | 81.19 | 82.83 | 92.66 | 44.11 | 66.27
ENT | 16.30 | 47.54 | 38.74 | 58.16 | 79.69 | 91.74 | 44.91 | 65.81
\+ Ours | 13.41 (-2.89) | 16.93 (-30.61) | 37.55 (-1.19) | 42.60 (-15.56) | 63.89 (-15.80) | 69.01 (-22.73) | 38.28 (-6.63) | 42.85 (-22.96)
TENT [57] | 12.56 | 27.80 | 36.04 | 45.26 | 68.53 | 80.93 | 39.04 | 51.33
\+ Ours | 12.39 (-0.17) | 14.94 (-12.86) | 36.18 (+0.14) | 39.62 (-5.64) | 59.90 (-8.63) | 63.31 (-17.62) | 36.16 (-2.88) | 39.29 (-12.04)
EATA [46] | 12.39 | 25.52 | 36.39 | 54.22 | 59.02 | 61.72 | 35.93 | 47.15
\+ Ours | 12.35 (-0.04) | 14.92 (-10.60) | 36.25 (-0.14) | 39.58 (-14.64) | 59.30 (+0.28) | 62.11 (+0.39) | 35.97 (+0.04) | 38.87 (-8.28)
SWR [9] | 10.76 | 29.32 | 34.21 | 44.79 | 60.34 | 65.18 | 35.10 | 46.43
\+ Ours | 10.74 (-0.02) | 27.52 (-1.80) | 34.23 (+0.02) | 41.52 (-3.27) | 58.50 (-1.84) | 62.94 (-2.24) | 34.49 (-0.61) | 44.00 (-2.43)

Table 2: Error rates of image classification after 1 round of adaptation
(i.e., short-term test-time adaptation). We note the performance gain by
reduced error rates.

###  2.3 Proposed Method

As shown in Fig. 5, we propose a simple yet effective sample selection method
using the confidence difference between $\tilde{y}^{c_{o}}$ and
$\hat{y}^{c_{o}}$. Our sample selection criterion is formulated as

$\Phi(\hat{y}^{c_{o}},\tilde{y}^{c_{o}})=\mathbb{1}\left(\hat{y}^{c_{o}}\geq\tilde{y}^{c_{o}}\right),$
(2)

where $\Phi(\cdot)$ is our sample selection criterion and $\mathbb{1}(\cdot)$
is the indicator function. Our total objective function using entropy
minimization is formulated as

$\mathcal{L}^{\text{main}}(x;\theta_{a})=\Phi(\hat{y}^{c_{o}},\tilde{y}^{c_{o}})\cdot
H(\hat{y_{i}})-\lambda_{\text{max}}H(\overline{y}).$ (3)

$H(p)=\Sigma^{C}_{k=1}p^{k}\log{p^{k}}$,
$\overline{y}=\frac{1}{N}\Sigma^{C}_{k=1}\hat{y_{i}}$, and $\lambda_{max}$ is
the scalar value for balancing the two loss values. Note that
$H(\overline{y})$ has been widely used in previous studies [9, 39, 29, 38, 3,
5] to prevent the model from making imbalanced predictions towards a certain
class.

Recent studies require the pre-deployment stage that obtains the necessary
information needed for each method by using the samples from the source data
before the adaptation phase [9, 46, 39]. However, we want to emphasize that
our method does not require such a pre-deployment stage as well as those
samples from the source distribution. Due to such an advantage, our method can
be easily applied to existing TTA methods without additional preparations.
Through extensive experiments, we demonstrate the wide applicability of our
method to existing TTA methods.

##  3 Experiments

Time | $t\xrightarrow{\hskip 199.16928pt}$ | Average
---|---|---
Method | Cityscapes | BDD-100K | Mapillary
Source [6] | 34.74 | 16.15 | 36.97 | 29.29
BN Adapt [43] | 40.77 | 25.21 | 39.10 | 35.03
TTN [39] | 46.28 | 28.07 | 45.46 | 39.94
TENT [57] | 46.73 | 29.59 | 35.69 | 37.34
\+ Ours | 46.76 (+0.03) | 30.55 (+0.96) | 43.42 (+7.73) | 40.24 (+2.90)
SWR [9] | 46.17 | 10.70 | 1.28 | 19.38
\+ Ours | 46.65 (+0.48) | 32.28 (+21.58) | 45.09 (+43.81) | 41.34 (+21.96)

(a)

(b)

Table 3: Semantic segmentation performance (mIoU) on continuously-changing
target domains with 1 round of adaptation. We evaluate with DeepLabV3Plus-
ResNet-50 [6] pretrained on GTAV dataset.

Method | BDD-100K | Mapillary | GTAV | SYNTHIA | Average
---|---|---|---|---|---
Round 1 | Round 10 | Round 1 | Round 10 | Round 1 | Round 10 | Round 1 | Round 10 | Round 1 | Round 10
Source [6] | 43.50 | 43.50 | 54.37 | 54.37 | 44.55 | 44.55 | 22.78 | 22.78 | 41.30 | 41.30
BN Adapt [43] | 43.60 | 43.60 | 47.66 | 47.66 | 43.22 | 43.22 | 25.72 | 25.72 | 40.05 | 40.05
TTN [39] | 48.43 | 48.43 | 57.28 | 57.28 | 46.71 | 46.71 | 26.41 | 26.41 | 44.71 | 44.71
TENT [57] | 48.90 | 47.57 | 57.94 | 53.36 | 48.14 | 17.91 | 26.88 | 13.36 | 45.47 | 33.05
\+ Ours | 48.90 | 48.88 (+1.31) | 57.94 | 56.49 (+3.13) | 48.28 (+0.14) | 47.98 (+30.07) | 26.90 (+0.02) | 25.62 (+12.26) | 45.51 (+0.04) | 44.74 (+11.69)
SWR [9] | 49.39 | 49.68 | 59.33 | 59.70 | 47.82 | 48.13 | 28.40 | 1.18 | 46.24 | 39.67
\+ Ours | 49.88 (+0.49) | 50.57 (+0.89) | 58.79 (-0.54) | 58.89 (-0.81) | 49.17 (+1.35) | 49.27 (+1.14) | 27.75 (-0.65) | 27.82 (+26.64) | 46.40 (+0.16) | 46.64 (+6.97)

Table 4: Semantic segmentation performance (mIoU) on a fixed target domain
with 10 rounds of adaptation. We use DeepLabV3Plus-ResNet-50 [6] pretrained on
Cityscapes dataset.

###  3.1 Experimental Setup

Datasets For the image classification task, we use the widely used corruption
benchmark datasets: CIFAR-10/100-C and TinyImageNet-C. We apply 15 different
types of corruptions (e.g., gaussian noise) to CIFAR-10/100 [30] and
TinyImageNet [31]. Pretrained models are trained on the clean train set and
adapted to the corrupted test set. For the open-set setting, we use SVHN [44]
for CIFAR-10/100-C, and ImageNet-O [20] for TinyImagenet-C, where we apply the
same corruption type as the original test sets. We term the datasets as SVHN-C
and ImageNet-O-C, respectively. We apply the identical corruption type in
order to construct open-set samples that are drawn from the same domain shift
but with unknown classes. For the semantic segmentation task under continually
changing domains, we use a model pretrained on GTAV [50], and evaluate it with
Cityscapes [10], BDD-100K [60], and Mapillary [45]. For semantic segmentation
with a fixed target domain with multiple rounds, we use the Cityscapes for the
source distribution and BDD-100K [60], GTAV [50], Mapillary [45], and SYNTHIA
[51] for the target distributions. Note that the semantic segmentation task
inherently includes open-set classes in the test set (e.g., traffic cones in
BDD100K not shown during training with Cityscapes).

#### Evaluation settings

Following the recent TTA studies, we evaluate TTA models under continuously
changing domains without resetting the model after each domain [58, 39, 46].
For the closed-set and open-set continual long-term TTA in the image
classification, we perform adaptation for 50 rounds to simulate a long-term
TTA with continuously changing domains. We report both TTA performances after
1 round (i.e., short-term TTA) and 50 rounds (i.e., long-term TTA). Note that
we evaluate predictions made during online model adaptation, not after
visiting the entire test set, strictly following the established TTA settings.
For the open-set TTA, we construct the mini-batch that includes an equal
number of closed-set samples (e.g., CIFAR-10-C, shot noise) and open-set
samples (e.g., SVHN-C, shot noise). Although included in the mini-batch, we
exclude open-set samples from the evaluation and only evaluate models with
closed-set samples. To the best of our knowledge, our work is the first paper
to conduct experiments with the open-set TTA. We report the error rates and
mean intersection of union (mIoU) for image classification and semantic
segmentation, respectively.

#### Baselines

We mainly compare our method with previous methods addressing noisy labels
[64] or improving pseudo-labeling performances in TTA [14, 46]. Note that ENT
denotes updating all parameters while TENT [57] only updates affine parameters
of the batch normalization layers, both utilizing the entropy minimization
loss function. Gray-shaded digits indicate the performance gain by applying
our method to each baseline model, and bold digits indicate the better
performance between the two methods.

#### Implementation details

For the image classification, we use the learning rate of $1e$-$3$ and
$1e$-$4$ for models updating only affine parameters (TENT [57], EATA [46], GCE
[64], Conjugate [14]) and all parameters (ENT, SWR [9]), respectively. We use
the batch size of 200 and Adam optimizer [28] for all experiments. For
experiments conducting small batch sizes in Table 8, we use the learning rate
of $1e$-$4$ and update models after 200 steps, following TTN [39]. For the
semantic segmentation, we use the learning rate of $1e$-$6$ and batch size of
2 following TTN. Regarding using TTN in semantic segmentation, we update the
test batch statistics in an online manner to further improve the segmentation
performance for all experiments. Further details on our experimental setup are
included in our supplementary.

Method | CIFAR-10 / SVHN-C | CIFAR-100 / SVHN-C
---|---|---
AUROC$\uparrow$ | FPR@TPR95$\downarrow$ | AUROC$\uparrow$ | FPR@TPR95$\downarrow$
MSP [19] | 51.87 | 92.39 | 60.69 | 87.96
Max Logit [18] | 54.68 | 90.31 | 64.88 | 85.45
Energy [40] | 54.68 | 90.30 | 64.87 | 85.46
Ours | 88.24 | 40.34 | 83.76 | 64.86

(a)

Method | CIFAR-10 / SVHN-C | CIFAR-100 / SVHN-C
---|---|---
AUROC$\uparrow$ | FPR@TPR95$\downarrow$ | AUROC$\uparrow$ | FPR@TPR95$\downarrow$
MSP [19] | 50.83 | 93.64 | 56.14 | 90.34
Max Logit [18] | 56.25 | 90.65 | 62.76 | 87.35
Energy [40] | 56.26 | 90.63 | 62.79 | 87.27
Ours | 83.50 | 54.46 | 82.17 | 73.16

(b)

Table 5: Utilizing the confidence difference for thresholding in open-set test
time adaptation. We use TENT [57] adapted to each target domain including
open-set classes (SVHN-C) for 50 rounds.

Method | CIFAR-10-C | CIFAR-100-C | CIFAR-10/100-C
---|---|---|---
Error Rate (%) | Error Rate (%) | Memory (MB) | Time (ms)
ENT | 88.16 | 77.56 | 1147 | 22.98
SWR [9] | 50.38 | 54.42 | 1155 | 47.97
TENT [57] | 65.53 | 63.78 | 556 | 18.38
EATA [46] | 55.92 | 74.03 | 559 | 37.04
TENT [57] \+ Ours | 14.94 | 40.60 | 565 | 26.62

Table 6: Comparisons on error rates (%), memory (MB), and time (ms). For the
time, we report the average time after 5000 trials on NVIDIA RTX A5000.

Method | ResNet50 [17] | WDR28 [61]
---|---|---
Closed | Open | Closed | Open
Source | 48.80 | 48.80 | 43.52 | 43.52
BN Adapt [43] | 16.01 | 16.89 | 20.43 | 23.61
TENT [57] | 61.69 | 83.62 | 56.00 | 77.72
\+ Ours | 15.28 (-46.41) | 16.99 (-66.63) | 20.16 (-35.84) | 23.70 (-54.02)
SWR [9] | 16.19 | 88.53 | 17.94 | 90.15
\+ Ours | 16.08 (-0.11) | 71.83 (-16.70) | 15.35 (-2.59) | 83.76 (-6.39)

(a)

Method | ResNet50 [17] | WDR28 [61]
---|---|---
Closed | Open | Closed | Open
Source | 48.80 | 48.80 | 43.52 | 43.52
BN Adapt [43] | 16.01 | 16.89 | 20.43 | 23.61
TENT [57] | 14.03 | 22.76 | 18.23 | 32.74
\+ Ours | 13.82 (-0.21) | 16.36 (-6.40) | 18.32 (+0.09) | 23.40 (-9.34)
SWR [9] | 13.81 | 45.58 | 16.62 | 83.08
\+ Ours | 13.80 (-0.01) | 43.35 (-2.23) | 15.73 (-0.89) | 75.89 (-7.19)

(b)

Table 7: Error rates of image classification on CIFAR-10-C using diverse
architectures.

Method | Learning rate | Std.$\downarrow$
---|---|---
0.005 | 0.001 | 0.0005 | 0.0001
Source | 76.71 | 76.71 | 76.71 | 76.71 | 0
TENT [57] | 99.51 | 89.91 | 75.02 | 63.83 | 15.79
\+ Ours | 64.14 | 60.04 | 59.59 | 58.76 | 2.40

(a)

Method | Batch size | Std.$\downarrow$
---|---|---
64 | 32 | 16 | 8
Source | 76.71 | 76.71 | 76.71 | 76.71 | 0
TENT [57] | 67.54 | 72.62 | 81.21 | 94.75 | 11.90
\+ Ours | 60.32 | 62.14 | 65.88 | 73.83 | 5.99

(b)

Table 8: Error rates of image classification on TinyImageNet-C with diverse
learning rates and batch sizes. Std. is the abbreviation of the standard
deviation.

###  3.2 Results

Image classification ††2Note that the performance variation of the source
model in Cityscapes is due to the order of data samples (e.g., challenging
ones in the later stage), not due to the adaptation. As shown in Table 1,
existing TTA models show a large performance degradation during the long-term
adaptation. This is mainly due to the confirmation bias, caused by the
unsupervised losses that inevitably include noisy losses. We significantly
improve the long-term performance of the existing four different TTA models in
both closed-set and open-set TTA. For example, we improve the error rate of
TENT [57] by an average of 24.23% and 49.38% in the closed-set and open-set
settings, respectively. Note that we do not use prior knowledge of whether the
target distribution includes open-set samples or not. Additionally, Table 2
shows that our method also generally improves the short-term TTA performances.

While previous studies focused on improving the performance of closed-set TTA
until now, our results show that they suffer from a large performance drop
when adapted with open-set classes included. We believe that this is a
practical setting since we can not guarantee that samples from the target
distributions are always drawn from the classes learned during the training
stage. Such results indicate that improving the TTA performance with open-set
classes is yet to be explored in the future.

Semantic segmentation Table 3 shows the semantic segmentation performance with
continuously changing domains. We evaluated a model pretrained on GTAV [50]
with real-domain datasets (Cityscapes [10], BDD-100K [60], and Mapillary [45])
in order to simulate the situation where real-world target datasets are not
available with only synthetic datasets provided. We observe that the
performance gain by applying our method increases as the adaptation proceeds.
For example, SWR [9] (Table LABEL:tab:continual_segmentation_quali \- red)
suffers from a large performance drop with the last target domain, Mapillary
(1.28 mIoU), while ours (Table LABEL:tab:continual_segmentation_quali \- blue)
shows a stable level of performance (45.09 mIoU). Regarding Table
LABEL:tab:continual_segmentation_quali, we evaluate models after certain steps
and show the average mIoU up to then. While the model without adaptation
(i.e., source) does not suffer from the error accumulation, it fails to bring
performance gain. On the other hand, our method not only brings performance
gain but also circumvents error accumulation by filtering the noisy losses.

Table 4 also reports the semantic segmentation performance with a fixed target
domain over multiple rounds of adaptation. We observe that applying our method
improves the performance of TENT [57] and SWR [9] by an average of 11.69 mIoU
and 6.97 mIoU, respectively, after 10 rounds. As aforementioned, performing
test-time adaptation in semantic segmentation needs to address not only the
wrong predictions but also the inherently included open-set classes in the
target distribution. Our method again improves TTA performance by effectively
discarding such noisy pixels. We believe such a filtering mechanism is
especially important in safety-critical applications in two aspects. First, it
prevents the performance drop caused by learning with noisy losses. Second,
when confronted with unknown objects, we could alarm a device immediately,
which could be the starting point for it to take a different action (e.g.,
autonomous vehicles swerving directions to avoid running into wild animals
unexpectedly shown on roads) [23].

##  4 Further Analysis

###  4.1 Utilizing Confidence Difference as Thresholds

We show that the confidence difference is an adequate metric to differentiate
between correct samples and noisy samples, given that a pretrained model is
adapting to a novel domain. For the evaluation, we train TENT [57] and compare
utilizing confidence difference as the thresholding metric with existing
prediction-based out-of-distribution (OoD) methods [19, 18, 40]. By setting
the correct samples as the positive samples, we analyze two different negative
samples: negative samples 1) including closed-set wrong samples and 2)
excluding closed-set wrong samples. The former case shows how well a given
metric differentiates between correct samples and noisy samples, including
both closed-set and open-set samples. The latter case evaluates how well a
given metric distinguishes between the correct samples and open-set samples
only. Table 5 shows that using confidence difference outperforms the existing
OoD metrics in both cases. In addition to the superior performance, another
advantage of using the confidence difference is that we can filter the noisy
samples immediately, while the existing OoD metrics need the entire test
samples in order to choose the threshold with the best AUROC score. Such a
result indicates that confidence difference can also be widely used to
distinguish out-of-distribution samples in future studies with adapted models.

###  4.2 Comparisons on Resource Costs

Along with the TTA performances, Table 6 compares the memory usage and the
time consumption of the baseline models and our method applied to TENT [57].
For the TTA performance, we average the long-term adaptation performance of
closed-set and open-set TTA for each dataset. For memory usage, we use the
official code of TinyTL [4] to calculate both the model parameters and the
intermediate activation size, following the previous studies [22, 59, 55]. The
time indicates the amount of time consumed for the forward process and the
backpropagation. Since we utilize the outputs of $\theta_{o}$, our method
accompanies an additional forward process. However, as shown, such an
additional forward process is negligible compared to the state-of-the-art
models. For example, our method applied to TENT brings a significant
performance gain with only half the memory and time compared to SWR [9].
Further details on resource costs, along with the results on semantic
segmentation, are included in our supplementary.

###  4.3 Applicability on Various Models

Since our method focuses on improving the pseudo-labeling quality of entropy
minimization, it does not rely on model architectures. Table 7 shows that
applying our method consistently outperforms the baseline models with ResNet50
[17] and WideResNet28 [61] that were used in previous TTA studies [9, 39].
Such results demonstrate that our method is widely applicable to various
architectures.

###  4.4 Robustness to Hyper-parameters

In real-world applications, we may not know an adequate learning rate before
encountering test samples or may not use an optimal batch size due to memory
constraints. In such a case, we need an approach with a stable performance
regardless of such hyper-parameters. Table 8 shows that our method is more
robust to such hyper-parameters compared to TENT [57], which is highly
dependent on them. Such results demonstrate the scalability of our method when
we do not know the optimal hyper-parameters.

##  5 Related Work

Test-Time Adaptation The main differences between TTA studies and other
studies addressing domain shifts such as domain generalization [66, 37, 26,
15, 42, 52] or unsupervised domain adaptation (UDA) [49, 7, 38, 41, 36] is
that TTA studies do not utilize 1) the source data during the adaptation stage
and 2) ground truth labels on the target distribution [57, 58, 46, 14, 9, 39].
Recent studies [58, 39, 12] show that TTA models suffer from a large
performance degradation with continually changing domains and a long-term
adaptation. To tackle such a challenging problem, this paper mainly evaluates
long-term adaptation with continually changing domains.

Noisy Signals in Test-Time Adaptation As aforementioned, one of the key
challenges in TTA is that the model is prone to utilizing wrong predictions.
Preventing the model from learning with noisy supervision has been studied
widely beyond TTA [64, 49, 21, 16, 1, 32, 48]. However, the main difference
between TTA and these studies is that TTA studies assume that we cannot
revisit the sample after performing adaptation with it. Such an assumption
limits from proposing methods that require knowledge of the full data
distributions [53, 1] or consistency of predictions for a given sample [49,
54]. Without such knowledge, we use the difference of confidence scores
between $\theta_{o}$ and $\theta_{a}$ by using the wisdom of crowds to improve
pseudo labeling.

##  6 Conclusion

This paper proposed a simple yet effective data sample selection that is
widely applicable to existing various test-time adaptation methods. Based on
the observation that signals from wrong samples fail to increase the
confidence values of the predicted labels even with entropy minimization, we
only select the samples that achieve higher confidence values with the
adaptation model compared to those with the original model. This is mainly due
to the wisdom of crowds, the dominant signals generally found in the correct
samples influencing signals of other samples. Our method improved TTA
performance on the existing TTA methods on both image classification and
semantic segmentation. Additionally, we proposed a novel evaluation setting,
an open-set TTA, which was overlooked until now even with its importance and
practicality. We hope our work inspires future researchers to conduct more
practical TTA research that improves both closed-set and open-set TTA.

Acknowledgement We would like to thank Kyuwoong Hwang, Sungrack Yun, Simyung
Chang, Hyunsin Park, Janghoon Cho, Juntae Lee, Hyoungwoo Park, Seokeon Choi,
Seunghan Yang, and Sunghyun Park of the Qualcomm AI Research team for their
valuable discussions.

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Figure 10: Identifying wrong predictions and open-set samples in test-time
adaptation (TTA). During the long-term adaptation, previous models not only
show a large performance degradation but also predict the open-set samples as
one of the pre-defined classes learned during the training phase. By filtering
out noisy ones, both wrong and open-set samples, we can (a) prevent
performance degradation and (b) identify unexpected obstacles to prevent
accidents immediately. Red boxes indicate the regions of pixels that include
misclassified predictions or open-set classes. In the fourth column, on top of
the prediction of the model trained through our method, we color pixels that
are filtered out by our method as black.

##  Appendix A Further Analysis on Semantic Segmentation

Fig. 10 shows how filtering out noisy samples is important in semantic
segmentation. As mentioned in the main paper, discarding noisy samples is
crucial in two aspects: we can 1) prevent significant performance degradation
caused by noisy samples and 2) immediately identify unknown objects that could
be highly dangerous if not detected. For example, TENT [57] predicts the
motorcycle (wrong prediction) or the guardrails (open-set predictions) as
roads in the first and the second rows, respectively. When applying TTA in
real-world applications (e.g., autonomous driving), such an issue could lead
to a serious accident. However, our method effectively identifies them
immediately (black pixels in the fourth column), which can prevent such
accidents.

Time | $t\xrightarrow{\hskip 199.16928pt}$ | Average
---|---|---
Method | Cityscapes | BDD-100K | Mapillary
TENT [57] | 46.73 | 29.59 | 35.69 | 37.34
TENT w/o open-set [57] | 47.04 | 31.12 | 38.66 | 38.94
\+ Ours | 46.76 (+0.03) | 30.55 (+0.96) | 43.42 (+7.73) | 40.24 (+2.90)

Table 9: Effect of removing open-set samples in semantic segmentation. We
filtered out open-set pixels using ground-truth labels for TENT. We observe
performance gain compared to the original performance of TENT.

Method | ImageNet-C | Average
---|---|---
Closed | Open
Source [61] | 81.99 | 81.99 | 81.99
BN Adapt [43] | 68.49 | 69.65 | 69.07
TENT [57] | 99.71 | 99.72 | 99.72
\+ Ours | 65.62 (-34.09) | 67.78 (-31.94) | 66.70 (-33.02)
SWR [9] | 65.20 | 68.40 | 66.80
\+ Ours | 64.35 (-0.85) | 66.33 (-2.07) | 65.34 (-1.46)

(a)

Method | ImageNet-C | Average
---|---|---
Closed | Open
Source [61] | 81.99 | 81.99 | 81.99
BN Adapt [43] | 68.49 | 69.65 | 69.07
TENT [57] | 95.79 | 97.53 | 96.66
\+ Ours | 60.82 (-34.97) | 64.33 (-33.20) | 62.58 (-34.08)
SWR [9] | 66.59 | 69.02 | 67.81
\+ Ours | 65.29 (-1.30) | 66.86 (-2.16) | 66.08 (-1.73)

(b)

Table 10: Comparisons on ImageNet-C. We note the performance gain by reduced
error rates.

Method | CIFAR-10-C | CIFAR-100-C |  Memory (MB) |  Time (ms)
---|---|---|---|---
(a) | (b) | (c) | (a) | (b) | (c) | (MB) | (ms)
TENT [57] | 56.00 | 45.84 | 45.84 | 45.20 | 42.34 | 42.34 | 556 | 18.38
CoTTA [58] | 31.28 | 75.97 | 83.19 | 41.40 | 94.52 | 97.43 | 36442 | 379.49
TENT + Ours | 20.16 | 14.10 | 14.10 | 33.39 | 38.62 | 38.62 | 565 | 26.62

(a)

Time | $t\xrightarrow{\hskip 199.16928pt}$ | Average
---|---|---
Method | Cityscapes | BDD-100K | Mapillary
TENT [57] | 46.73 | 29.59 | 35.69 | 37.34
CoTTA [58] | 41.03 | 26.42 | 40.03 | 33.23
TENT+ Ours | 46.76 (+0.03) | 30.55 (+0.96) | 43.42 (+7.73) | 40.24 (+2.90)

(b)

Table 11: Comparison between our method and CoTTA [58]. We show the results of
our method applied to TENT. We perform better than CoTTA even with a
substantially smaller amount of memory usage and time consumption.

Table 9 shows that open-set samples degrade the performance of TTA models in
semantic segmentation. For the analysis, we compare the performance of TENT
[57] and that of TENT trained without the backpropagation of the open-set
pixels. We use the ground truth labels and filter out the open-set pixels. As
shown, TENT achieves better performance without the backpropagation of the
open-set pixels compared to the original performance. Such a result again
demonstrates that addressing open-set samples is crucial for practical TTA.
Note that our approach still outperforms TENT adapted with open-set samples
filtered out after a long-term adaptation (e.g., Mapillary). This is mainly
due to the fact that our method discards the wrong predictions well in
addition to the open-set samples.

##  Appendix B Comparisons on ImageNet-C

In Table 10, we also verify the effectiveness of our method on a large-scale
dataset, ImageNet-C [imagenetC]. Due to the fact that experimentation on
ImageNet-C is time consuming, we simulate the long-term adaptation with 10
rounds instead of the 50 rounds used in the main paper. We evaluate under
continuously changing target domains without resetting the model between each
domain. We use the batch size of 64 and the learning rate of 0.00025 with the
SGD optimizer [sgd], following the previous studies [57, 39, 9]. We observe
that our method again consistently improves the TTA performance on existing
baseline models in closed-set and open-set settings with short-term and long-
term adaptation. Regarding SWR [9], we observe a significant performance drop
of SWR when utilizing the adapted model of the previous iteration for the
regularization. Therefore, we use the source pretrained original model,
$\theta_{o}$, for the regularization. Other hyper-parameters are set as the
default values.

##  Appendix C Comparisons with CoTTA

We also compare our method with CoTTA [58], another seminal work in the
continual test-time adaptation. Table 11 compares the performances of image
classification and semantic segmentation and the resource costs between CoTTA
and our method applied to TENT [57]. As shown, although our method utilizes a
significantly smaller amount of memory usage and time consumption, we achieve
better performance in both image classification and semantic segmentation. We
describe the results in detail.

Method | Memory (MB) | Time (ms)
---|---|---
TENT [57] | 2714 | 529
SWR [9] | 5969 | 625
CoTTA [58] | 20276 | 4499
TENT [57] \+ Ours | 3036 | 685

Table 12: Comparisons on memory usage (MB), and time consumption (ms) on
semantic segmentation. We evaluate with DeepLabV3Plus-ResNet-50 [6]. For
memory usage, we use the batch size of 2. For the time, we report the average
time after 5000 trials with the image resolution of 3$\times$800$\times$1455
on NVIDIA RTX A5000.

###  C.1 Image Classification

We observe that CoTTA [58] shows performance variations depending on the
hyper-parameter $p_{th}$, which is a threshold to decide whether to use
ensembled predictions or a single prediction in CoTTA. However, we found it
challenging to find adequate $p_{th}$ for CoTTA with the model architecture
used in our work (i.e., WideResNet40 [61] for both CIFAR-10-C and
CIFAR-100-C). Although the supplementary of CoTTA illustrates how to find
$p_{th}$, we could not obtain identical values by using the architecture used
in CoTTA even with the description. Therefore, we report the comparisons
between CoTTA and our method with the following experimental setups: a)
architectures used in the CoTTA paper (i.e., WideResNet28 [61] for CIFAR-10-C
and ResNeXt-29 for CIFAR-100-C) with their default $p_{th}$ values, b)
architectures used in our main paper with their default $p_{th}$ values, c)
architectures used in our main paper with $p_{th}$ values we found by
following the description of the supplementary of CoTTA. Table
LABEL:tab:supple_cotta_image shows that our method outperforms CoTTA in all
three cases even with a substantially smaller amount of memory usage and time
consumption. For the experiments, we use the official repository of
CoTTA222https://github.com/qinenergy/cotta.

###  C.2 Semantic Segmentation

Regarding semantic segmentation, we evaluate CoTTA with continuously changing
target domains with a model pretrained on GTAV, as done in the main paper.
While TENT [57] and our method show performance gains by using TTN [39], CoTTA
achieves better performance by utilizing batch normalization with the test
statistics (i.e., TBN) than by using TTN. Therefore, we report the performance
of CoTTA using the TBN and the results of TENT and ours using TTN. In Table
LABEL:tab:supple_cotta_segment, we again observe that our method outperforms
CoTTA with real-domain shifts in semantic segmentation.

Additionally, we compare the memory usage and time consumption of our method
applied to TENT and other baseline models on semantic segmentation in Table
12. As shown, our method accompanies a negligible amount of resource cost. For
example, while our method outperforms CoTTA, we accompany a substantially
smaller amount of resource cost compared to CoTTA.

##  Appendix D Further Details on Experimental Setup

###  D.1 Datasets

Image classification For constructing SVHN-C and Imagenet-O-C, we apply
corruption types used for CIFAR-10/100-C and TinyImagnet-C by using the
official code333https://github.com/hendrycks/robustness of Hendrycks
[imagenetC]. Since the image sizes of Imagenet-O [20] and TinyImageNet [31]
are different, we resize the resolution of Imagenet-O images to 64$\times$64\.
Among the 5 severity levels, we use corruption level 5, the most corrupted
version. Fig. 12 shows the example images of the datasets used in our work.

Semantic segmentation For the experiments with continuously changing domains,
we use the train sets of each target domain in order to conduct experiments
with a long-term adaptation without using multiple rounds. Note that each
target domain includes a different number of images. For example, Cityscapes,
BDD-100K, and Mapillary include 2975, 7000, and 18000 images, respectively.
Due to this fact, for showing the mIoU changes in Table 3b of the main paper,
we evaluate models an equal number of times (i.e., 20 times) for each target
domain, not after certain steps. For the experiment with a fixed target domain
over multiple rounds, we use the validation sets of each target domain.

###  D.2 Baselines

Conjugate [14] Conjugate pseudo labeling was recently proposed on the
observation that conjugate functions are approximate to the optimal loss
function. We use the official
codes444https://github.com/locuslab/tta_conjugate of Conjugate [14].

Figure 12: Examples of datasets used in our work. We use CIFAR-10-C and SVHN-C
for the images. In the closet-set TTA, all images in the mini-batch only
include the covariate shift (i.e., domain shift). On the other hand, in the
open-set TTA, half of the images in the mini-batch only include covariate
shift while the other half includes both covariate shift and semantic shift
(i.e., open-set samples).

GCE [64] Generalized Cross Entropy (GCE) loss was first introduced to address
the noisy labels in image classification. It emphasizes the learning of
correct samples by imposing high weights on the gradients of the samples
achieving low loss values, which are highly likely to be correctly annotated.
Following Conjugate [14], we use GCE as the baseline model to show that simply
applying existing noisy-labeling studies does not guarantee preventing the
error accumulation in TTA. Since the official repository of Conjugate includes
GCE codes, noted as RPL, we use the same codes in our work.

EATA [46] EATA555https://github.com/mr-eggplant/EATA filters out samples that
achieve loss values higher than a pre-defined static threshold and utilizes
the fisher regularization to prevent catastrophic forgetting. For the fisher
regularization, the original paper utilizes the test set of the source
distribution to obtain the weight importance $w(\theta)$. However, we believe
that such an assumption is not valid since the currently widely used corrupted
test sets apply the corruptions to the test samples of the source
distribution. In other words, such an approach necessitates the test samples
to obtain the weight importance before encountering the test samples.
Therefore, we use the train set of the source distribution to obtain the
weight importance. For the fisher coefficient, we use 1 for CIFAR-10/100-C and
2000 for TinyImageNet-C, which are the default values reported in the main
paper. For applying our method to EATA, we only replace the filtering method
and utilize the fisher regularization.

SWR [9] SWR proposes 1) updating domain-sensitive weight parameters more than
the insensitive ones and 2) aligning the prototype vectors of the source and
the target distributions [9]. Since SWR does not have an official repository,
we re-implemented the codes and report the results.

###  D.3 Implementation Details

Image classification For the image classification on CIFAR-10/100-C, we mainly
use WideResNet40 [61] which applied the AugMix [augmix] during the pre-
training stage, following the previous recent TTA studies [39, 9, 55]. The
pretrained model is available from RobustBench [robustbench]. For the
TinyImageNet-C, we use ResNet50 [17]. We pretrained ResNet50 for 50 epochs
with a batch size of 256 and a learning rate of 0.01 with cosine annealing
applied using the SGD optimizer [sgd]. We set $\lambda_{max}=0.5$ for
experiments on SWR and $\lambda_{max}=0.25$ for the rest of the experiments.

Semantic segmentation For all semantic segmentation experiments which utilize
the backpropagation, we use TTN [39] since it brings further performance gain
compared to using TBN. For applying our method on semantic segmentation, we
use a relaxed version: we select pixels achieving
$\hat{y}^{c_{o}}-\tilde{y}^{c_{o}}\geq-0.2$. For applying our method on SWR,
we reduce the coefficient of the mean entropy maximization loss
($\lambda_{max}$) from 0.5 to 0.2. The main reason is that the mean entropy
maximization works as regularization and reduces the effect of entropy
minimization loss. However, since our work improves the quality of entropy
minimization, the mean entropy maximization rather hampers further performance
gain from our method. By reducing the coefficient of mean entropy
maximization, our method improves the semantic segmentation performance. Such
an observation again demonstrates that our method improves the quality of the
entropy minimization loss. We set other hyper-parameters as the default
values.

##  Appendix E Further Details on Resource Costs

We illustrate how we compute the resource costs including memory usage and
time consumption. For memory usage, as mentioned in the main paper, we use the
official code provided by TinyTL [4]. Note that the activation size occupies
memory usage more than the parameter size [4, 55]. For ENT, which updates all
parameters, we add the parameter size and activation size of all parameters.
For TENT [57], which updates the affine parameters in the batch normalization
layers, we only add the parameter size and activation size of the affine
parameters. For SWR [9], which updates all parameters and utilizes an
additional model for the regularization, we add the parameter size of the
whole model parameters in addition to the memory usage of ENT. For EATA [46],
which also utilizes an additional model for the fisher regularization, we only
add the parameter size of the affine parameters in addition to the memory
usage of TENT. For our method applied to TENT, in addition to the memory usage
of TENT, we add 1) the parameter size of all parameters and 2) the parameter
size of the output tensors. We add the parameter size of all parameters since
we need the whole model parameters in order to compute $\tilde{y}$. Also,
since we utilize $\tilde{y}$, we add the memory of the output tensors that is
negligible compared to the parameter size of the whole model.

Method | CIFAR-10-C | CIFAR-100-C | TinyImageNet-C
---|---|---|---
Source [61] | 18.27 | 46.75 | 76.71
BN Adapt [43] | 14.49 | 39.26 | 61.90
TENT [57] | 45.84 | 42.34 | 98.10
\+ Ours (logit) | 33.46 (-12.38) | 72.08 (+29.74) | 92.24 (-5.86)
\+ Ours (softmax) | 14.10 (-31.74) | 38.62 (-3.72) | 60.87 (-37.23)

Table 13: Variant of our method. We observe that utilizing the softmax outputs
outperforms utilizing the logit values.

##  Appendix F Variant of Proposed Method

To compare the prediction values between $\theta_{a}$ and $\theta_{o}$, our
method utilizes the probability values of the softmax outputs. In Table 13, we
also analyze our method by using the logit values instead of the softmax
values. We observe that utilizing logit values fails to bring large
performance gains compared to using the softmax values. The main reason is
that the logit values generally increase regardless of the correct or wrong
samples. However, such an issue is not found in the softmax outputs since the
values are normalized to sum-to-one vectors.