Persistent Test-time Adaptation in Recurring Testing Scenarios
Authors: Trung-Hieu Hoang
MinhDuc Vo
Minh N. Do
Venue: NeurIPS 2024 poster (2024)
Keywords: domain adaptation
test-time adaptation
continual adaptation
performance degradation
self-supervised learning
Primary area: other
TL;DR: We conduct a simple-model theoretical analysis, introduce a benchmark and a baseline approach to address the gradual performance degradation of continual test-time adaptation methods.

Abstract: Current test-time adaptation (TTA) approaches aim to adapt a machine learning model to environments that change continuously. Yet, it is unclear whether TTA methods can maintain their adaptability over prolonged periods. To answer this question, we introduce a diagnostic setting - **recurring TTA** where environments not only change but also recur over time, creating an extensive data stream. This setting allows us to examine the error accumulation of TTA models, in the most basic scenario, when they are regularly exposed to previous testing environments. Furthermore, we simulate a TTA process on a simple yet representative $\epsilon$-**perturbed Gaussian Mixture Model Classifier**, deriving theoretical insights into the dataset- and algorithm-dependent factors contributing to gradual performance degradation. Our investigation leads us to propose **persistent TTA (PeTTA)**, which senses when the model is diverging towards collapse and adjusts the adaptation strategy, striking a balance between the dual objectives of adaptation and model collapse prevention. The supreme stability of PeTTA over existing approaches, in the face of lifelong TTA scenarios, has been demonstrated over comprehensive experiments on various benchmarks. Our project page is available at [https://hthieu166.github.io/petta](https://hthieu166.github.io/petta).

Persistent Test-time Adaptation in Recurring Testing Scenarios
Trung-Hieu Hoang
Duc Minh Vo
Minh N. Do
Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign The University of Tokyo
Vin Iillnnoos Smart Health Center, VinUniversity Chthieu, minhdo ( 0illlm0illlllii www.ioii..eeeit. edu vmduc@nlab. ci i.u-tokyo ac.
Abstract
Current test-time adaptation (TTA) approaches aim to adapt machine learn- ing model to environments that change continuously. Yet, unclear whether TTA methods can maintain their adaptability over prolonged periods. To answer this question, we introduce diagnostic setting recurring TTA where envi- ronments only change but also recur stream. This setting allows us to examine the error accumulation of TTA models, in the most basic scenario, when they are regularly exposed previous testing environments. Furthermore, we simulate a TTA process on a simple yet repre- sentative e-perturbed Gaussian Mixture Model Classifier, deriving theoretical insights into the dataset- and algorithm-depende performance degradation. Our investigation leads
time, creating extensive data
factors contributing to gradual propose persistent TTA (PeTTA), which senses when the model is diverging towards collapse and adjusts the adaptation strategy. striking a balance between the dual objectives of adaptation and model collapse prevention. The supreme stability of РeТTA over existing approaches, in the face of lifelong TTA scenarios, has been demonstrated over comprehensive experiments on various benchmarks. Our project page is available at https:
github. io/petta.
Introduction
Machine learning (ML) models have demonstrated significant achievements in various areas [18, 38, 47, 23]. Still, they are inherently susceptible to distribution-shifi [46, 13, 48, 21, 6] (also known as the divergence between the training and testing environments), leading to a significant degradation in model performance. The ability to deviate from the conventional testing setting appears as a crucial aspect in boosting MI models' adaptability when confronted with a new testing environment that has been investigated [30, 53, 14]. Among common domain generalization methods [58, 24, 1], test-time adaptation (TTA) ) takes the most challenging yet rewarding path that leverages unlabeled data available at test time for self-supervised adaptation prior to the final inference [57, 39, 8, 41, 59].
Early TTA studies have concentrated on simply ideal adaptation scenario where the test samples come from fixed single domain [57, 39, 41]. As result, such an assumption is far from the ever- changing and complex testing environments. To confront continually changing environments [59, 12]. Yuan el al. [61] proposed pructical 77A scenario where distribution changing and correlative sampling occur [15] simultaneously. Though practical TTA is more realistic than what the previous assumptions have made, it still assumes that any environment only appears once in the data stream, condition which does not hold true. Taking a surveillance camera as an example, it might accom- modate varying lighting conditions recurringly day after day (Fig. 1-left). Based on this reality, hypothesize that the recurring those conditions may reveal the error accumulation phenomenon in TTA, resulting in performance degradation over long period. To verify hypothesis, we simulate
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Figure 1: Recurring Test-time Adaption (TTA). (left) Testing environments may change recurringly and preserving adaptability when visiting the same testing condition is not guaranteed. (right) The testing error of RoTTA [61] progressively raises (performance degradation) and exceeds the error of the source model (no TTA) while our PeTTA demonstrates its stability when adapting to the test set of CIFAR-IO-C [19] 20 times. The bold lines denote the running mean and the shaded lines in the background represent the testing error on each domain (excluding the source model, for clarity).
recurring testing environment and observe the increasing error rate by recurringly adapting to the test set of CIFAR-10-4 C [19] multiple times. We showcase the testing error of RoTTA [61] after 20 cycles of adaptation in Fig. I-right. As expected, RoTTA can successfully adapt and deliver encouraging outcomes within the first few passes. However, this advantage is short-lived as our study uncovers significant issue: TTA approaches in this setting may experience severe and persistent degradation in performance. Consequently, the testing error of RoTTA gradually escalates over time and quickly surpasses the model without adaptation. This result confirms the risk of TTA deployment in our illustrative scenario, Therefore, ensuring sustainable quality is crucial for real-world applications, especially given the recurring nature of testing environments.
algorithm might work well in the first place and gradually degenerate.
This study examines whether the adaptability of TTA algorithm persists over an extended testing stream. Specifically, in the most basic scenario, where the model returns to previously encountered testing environment after undergoing various adjustments. We thus propose a more general testing scenario than the practical TTA [61], namely recurring 7TA, where the environments not only change gradually but also recur in a correlated manner over time. We first analyze a simulation using the --perturbed Gaussian Mixture Model Classifier (e-GMMC) on synthesized dataset and derive theoretical analysis to confirm our findings, offering insights to tackle similar issues in deep neural networks. The analysis provides hints for reasoning the success of many recent robust continual TTA approaches [61, 12, 59, 15] and leading us to propose a simple yet effective baseline to avoid performance degradation, namely Persistent TTA (PeTTA). PeTTA continuously monitors the chance of collapsing and adjusts the adaptation strategy on the fly, striking a balance between the two objectives: adaptation and collapse prevention. Our contributions can be summarized as follows:
First, this work proposes testing scenario recurring TTA, simple yet sufficient setup for diagnosing the overlooked gradual performance degradation phenomenon of TTA.
Second, we formally define the phenomenon of TTA collapsing and undertake a theoretical analysis on an e-GMMC, shedding light on neeenett and algorithm-dependen factors that contribute the error accumulation during TTA processes.
Third, we introduce persistent TTA (PeTTA) simple yet effective adaptation scheme that surpasses all baseline models and demonstrates persisting performance.
For more context on related work, readers are directed to visit our discussions in Appdx. A.
2 Background
Test-time Adaptation (TTA), A TTA algorithm operates on an ML classifier fa X' + 3 with parameter 0, E O (parameter space) gradually changing over time (1 E T) that maps an input image E X' to a category (label) y E y. Let the capital letters (X,, Yt) E X x y denote a pair of random variables with the joint distribution Pr y) E Pa, E T. Here, Pa belongs to collection of D sets of testing scenarios (domains) (Pa) d= The covariate shift [46] is assumed: Px(x) and Py (xx)

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could be different but Pi x)) (r(l)z holds Vt øt. At model trained on Po E Po (source dataset). The model then explores an online stream of testing data. For each /. >> 0, it receives X, (typically in form of a batch of N₂ testing samples) for adapting itself fi-1 -> ft before making the final prediction fa (Xt).
0, 6g is initialized by a supervised
TTA with Mean Teacher Update. To achieve a stable optimization process, the main (teacher) model ft are updated indirectly through student model with parameters 0g [57, 61, 12, 15, 55J. At first, the teacher model in the previous step introduces a pseudo label [28] Y, for each X₃:
= fi-
(I)
With a classification loss LCLS (e.g., cross-entropy [16]), and model parameters regularizer R, the student model is first updated with a generic optimization operator Optim, followed by an exponential moving average (EMA) update of the teacher model parameter 0:-1:
0/ Optim Ep. [CCLs Xxxx0;))) +AR(0)) ee
(2)
0. = (1 - a0)0,-1 + all;
with a E (0, 1) the update rate of EMA, and E IR+ the weighting coefficient of the regularization term, are the two hyper-parameters.
Practical TTA. In practical TTA [61], two characteristics of the aforementioned distribution of data stream are noticeable. Firstly, P/'s can be partitioned by l which {P:) C Pa. Here, each partition of consecutive steps follows the same underlying distribution which will change continually through D domains [59] (Pi + P2... + PD). Secondly, the category distribution in each testing batch is temporally correlated [15]. This means within a batch, a small subset of categories is dominant over others, making the marginal distribution (r()) 0, Vy V C ) even though the category distribution over all batches are balanced. Optimizing under this low intra- matra-batch diversity
(D4) < ()1) situation can slowly degenerate the model [7].
3 Recurring TTA and Theoretical Analysis
This section conducts a theoretical analysis on a concrete failure case of simple TTA model. The results presented at the end of Sec. 3.2 will elucidate the factors contributing to the collapse (Sec. 3.1), explaining existing good practices (Sec. 3.3) and give insights into potential solutions (Sec. 4).
3.1 Recurring TTA and Model Collapse
Recurring TTA. To study the gradual performance degradation (or model collapse), we propose a new lesting scenario based on practical TTA [61]. Conducting a single pass through D distributions, as done earlier studies [61, 59], may not effectively identify the degradation. To promote consistency, recurring TTA performs revisiting the previous distributions K times to compare the incremental error versus the previous visits. For example, a sequence with K could be P1 + P2 > PD - P1 + P2 + + Pp. Appdx. D extends our justifications constructing recurring TTA. Definition 1 (Model Collapse). A model is said to be collapsed from step T E T. T < ec if there exists non-empty subset of categories y C. y such that Pr(Y, E У} > 0 but the marginal Pr(ý, 5} converges to zero in probability:
lim Pr(t E vD}- =0
Here, upon collapsing, a model tends to ignore almost categories in ÿ. As it is irrecoverable once collapsed, the only remedy would be resetting all parameters back 6o.
3.2 Simulation of Failure and Theoretical Analysis
Collapsing behavior varies across datasets and the adaptation processes. Formally studying this phenomenon on a particular real dataset and TTA algorithm challenging. Therefore, we propose a theoretical analysis on e-perturbed binary Gaussian Mixture Model Classifier (e-GMMC) that shares the typical characteristics by construction and demonstrates the same collapsing pattern in action (Sec. 5. as observed on real continual TTA processes (Sec. 5.3).
3

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Figure 2: e-perturbed binary Gaussian Mix- ture Model Classifier, imitating a continual TTA algorithm for theoretical analysis. Two main components include a pseudo-label predictor (Bq. 1), and mean teacher up- date (Eqs. 2, 3). The predictor is perturbed for retaining a false negative rate of e to simulate an undesirable TTA testing stream.
Pseudo-label Predictor
Mean-teacher Update
Y, argmax Pr. ( be
6 Optin Pp foous (Fr, X.:0))
p'ze
6. (1 0101-1 ab;
B
Simulated Testing Stream. Observing a testing stream with (X2, Y2) E X x y= R x {0, 1] and the underlying joint distribution (r, y) Py.t -Nxx;; Myy oz)) The main task is predicting X2 was sampled from cluster 0 or (negative or positive). Conveniently, let pa. - P.(x)) Pr(Y, = y) and Py. Pr(Y, y) be the marginal distribution of the true label Y; and pseudo label &,
GMMC and TTA. GMMC first implies an equal prior distribution by construction which is desirable for the actual TTA algorithms (e.g., category-balanced sampling strategies in [61, 15]). Thus, it simplifies ft into a maximum likelihood estimation (z) = argmax,ey> Pr(xly: 0,) with
Pr(æ|y; 0₂) = (z; Py,e, (. The goal is estimating set of parameters 8g = (Ay,e. o hey. A perfect classifier 0g = {uy, ai vyJvey is initialized at = O. For the consecutive steps, the simplicity of GMMC allows solving the Optim (for finding O, Eq. 2) perfectly by computing the empirical mean
and variance of new samples, approximating Ep,, The mean teacher update (Eq. 3) for GMMC is:
{ (1 a) + &BP, [x11x]] if = otherwise
Py.
(4)
(ay,t-1
The update of o,, is similar. Y, = f(-((X)) can be interpreted as a pseudo label (Eq. 1).
c-GMMC. Severe distribution shifts or low intra-batch category diversity of recurring TTA/practical TTA both result in an increase the error rate of the predictor. Instead of directly modeling the dynamic changes of Py, (which can be complicated depending the dataset), we study an e e-pertubed GMMC (e (MMMM,C) where Py, is assumed to be static (defined below) and the pseudo- label predictor of this model is perturbed to simulate undesirable effects of the testing stream on the predictor. Two kinds of errors appear in a binary classifier [4]. Let
c = Pr(Y, 1 0}
be the false negative rate (FNR) of the model at step Without loss of generality, we study the increasing type II collapse of C-MMCC By intentionally flipping the true positive pseudo labels in simulation, an FNR of & maintained (Fig. 2).
Assumption (Static Data Stream). The marginal distribution of the true label follows the same Bernoulli distribution Ber(Po): Po,: Po. (P1. P1 = 1 Pu), Vt E T.
Lemma 1 (Increasing FNR). Under Assumption 1, binary e- C-MMCC would collapsed (Def. 1) with limpi, = 0 (or lim fom = 1, equivalently) if and only if lime, Pt.
b-7
Lemma states the negative correlation between 인1, and ez. Unsurprisingly, towards the collapsing point where all predictions are zeros, the FNR also increases at every step and eventually reaches the highest possible FNR of pı-
Lemma 2 (e-GMMC After Collapsing). For a binary eGGMMC model, with Assumption 1, if limm,, 0 (collapsing), the cluster 0 in GMMC converges in distribution to single-cluster GMMC t-
with parameters:
Nioo, 2 4 Nooooo + P1#1, Poot pio + PoPi(Po 21)2).
Lemma 2 states the resulting e- GMMC after collapsing. Cluster 0 now covers the whole data distribution (and assigning label 0 for all samples). Furthermore, collapsing happens when foo. moves toward μ1. We next investigate the factors and conditions for this undesirable convergence.

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Theorem 1 (Convergence of c- GMMC). For binary e- e-GMMC model, with Assumption 1, let the distance from Fo, toward Y1 is de do-+1= pr [o,t] -1ll then:
(IN0 p1l
d9-1
d d S a po
1 1-6
From Thm. 1, we observe that the distance d- converges (also indicating the convergence to the
distribution in Lemma 2) if df < d=3. The model collapse happens when this condition holds for a sufficiently long period.
Corollary 1 (A Condition for e-GMCC Collapse). With fixed po. a, po: M1, e- CMGMCC collapsed if there exists a sequence of [E)T-a, (T 2 A, > 0) such that:
d
Pi 2 4
tE
p1
Corollary introduces a condition c-GMMC collapse. Here, et 's are non-decreasing, lime; = Pi- 1-
Remarks. Thm. 1 concludes two sets of factors contributing to collapse: (i) data-dependent factors: the prior data distribution (Po), the nature difference between two categories (1/in - l):): and (ii) setorithn-dependde
factors: the update rate (a)), the FNR at each step (c2). c-GMMC analysis sheds light on explaining model collapse on real datasets (Sec. 5.3), reasons the existing approaches (Sec. 3.3) and motivates the development of our baseline (Sec. 4).
3.3 Connection to Existing Solutions
Prior TTA algorithms have already incorporated implicit mechanisms mitigate model collapse. The theoretical results in the previous section explain the rationale behind these effective strategies.
Regularization Term for 8,. Knowing that fu is always well-behaved, attempt is restricting
the divergence of 0g from do e.g. using R(0,) _ |10g - 0 regularization [40]. The key idea is introducing a penalty term avoid an extreme divergence happening in Thm. 1.
Memory Bank for Harmonizing (z)). Upon receiving X,, samples in this batch are selectively updated memory bank M (which already contains subset of some instances of Xp, t in the previous steps). By keeping a balanced number of samples from each category, distribution P.M (y) of samples in M is expected have less entries than (y(y), making the optimization step over P.M more desirable. From Thm. 1, M moderates the extreme value of the category distribution (po term) which typically appears on batches with low intra-batch category diversity.
4 Persistent Test-time Adaptation (PeTTA)
Now we introduce our Persistent TTA (PeTTA) approach. Further inspecting Thm. 1, while ér (Eq. 5) is not computable without knowing the true labels, the measure of divergence from the initial distribution (analogously to df term) can provide hints to fine-tune the adaptation process.
Key Idea. A proper adjustment toward the TTA algorithm can break the chain of monotonically increasing e 's in Corollary to prevent the model collapse. In the mean teacher update, the larger value of X (Fq. 2) prioritizes the task of preventing collapse on hand but also limits its adaptability to the new testing environment. Meanwhile, a (Eq. 3) controls the weight on preserving versus changing the model from the previous step. Drawing inspiration from the iitooratioo-epllitaattii tradeoff (49, 25] encountered in reinforcement learning [54], we introduce mechanism for adjusting À and a on the fly, balancing between the two primary objectives: adaptation and preventing model collapse. Our strategy is prioritizing collapse prevention (increasing X) and preserving the model from previous steps (decreasing a) when there is a significant deviation from ty.
In [40, 61, 59], À and a were fixed through TTA environments and the lack of validation set [62]. Furthermore, Thm. I suggests the convergence rate quickly escalates when 6, increases, making constant X, a insufficient to prevent collapse.
tuning. This is suboptimal due to varying
Sensing the Divergence of fg. We first equip PeTTA with mechanism for measuring its divergence from Og. Since f(( argmax yey Pr(y) æx; 0z), we can decompose Pr(y]æ; 0.) = [h (de. (x))) with Φo, (.) is a O--parameterized deep feature extractor followed by a fixed classification head (a linear and softmax layer) h(.)) The operator [ily extracts the 7d component of vector.
5

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Since h(-)) remains unchanged, instead of comparing the divergence in the parameter space (9) or between the output probability Pr(y) 0:) and Pr(y] [z; Bn), we suggest an inspection over the feature embedding space that preserves maximum amount of information in our case (data processing inequality (9).. Inspired by [31] and under Gaussian assumption, the Mahalanobis distance of the first moment of the feature embedding vectors is compared. Let &e, (a:), we keep track of collection of the running mean of feature vector z: {pr} hey in which Pi is EMA updated with vector Z if fx) y. The divergence of 0r at step é, evaluated on class y is defined as:
7p = 1 - cxp (-(0! "0)" (ES) (A -m?)+)))
(6)
where p and Σg are the pre-computed empirical mean and covariant matrix of feature vectors in the source dataset (Po). The covariant matrix here is diagonal for simplicity. In practice, without directly accessing the training set, we assume a small set of unlabeled samples can be drawn from the source distribution for empirically computing these values (visit Appdx. E,4 for further details).
Here, we implicitly expect the independence of each entry in Z and TTA approaches learn align feature vectors of new domains back to the source domain (Po). Therefore, the accumulated statistics of these feature vectors at each step should be concentrated near the vectors of the initial model. The value of r E [0,1] is close to O when 0, Og and increases exponentially as pi diverging from pn
Adaptive Regularization and Model Update. With co, Ap are initial values, utilizing 7 derived in Eq. 6, a pair of (Ae, o4) is adaptively chosen at each step:
5i = { (0 =1-11,.- .N.. Ni.:
7
V
D
DEV.
A To Ao.
at = (1 - 7c) a..
Di is a set of unique pseudo labels in testing batch (2,"0) is the 1hh realization of Pt).
Anchor Loss. Penalizing the divergence with regular vector norms in high-dimensional space (+) is insufficient (curse of dimensionality [5, 51]), especially with a large model and limited samples. Anchor loss LAL can nail down the similarity between ft and fo in the probability space [32, 12]:
CAL
Σ yEY
Pr(viXe; 0o) log Pr(v)Xx;0),
which is equivalent to minimizing the KL divergence DEL (Pr(v)X,; 0o)1 Pr())
Persistent TTA. Having all the ingredients. we design our approach, PeTTA. following the convention setup of the mean teacher update, with the aategory-balancee memory bank and the robust batch normalization layer from [61]. Appdx. E. introduces the pseudo code of PeTTA For LeLS, either the self-training scheme [12] or the regular cross-entropy [16] is adopted. With R(), cosine similarity or L.2 distance are both valid metrics for measuring the distance between and 6o the parameter space. Fisher regularizer coefficient [40, 27] can also be used, optionally. To sum up, the teacher model update of PeTTA is an elaborated version of EMA with de, ar (Eq. 7) and CAL (Eq. 8):
02 Optim Ep, [COLs (+:)X:: + CAL (Xx:0)) + AQR)()), Ee
0₂ (1 ai 0 + a.8,
5 Experimental Results
5.1 c-MMC Simulation Result
Simulation Setup. A total of 6000 samples from two Gaussian distributions: NN = 0, oî = and (14 2, o 1) with Po P1 are synthesized and gradually released in a batch of B = 10 samples. For evaluation, an independent set of 2000 samples following the same distribution is used for computing the prediction frequency, and the false negative rate (FNR). Care-GMMM update follows Eq. 4 with a = 5e 2, To simulate model collapse, the predictor is intercepted and 10% of the true-postive pseudo labels at each testing step are randomly flipped (Corollary 1).
Simulation Result. action, both the likelihood of predicting class 0 (Fig. 3a-left) and the e, (Eq. 5) (Fig. 3c-right, solid line) gradually increases over time as expected (Lemma 1). After collapsing,

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c-GMMC
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Figure 3: Simulation result on e-perturbed Gaussian Mixture Model Classifier (c-GMMC) and GMMC (perturbed-free) (a) Histogram of model predictions through time. A similar prediction frequency pattern is observed on CIFAR-10-C (Fig. 5a-left). (b) The probability density function of the clusters after convergence versus the true data distribution. The initial two clusters of c-GMMC collapsed into single cluster with parameters stated in Lemma 2. In the perturbed-free GMMC converges to the true data distribution. (c) Distance toward μr (EE,, [ ph and false- negative rate (<)) in simulation coincides with the result in Thm. 1 (with 6, following Corollary 1).
c-GMMC merges the two initial clusters, resulting in single one (Fig. 3b-left) with parameters that match Lemma 2. The distance from Ao. (initialized at (o) towards μi converges (Fig. 3e-left, solid line), coincided with the analysis in Thm. 1 when é, is chosen following Corollary (Fig. 3c, dashed line). GMMC (perturbed-fres stably produces accurate predictions (Fig. 3a-right) and approximates the true data distribution (Fig. 3b-right). The simulation empirically validates our analysis (Sec. 3.2), confirming the vulnerability of TTA models when the pseudo labels are inaccurately estimated.
5.2 Setup Benchmark Datasets
Datasets. We benchmark the performance on four TTA classification tasks. Specifically, CIFARIO + CIFARIO-C, CIFAR 100 - CIFAR100-C, and ImageNet > ImageNet-C [19] are three corrupted images classification tasks (corruption level 5, the most severe). Additionally,
incorporate DomainNet [44] with 126 categories from four domains for the task real + clipart, painting, sketch.
Compared Methods. Besides PeTTA, the following algorithms are investigated: CoTTA [59], EATA [40], RMT [12], MECTA [22], RoTTA [61], ROID [37] and TRIBE [52]. Noteworthy, only RoTTA is specifically designed for the practical TTA setting while others fit the continual TTA setting in general. A parameter-fin free approach: LAME [7] and a based approach (i.e., reverting the model to the source model after adapting every 1, 000 images): RDumb [45] are also included.
Recurring TTA. Following the practical TTA setup, multiple testing scenarios from each testing set will gradually change from one to another while the Dirichlet distribution (Dir(0.1) for CIFAR 10- C, DomainNet, and ImageNet-C, and Dir(0.01) for CIFAR 100-C) generates category temporally correlated batches of data. For all experiments, set the number of revisits K 20 (times) this number is sufficient to fully observe the gradual degradation on existing TTA baselines.
Implementation Details. We use PyTorch [43] for implementation. RobustBench [10] and torchvision [35] provide pre-trained source models.
choices are kept as close as possible to the original selections of authors. Visit Sec. G for more implementation details. Unless otherwise noted, for all PeTTA experiments, the EMA update rate for robust batch normalization [61] and feature embedding statistics is set to 5c oo 1 and cosine similarity regularizer is used. CIFAR 101000-C and ImageNet-C we use the self-training loss in [12] for LCLS and X = 10 while the regular cross-entropy loss [13] and A (severe domain shift requires prioritizing
7

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Table I: Average classification enror of the task CIFAR-10 CIFAR-IO-C in recurring 77A. The lowest error is in bold, average value across 5 runs (different random seeds) is reported for PeTTA.
Recursing 7TA 1822
Methed
10
11
12
13
15
16
17
18
19
20
Avg
Source
45.5
43.5
LAME [7]
31.1
31.1
$7.0
38.6
89.7
82.6
99.9
73.5
75.4
73.3
73.6
75.7
72.2
39.1
98.3
RoTTA
33.6
46.2
54.1
59.4
31.6
72.7
72.7
TRIDE
17.3
PeTTA (0WPS)
24.3
23.0
22.6
22.4
22.4
22.5
22.3
22.5
22.8
22.8
22.6
22.7
22.7
22.9
22.6
22.7
22.6
22.8
22.9
23.0
22.8
adaptability) are applied in DomainNet experiments. In Appdx. E5,, we provide a sensitivity analysis on the choice of hyper-parameter X in PeTTA.
5.3 Result- Benchmark Datasets
Recurring TTA Performance. Fig. 1-right presents the testing error on CIFAR-1O-C in recurring TTA setting. RoTTA [61] exhibits promising performance the first several visits but soon raises and eventually exceeds the source model (no TTA). The classification error of compared methods on DIFAR-1O-CIFAA 10-C, and ImageNet - ImageNet-C [19] tasks are shown in Tab. I, and Tab. 2. Appdx. F.1 provides the results the other two datasets. The observed performance degradation of CoTTA [59], EATA [40], ROTTA [61], and TRIBE [52] confirms the risk of error accumulation for extensive period. While RMT [12], MECTA [22], and ROID [37] remain stable, they failed to adapt to the temporally correlated test stream at the beginning, with a higher error rate than the source model. LAME [7] (parameter-free TTA) and RDumb [45] (reset-based TTA) do not suffer from collapsing. However, their performance is lagging behind, and knowledge accumulation is limited in these approaches that could potentially favor higher performance as achieved by РеТТA. Furthermore, LAME [7] is highly constrained by the source model, and selecting precise reset frequency in RDumb [45] challenging practice (see Appdx. F.3 for a further discussion).
In average, PeTTA outperforms almost every baseline approaches and persists across 20 vis- its over the three datasets. The only exception is the case of TRIBE [52] on CIFAR-10- C. While this state-of-the-arr model provides stronger adaptability, outweighing the PeTTA, and baseline RoTTA [61] in several recurrences, the risk of the model collapsing still presents in TRIBE [52]. This can be clearly observed when increase the observation period to 40 recur- ring visits in Fig. 4. As the degree of freedom for adaptation in PeTTA is more constrained, takes bit longer for adaptation but remains sta- ble afterward. Fig. 5b-bottom exhibits the con- fusion matrix at the last visit with satisfactory accuracy. The same results are also observed when shuffling the order of domain shifts within each recurrence (Appdx. D.3), or extending the number of recurrences to 40 visits (Appdx. F.4).
24
Classficcttonn Error
22
20
18
PeTTA (ours)
16
TRIBE [52]
10
20
30
40
Recurring TTA Visit
Figure 4: Classification error of TRIBE [52] and PeTTA (ours) of the task task in recurring TTA with 40 visits.
Continuously Changing Corruption (CCC) [45] Performance. Under CCC [45], Tab. 3 reveals the supreme performance of PeTTA over RoTTA [61] and RDumb [45]. Here, we report the average classification error between two consecutive adaptation step intervals. An adaptation step in this table corresponds to mini-batch of data with 64 images. The model is adapted 80, 000 steps total with more than 5.1M images, significantly longer than 20 recurring TTA visits. Undoubtedly. РeТTA still achieves good performance where the corruptions are algorithmically generated, non-cyclic with two more corruption types can happen simultaneously. This experiment also empirically justifies the construction of our recurring TTA as a diagnostic tool (Appdx. D.2) where similar observations concluded on the two settings. Obviously, our recurring TTA is notably simpler than CCC [45].

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Table 2: Average classification error of the task ImageNet - ImageNet-C in recurring TTA scenario.
Recurring 77A v888
Mechod
1
7
10
12
13
14
15
16
17
18
19
20
AAg
Source
82.0
82.0
LAME [7]
30.9
COTTA [:9]
98.6
99.1
99.4
94.4
99.5
99.5
99.5
99.5
99.6
99.7
99.6
99.6
99.6
99.6
99.6
99.6
99.6
99.5
99.7
99.7
99.5
EAIA [40]
60.4
-9.3
63.3
72.6
79.1
81.3
88.7
92.7
95.2
96.9
97.7
98.1
98.8
95.6
98.7
98.8
98.3
98.9
98.9
99.0
89.0
RMT [12]
12.3
11.0
69.9
09.1
58.8
68.5
68.4
53.3
70.0
70.2
70.1
70.2
72.8
76.8
75.6
75.1
75.1
75.2
74.8
74.3
71.3
MECTA [22)
772
$2.5
86.1
$7.9
$3.9
89.8
$9.8
89.9
90.0
90.4
90.6
90.3
90.7
90.8
90.8
90199
90.3
90.8
90.7
90.8
89.0
RoTTA [61]
68.3
62.1
61.8
64.5
08.4
75.4
$2.7
95.1
95
96.6
97.1
97.9
96.3
99
99.1
99.3
99.4
99.5
99.6
87.9
KDamh
72.2
73.0
72.8
72.2
78.3
72.7
71.
73.0
73.2
73
72.0
727
72.6
73.3
73.1
72.8
ROID
62.3
62.5
62.1
63.4
62.4
62.3
62.6
62.5
62.5
62.4
62.4
62.4
62.5
62.4
TRIRE
63.6
64.0
64.9
67.8
09.6
73.5
75.5
79.8
85.0
99.4
99.8
99.9
90.8
99.8
99.9
99.9
99.9
PVTTA (owrz/"1
65.3
61.7
59.8
39.1
59.4
59.6
99.8
59.3
99.4
60.0
60.3
61.0
60.7
60.4
60.6
60.7
60.8
60.7
60.4
60.2
60.5
Table 3: Average classification error on CCC [45] setting. Each column presents the average error within an adaptation interval (e.g., the second column provides the average error between the 6701 and 13400 adaptation steps). Each adaptation step here is performed on a mini-batch of 64 images.
CCC [CC Adoptation Step
Method
6700
13400
20100
26800
33500
40200
46900
53600
60200
66800
73400
80000
Avg
Source
0.83
0.83
0.83
0.83
0.84
0.84
0.83
0.83
RoTTA [61]
0.70
0.85
0.92
0.96
0.98
1.00
1,00
1.00
1.00
1.00
1.00
1.00
0.95
RDumb [45]
0.78
0,74
0.75
0.77
0.75
0.72
0.75
0.77
0.75
0.74
0.75
0.75
0.75
PeTTA (ours)
0.67
0.63
0.62
0.65
0.65
0.64
0.64
0.68
0.63
0.63
0.65
0.65
0.64
RoTTA [61]
PeTTA (ours)
RoTTA [61] 20 visit
Prediction Frequency
0.8
0.8
0.23
9.12
0.6
0.6
True label
3
6.3 3:30
9.21
0.4
0.4
0.52
3-2
2.05 0.E0
0:00 0.00
0:27 944
9.16
0.3 (.0
0.2
0.2
0.28 050
10
15
20
15
20
Visits
(a)
Visits
PeTTA (ours)- 20th visit
ship
0:33 0.t 304052
5:0
6:02
000
0.0
truck
207055022
stip
airplanc
Truc label
cat
Scot
0:44
freq
-.44
003
78.01
9.0
0.00
0.43
5.00
A1 8:35 ao0n
6:11
Misclassification rate collapsed RoTTA
Inter-category cosine similarity (source model)
Predicted label
(b)
task. (a) Histogram of model predictions (10 labels are color-coded). PeTTA achieves persisting performance while RoTTA [61] degrades. (b) Confusion matrix at the last visit, RoTTA classifies all samples into few categories (e.g., O: airplane, 4: deer). (c) Force-directed graphs showing (left) the most prone misclassification pairs (arrows indicating the portion and pointing from the true to the misclassified category): (right) similar categories tend to be easily collapsed. Edges denote the average cosine similarity of feature vectors (source model), only the highest similar pairs are shown. Best viewed in color.
Figure 5: Recurring TTA (20 visits) on CIFAR-IO-CCIAARI
The rise in classification error (Fig. 1-right) can be reasoned by the prediction frequency of RoTTA [61] in an recurring TTA setting (Fig. Sa-left). Similar to c-GMMC, the likelihood of receiving predictions on certain categories gradually increases and dominates the others. Further inspecting the confusion matrix of a collapsed model (Fig. 5b-top) reveals two major groups of categories are formed and a single category within each group represents all members, thereby becoming dominant. To see this, Fig. 5c-left simplifies the confusion matrix by only visualizing the
Collapsing Pattern.

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Table 4: Average (across 20 visits) error of multiple variations of РeTTA: without (w/o) R(Ø), CAL: LAL only; fixed regularization coefficient \; adaptive coef-
Table 5: Average (across 20 visits) error of PeTTA. PeTTA favors various choices of reg- ularizers R(0): L2 and cosine similarity in
ficient d2, update rate or; using anchor loss LAL.
conjunction with Fisher [27, 40] coefficient.
Method
CF-10-C
CF-100-C
DN
IN-C
Baseline w/o R(0), CAL
42,6
63.0
77.9
9344
Method
CF-10-C
CF-100-C
DN
IN-C
R(V)
Fisber
R(9) fixed À 0.
43.3
65.0
80.0
92.5
R(9) fixed À
42.0
64.6
66.6
92.9
L.2
3
23.0
35.6
43.1
70.8
22.7
36.0
43.9
70.0
CAL only
25.4
56.5
47.5
68.1
PeTTA Au
27.1
55.0
59.7
92.7
Cosine
/
x
22.8
35.1
42,9
63.8
60.5
PeTTA A +
23.9
41.4
44.5
75.7
PeTTA fAL
26.2
36.3
43.2
62.0
CF: CIFAR, DN: DomainNet, IN: ImageNet
PeTTA + + CAL
22.8
35.1
42.9
600.5
top prone-to tomne-tt-mmillsssfiee pair of categories. Here, label deer is used for almost every living animal while airplane represents transport vehicles. The similarity between categories in the feature space of the source model (Fig. 5c-right) is correlated with the likelihood of being merged upon collapsing. As distance in feature space analogous |M0 |nl (Thm. 1), closer clusters are at higher risk of collapsing. This explains and showcases that the collapsing behavior is predictable up to some extent.
5.4 Ablation Study
Effect of Each Component. Tab. 4 gives an ablation study on PeTTA, highlighting the use of a regularization term (R(0)) with fixed choice of X, a not only fails mitigate model collapse but may also introduce a negative effect (rows 2-3). Trivially applying the anchor loss (CAl.) alone is also incapable of eliminating the lifelong performance degradation in continual TTA (row 4). Within PeTTA, adopting the adaptive A scheme alone (row 5) or in conjunction with either a or anchor loss CAL (rows 6-7) partially stabilizes the performance. Under the drastic domain shifts with larger size of categories or model parameters (e.g., on CIFAR- a adjustment limits the ability of PeTTA to stop undesirable updates while common regularization term without CAL is insufficient to guide the adaptation. Thus, leveraging all elements secures the persistence of РeTTA (row 8).
DomainNet, ImageNet-C), restricting
Various Choices of Regularizers. The design of PeTTA is not coupled with any specific regu- larization term. Demonstrated in Tab. 5, PeTTA works well for the two common choices: 1.2 and cosine similarity. The conjunction use of Fisher coefficent [27, 40] for weighting the model parameter importance also studied. While the benefit (in terms of improving accuracy) varies across datasets, PeTTA accommodates all choices, as the model collapse is not observed in any of the options.
6 Discussions and Conclusion
On Potential Risk of TTA in Practice. We provide empirical and theoretical evidence the risk of deploying continual TTA algorithms. Existing studies fail to detect this issue with a single pass per test set. The recurring TTA could be conveniently adopted as straightforward evaluation, where its challenging test stream magnifies the error accumulation that a model might encounter in practice.
Limitations. PeTTA takes one step toward mitigating the gradual performance degradation of TTA. Nevertheless, a complete elimination of error accumulation cannot guaranteed rigorously through regularization. Future research could delve deeper into expanding our efforts to develop an algorithm that achieves сгтог accumulation-fin free by construction. Furthermore, as tackling the challenge of the temporally correlated testing stream is not the focus of PeTTA using small memory bank as in [61, 15] is necessary. It also assumes the features statistics from the source distribution are available (Appdx. E.3, E.4). These constraints potentially limit its scalability in real-world scenarios.
Conclusion. Towards trustworthy and reliable TTA applications, we rigorously study the performance degradation problem of TTA. The proposed recurring TTA setting highlights the limitations of modern TTA methods, which struggle to prevent the error accumulation when continuously adapting to demanding test streams. Theoretically inspecting a failure case of e- GMMMC paves the road for designing РeTTA- a simple yet efficient solution that continuously assesses the model divergence for harmonizing the TTA process, balancing adaptation, and collapse prevention.
10

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Acknowledgements
This work was supported by the Jump ARCHES Endowment through the Health Care Engineering Systems Center, JSPS/MEXT KAKENHI 2024K20830, ROIS NIT Open Collaborative Research 2024-24$1201, in part by the National Institute of Health (NIH) under Grant R01 A1139401, and in part by Vingroup Innovation Foundation under Grant VINIF:221.DA0012
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Persistent Test-time Adaptation in Recurring Testing Scenarios Technical Appendices
Table of Contents
A Related Work
16
B Proof of Lemmas and Theorems
16
Proof Lemma
B.2 Proof of Lemma 2.
B.3 Proof of Theorem and Corollary 1.
18
C Further Justifications Gaussian Mixture Model Classifier
19
D Further Justifications on the Recurring Testing Scenario
20
D.1 Recurring TTA Follows the Design of Practical TTA Stream Recurring TTA as Diagnostic Tool.
20
20
D.3 Recurring TTA with Random Orders
20
E. Further Justifications Persistent TTA (PeTTA)
21
E.1
Pseudo Code
21
E.2
Anchor Loss
22
E.3
The Use of the Memory Bank
23
Empirical Mean and Covariant Matrix Feature Vectors on Source Dataset
23
E.5
Novelty of PeTTA .............
24
F
Additional Experimental Results of PeTTA
24
E11
Performance of РсTTA Versus Compared Methods
24
F2
An Inspection of PeTTA
25
F3
Does Model Reset Help?
25
PeTTA with Recurring Visits
27
F5
The Sensitivity Hyper-paranee parameter Choices in PeTTA
..................
27
F6
More Details Ablation Study
27
F7
More Confusion Matrices in Recurring TTA Setting
29
G Experimental Details
29
G.I Computing Resources
29
G.2 Experiments on CCC Testing Stream
29
G.3 Test-time Adaptation Methods
29
G.4 The Use of Existing Assets
15

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A
Related Work
Towards Robust and Practical TTA. While forming the basis, early single-target TTA ap- proaches [53, 57, 39, 41, 33] is far from practice. Observing the dynamic of many testing envi- ronments, continual TTA setting is proposed where an ML model continuously adapts to sequence of multiple shifts [36, 59]. Meanwhile, recent studies [15, 7] point out that the category distribution realistic streams highly temporally correlated. Towards real-world TTA setting, Yuan et al. [61] launch the practical TTA which considers the simultaneous occurrence of the two aforementioned challenges.
For robust and gradual adaptation, an update via the mean teacher [55] mechanism is exploited in many continual TTA algorithms [59, 61, 12, 22]. To moderate the temporally correlated test stream, common approaches utilize a small memory bank for saving a category-balancee subset of testing samples [15, 61], inspired by the replay methods [50, 2] to avoid forgetting in the task of continual learning [34, 3, 11J. Our study emphasizes another perspective: beyond supreme performance, desirable TTA should also sustain it for an extended duration.
Temporal Performance Degradation. By studying the quality of various ML models across multiple industry applications [56, 60] the issue of AI "aging" with the temporal model degradation progress. even with data coming from stable process has been confirmed. In TTA, the continuous changes of model parameters through gradient descent aggravate the situation, also recently noticed in [45]. Apart from observation, we attempt to investigate and provide theoretical insights towards the mechanism of this phenomenon.
Accumulated Errors in TTA. In TTA, the issue of accumulated error has been briefly acknowledged. Previous works strive avoid drastic changes model parameters good practice. Up some degree, it helps to avoid performance degradation. Nevertheless, it is still unclear whether their effectiveness truly eliminates the risk. To preserve in in-distritution performance, regularization [27, 40] or replaying of training samples at test-time [12] have been used. Other studies explore reset (recovering the initial model parameters) strategies [59, 45], periodically or upon the running entropy loss approaches a threshold [41]. Unfortunately, knowledge accumulated in the preceding steps will vanish, and bad heuristic choice of threshold or period leads to highly frequent model resets. Noteworthy, tuning those hyper-parameters is exceedingly difficult due the unavailability of the validation set [62]. LAME [7] suggests a post-processimg step for adaptation (without updating the parameters). This approach, however, still limits the knowledge accumulation. Our PeTTA is reset-free by achieving an adaptable continual test-time training.
B Proof of Lemmas and Theorems
In this section, we prove the theoretical results regarding the e-perturbed Gaussian Mixture Model Classifier (c (c-MMCC) introduced in Sec. 3.2. We first briefly summarize the definition of model collapse and the static data stream assumption:
Definition 1 (Model Collapse). A model is said to be collapsed from step E T,T T < oc if there exists non-empty subset of categories y C y such that Pr(Y, E y} > 0 but the marginal Pr(ý, } converges to zero in probability:
lim Pr(t E E 0. (-T
Assumption 1 (Static Data Stream). The marginal distribution of the true follows the same Bernoulli distribution Ber(po): Po,: = Po. (p1,: = P1 = =-
) Vt E T.
Preliminary. Following the same set of notations introduced in the main text, recall that we denoted Py,t _ Pr(Y, = :), P.. _ Pr(ý, = ツ} (marginal distribution of the true label Y, and pseudo label Y, receiving label y. respectively) and e, = Pr(Y, 1|9 0} (the false negative rate (FNR) of
16

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c- GMGMC). At testing step t, we obtain the following relations:
Ep. X (1 =) + 6x/01
(9)
En, X
11,,
(10)
V. (X,lÝ; =0 0) =(1 -11-e, e.o + te(1 e:) m1)²,
(11)
Varp, (xi 1 oj.
(12)
In addition, under Assumption 1, the marginal distribution (x( (also referred as data distribution in our setup) is:
P((x) (o; + PI + pioî + PoP1 (po - m1)2) E T.
(13)
B.1 Proof of Lemma
Lemma 1 (Increasing FNR). Under Assumption 1, a binary e-GMMC would collapsed (Def. 1)
with limp1, 0 (or limpu. 1, equivalently) if and only if lime; P1- 1-
Proof. Under Assumption 1, we have Ep, [X,] = PoHo + (1 po) Also note that:
Ep [X,] Ep,
Ep. [1111]]
Er, [x,Ii o] io.t + Er: [xll; 1
pi.
(14)
[(1 -Ee)uo + exjil + mi p0,z)
[(1 ( po) no + [1 - P1 (t) 1
Po/so + (1 P1)
where the second equality follows Eqs. 9-10. Therefore:
(15)
62
Eq. 15 shows positive correlation between Po,, and 6. Given lim = Pi, taking the limit introduces: t-+
limpo, lim Pd po 1.
6
p1
Similarly, having Jim the false negative rate when t - is:
lime, = Po P1-
Since Po. + fI.t 1, fi. = 0, equivalently. Towards the collapsing point, the model tends to
[-t predict a single label (class in the current setup). In addition, the FNR of the model e also raises correspondingly.
B.2 Proof of Lemma
Lemma 2 (e-GMMC After Collapsing). For binary e-GMMC model, with Assumption 1, if lim61. (collapsing), the cluster in GMMC converges in distribution single-cluster GMMC
(-r
with parameters:
Nfo, 0o,)) 4 N + P1P1, Poo piot + PoPi(po -1)3).
Proof. From Eqs. 10, under the increasing type II collapse of GMMC setting, the perturbation does not affect the approximation of μn. Meanwhile, when e, increases, one expect that foo,c
17

---

moves further away from fi toward Frist, the mean teacher model of GMMC (Eq. 4, main text) gives:
Ep. Po...Y, = 1 = Ep, [Ao,e-1]
Ep. / = 0] = =(1 (1- - a BP,, [Po,0-188 =0 o] + alEp, [x0l = of
= (1 AP | + [.p, Xi 0))
Ep. M = 1 = (1 - PP | 8 = 1 + a.Ep, [x,|{==1 =1
(1 a)BP. p.... [ + Ep.. X 11)
Ep, (22,0)) 0] Ep.., [
By defining Vy = Ep, Wy we obtain the following recurrence relation between do,t and u0
u Ep. Auu.1 o/ io.. Ep. Ro.. 1] $1.e
(1 a)mo,,-1 + alEp, [x,1r: = 01) io.t 101,
[11 a) P1.el 10,1-1 + o P. [x,1D= 0|
(1 afp,:) + BBr [ [x,18]= =0 0]
(1 o u0 + a [(1 E<) + expr).
(16)
Given limpo, 1, it follows that lim Pı by Lemma From this point:
u0.1 (1 a0 + a (pono + P1))
<L
Taking the limit t oc:
lim uo,t lim (1 ax)u0.t-1111111011000000 (polio + p1111)
10000
lim (1 ax) io.00 + a Σ - a)² (poHo + PIHI)
1000
lim (1 Pooo + (1 - (PUHo + P1/4))
1+00
= Poleo + P1
The second equation is obtained by solving the recurrence relation. When limmn, = 1, (igxlve(o.) (-+
becomes a deterministic values. Hence, giving uy, Er, [py.e] = io. 7) and
lim fo.: lim u0 = Potto + P1
(17)
8000 t1000
Repeating the steps above with Eqs. 11-12 in place of Eqs. 9-10, we obtain similar result for of.e
lim 00, po p1od + PoP1 (No p1)
(18)
By Lévy's continuity theorem (p. 302, [421), from Eqs. 17-18, when t + co, the estimated distribution of the first cluster when collapsing.
Mousterr((ss ((;; ō..)) converges the whole data distribution Px(x)) (Eq. 13)
B.3 Proof of Theorem 1 and Corollary 1.
Theorem I (Convergence of GMMC). For binary e-GMMC model, with Assumption let the distance from fo, toward Ln is d0-+1 = pEp, [0o.1 - -1), then:
0-1 0-1 < a Po (wo Hil
d
18

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Proof. Substituting Eq. 15 into Po,e of Eq. 16 gives:
1
apo
apo
[[L - ())0 + ep11
40,1-1
E2
Hence, we have the distance from Ma,e toward /112
oPo
appc:/41
No,t-1 + 7 apoo40 1 ct
M4
Et
apo
apoc//i
(u0,1-1 (1) + apooo
Et
1 6
6
0po/11 (1 6x)
(10 (1) apoop
Ct
Ce
apo
(u0,1-1 (1) app pam
Et
apo
|10 1uر + 10 p1/-
The last inequality holds due to the triangle inequality. Equivalently,
I0, - A1l 10,0-1 Mil & a Po (Ino 11l d0,1-1 H1
Let dg-+1 = EEp [Ã0.1] Mil. we conclude that:
251 - --111 S aa. po (11o m1l
Corollary 1 (A Condition for e e-GMMC Collapse). With po, a, po, Mi, G-GMMC is collapsed there exists a sequence of (EITT-A, 2 A, 0) such that:
d==1
P1
Im0 pil
-E
Proof. Initialized at Po. e-GMCC is collapsing when fo, converges to the mid-point PoP4 + P1
(Lemma 2), i.e., moving closer to fr. From Thm. 1, the distance towards μl di dp_t!
|M0 - Ail Uo,t-1 M1
|
u0,111 11l
< + |10 - mil <
E,
1-
Ng
iil
When there exists this sequence (ct) (r 2 A, > 0) it follows that d < di and > E is guaranteed Vt E [r -
[r, 7). Hence, limo P1 (model collapsed, by Lemma 1). t-+7
C Further Justifications on Gaussian Mixture Model Classifier
One may notice that in c-GMMC (Sec. 4.2), the classifier is defined f((x) = argmax yey Pr(xly: 0s) (maximum likelihood estimation) while in general, fe:x:) argmaxyey) Pr(y) (r; 0.) (maximum posterior estimation), parameterized by a neural network. In this case, since the equal prior (i.e., Pr(y; 0c) = Pr(y': (.), Vy. E C) is enforced in e-GMCC, the two definitions are equivalent.
Proof.
Having:
Pr(z y 0.) Pr(y: 0.)
argmax 2r(y(2;0)) argmax yey Eyey Pr(xly': 0,) Prly'se)
argmax vey Pr(cly: 0.).
We conclude that the two definitions are equivalent. fact, is well-known that maximum likelihood estimation is special case of maximum a posterior estimation when the prior is uniform.
D
19

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D Further Justifications on the Recurring Testing Scenario
D.1 Recurring TTA Follows the Design of a Practical TTA Stream
Note that in recurring TTA, besides the recurrence of envirouments (or corruptions) as in [59, the distribution of class labels is also temporally correlated (non-i.i.d.) as suggested by [15, 61] reflect the practical testing stream better. In short, recurring TTA is formed by recurring the environments of practical TTA scenario introduced in [61] multiple times (readers are encouraged visit the original paper for additional motivations on this scenario).
D.2 Recurring TTA as a Diagnostic Tool
Noticeably, CoTTA [59] also performed 10-round repetition across multiple domain shifts to simulate a lifelong TTA testing stream just like our recurring TTA. However, the key difference is CoTTA assumes the distribution of class labels is i.i.d., which does not hold in many real-life testing scenarios as argued in [15, 61J. Our recurring TTA lifts this assumption and allows temporally correlated (non-i.i.d.) label distribution (more challenging, more practical). This extension allows recurring TTA to spot the risk of model collapse on CoTTA [59] and other methods. The pver-simpliccty of the repeating scheme in CoTTA for spotting performance degradation is also suggested in [45]. Clearly, it seems not to be a problem at first glance in Tab. 5 of [59] (CoTTA's 10-round repetition), but in fact, the risk in COTTA remains, as explored in our scenario and also on CCC [45].
The construction of our recurring TTA is notably simple a technical effort to extend the testing stream. However, this simplicity is on purpose, serving as diagnostic tool for lifelong continual TTA. Counterintuitively. our experiments on four different tasks with the latest methods verify that even if the model is exposed to the same environment (the most basic case), their adaptability and performance are still consistently reduced (demonstrated visually in Fig. 1, quantitatively in Sec. 5.3).
We believe that the extensive testing stream by recurrence in setup is simple yet sufficient scenario to demonstrate the vulnerability of existing continual TTA methods when facing the issue of model collapse (compared to CCC [45], notably more complicated scenario than our recurring TTA). Indeed, recurring shifts are sufficient to show this failure mode and any lifelong TTA method should necessarily be able to handle recurring conditions.
D.3 Recurring TTA with Random Orders
Recall that in Sec. 3.1, recurring TTA is constructed by repeating the same sequence of D distributions K times. For example, a sequence with K 2 could be Pi Pa - - Pn + Pi - P2 + ..... ...... Pp. For simplicity and consistency that promote reproducibility. the same order of image corruptions (following [61]) is used for all recurrences. This section presents supplementary experimental findings indicating that the order of image corruptions within each recurrence, indeed, does not affect the demonstration of TTA model collapse and the performance of our PeTTA
We refer to the setting same-order as using one order of image corruptions in [61] for all recurrences (specifically, on CIFAR- 1AARR-10/10---6 and ImageNet-C: motion - snow fog + shot + defocus + contrast > zoom - brightmess + frost + elastic + glass + gaussian ipeg impulse). Conversely, in random-order, the order of image corruptions is randomly shuffled at the beginning of each recurrence. Hence, the corruption orders across K recurrences are now entirely different. We redo the experiment of the second setting three times (with different random seeds = 0, 1, 2). Nevertheless, different TTA methods are ensured to be evaluated on the same testing stream, since it fixed after generation. Without updating its parameters, the performance of the source model is trivially independent of the order of corruptions.
Experiment Setup.
+ pixelated
Experimental Result. The experimental results are visualized in Fig. 6. The first column plots the experiments under the same-order, while the remaining three columns plot the experiments in the random-order setting, with varying random seeds. Note that the message conveyed by each sub-figure entirely matches that of Fig. 1-right.
Discussions. Clearly, similar collapsing pattern is observed in all three TTA tasks, with three combinations of 20 image corruption orders. This pattern also matches the easiest setting using the same order of image corruptions promoted in recurring TTA.
20

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Same order
Randon-order (seed=0)
Random-order (seed=]]
Random-order (seed=2)
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0.5
Testing
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30
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20
Rocurring TTA
Recurring TTA
Recurring TTA
Recurring visit
(a) CIFAR-10 CIFAR-IO--- 10-C task.
Same-order
(seed=0)
(xeed!=])
Random (sced=2)
1.0
1.0
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Recurring TTA visit
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Recurring TTA visit
(b) CIFAR-100 CIFAR-100-CC task.
Same order
Randem-order (seed=0)
Randon-order (seed=1)
Randem-order (seed=2)
1.0
1.0
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Rocurring TTA visit
Recurring TTA visit
Recurring TTA visit
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(c) ImageNet
ImageNet-C task.
Figure 6: Recurring TTA with different order corruptions. This figure plots the testing error of two TTA approaches: RoTTA [61], and, PeTTA-- (ours), and source model- x- as a reference performance under our recurring TTA (with 20 visits) across three TTA tasks. On the same-order experiments (column 1), the same order of image corruptions is applied for all 20 visits. Meanwhile, in candom-order this order is reshuffled the beginning of each visit (columns 2-4). candom-order experiments are redone three times with different random seeds. Here, we empirically validate that using the same order of domain shifts (image corruptions) in our recurring TTA sufficient to showcase the model collapse and evaluate the persistence of our РеTTA. Best viewed in color.
E
Further Justifications on Persistent TTA (PeTTA)
E.1
Pseudo Code
We summarize the key steps of our proposed PeTTA in Alg. 1, with the key part (lines 4-13) highlighted in blue. Our approach fits well in the general workflow TTA algorithm, enhancing the regular mear-teucher update slep. Appdx. E.5 elaborates more on our contributions in PeTTA, distinguishing them from other components proposed in previous work. The notations and definitions of all components follow the main text (described in detail in Sec. 4). On line 8 of Alg. 1,
21

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Algorithm Persistent TTA (PeTTA)
Input: Classification model ft and its deep feature extractor de,, both parameterized by 0, e. Testing stream {Xx] initial model parameter (fb). initial update rate (ao). regularization term coefficient (Ap), empirical mean yeyr and covariant matrix ((2) Jyey) of feature vectors the training set, fi EMA update rate (V).
P < Po: Vy E ;
// Initialization
for f E [1, .......
,T ,
Y t fX-1
// Obtaining pseudo- labels for all samples in X
// Persistent TTA (PeTTA)
D 11, N,::
// Set of (unique) pseudo labels in X
Te 0;
for y E 34 do
7i' exp (- 1²)²² (E%) (pl (ox--(a) // Divergence sensing term
1D)):
category y
" - + 72
11 Average divergence sensing term for step t vÃo (X, = -): EMA update P for samples with
[D
10
P.. t (1-1) v)
Y, = y
11
end
d f- ÿt Ao
Computing adaptive regularization term coefficient
13
ar + (1 7r)) on
// Computing adaptive update rate
14
// Regular Nean-teacher Update
15
8, Optim Ep. LCLS (x; + CAL (X1;0) (+)(
// Student model
update
0, + (1 a, 0r))0-11 -0--1+
// Teacher model update
17
// Final prediction
yeild f((X:):
// Returning the final inference with updated model fa
19 end
shorthand notation, 0o,,, (X, y) denotes the empirical mean of all feature vectors of x(() (extracted by Φo._, (x,") ) 8((0) y. ,, , in the current testing batch.
E.2
Anchor Loss
KL Divergence Minimization-based Interpretation of Anchor Loss. In Sec. we claimed that minimizing the anchor loss CAL is equivalent to minimizing the relative entropy (or KL divergence) between the output probability of models parameterized by 6g and 0.
Proof. Having:
Pr(y]X,;00))
DrL (Pr()) = Σ Pr(y]XX 00) log
Pr(y/Xe:00
yEy
Pr(v 0o) log Pr Xx; 0) H (Pr(v) X1; 00))
yEY
(AI((x)))
Hence,
argmin (X,;0) = argmin DEL (Pr(y)Xx:0)))) Pr(y)) Xx; 0)).
850
22

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Intuitively, a desirable TTA solution should be able to adapt to novel testing distributions on the one hand, but should not significantly diverge from the initial model. CAI. fits this purpose, constraining the KI. divergence between two models at each step.
Connections between Anchor Loss and Regularizer Term. While supporting the same objective (collapse prevention by avoiding the model significantly diverging from the source model), the major difference between Anchor loss (CAl) and the Regularizer term (R(0) that the anchor loss operates on the probability space of model prediction while the regularizer term works on the model parameter spaces. Tab. 4 (lines and 5) summarizes the ablation study when each of them is eliminated. We see the role of the regularization term is crucial for avoiding model collapse, while the anchor loss guides the adaptation under the drastic domain shift. Nevertheless, fully utilizing all components is suggested for maintaining TTA persistence.
E.3 The Use of the Memory Bank
The size of Memory Bank. The size of the memory bank in PeTTA is relatively small, equal the size of one mini-batch for update (64 images, specifically).
The Use of the Memory Bank in PeTTA is Fair with Respect To the Compared Methods. Our directly comparable method RoTTA [61] also takes this advantage (referred to as eategory-balancee sampling, Sec. 3.2 of [61]). Hence, the comparison between PeTTA and RoTTA is fair in terms of additional memory usage. Noteworthy, the use of a memory bank a common practice in TTA literature (e.g.. [15, 8, 61]), especially in situations where the class labels are temporally correlated or non-i.i.d. distributed (as we briefly summarized in Appdx. A - Related Work section). COTTA [59], EATA [40] and MECTA [22] (compared method) assume labels are i.i.d. distributed. Hence, memory bank is unnecessary, but their performance under temporally correlated label distribution has dropped significantly as a trade-off. The RMT [12] (compared method) does not require memory bank but it needs to cache a portion of the source training set for replaying (Sec. 3.3 in [12]) which even requires more resources than the memory bank.
Eliminating the Need for Memory Bank. As addressing the challenge of temporally correlated label distribution on the testing stream is not the focus of PeTTA we have conveniently adopted the use of the memory bank proposed in [61]. Since this small additional memory requirement not universally applied in every real-world scenario, we believe that this is reasonable assumption, and commonly adopted in TTA practices. Nevertheless, exploring alternative ways for reducing the memory size (c.g., storing the embedded features instead of the original image) would be an interesting future direction.
E.4 Empirical Mean and Covariant Matrix of Feature Vectors on the Source Dataset
Two Ways of Computing 1e and Σg in Practice. One may notice that in PeTTA, computing 7 requires the pre-computed empirical mean (1²) and covariance (E of the source dataset. This requirement may not be met in real-world situations where the source data is unavailable. In practice, the empirical mean and covariance matrix computed on the source distribution can be provided in the following two ways:
1. Most ideally, these values are computed directly by inference on the entire training set once the model is fully trained. They will be provided alongside the source-distribution pre-trained model as a pair for running TTA.
2. With only the source pre-trained model available, assume we can sample set of unlabeled data from the source distribution. The (pseudo) labels for them are obtained by inferring from the source model. Since the source model is well-performed in this case, using pscudo is approximately as good as the true label.
Accessing the Source Distribution Assumption in TTA. In fact, the second way is typically
assumed to be possible in previous TTA methods such as EATA [40], and MECTA [22] (a compared method) to estimate Fisher matrix (for anti-forgetting regularization purposes). Our work- - PeTTA follows the same second setup the previous approaches mentioned above. A variation of RMT [12] (a compared method) approach even requires having the fully labeled source data available at test-time for source replaying (Sec. 3.3 of [12]). This variation is used for comparison in experiments.
23

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We believe that having the empirical mean and covariant matrix pre-computed on a portion of the source distribution PeTTA reasonable assumption. Even in the ideal way, revealing the statistics might not severely violate the risk of data privacy leakage or require notable additional computing resources.
Number of Samples Needed for Computation. To elaborate more on the feasibility of setting (2) mentioned above, we perform small additional experiment on the performance of PeTTA while varying the number of samples used for computing the empirical mean and covariant matrix on the source distribution. In this setting, we use the test set of CIFAR-IO, CIFAR-100, DomainNet validation set of ImageNet (original images, without corruption, or the real domain test set of DomainNet),, representing samples from the source distribution. The total number images is 10, 000 in RIFAR-10/A00 50, 000 in ImageNet, and 69, 622 in DomainNet. We randomly sample 25%, 50%, 75%, and 1 100% of the images this set to run РeTTA for 20 rounds of recurring. The result is provided in Tab. 6 below.
Table 6: Average classification error of PeTTA (across 20 visits) with varying sizes of source samples used for computing feature empirical mean (142) and covariant matrix (E)).
TTA Task
25%
50%
75%
100 %
CIFAR- 10 CIFAR-IO-G
22.96
22.99
23.03
22.75
CIFAR-100 CIFAR-100-C
35.01
35.11
35.09
35.15
DomainNet: real clip + paint sketch
43.18
43.12
43.15
42,89
ImageNct ImageNet-C.
61.37
59.68
61.05
60.46
The default choice of PeTTA is using 100% samples of the validation of the source dataset. However, we showcase that it is possible to reduce the number of unlabeled samples from the source distribution to compute the empirical mean and covariant matrix for PeTTA, without significantly impacting its performance.
E.5
Novelty of PeTTA
PeTTA is composed of multiple components. Among them, the anchor loss is existing idea (examples of previous work utilizing this idea are [32, 12]). Similarly, the mean-teacher update; and regularization are well-established techniques and very useful for the continual or gradual TTA scenario. Hence, we do not aim to improve or alternate these components.
Nevertheless, the novelty of our contribution is the sensing of the divergence and adaptive model update, in which the importance of minimizing the loss (adaptation) and regularization (collapse prevention) is changed adaptively. In short, propose harmonic way of combining those elements adaptively to achieve a persistent TTA process.
The design of PеTTA draws inspiration from a theoretical analysis (Sec. 3.2), empirically surpassing both the conventional reset-based approach [45] (Appdx. F.3) and other continual TTA approaches [61, 12, 59, 22, 7] on our proposed recurring TTA (Sec. 3.1, Appdx. F.1), as well as the previously established CCC [45] benchmark.
F
Additional Experimental Results of PeTTA
F.1 Performance of PeTTA Versus Compared Methods
Performance on CIFAR-100-C and Domainnet Datasets. Due to the length constraint, the classification errors the tasks CIFAR- 100 -O0--CIFF---0--C and real clipart, painting. sketch of DomainNet are provided in Tab. and Tab. 8. To prevent model collapse, the adaptability of PeTTA more constrained. As result, it requires more time for adaptation initially (e.g.. in the first visit) but remains stable thereafter. Generally, consistent trends and observations are identified across all four TTA tasks.
Standard Deviation of PeTTA Performance Across Multiple Runs. For PeTTA experiments marked with (*) in Tab. 1, Tab. 2, Tab. 7, and Tab. 8, the average performance across five independent runs with different random seeds is reported. Due the space constraint, the corresponding standard deviation values now reported in Tab. 9. Generally, the average standard deviation across runs
24

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Table 7: Average classification error of the task CIFAR-100 CIFAR-100-C in recurring TTA scenario. The lowest error is highlighted in bold, (+) average value across 5 runs (different random seeds) reported for PeTTA.
Recurring
Method
LAME [?]
40.5
40.5
CaTTTA [59]
53.4
58.4
63.4
67.6
71.4
74.8
18.2
81.1
84.0
96.7
55.8
90.7
92.3
93.5
94.7
95.6
96.3
97.0
97.3
97.6
83.1
EATA [40]
88.5
95.0
96.8
97.3
97.4
97.2
97.2
97.3
97.4
97.5
97..5
97.5
97.6
97.77
97.7
97.77
97.5
97.8
97.7
97.7
96.9
RMT 1120
53.5
48.5
47.9
47.4
47.3
47.1
46.9
46.9
46.5
46.8
45.7
46.5
46.5
46.6
46.5
46.5
46.3
46.5
46.3
46.5
47.1
MECTA [22]
44.8
44.3
44.6
43.1
44.8
44.2
44.4
43.8
43.8
43.9
44.6
43.5
44.4
44.6
43.9
64.2
43.8
44,4
44.9
44.2
44.2
KoTTA [61]
35.5
25.2
38.5
41.9
45.3
49.2
$2.0
55.2
58.1
61.5
64.6
67.5
70.7
73.2
75.4
77.1
79.2
$1.5
82.8
54.5
61.4
R.annd [45]
36.7
36.7
36.6
36.5
36.7
36.8
96.7
56.5
36.6
36.5
36.7
36.6
56.5
36.7
36.5
36.6
36.6
56.7
36.6
36.5
36.6
RDID [37]
76.4
76.4
76.2
76.2
76.3
76.1
75.9
76.1
76.3
76.3
76.6
76.3
76.8
76.
76.6
76.3
76.2
76.0
75.9
76.0
76.3
TRIBE 1:23
33.8
33.3
35.3
34,9
35.3
38.1
37.1
57.2
37.
36,1
39.2
41.
43.0
43.1
45.1
45.1
45.0
44.9
44.9
44.9
39.6
352
35.3
552
352
352
Table 8: Average classification error of the task real clipart - painting - sketch on DomainNet dataset in recurring TTA scenario.
Apiradlc 77A via
Method
Avg
LAME [?]
CUTTAA
98.5
RMT
76.1
76.0
MECTA
98.4
99.0
RoTTA
68.0
82.6
44.3
44.3
44.3
44.4
(oury)"
42.
stays within +..1% % for small datasets (CIFAR-IO-C, CIFAR-100-C) and +-0.5% for larger datasets (ImageNet-C, DomainNet).
Table 9: Mean and standard deviation classification error of PeTTA the four datasets: CIFAR-10-C (CF-10-C), CIFAR-100-C (CF-100-C), DomainNet (DN), and ImageNet-C (IN-C) with recurring T7A scenario. Each experiment run times with different random seeds.
Recurring TTA
Datusel
6
8
10
12
14
16
18
20
Avg
CF-10-C
24.3
23.0
22.4
22.4
22.5
22.5
22.8
22.7
22.9
22.7
22.8
23.0
+0.4
+-0.3
+0.
0.3
0.3
0.3
0.4
0.2
+03
0.4
0.4
40.2
+0.1
0.3
+0.5
4..2
0.2
0.3
+0.4
00.5
+0.1
CH-100-C
34.4
35.0
35.2
10.4
10.4
102
20.2
10.
10.1
202
10.
10.1
0.2
10.1
20.
102
0.
10.1
20.1
10.1
20.1
10.2
202
10.1
42.6
42.
42.
430
430
43.0
42.9
10.1
10.11
10.2
10.2
10.3
10.3
103
10.4
10.
10.4
104
10.4
10.4
0.3
103
10.2
10.
10.3
1033
10.3
10.3
IN-C
61.7
59.1
59.6
59.3
60.0
61.0
60.7
60.4
60.7
60.7
60.2
+0.6
+0.5
+0.5
+0.5
+1.
+11
+1.
+0.5
+0.8
0.9
+0.
+0.8
+0.9
0.8
0..9
+0.88
110
+0.6
+0.6
+0.7
+0.5
F.2
An Inspection of PeTTA
In Fig. 7, we showcase an inspection of our PeTTA on the task CIFAR-I0 + CIFAR-IO-C [19] in a typical recurring TTA with 20 visits. Specifically, the visualizations of PeTTA parameters (ft- Nx, and az), adaptation losses (CCLS, LAL) and regularization term (R(#)) are provided. Here, we observe the values of adaptive parameters X and a, continuously changing through time, as the testing scenarios evolve during recurring TTA. This proposed mechanism stabilizes the value of the loss functions, and regularization term, balancing between the two primary objectives: adaptation and preventing model collapse. Thus, the error rale persists result. A similar pattern is observed on other datasets (CIFAR- CCIFAR-100-CI [19] and DomainNet [44]).
Does Model Reset Hclp?
Experiment Setup. We use the term "model reset" to represent the action of "reverting the current TTA model to the source model". This straightforward approach named RDumb [45]. We thoroughly conducted experiments to compare the performance of RDumb with PeTTA The implementation RDumb this setting as follows. We employ RoTTA [61] as the base test-time adaptor due to the characteristies of the practical TTA [61] stream. The model (including model
25

---

parameters, the optimizer state, and the memory bank) is reset after adapting itself to T images. For each dataset, three values of this syper-parameter T are selected:
T 1, 000: This is the value selected by the RDumb's authors [45]. Unless specifically stated, we use this value when reporting the performance of RDumb [45] in all other tables.
T = 10, 000 (EFAR-10/100-C), T = 5, 000 (ImageNet-C) and = 24, 237 (Domain- Net). This value is equal to the number of samples in the test set of a single corruption rype, i... the model is reset exactly after visiting each P; 's (see Sec. 3.1 for notations). For DomainNet [44], since the number of images within each domain is unequal, the average number of images is used instead.
T 150, 000 (CIFAR- SCIFAR-10/100-0- 10/100--), T 75, 000 (ImageNet-C) and T 72, 712 (Domain- Net). This number is equal to the number of samples in one recurrence of our recurring TTA, i... the model is reset exactly after visiting Pi -.... - Pp. Here, D 15 ttppss of corruptions [19] for CIFAR-I0/100- and ImageNet-C and D = for DomainNet (clipart, painting, sketch). For example, the model is reset 20 times within a recurring 7TA setting with 20 recurrences under this choice of T.
The second and the last reset scheme could be interpreted as assuming the model has access to oracle model with capability of signaling the transitions between domains, or recurrences. Typically, this is an unrealistic capability in real-world scenarios, and a desirable continual TTA algorithm should be able to operate independently without knowing when the domain shift happening.
Experimental Results. An empirical comparison between RDumb [45] and our PeTTA are reported in Tab. 10, Tab. 11, Tab. 12 and Tab. 13 for all four tasks.
Table 10: Average classification error comparison between RDumb [45] (a reset-based approach) with different reset frequencies and our PeTTA on CIFAR-10 + CIFAR-101 task.
Recoriing
mar
Resel Every
12
15
Avg
26.1
24.1
24.8
24.7
PeTTA (
243
23.0
2266
23.4
224
23.5
223
22.5
238
228
33,7
227
229
22.
22.7
22.6
228
22.9
230
22.8
Table 11: Average classification error comparison between RDumb [45] (a reset-based approach) with different reset frequencies and our PeTTA on CIFAR-100-6 dataset.
Recoriing risit
Every
1000
36.7
36.7
36.5
10000
43.4
150000
35.3
(M
Table 12: Average classification error comparison between RDumb [45] (a reset-based approach) with different reset frequencies and our PeTTA on NomainNet dataset.
Recarding
Reset Every
12
15
1000
44.3
44.4
463
44.
46.
44.
463
44.5
46.4
442
46.3
44.
44.
463
44.
463
44.2
46.3
44.4
46.3
44.3
T-
2:337
44.1
44.3
439
44.
46.1
44.3
463
44.4
44
44.1
44.0
44.
44.1
44.0
44.0
443
44.1
46.1
44.1
44.1
44.1
7- 72712
44.
24.3
440
443
44.1
44.3
44.
44.4
44.
44.1
44.0
44.1
442
44.1
44.1
44.1
44.1
44.0
44.0
44.1
44.2
PeTTA (
438
433
42.8
428
42.9
43.0
4300
428
42.9
Discussions. Across datasets and reset frequencies, PeTTA approach is always better than RDumb [45]. The supreme performance holds even when RDumb has access to the oracle information that can reset the model exactly at the transition between each domain shift recurrence. Importantly, this oracle information typically unavailable in practice.
A slight abuse of notation. T here is the number of images between consecutive resets, following the notation on See. [45]. not the sample indices in our notations. 'A subset of 5, 000 samples from ImageNet-C are selected following RobustBench [10] for consistent evaluation with other benchmarks.
26

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Table 13: Average classification comparison between RDumb [45] (a reset-based approach) with different reset frequencies and our PeTTA on ImageNet- C- dataset.
Recorning
Every
(owzg)
65.3
Noteworthy, it is clear that the performance RDumb varies when changing the choice of the reset frequency. For given choice of T, the better performance one dataset does not guarantee the same performance on other datasets. For example, = 1, 0.0 the best empirical value found by RDumb authors [45] on CCC, does not give the best performance on our recurring TTA scenario; the second choice T negatively impact the performance on many tasks; the third choice gives the best results, but knowing this exact recurrence frequency of the testing stream is unrealistic. The result highlights the challenge in practice when tuning this parameter (too slow/frequent), especially the TTA setting where validation set is unavailable. Our PeTTA, in contrast, is reset-free.
F.4 PeTTA with 40 Recurring Visits
To demonstrate the persistence of PeTTA over an even longer testing stream, in Tab. 14 and Fig. we provide the evaluation results of PeTTA on recurring with 40 recurrences.
F5 The Sensitivity of Hyper-parameter Choices in PeTTA
Table 15: Sensitivity PeTTA with different choices of An.
Dataset
le
5e
A
lel
Àg
5e
Ap
1e2
-O-C
22.9
22.7
22.8
23.2
24.1
CIFAR 100-C
35.7
35.3
35.1
35.6
36.
ImageNet-C
61.2
61.0
60.5
61.3
62.4
There are two hyper-parameters in PeTTA: og and Ag- The initial learning rate cip = 1e-3 is used for all experiments. We do not tune this hyper-parameter, and the choice of is universal across all datasets, following the previous works/compared methods (c.g., RoTTA [61]. CoTTA [59]).
Since Ao is more specific to PeTTA, we included sensitive analysis with different choices of Ao on РeTTA, evaluated with images from RIFAR-10/100--1 and ImageNet-C in Tab. 15. Overall, the choice of Ag is not extremely sensitive, and while the best value is 1e on most datasets, other choices such as 5e or 5e also produce roughly similar performance. Selecting Àg intuitive, the larger value of λo stronger prevents the model from collapsing but also limits its adaptability as trade-off.
In action, Åo mechanism in PeTTA, meaning it does not require careful tuning. More generally, this hyper- parameter can be tuned similarly the hyper-parameters of other TTA approaches, via additional validation set, or some accuracy prediction algorithm [29] when labeled data is unavailable.
initial value and will be adaptively scaled with the sensing model divergence
F.6 More Details on the Ablation Study
We provide the detailed classification error for each visit in the recurring TTA setting of each row entry in Tab. 4 (PeTTA Ablation Study): Tab. 16, Tab. 17, Tab. 18, Tab. 19; and Tab. 5 (PeTTA with various choices of regularizers): Tab. 20, Tab. 21, Tab. 22, Tab. 23.
Fig. presents an additional examination of the ablation study conducted on the task CIFAR- + CIFAR- 100-C [19] for our PeTTA approach. We plot the classification error (top) and the value of 7t (bottom) for various РeTTA variations. As the model diverges from the initial state, the value of 7i increases. Unable to adjust aG or constraint the probability space via CAI. limits the ability of PeTTA to prevent model collapse. In all variations with the model collapse in ablation studies, the rapid saturation of 7, is all observed. Therefore, incorporating all components in PeTTA is necessary.
27

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Table 16: Average classification error of multiple variations of PeTTA. Experiments CIFARIO CIFARIO-C [19] task.
Aptoodl
Method
Baseline R(())
23.5
334
156
38,0
46.0
42.6
Rig hxed 0.1.c
23.5
24.0
21.2
29.8
534
153
37.9
40.5
43.3
45.3
45.8
49.3
50.9
510
52.1
53.2
56,0
54.8
56,9
57.6
43.3
RL9 haed
23.5
23.6
26.2
25,4
31.5
33.
36.4
38.7
41.1
43.1
44.8
47.5
49.3
48.5
50.9
52.1
53.1
54.2
55.6
57.0
42.0
2499
25.3
26,4
27.2
26.5
27.2
17.7
27.8
25.0
27.4
270
27.6
27.1
PeTTA- A
25.5
34.5
13.7
23.1
23.2
12.4
23.3
23.2
23.3
36.1
23.9
24.5
34.3
34.0
23.8
23.9
238
34.1
34.6
34.7
23.0
PeTTA- A Eat
23.3
23.0
24.6
25.3
26.2
25.9
26.4
26.6
26.9
26.6
26.7
26.7
36.7
26.8
26.8
27.2
26.9
26.9
26.8
27.0
26.2
PcTTA n. Fax
24.3
230
22.6
22.4
13.
22.5
22.3
22.5
22,8
22.8
22.6
22.7
22.7
22.9
22.6
22.7
22.6
22.8
22.9
23,0
21.8
Table 17: Average classification error of multiple variations of PeTTA. Experiments CIFAR-100 + CIFAR100-C [19] task.
Epiindie
Method
Bascline w/o (2(9)
40.2
46.3
51.2
54.
57.3
99.4
61.3
62.6
63.9
65.1
66.3
67.1
68.1
68.9
60.6
70.3
71.1
71.6
72.4
72.9
63.0
129) fezel 0.13g
40.5
46.1
51.5
55.1
56.2
60.5
62.6
64.2
65.7
67.3
68.6
69.5
70.6
71.8
72.5
73.4
74.
14.9
75.
76.5
63.0
R19 lival
41.8
47.6
52.5
56.1
56.9
60.7
62.5
63.9
65.0
66.2
67.1
68.3
69.5
70.3
71.4
72.4
73.
74.1
75.0
73.6
64.6
PcTTA- PeTTA- A
39.4
43.4
46.5
49.1
51.0
53.6
53.8
54.7
55.7
56.5
57.1
57.7
58.3
58.8
59.3
59
80.6
61.0
61.6
62.1
55.0
19.4
40.1
40.8
40.7
41.2
41.5
41.4
41.6
41.5
41.5
41.7
41.5
41.8
R1.7
41.8
42.0
41.9
41.9
42.0
61.8
41.4
PeTTA- A Cas
16.2
35.6
35.7
16.1
36.2
30.4
16.4
58.5
10.3
76.2
35.6
16.3
36.5
36.6
16.3
36.6
36.5
16.3
36.3
16,1
36.1
PeTTA by oz Cat.
358
34.4
54.7
35,0
35.1
35.1
35.2
35.3
35.3
35.3
35.2
35.3
35.2
35.2
35.1
35.2
35.3
35.2
35.2
35.3
35.1
Table 18: Average classification error of multiple variations of PeTTA. Experiments on real + clipart, painting, sketch task from DomainNet [44] task.
Recuring TMA
Method
10
14
15
16
20
Ang
Usseline wi R(9)
523
69.0
68.5
38.0
69.8
10.5
718
734
75.6
776
7588
51.
828
84.5
53.9
87.4
W5.5
89.9
908
V2.1
77.9
R19) fised 0.13g
525
70.0
09.5
70.0
71.1
72.
76.6
76.1
77.8
30.4
81.9
53.5
85.2
87.2
59.1
90.2
91.5
93.1
95.1
94.9
80.0
R19) Beed A- Ag
54.6
698
63.7
360
61.7
76.4
70.4
02.
58.2
7600
73.5
06.3
58.6
62.3
50.3
75.5
60.0
59.9
59.3
78.3
66.6
PeTTA A
49.2
64.5
62.4
20.
59.6
58.6
57.
57.8
57.6
57.7
55.0
58.5
5900
59.5
59.8
61.1
62.0
62.6
63.6
649
59.7
PaTTA- A
43.
42.5
42.3
42.3
42.6
41.8
43.1
43.7
43.0
44.3
44.6
45.1
45.4
45.7
45.7
46.
46.1
46.2
46.5
46.4
44.5
PeTTA- A Eat
43.6
42.5
43.5
42.6
42.9
43,0
43.3
43.4
43.1
43.2
43.1
43.3
43.3
43.2
43.2
43.9
43.7
43.0
45.
43.5
43.2
fat.
42.9
42.9
Table 19: Average classification error of multiple variations of PeTTA, Experiments on ImageNet ImageNet-C [19] task.
Recarring 7TA rill
Method
5
6
10
11
12
14
15
16
17
19
20
Avg
Bascline wio R(()
66.9
61.9
72.7
03.6
97.4
97.8
95.0
95.2
98.3
98.3
95.4
95.4
98.5
98.5
95.6
98.5
98.6
95.6
95.7
96.7
93.4
R(9)
(.I.A,
00.3
96.9
07.9
98.7
R(9) fina
97.0
97.7
97.9
PeTTA-X: A
63.0
62.1
76.3
96.7
97.0
06.9
96.9
96.9
97.0
97.1
07.0
97.2
97.0
97.1
97.1
97.0
97.0
97.0
07.0
97.0
92.7
PeTTA- X
6488
30.5
76.6
75.8
73.5
75.8
76.
16.2
70.2
78.3
70.7
71.0
76.9
714
71.1
77.3
72.2
71.4
71.4
73.7
PeTTA- X Car.
648
61.1
59.8
60.4
08.4
61.2
61.2
61.6
61.
02.1
61.3
60.1
62.1
62.8
62.7
02.1
60.0
PeTTA (owrs!"))
63.3
61.7
59.1
59.4
59.6
59.8
400
68.3
63.0
60.7
68.6
60.7
60.8
60.7
68.4
60.3
60.5
Table 20: Average classification enror of РеТTA with various choices of regularizers, Experiments on
CIFAR-10 CIFAR- -IFAR-IO-C [19] task.
Eptrodic voll
Method
7
17
19
L2
24.8
23.2
2300
22.7
22.7
22.7
22.5
22.2
218
L2sPisher
25.
23.7
22.5
22.3
22.3
228
22.
23.9
226
23.3
22.7
Cosine
24.3
23,0
22.6
22.4
22.4
22.5
22.3
23.5
328
22.8
12.6
22.7
20.7
32.9
22.6
22.7
22.
22.8
22.9
23.0
22.8
Fishr
23.8
216
21,0
210
21.
22.4
22.6
226
22.8
23.3
22.6
Table 21: Average classification error of PeTTA with various choices of regularizers. Experiments on CIFAR-100 - CIFAR-100-C (19 [19] task.
Recurring TTA vbir
Method
4
5
7
9
10
11
12
14
13
16
17
19
20
Avg
1.2
35.5
35.7
35.6
35.4
35.6
35.5
35.7
35.7
3588
35.4
35.5
L24Pisher
36.8
25.4
35.4
35.3
35,9
36.0
25.9
35.9
35.9
35.8
36.1
36.1
36.1
36.1
36.1
36.1
36.2
36.11
36.0
33.9
36.0
Cosine
34.4
35.0
35.1
35.2
35.1
36.2
Fisher
35.2
35.
36
360
36.0
360
360
35.
359
360
28

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Table 22: Average classification of PeTTA with various choices of regularizers. Experiments real - clipart, painting, sketch task from DomainNet [44] dataset.
Recurting
Method
19
Avg
124Ficher
Table 23: Average classification enror of PeTTA with various choices of regularizers. Experiments on
ImageNet ImageNet-C [19] task.
Recariing TTA sint
Method
7
11
12
13
14
16
17
19
20
Avg
12
70.8
72.2
71.5
69.8
22.3
69.3
703
70.5
70.0
70.8
70.2
72.1
71.4
10.8
70.9
70.9
69.7
71.0
71.1
7044
20.8
124Ficher
70.5
70.0
69.5
69.4
09.6
69.9
89
69.3
72.
70.4
71.0
70.5
71.7
71.5
71.3
68
68.6
68.8
68.7
88.
70.0
Cosine
65.3
61.7
918
59.1
99.4
59.6
59.8
913
994
68.0
68.3
618
60.7
68A
09.6
64.7
608
68.7
60.
68.2
68.5
Fisher
68.1
61.7
60.9
612
61.9
62.6
62.8
63.2
642
63.4
64.3
64.
63.9
64.3
858
65.5
64.9
65.0
65.2
65.2
63.8
E.7 More Confusion Matrices in Recurring TTA Setting
For the task CIFAR-I0 - CIFAR-IO-C [19] in recurring TTA setting (with 20 visits), we additionally showcase the confusion matrix of RoTTA [61] (Fig. 10) and our proposed РeTTA (Fig. 11) each visit. Our РеТTА persistently achieves competitive performance across 20 visits while RoTTA [61] gradually degrades.
G Experimental Details
G.1 Computing Resources
A computer cluster equipped with an Intel(R) Core(TM) 3.80GHz 17- CPU, 64 GB RAM, and one NVIDIA GeForce RTX 3090 GPU (24 GB VRAM) is used for our experiments.
G.2 Experiments on CCC Testing Stream
In this section, we further evaluate the performance of our PeTTA on the testing data stream of Continuous Changing Corruption (CCC) [45] setting. Here we use the baseline accuracy 20%, transition speed 1000, and random seed 44,3 The compared methods are source model (ResNet 50), PeTTA, RoTTA [61], and RDumb [45]. Noteworthy, different from recurring TTA, the class labels here i..dd distributed. The adaptation configuration of РсТTA follows the same settings used on ImageNet-C, while the same setting introduced in Sec. F3, with T 1000 used for RDumb [45].
G.3 Test-time Adaptation Methods
Pre-trained Model Source Distribution. Following previous studies [57, 61, 12, 59], only the batch norm layers are updated. As stated in Sec. 5.2, RobustBench [10] and torchvision [35] provide pre-trained models trained source distributions. Specifically, for ImageNet-C and Do- mainNet experiments, a ResNet50 model [17] pre-trained on ImageNet V2 (specifically, checkpoint ReaNet50 Beellett__Weights._.. IMAGENET1K
torchvision) is used. From RobustBench, the model ResNNett [20] are adopted for CIFARIO-G
with checkpoint Standard and and CLFARR100-CE experiments, respectively. Lastly, experiments on DomainNet dataset utilize the checkpoint
provided in AdaContrast [8] study.
Optimizer. Without specifically stated, Adam [26] optimizer with learning rate equal 1 and 3 (0.9, 0.999) is selected as a universal choice for all experiments.
More Details on PеTTA. Since designing the batch normalization layers, and the memory bank is not the key focus of PeTTA, we conveniently adopt the implementation the Robust Batch Norm layer and the Category- balanced Sampling strategy using memory bank introduced in RoTTA [61].
29

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G.4 The Use of Existing Assets
Many components of PeTTA is utilized from the official repository of ROTTA [61] $ and RMT [12]. These two assets are released under MIT license. All the datasets, including CIFAR-10-C CIFAR- 100-C and ImageNet-C [19] are publicly available online, released under Apache-2.0 license." DomainNet dataset [44] (cleaned version) is also released for research purposes.
hnstpe://githab.ccom//ee htw.a//...cce/eeeeccrca//ooaaaaa
hhtpps:/aai bu.eu////D//
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1,00
0.80
0.40
1000
20000
30000
40000
Test-time udaptation step (+)
10.00
8.00
6.00
4.00
0
10000
20000
30000
40000
adapeation step (f)
1e-3
0.75
0.50
0.25
0.00
1000
20000
30000
400000
udaptation step (+)
4.00
3.00
2.00
10000
200000
30000
400000
udaptation step (+)
10.00
7.50
5.00
2.50
0.00
30000
40000
Teat-time adapcation step (f)
2.00
1.00
10000
20000
30000
40000
adaplation step (()
1.00
Testin etror
0.75
0.50
M
0.25
0.00
20000
40000
Test-time udaptation step (+)
Figure 7: An inspection of PeTTA on the task CIFAR-10 - CIFAR 10-C [19] in a recurring with 20 visits (visits separated by the vertical dashed lines). Here, we visualize (rows 1-3) the dynamic of PeTTA adaptive parameters (5, At, (r),, (rows 4-5) the value of the loss functions (CCLs. CAL) and (row 6) the value of the regularization term (R(O)) and (row 7) the classification error rate each step. The solid line in the foreground of each plot denotes the running mean. The plots show an adaptive change of λτ, Or through time РeTTA, which stabilizes TTA performance, making РеTТA achieve a persisting adaptation process all observed values across 20 visits.
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Testing enor
50
CIFAR-10-C
CIFAR 100
40
ImageNetc
Total Visits
CF-10-C
CF-100-C
IN-C
visits
22.8
30
40 visits
22.9
35.1
61.0
Table 14: Average testing error of PeTTA in recurring TTA with 20 and 40 visits. PеTTA demonstrates its persistence over extended testing time horizon beyond the 20th visit milestone (Fig. 8's horizontal dashed line).
10
20
30
40
Recurring IIA visit
Figure 8: Testing error of PeTTA with 40 recurring TTA visits.
PoTTA
PeTTA
CAL
+a,
Baseline wio (6)
R(d) fixed == 0.1Ae
R(5) fixed
0.80
0.70
Tosling Eiror
0.50
0.40
0.30
10000
20000
30000
40000
udaptation step (+)
PeTTA A
PeTTA A
PoTTA d CAT.
PoTTA A
CAT.
1.00
0.80
0.60
0.40
0.20
20000
30000
40000
Test-Lime adaptation stop (4)
Figure 9: An inspection on the ablation study of multiple variations PeTTA on the task CIFAR 100 + CIFAR-100- [19] in episodic TTA with 20 visits (visits separated by the vertical dashed lines), (top): testing error of multiple variations of PeTTA. The performance of PeTTA without (w/o) R()), fixed regularization coefficient (A Ao/0.11a) degrades through time (the top lines). The degradation of PeTTA -A, is still happening but at a slower rate (justification below). The performance of the other three variations persists through time with PeTTA -λ₃ + + CAL achieves the best performance. (bottom): changes of 7, in multiple variations of PeTTA. When limiting the degree of freedom in adjusting ar lacking of supervision from CAL (e.g., PeTTA -λ, di, PeTTA - + LAL. and especially PеTTA -Xt), the value 7t, unfortunately, escalates and eventually saturated. After this point, PeTTA has the same effect using fixed regularization coefficient. Therefore, fully utilizing all components is necessary to preserve the persistence of РeТTA. Best viewed in color.
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2
visit
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Figure 10: The dynamic of the confusion matrix of RoTTA [61] in episodic TTA with 20 visits.
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2
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Figure 11: The dynamic of the confusion matrix of PTTTA (ours) in episodic TTA with 20 visits.
34

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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: We have highlighted the three main claims and contributions of our work in both the abstract (highlighted in bold font) and the introduction section (listed as bullet points).
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The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or
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The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
is fine to include aspirational goals as motivation as long as is clear that these goals are not attained by the paper.
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Justification: We have discussed the limitations and potential future work of our study in Sec. Specifically, three main limitations are included: (1) Collapse prevention can not be guaranteed through regularization, PeTTA requires (2) the use of a relatively small memory bank is available and (3) the empirical mean and covariant matrix of feature vectors on the source dataset is computable. We also include discussions in Appdx. E.3 and Appdx. E.4 to further elaborate (2), and (3) respectively.
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3. Theory Assumptions and Proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
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[Yes]
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The answer NA means that the paper does include theoretical results. All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
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Does the paper fully disclose all the information needed to reproduce the main ex- perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided not)?
Question:
Answer: [Yes]
Justification: This study propose new TTA approach PeTTA. full description of this approach is given in Sec. with its pseudo-code provided in Appdx. E. The implementation PeTTA in Python is also attached as supplemental material. Additionally, Sec. 5.2 and Appdx. G are dedicated to providing further implementation details for reproducing the main experimental results. Lastly, the construction of recurring TTA is notably simple, and can be easily extended to other TTA streams. Its configuration on each tasks is described in the Recurring TTA paragraph of Sec. 5.2.
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36

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(d)
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37

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Answer: [Yes]
Justification: Due to the limited computing resources, we only extensively evaluate the performance of proposed method (PeTTA) across 5 independent runs, with different random seeds. Specifically, the mean values in runs are reported in Tab. 1, Tab. 2, Tab. 7, and Tab. 8. The corresponding standard deviation values provided in Appdx. F.1.
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Reviews:
[{"review_id": "WCUALOc799", "reviewer": "Reviewer_rAs4", "summary": "The authors proposed a practical TTA scenario called recurring TTA and, within this scenario, suggested the best-performing TTA methodology, which they named persistent TTA (PeTTA), measured by various benchmark performances.", "questions": "1. The authors said EATA is the baseline. However, I couldn't find any results of EATA.\n2. It seems that the images on the left and right in Figure 4(c) are switched.\n3. How about applying only $\\mathcal{L}\\_\\mathrm{AL}$ without $\\lambda$ in Table 3?", "limitations": "Yes", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The proposed recurring TTA scenario reflects the challenging and practical situation well.\n2. PeTTA is a simple yet effective solution. Especially, PeTTA excels in collapse prevention.", "weaknesses": "1. I think Corollary 1 is not a rigorous condition for model collapse. Of course it guarantees the decrease of distance ($d_t^{0 \\rightarrow 1}$), the distance could converge to another value because $\\lim_{t \\rightarrow \\tau}\\epsilon_t$ is not always $p_1$. Just saying 'The model collapse happens when this condition holds for a sufficiently long period' is not proper.\n2. I think the role of $\\mathcal{L}\\_\\mathrm{AL}$ is similar to $\\mathcal{R}(\\theta')$, but $\\mathcal{L}\\_\\mathrm{AL}$ is not controlled by $\\lambda$. However, in Table 3, adding $\\mathcal{L}\\_\\mathrm{AL}$ leads to significant performance improvement in several benchmarks. The authors should provide the difference between $\\mathcal{L}\\_\\mathrm{AL}$ and $\\mathcal{R}(\\theta')$, and explain why they chose the current design.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Comments on the Weaknesses Part:**\n1. In Lemma 1, we mathematically showed that under Assumption 1, we have $\\lim_{t \\rightarrow \\tau}\\epsilon_t = p_1$.  Furthermore, the convergence of $\\epsilon_t$ to $p_1$, and the collapsing behavior when $\\epsilon_t$ selected following Collary 1 are both empirically validated through a numerical simulation in Sec. 5.1. Since the rate of collapsing depends on various factors, including both data-dependent and algorithm-dependent, the Corollary 1 here holds as a generic condition for model collapsing. Nevertheless, we will further explore specific conditions/settings that can make this statement more rigorous in future work. \n\n\n2. The motivation, and reasoning for our design choice behind the anchor loss and the regularization term are detailed in lines 684-691 of Appendix E.2. While using anchor loss is beneficial on many benchmarks, it is not sufficient on its own to achieve PeTTA’s performance, as shown in Table I of the rebuttal PDF. In response to reviewer MNYK, we expanded our evaluation to include TRIBE (AAAI'24), a more recent robust TTA algorithm that also utilizes a concept similar to anchor loss. Despite demonstrating better adaptability, this method is still prone to model collapse over a longer time horizon, necessitating the exploration of additional strategies beyond a simple anchor loss.\n\n**Comments on the Questions Part:**\n1. All tables show the performance of MECTA with EATA backbone (line 242). MECTA is an advanced version of EATA, with all the components preserved, and the batch normalization blocks are replaced with MECTA blocks, showing higher efficacy. For completeness, the experiments of a standalone EATA adapter are also included in Table III-V of this rebuttal PDF. In short, *EATA still suffers from performance degradation in the recurring TTA setting just like MECTA and other methods*.\n\n2. Yes, thank you for catching that. This figure will be updated in the revised paper.\n\n3. In Table I, we provided an additional ablation study where a baseline model is trained with and without the anchor loss (no regularization). The results show that while having initial benefits on some benchmarks, *trivially applying the anchor loss alone is incapable of eliminating* the lifelong performance degradation in continual TTA.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pTmuY5ophe", "reviewer": "Reviewer_oiWU", "summary": "The paper provides theoretical and empirical analyses on the error accumulation and model collapse in continuous TTA scenarios. From the analyses, the authors discover the risk of using constant key hyperparameters ($\\alpha$ and $\\lambda$ in RoTTA) and periodic reset of model parameter (in RDumb). They propose Persistent Test-time Adaptation (PeTTA) with adaptive $\\alpha_t$ and $\\lambda_t$ depending on distribution shifts and anchor loss $\\cal{L}_{AL}$. The proposed method shows lower and stabler error over the continuous distribution shifts.", "questions": "The following questions include the concerns in the weakness.\n\n- Can PeTTA  work under scenarios of (i) non-cyclic domain shift; and (ii) label distribution shift?\n\n- Can you provide further results on longer time horizon than 20? I want check if the error accumulation is fully resolved by PeTTA.\n\n- Is the performance of PeTTA sensitive to the choice of $\\alpha_0$ and $\\lambda_0$? If so, how can we select them in practice?\n\n- How did you choose the hyperparameters of PeTTA for the experiment result? Were the hyperparameters of other algorithms tuned in the same way of PeTTA?\n\n- Why there is no balancing hyperparameter in front of $\\cal{L}_{AL}$?\n\n- Is PeTTA runnable without the access of source dataset?", "limitations": "- In the current manuscript, the limitation of this work is not described in detail. It would be great if you can provide more specific description on the direction of further improvement.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper considers a practical scenario of TTA, continuous TTA.\n\n - The theoretical analysis explains an interesting outcome of error accumulation in TTA, called model collapse: a collapsed model is prone to misclassifying several classes as a few classes. This is also verified in an experiment.\n\n - Based on the findings of model collapse, the authors propose a mechanism of detecting divergence of $\\theta_t$ and adaptively selecting $\\alpha_t$ and $\\lambda_t$, which is well justified and verified.\n\n - The authors provide an extensive set of experiments demonstrating their findings and the superiority of the proposed method, compared to the state of the arts: RoTTA and RDumb.", "weaknesses": "- The paper focuses solely on recurring TTA scenarios, potentially overlooking other types of domain shifts such as non-cyclic domain shifts and label distribution shifts. The proposed method appears too specific to recurring TTA scenarios and may be prone to failures, particularly under label distribution shifts.\n\n- The sensitivity of hyperparameter choices, particularly $\\alpha_0$ and $\\lambda_0$, is not studied. Additionally, there is insufficient justification for the hyperparameter choices of other algorithms. This raises concerns about the empirical study's claim of the proposed method's superiority.\n\n- The proposed method requires access to the source dataset, as indicated in Equation 6.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Comments on the Weakness Part:**\n1. We would like to emphasize that recurring TTA serves as a *diagnostic tool* for catching the lifelong performance degradation of continual TTA, and even *in this simplest case, several SOTA continual TTA algorithms fail to preserve their performance*. This raises awareness in the community when evaluating their methods (visit Appendix D.2. for further discussion). Extending our recurring TTA to include more challenging scenarios and complex types of shifts is necessary, but should be addressed in the following work.\n2. We have provided additional justifications and discussions on the choices of hyper-parameters for consideraiton. See the comments below.\n3. See the comments below.\n\n**Comments on the Questions Part:**\n1. Yes, (i) our method can handle non-cyclic domain shift. PeTTA does not make any assumptions or utilize this property of the testing stream. As an example, it achieves good performance on CCC [1]  where the corruptions are algorithmically generated, non-cyclic with two or more corruption types can happen simultaneously - see the Appdx. F.4. Regrading (ii), for all experiments, the label distribution is temporally correlated (non-iid) following [2, 3]. The class distribution within each data batch can be highly imbalanced, with some classes dominating the others. The robustness of our PeTTA with label distribution shifts is demonstrated up to this extent.\n\n2. Absolutely, PeTTA is experimented with 40 recurrences in Tab. II and Fig.I (a). The experimental results confirm the persistence of PeTTA. Additionally, the performance of PeTTA over an extended time horizon is presented in Table 13 (Appendix F4). In this case, the model is adapted to over 5.1 million images, which is significantly more than the default 20 recurrences.\n\n3. In PeTTA, $\\alpha_0=1e^{-3}$ is the initial learning rate for adaptation. We do not tune this hyper-parameter, and the choice of $\\alpha_0$ is universal across all datasets, following the previous works/compared methods (e.g., RoTTA, CoTTA). \n\nSince $\\lambda_0$ is more specific to PeTTA, we included a sensitive analysis with different choices $\\lambda_0$ on CIFAR-10/100-C and ImageNet-C in Table VI for the sake of time. Overall, the choice of $\\lambda_0$ is not extremely sensitive, and while the best value is $1e^1$ on most datasets, other choices such as $5e^0$ or $5e^1$ also produce roughly similar performance.  Selecting $\\lambda_0$ is intuitive, the larger value of $\\lambda_0$ stronger prevents the model from collapsing but also limits its adaptability as a trade-off.\n\nIn action, $\\lambda_0$ is just an initial value and will be adaptively scaled with the sensing model divergence mechanism in PeTTA, meaning it does not require careful tuning. More generally, the choice of this hyper-parameter can be tuned similarly to the hyper-parameters of other TTA approaches, via the use of an additional validation set, or some accuracy prediction algorithm [4] when labeled data is not available.  \n\n4. Except for the recurring testing condition, each recurrence follows the standard continual TTA established in previous studies. Hence, for all compared methods, we use the best parameters provided by the authors. The performance after the first visit is manually verified to ensure the reproducibility of the original work. Noteworthy, the primary observation of this work is to determine how long these approaches can sustain their initial performance.\n\n5. Since the purpose of the anchor loss is to guide the adaptation under the drastic domain shift (Lines 684-691, Appendix E.2), we empirically found out that it is unnecessary to introduce an additional hyper-parameter and let the adaptive regularization term take the leading role of collapse prevention.\n\n6. No, PeTTA requires sampling from the source dataset to perform. Nevertheless, Appdx. E4 demonstrates that only a small number of samples is required for reliably estimating the empirical mean and covariance matrix. PeTTA relies solely on the typical assumptions used in the other methods it is compared against (e.g., EATA, RMT). Please visit Appendix E.4 for our discussions on the feasibility of accessing the source dataset.\n\n[1] Robust Test-Time Adaptation in Dynamic Scenarios, CVPR'23.\n\n[2] RDumb: A simple approach that questions our progress in continual test-time adaptation, NeurIPS'22.\n\n[3] Gong et al., NOTE: Robust Continual Test-time Adaptation Against Temporal Correlation, NeurIPS'22.\n\n[4] Lee et al., AETTA: Label-Free Accuracy Estimation for Test-Time Adaptation, CVPR’24.\n\n**Comments on the Limitations Part:**\t\nWe thank the reviewer for the suggestion. Appendix E elaborates some limitations mentioned in Section 6 in detail. The direction of further improvement will be included in the revised paper.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "485j9vzHPD", "reviewer": "Reviewer_MNYK", "summary": "This paper investigates the risk behind long-term test time adaptation. To achieve this, the authors simulate a long-term test data stream called Recurring Test-Time Adaptation by repeating a single period continual TTA setting for 20 times, and propose a Persistent TTA (PeTTA). Within the proposed algorithm, the authors calculate the Mahalanobis distance between the feature distributions at the current time slot and before test-time adaptation to weight the regularization term and model updated momentum. Moreover, the authors also leverage an Anchor Loss in the probability space to further avoid over-adapting process. To validate the proposed method, the authors conduct the experiments on several TTA benchmark datasets, e.g. CIFAR10-C and ImageNet-C. The results demonstrate the effectiveness of the proposed method which provides stable adapting process in long-term test time adaptation scenarios.", "questions": "See the weakness section above.\nBy the way, how about the performance of feature alignment based TTA method e.g. [D] in the recurring TTA scenario.\n\n[D] Revisiting Realistic Test-Time Training: Sequential Inference and Adaptation by Anchored Clustering Regularized Self-Training. TPAMI 2024.", "limitations": "There is not nearly enough technical novelty, and there is a lack of discussion and comparison of some of the competing methods that should be compared.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. This paper investigates a realistic and valuable TTA setting, whereas the test data stream is enough long and most of the existing TTA methods based on self-training self-supervised objectives produce poor results.\n2. This paper is well-written and sounds interesting.\n3. The authors attempted to theorize the causes of long-term TTA failures.", "weaknesses": "1. The novelty is incremental. Methodologically, the Anchor Loss proposed in this paper is similar to that used in [A] (compare eq.8 in the manuscript with eq. 5 in [A]), only replacing L2 distance between two probabilities to Cross Entropy. In the ablation study, I observe most of the promotion is provided by this Anchor Loss module on CIFAR100, DN and IN-C datasets. Based on these, I suggest the authors to provide some experimental comparisons on these two TTA method and more discussions on the differences between them.\n2. The danger brought by self-training based TTA is well-known [B], while there is no denying that the author may be correct in making the theoretical analysis in the manuscript. There are many TTA methods proposed to alleviate this issue, for example, Balanced BatchNorm proposed by [A] alleviates the impact from the biased class to batchnorm, class diversity weighting utilized in [C] avoid the accumulation of prediction bias during test time. Unfortunately, they are not discussed and compared in the experiments of the manuscript.\n\n[A] Towards Real-World Test-Time Adaptation: Tri-Net Self-Training with Balanced Normalization, AAAI 2024.\n\n[B] On Pitfalls of Test-time Adaptation, ICLR 2023.\n\n[C] Universal Test-time Adaptation through Weight Ensembling, Diversity Weighting, and Prior Correction, WACV 2024.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Comments on the weaknesses part:**\n1. *We respectfully disagree with the comment about the novelty since this study is not all about the anchor loss*. Indeed, we *acknowledged the anchor loss is not a new idea or our novel point* on line 751 (Appendix E5) and will include a proper citation to the anchor network in [A] for completeness. The anchor loss alone is insufficient to avoid the model collapsing as we responded to Reviewer rAs4. Nevertheless, the main contributions of this paper are the *theoretical analysis* and the *sensing of divergence for an adaptive model update* in PeTTA. We also acknowledged that the model collapsing is observed in previous studies, as an inspiration for improving the performance but to the best of our knowledge, there is no theoretical investigation on this phenomenon. The suggestion of comparing PeTTA and [A] is interesting. We conducted additional experiments and provided the discussion in the following comment.\n\n2. We sincerely appreciate the reviewer for suggesting recent methods such as ROID (WACV'24) [C] and TRIBE (AAAI'24) [A]. We have benchmarked their performance in Table III-V. Even though TRIBE - a SOTA model can provide stronger adaptability, outweighing the PeTTA model, and baseline RoTTA in the first several recurrences, *the risk of the model collapsing still presents in TRIBE* when increasing the observation period as demonstrated in the case of CIFAR-10-C (Fig. I(b)). Nevertheless, this result *underscores the importance of our proposed recurring TTA* setting for an extended testing stream evaluation. We also experimented with the class diversity weighting utilized in ROID [C], unfortunately, it cannot handle the temporally correlated testing stream in the recurring/practical TTA as PeTTA, and tend to be collapsed at the beginning. This is consistent with the finding by ROID's authors in Tab. 4 - ablation study of [C]. *ROID with class diversity weighting cannot handle the practical TTA scenario and falls behind PeTTA on all benchmarks*.\n\nSince PeTTA utilizes RoTTA as a baseline approach, TRIBE - a much better baseline would be interesting for us to combine the amazing adaptability of TRIBE and the collapsing prevention of PeTTA to create a stronger baseline in future work. Overall, further evaluations of PeTTA against the most recent approaches in this rebuttal highlight (1) the value of recurring TTA in spotting the lifelong performance degradation of continual TTA and (2) the novel design of the sensing model divergence and adaptive update in PeTTA, inspired by a theoretical analysis to address the collapsing phenomenon of the TTA model. The revised paper will include a discussion of these methods and an experimental comparison with them as outlined in the rebuttal PDF.\n\n**Comments on the questions part:**\n\nWe appreciate the reviewer's suggestion regarding the feature-based alignment TTA mentioned in [D]. While sharing several similarities in the design, this work does not directly study the performance degradation of continual TTA methods as PeTTA. Nevertheless, evaluating the performance of this approach under our recurring TTA setting is interesting and straightforward due to the simplicity of our setting. We will mention [D] in the revised paper and leave the evaluation for future work due to the length of the rebuttal period and since [D] has just officially appeared on the 8/2024 issue of TPAMI 2024.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "PC1Mbduoc1", "reviewer": "Reviewer_JUpp", "summary": "The paper proposes a novel method called Persistent Test-time Adaptation (PeTTA), aimed at addressing the gradual performance degradation of models when used for long-term test-time adaptation (TTA). Traditional TTA methods adapt to continuously changing environments but fail to account for the cumulative errors and performance drops that can occur when these environments reoccur over time. The study demonstrates, through the simulation of a simple Gaussian Mixture Model classifier, how TTA methods might gradually fail in such recurring environments. PeTTA achieves a balance between adaptation and preventing model collapse by monitoring the model's tendency to crash and adjusting the adaptation strategy accordingly, significantly improving the model's stability and performance in long-term testing.", "questions": "Please see the weaknesses.", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "(1)\tThe study introduces a new test scenario—recurring TTA—which realistically simulates conditions that might be encountered in real-world applications. This setup is more practical than traditional static test scenarios and helps reveal potential long-term issues with TTA methods.\n(2)\tBy using simulations with Gaussian Mixture Model classifiers, the study thoroughly analyzes the reasons for performance degradation in TTA methods and proposes a theoretically supported solution. PeTTA's stable performance across multiple benchmarks validates its effectiveness.", "weaknesses": "(1)\tAlthough the recurring test scenario has theoretical significance, in practical applications, environmental changes involve more than just lighting conditions; factors such as shooting angles and weather conditions also impact the captured data. Therefore, a simple Gaussian Mixture Model classifier may not fully simulate the variations in real complex scenarios.\n(2)\tWhile virtual data experiments provide theoretical support, whether their results fully apply to real scenarios requires further validation. Using real images for experiments in theoretical analysis would better validate the practical applicability of the PeTTA method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "1. We agree with the reviewer that the real-world scenarios are significantly more complicated. Despite its simplicity, our $\\epsilon$-GMMM model can *empirically demonstrate the behavior of a collapsing TTA model on a real-world CIFAR-10-C dataset* (see the similarity between Fig. 3(a) and Fig. 4(a)). To the best of our knowledge, this is the first attempt to establish a theoretical foundation for studying the collapse of TTA in the simplest case, promoting future research on the theoretical aspects of the collapsing TTA model. \n\n2. Yes, a theoretical study on real images would certainly validate their practical applicability. However, the challenge of modeling real images lies in the difficulty of theoretically analyzing the effect of added noise as it propagates through a highly complex machine-learning model. The idea behind $\\epsilon$-GMMM is to simplify this complex process where the update rule at each step is rigorously defined. Nevertheless, the *inspiration gained from this simple study* facilitates the development of PeTTA, which *demonstrated its ability to meet several real-world continual TTA benchmarks.*", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank the reviewers for their insightful comments and valuable feedback on our work. During the rebuttal period, we have extensively conducted additional experiments: benchmarking the performance of EATA and the most recent continual TTA methods (ROID (WACV’24), and TRIBE (AAAI’24)). The persistence of PeTTA is further justified over a recurring TTA with 40 visits (twice longer than previous experiments). Furthermore, we provided a discussion on the role of anchor loss $\mathcal{L}_{AC}$ and sensitivity analysis on the choice of $\lambda_0$ - a PeTTA’s hyper-parameter. Lastly, we offered a point-by-point response to each reviewer's comment below.

Meta-review: The authors present a method for avoiding model collapse in continuous test-time adaptation. The approach is validated on a variety of small and larger-scale datasets from CIFAR to ImageNet scale, and backed up by theoretical investigation. The reviewers found the approach "simple yet effective", and positively note the clarity and relevance of the paper. Following the author rebuttal, all reviewers suggest at least borderline acceptance.

I agree with the reviewers and recommend acceptance, on the condition of including the additional results provided during the rebuttal into the manuscript. I would also recommend the authors to revisit whether some of the appendix tables containing results on larger scale datasets (like CCC), which stress the practical applicability of the method on real-world datasets, could be moved into the main paper. Likewise, the authors should consider how to integrate the most important results from their rebuttal document into the paper, vs. defaulting to moving all new results to the supplement. Again, I would aim to stress in particular larger scale datasets in the main paper.
Decision: Accept (poster)