Title: Persistent Test-time Adaptation in Recurring Testing Scenarios
Venue: NeurIPS 2024
Accepted: yes
Keywords: domain adaptation

test-time adaptation

continual adaptation

performance degradation

self-supervised learning
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

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 ϵ -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 .

1 Introduction

Machine learning (ML) models have demonstrated significant achievements in various areas [18, 38, 47, 23]. Still, they are inherently susceptible to distribution-shift [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 ML 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 a simply ideal adaptation scenario where the test samples come from a fixed single domain [57, 39, 41]. As a result, such an assumption is far from the everchanging and complex testing environments. To confront continually changing environments [59, 12], Yuan et al. [61] proposed a practical TTA 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, a condition which does not hold true. Taking a surveillance camera as an example, it might accommodate varying lighting conditions recurringly day after day (Fig. 1-left). Based on this reality, we hypothesize that the recurring of those conditions may reveal the error accumulation phenomenon in TTA, resulting in performance degradation over a long period. To verify our hypothesis, we simulate a

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recurring testing environment and observe the increasing error rate by recurringly adapting to the test set of CIFAR-10-C [19] multiple times. We showcase the testing error of RoTTA [61] after 20 cycles of adaptation in Fig. 1-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 a 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, as an algorithm might work well in the first place and gradually degenerate. Therefore, ensuring sustainable quality is crucial for real-world applications, especially given the recurring nature of testing environments.

This study examines whether the adaptability of a TTA algorithm persists over an extended testing stream. Specifically, in the most basic scenario, where the model returns to a previously encountered testing environment after undergoing various adjustments. We thus propose a more general testing scenario than the practical TTA [61], namely recurring TTA , where the environments not only change gradually but also recur in a correlated manner over time. We first analyze a simulation using the ϵ -perturbed Gaussian Mixture Model Classifier ( ϵ -GMMC) on a synthesized dataset and derive a 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 a testing scenario - recurring TTA , a 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 ϵ -GMMC, shedding light on dataset-dependent and algorithm-dependent factors that contribute to the error accumulation during TTA processes.

Third, we introduce persistent TTA (PeTTA) - a simple yet effective adaptation scheme that surpasses all baseline models and demonstrates a 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 f t : X → Y with parameter θ t ∈ Θ (parameter space) gradually changing over time ( t ∈ T ) that maps an input image x ∈ X to a category (label) y ∈ Y . Let the capital letters ( X t , Y t ) ∈ X × Y denote a pair of random variables with the joint distribution P t ( x , y ) ∈ P d , t ∈ T . Here, P d belongs to collection of D sets of testing scenarios (domains) {P d } D d =1 . The covariate shift [46] is assumed: P t ( x ) and P t ′ ( x )

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could be different but P t ( y | x ) = P t ′ ( y | x ) holds ∀ t = t ′ . At t = 0 , θ 0 is initialized by a supervised model trained on P 0 ∈ P 0 (source dataset). The model then explores an online stream of testing data. For each t > 0 , it receives X t (typically in form of a batch of N t testing samples) for adapting itself f t -1 → f t before making the final prediction f t ( X t ) .

TTA with Mean Teacher Update. To achieve a stable optimization process, the main ( teacher ) model f t are updated indirectly through a student model with parameters θ ′ t [57, 61, 12, 15, 55]. At first, the teacher model in the previous step introduces a pseudo label [28] ˆ Y t for each X t :

With a classification loss L CLS (e.g., cross-entropy [16]), and a 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 θ t -1 :

with α ∈ (0 , 1) - the update rate of EMA, and λ ∈ R + - 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 t 's can be partitioned by t d 's in which { P t } t d t = t d -1 ⊂ P d . Here, each partition of consecutive steps follows the same underlying distribution which will change continually through D domains [59] ( P 1 →P 2 · · ·→P D ). 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 P t ( y ) = 0 , ∀ y ̸∈ Y t ⊂ Y even though the category distribution over all batches are balanced. Optimizing under this low intra-batch diversity ( |Y t | ≪|Y| ) 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 a 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 testing scenario based on practical TTA [61]. Conducting a single pass through D distributions, as done in earlier studies [61, 59], may not effectively identify the degradation. To promote consistency, our recurring TTA performs revisiting the previous distributions K times to compare the incremental error versus the previous visits. For example, a sequence with K = 2 could be P 1 →P 2 →··· → P D →P 1 →P 2 →··· → P D . Appdx. D extends our justifications on constructing recurring TTA.

Definition 1 ( Model Collapse ) . A model is said to be collapsed from step τ ∈ T , τ < ∞ if there exists a non-empty subset of categories ˜ Y ⊂ Y such that Pr { Y t ∈ ˜ Y} > 0 but the marginal Pr { ˆ Y t ∈ ˜ Y} converges to zero in probability:

Here, upon collapsing, a model tends to ignore almost categories in ˜ Y . As it is irrecoverable once collapsed, the only remedy would be resetting all parameters back to θ 0 .

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 a TTA algorithm is challenging. Therefore, we propose a theoretical analysis on ϵ -perturbed binary Gaussian Mixture Model Classifier ( ϵ -GMMC) that shares the typical characteristics by construction and demonstrates the same collapsing pattern in action (Sec. 5.1) as observed on real continual TTA processes (Sec. 5.3).

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Figure 2: ϵ -perturbed binary Gaussian Mixture Model Classifier, imitating a continual TTA algorithm for theoretical analysis. Two main components include a pseudo-label predictor (Eq. 1), and a mean teacher update (Eqs. 2, 3). The predictor is perturbed for retaining a false negative rate of ϵ t to simulate an undesirable TTA testing stream.

Simulated Testing Stream. Observing a testing stream with ( X t , Y t ) ∈ X × Y = R ×{ 0 , 1 } and the underlying joint distribution P t ( x, y ) = p y,t · N ( x ; µ y , σ 2 y ) . The main task is predicting X t was sampled from cluster 0 or 1 (negative or positive). Conveniently, let p y,t ∆ = P t ( y ) = Pr( Y t = y ) and ˆ p y,t ∆ = Pr( ˆ Y t = y ) be the marginal distribution of the true label Y t and pseudo label ˆ Y t .

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 f t into a maximum likelihood estimation f t ( x ) = argmax y ∈Y Pr( x | y ; θ t ) with Pr( x | y ; θ t ) = N ( x ; ˆ µ y,t , ˆ σ 2 y,t ) . The goal is estimating a set of parameters θ t = { ˆ µ y,t , ˆ σ 2 y,t } y ∈Y . A perfect classifier θ 0 = { µ y , σ 2 y } y ∈Y is initialized at t = 0 . For the consecutive steps, the simplicity of GMMCallows solving the Optim (for finding θ ′ t , Eq. 2) perfectly by computing the empirical mean and variance of new samples, approximating E P t . The mean teacher update (Eq. 3) for GMMC is:

The update of ˆ σ 2 y,t is similar. ˆ Y t = f t -1 ( X t ) can be interpreted as a pseudo label (Eq. 1).

ϵ -GMMC. Severe distribution shifts or low intra-batch category diversity of recurring TTA/practical TTA both result in an increase in the error rate of the predictor . Instead of directly modeling the dynamic changes of p y,t (which can be complicated depending on the dataset), we study an ϵ -pertubed GMMC ( ϵ -GMMC), where p y,t is assumed to be static (defined below) and the pseudolabel 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

be the false negative rate (FNR) of the model at step t . Without loss of generality, we study the increasing type II collapse of ϵ -GMMC . By intentionally flipping the true positive pseudo labels in simulation, an FNR of ϵ t is maintained (Fig. 2).

Assumption 1 ( Static Data Stream ) . The marginal distribution of the true label follows the same Bernoulli distribution Ber( p 0 ) : p 0 ,t = p 0 , ( p 1 ,t = p 1 = 1 -p 0 ) , ∀ t ∈ T .

Lemma 1 ( Increasing FNR ) . Under Assumption 1, a binary ϵ -GMMC would collapsed (Def. 1) with lim t → τ ˆ p 1 ,t = 0 (or lim t → τ ˆ p 0 ,t = 1 , equivalently) if and only if lim t → τ ϵ t = p 1 .

Lemma 1 states the negative correlation between ˆ p 1 ,t and ϵ t . 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 p 1 .

Lemma 2 ( ϵ -GMMC After Collapsing ) . For a binary ϵ -GMMC model, with Assumption 1, if lim t → τ ˆ p 1 ,t = 0 (collapsing), the cluster 0 in GMMC converges in distribution to a single-cluster GMMC with parameters:

Lemma 2 states the resulting ϵ -GMMC after collapsing. Cluster 0 now covers the whole data distribution (and assigning label 0 for all samples). Furthermore, collapsing happens when ˆ µ 0 ,t moves toward µ 1 . We next investigate the factors and conditions for this undesirable convergence.

Theorem 1 ( Convergence of ϵ -GMMC ) . For a binary ϵ -GMMC model, with Assumption 1, let the distance from ˆ µ 0 ,t toward µ 1 is d 0 → 1 t = | E P t [ˆ µ 0 ,t ] -µ 1 | , then:

From Thm. 1, we observe that the distance d 0 → 1 t 's converges (also indicating the convergence to the distribution in Lemma 2) if d 0 → 1 t < d 0 → 1 t -1 . The model collapse happens when this condition holds for a sufficiently long period.

Corollary 1 ( ACondition for ϵ -GMMCCollapse ) . With fixed p 0 , α, µ 0 , µ 1 , ϵ -GMMCiscollapsed if there exists a sequence of { ϵ t } τ τ -∆ τ ( τ ≥ ∆ τ > 0 ) such that:

Corollary 1 introduces a condition ϵ -GMMC collapse. Here, ϵ t 's are non-decreasing, lim t → τ ϵ t = p 1 .

Remarks. Thm. 1 concludes two sets of factors contributing to collapse: (i) data-dependent factors : the prior data distribution ( p 0 ), the nature difference between two categories ( | µ 0 -µ 1 | ); and (ii) algorithm-dependent factors : the update rate ( α ), the FNR at each step ( ϵ t ). ϵ -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 to mitigate model collapse. The theoretical results in the previous section explain the rationale behind these effective strategies.

Regularization Term for θ t . Knowing that f 0 is always well-behaved, an attempt is restricting the divergence of θ t from θ 0 , e.g. using R ( θ t ) ∆ = ∥ θ 0 -θ t ∥ 2 2 regularization [40]. The key idea is introducing a penalty term to avoid an extreme divergence as happening in Thm. 1.

Memory Bank for Harmonizing P t ( x ) . Upon receiving X t , samples in this batch are selectively updated to a memory bank M (which already contains a subset of some instances of X t ′ , t ′ < t in the previous steps). By keeping a balanced number of samples from each category, distribution P M t ( y ) of samples in M is expected to have less zero entries than P t ( y ) , making the optimization step over P M t more desirable. From Thm. 1, M moderates the extreme value of the category distribution ( p 0 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 ϵ t (Eq. 5) is not computable without knowing the true labels, the measure of divergence from the initial distribution (analogously to d 0 → 1 t -1 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 ϵ t 's in Corollary 1 to prevent the model collapse. In the mean teacher update, the larger value of λ (Eq. 2) prioritizes the task of preventing collapse on one hand but also limits its adaptability to the new testing environment. Meanwhile, α (Eq. 3) controls the weight on preserving versus changing the model from the previous step. Drawing inspiration from the exploration-exploitation tradeoff [49, 25] encountered in reinforcement learning [54], we introduce a mechanism for adjusting λ and α on the fly, balancing between the two primary objectives: adaptation and preventing model collapse . Our strategy is prioritizing collapse prevention (increasing λ ) and preserving the model from previous steps (decreasing α ) when there is a significant deviation from θ 0 .

In [40, 61, 59], λ and α were fixed through hyper-parameter tuning. This is suboptimal due to varying TTA environments and the lack of validation set [62]. Furthermore, Thm. 1 suggests the convergence rate quickly escalates when ϵ t increases, making constant λ, α insufficient to prevent collapse.

Sensing the Divergence of θ t . We first equip PeTTA with a mechanism for measuring its divergence from θ 0 . Since f t ( x ) = argmax y ∈Y Pr( y | x ; θ t ) , we can decompose Pr( y | x ; θ t ) = [ h ( ϕ θ t ( x ))] y , with ϕ θ t ( · ) is a θ t -parameterized deep feature extractor followed by a fixed classification head (a linear and softmax layer) h ( · ) . The operator [ · ] y extracts the y th component of a vector.

Since h ( · ) remains unchanged, instead of comparing the divergence in the parameter space ( Θ ) or between the output probability Pr( y | x ; θ t ) and Pr( y | x ; θ 0 ) , we suggest an inspection over the feature embedding space that preserves a 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 z = ϕ θ t ( x ) , we keep track of a collection of the running mean of feature vector z : { ˆ µ y t } y ∈Y in which ˆ µ y t is EMA updated with vector z if f t ( x ) = y . The divergence of θ t at step t , evaluated on class y is defined as:

where µ y 0 and Σ y 0 are the pre-computed empirical mean and covariant matrix of feature vectors in the source dataset ( P 0 ). 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 to align feature vectors of new domains back to the source domain ( P 0 ). Therefore, the accumulated statistics of these feature vectors at each step should be concentrated near the vectors of the initial model. The value of γ y t ∈ [0 , 1] is close to 0 when θ t = θ 0 and increases exponentially as ˆ µ y t diverging from µ y 0 .

Adaptive Regularization and Model Update. With α 0 , λ 0 are initial values, utilizing γ y t derived in Eq. 6, a pair of ( λ t , α t ) is adaptively chosen at each step:

ˆ Y t is a set of unique pseudo labels in a testing batch ( ˆ Y ( i ) t is the i th realization of ˆ Y t ).

Anchor Loss. Penalizing the divergence with regular vector norms in high-dimensional space ( Θ ) is insufficient (curse of dimensionality [5, 51]), especially with a large model and limited samples. Anchor loss L AL can nail down the similarity between f t and f 0 in the probability space [32, 12]:

which is equivalent to minimizing the KL divergence D KL (Pr( y | X t ; θ 0 ) ∥ Pr( y | X t ; θ )) .

Persistent TTA. Having all the ingredients, we design our approach, PeTTA, following the convention setup of the mean teacher update, with the category-balanced memory bank and the robust batch normalization layer from [61]. Appdx. E.1 introduces the pseudo code of PeTTA. For L CLS , either the self-training scheme [12] or the regular cross-entropy [16] is adopted. With R ( θ ) , cosine similarity or L2 distance are both valid metrics for measuring the distance between θ and θ 0 in 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 λ t , α t (Eq. 7) and L AL (Eq. 8):

5 Experimental Results

5.1 ϵ -MMCSimulation Result

Simulation Setup. A total of 6000 samples from two Gaussian distributions: N ( µ 0 = 0 , σ 2 0 = 1) and N ( µ 1 = 2 , σ 2 1 = 1) with p 0 = p 1 = 1 2 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). ϵ -GMMCupdate follows Eq. 4 with α = 5 e -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. In action, both the likelihood of predicting class 0 (Fig. 3a-left) and the ϵ t (Eq. 5) (Fig. 3c-right, solid line) gradually increases over time as expected (Lemma 1). After collapsing,

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ϵ -GMMC merges the two initial clusters, resulting in a single one (Fig. 3b-left) with parameters that match Lemma 2. The distance from ˆ µ 0 ,t (initialized at µ 0 ) towards µ 1 converges (Fig. 3c-left, solid line), coincided with the analysis in Thm. 1 when ϵ t is chosen following Corollary 1 (Fig. 3c, dashed line). GMMC (perturbed-free) 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, CIFAR10 → CIFAR10-C, CIFAR100 → CIFAR100-C, and ImageNet → ImageNet-C [19] are three corrupted images classification tasks (corruption level 5, the most severe). Additionally, we 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-free approach: LAME [7] and a reset-based approach (i.e., reverting the model to the source model after adapting to 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 CIFAR10C, DomainNet, and ImageNet-C, and Dir(0 . 01) for CIFAR100-C) generates category temporally correlated batches of data. For all experiments, we set the number of revisits K = 20 (times) as 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. Hyper-parameter 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 5 e -2 ; α 0 = 1 e -3 and cosine similarity regularizer is used. On CIFAR10/100-C and ImageNet-C we use the self-training loss in [12] for L CLS and λ 0 = 10 while the regular cross-entropy loss [13] and λ 0 = 1 (severe domain shift requires prioritizing

adaptability) are applied in DomainNet experiments. In Appdx. F.5, we provide a sensitivity analysis on the choice of hyper-parameter λ 0 in PeTTA.

5.3 Result - Benchmark Datasets

Recurring TTA Performance. Fig. 1-right presents the testing error on CIFAR-10-C in recurring TTA setting. RoTTA [61] exhibits promising performance in the first several visits but soon raises and eventually exceeds the source model (no TTA). The classification error of compared methods on CIFAR-10 → CIFAR-10-C, and ImageNet → ImageNet-C [19] tasks are shown in Tab. 1, and Tab. 2. Appdx. F.1 provides the results on 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 an 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 a higher performance as achieved by PeTTA. Furthermore, LAME [7] is highly constrained by the source model, and selecting a precise reset frequency in RDumb [45] is challenging in practice (see Appdx. F.3 for a further discussion).

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In average, PeTTA outperforms almost every baseline approaches and persists across 20 visits over the three datasets. The only exception is at the case of TRIBE [52] on CIFAR-10C. While this state-of-the-art 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 we increase the observation period to 40 recurring visits in Fig. 4. As the degree of freedom for adaptation in PeTTA is more constrained, it takes a bit longer for adaptation but remains stable afterward. Fig. 5b-bottom exhibits the confusion 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).

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 a mini-batch of data with 64 images. The model is adapted to 80 , 000 steps in total with more than 5 . 1 Mimages, significantly longer than 20 recurring TTA visits. Undoubtedly, PeTTA still achieves good performance where the corruptions are algorithmically generated, non-cyclic with two or 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 are concluded on the two settings. Obviously, our recurring TTA is notably simpler than CCC [45].

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9

10

11

12

13

14

15

16

17

18

19

20

Avg

Source

82.0

82.0

LAME [7]

80.9

80.9

CoTTA [59]

98.6

99.1

99.4

99.4

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

99.6

99.7

99.7

99.5

EATA [40]

60.4

59.3

65.4

72.6

79.1

84.2

88.7

92.7

95.2

96.9

97.7

98.1

98.4

98.6

98.7

98.8

98.8

98.9

98.9

99.0

89.0

RMT [12]

72.3

71.0

69.9

69.1

68.8

68.5

68.4

68.3

70.0

70.2

70.1

70.2

72.8

76.8

75.6

75.1

75.1

75.2

74.8

74.7

71.8

MECTA [22]

77.2

82.8

86.1

87.9

88.9

89.4

89.8

89.9

90.0

90.4

90.6

90.7

90.7

90.8

90.8

90.9

90.8

90.8

90.7

90.8

89.0

RoTTA [61]

68.3

62.1

61.8

64.5

68.4

75.4

82.7

95.1

95.8

96.6

97.1

97.9

98.3

98.7

99.0

99.1

99.3

99.4

99.5

99.6

87.9

RDumb [45]

72.2

73.0

73.2

72.8

72.2

72.8

73.3

72.7

71.9

73.0

73.2

73.1

72.0

72.7

73.3

73.1

72.1

72.6

73.3

73.1

72.8

ROID [37]

62.7

62.3

62.3

62.3

62.5

62.3

62.4

62.4

62.3

62.6

62.5

62.3

62.5

62.4

62.5

62.4

62.4

62.5

62.4

62.5

62.4

TRIBE [52]

63.6

64.0

64.9

67.8

69.6

71.7

73.5

75.5

77.4

79.8

85.0

96.5

99.4

99.8

99.9

99.8

99.8

99.9

99.9

99.9

84.4

PeTTA

(ours)

(

∗

)

65.3

61.7

59.8

59.1

59.4

59.6

59.8

59.3

59.4

60.0

60.3

61.0

60.7

60.4

60.6

60.7

60.8

60.7

60.4

60.2

60.5

0.46

0.44

0.4

0.43

0.46

0.47

0.44

0.43

0.48

0.4

0.43

0.43

0.41

airplane

bird

cat

dog

frog

ship

auto

deer

horse

truck

0.13

0.34

0.44

0.32

0.14

0.44

0.51

0.46

0.46

0.34

airplane

bird

cat

dog

frog

ship

auto

deer

horse

truck

Inter-category cosine

similarity (source model)

Misclassification rate of

collapsed RoTTA

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.69

0

0

0

0.08

0

0.04

0

0.13 0.05

0.34 0.17

0

0

0.1

0

0.03

0

0.23 0.13

0.24

0

0.1

0.03 0.44

0

0.11

0

0.07 0.01

0.21

0

0

0.11 0.32

0

0.21

0

0.14 0.01

0.14

0

0

0.01 0.71

0

0.06 0.01 0.06 0.01

0.21

0

0

0.07 0.44

0

0.16

0

0.1

0.01

0.05

0

0

0.08 0.51

0

0.25

0

0.1

0.01

0.17

0

0

0.03 0.46

0

0.04 0.21 0.06 0.04

0.46

0

0

0.01 0.06

0

0.03

0

0.41 0.03

0.34

0

0

0

0.12

0

0.03

0

0.18 0.32

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.73 0.01 0.06 0.04 0.02

0

0.03 0.01 0.08 0.02

0.01 0.88 0.01 0.01

0

0

0.02

0

0.02 0.05

0.04

0

0.75 0.07 0.05 0.02 0.05 0.01 0.01

0

0.01

0

0.06 0.72 0.05 0.04 0.06 0.02 0.01 0.01

0.02

0

0.06 0.07 0.76 0.01 0.05 0.02 0.01

0

0

0

0.07 0.19 0.05 0.59 0.05 0.02 0.01 0.01

0

0

0.03 0.07 0.02 0.01 0.84

0

0.01 0.01

0.01

0

0.06 0.06 0.08 0.02 0.02 0.74

0

0.01

0.04 0.02 0.02 0.02 0.01

0

0.03

0

0.84 0.02

0.01 0.05 0.02 0.03 0.01

0

0.02 0.01 0.04 0.82

1

5

10

15

20

0

0

.

2

0

.

4

0

.

6

0

.

8

1

Visits

1

5

10

15

20

0

0

.

2

0

.

4

0

.

6

0

.

8

1

Visits

RoTTA [61]

PeTTA

(ours)

PeTTA

(ours)

- 20

th

visit

RoTTA [61] - 20

th

visit

Predicted label

(a)

(b)

(c)

True label

True label

Prediction Frequency

Collapsing Pattern. The rise in classification error (Fig. 1-right) can be reasoned by the prediction frequency of RoTTA [61] in an recurring TTA setting (Fig. 5a-left). Similar to ϵ -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

CF: CIFAR, DN: DomainNet, IN: ImageNet top prone-to-misclassified 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 is analogous to | µ 0 -µ 1 | (Thm. 1), closer clusters are at a 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 ( θ ) ) with a fixed choice of λ, α not only fails to mitigate model collapse but may also introduce a negative effect (rows 2-3). Trivially applying the anchor loss ( L AL ) alone is also incapable of eliminating the lifelong performance degradation in continual TTA (row 4). Within PeTTA, adopting the adaptive λ t scheme alone (row 5) or in conjunction with either α t or anchor loss L AL (rows 6-7) partially stabilizes the performance. Under the drastic domain shifts with a larger size of categories or model parameters (e.g., on CIFAR-100-C, DomainNet, ImageNet-C), restricting α t adjustment limits the ability of PeTTA to stop undesirable updates while a common regularization term without L AL is insufficient to guide the adaptation. Thus, leveraging all elements secures the persistence of PeTTA (row 8).

Various Choices of Regularizers. The design of PeTTA is not coupled with any specific regularization term. Demonstrated in Tab. 5, PeTTA works well for the two common choices: L2 and cosine similarity. The conjunction use of Fisher coefficent [27, 40] for weighting the model parameter importance is 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 a Potential Risk of TTA in Practice. We provide empirical and theoretical evidence on 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 a 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 be guaranteed rigorously through regularization. Future research could delve deeper into expanding our efforts to develop an algorithm that achieves error accumulation-free by construction. Furthermore, as tackling the challenge of the temporally correlated testing stream is not the focus of PeTTA, using a 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 ϵ -GMMC paves the road for designing PeTTA- a simple yet efficient solution that continuously assesses the model divergence for harmonizing the TTA process, balancing adaptation, and collapse prevention.

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Longhui Yuan, Binhui Xie, and Shuang Li. Robust test-time adaptation in dynamic scenarios. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 1592215932, 2023.

Hao Zhao, Yuejiang Liu, Alexandre Alahi, and Tao Lin. On pitfalls of test-time adaptation. In ICLR 2023 Workshop on Pitfalls of limited data and computation for Trustworthy ML , 2023. URL https: //openreview.net/forum?id=0Go_RsG_dYn .

Persistent Test-time Adaptation in Recurring Testing Scenarios Technical Appendices

Table of Contents

A Related Work

Towards Robust and Practical TTA. While forming the basis, early single-target TTA approaches [53, 57, 39, 41, 33] is far from practice. Observing the dynamic of many testing environments, a continual TTA setting is proposed where an ML model continuously adapts to a sequence of multiple shifts [36, 59]. Meanwhile, recent studies [15, 7] point out that the category distribution realistic streams is 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 a 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-balanced subset of testing samples [15, 61], inspired by the replay methods [50, 2] to avoid forgetting in the task of continual learning [34, 3, 11]. Our study emphasizes another perspective: beyond a supreme performance, a 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 a stable process has been confirmed. In TTA, the continuous changes of model parameters through gradient descent aggravate the situation, as 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 to avoid drastic changes to model parameters as a good practice. Up to some degree, it helps to avoid performance degradation. Nevertheless, it is still unclear whether their effectiveness truly eliminates the risk . To preserve in-distribution 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 a bad heuristic choice of threshold or period leads to highly frequent model resets. Noteworthy, tuning those hyper-parameters is exceedingly difficult due to the unavailability of the validation set [62]. LAME [7] suggests a post-processing 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 ϵ -perturbed Gaussian Mixture Model Classifier ( ϵ -GMMC) 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 τ ∈ T , τ < ∞ if there exists a non-empty subset of categories ˜ Y ⊂ Y such that Pr { Y t ∈ ˜ Y} > 0 but the marginal Pr { ˆ Y t ∈ ˜ Y} converges to zero in probability:

Assumption 1 ( Static Data Stream ) . The marginal distribution of the true label follows the same Bernoulli distribution Ber( p 0 ) : p 0 ,t = p 0 , ( p 1 ,t = p 1 = 1 -p 0 ) , ∀ t ∈ T .

Preliminary. Following the same set of notations introduced in the main text, recall that we denoted p y,t ∆ = Pr { Y t = y } , ˆ p y,t ∆ = Pr { ˆ Y t = y } (marginal distribution of the true label Y t and pseudo label ˆ Y t receiving label y , respectively) and ϵ t = Pr { Y t = 1 | ˆ Y t = 0 } (the false negative rate (FNR) of

ϵ -GMMC). At testing step t , we obtain the following relations:

In addition, under Assumption 1, the marginal distribution P t ( x ) (also referred as data distribution in our setup) is:

B.1 Proof of Lemma 1

Lemma 1 ( Increasing FNR ) . Under Assumption 1, a binary ϵ -GMMC would collapsed (Def. 1) with lim t → τ ˆ p 1 ,t = 0 (or lim t → τ ˆ p 0 ,t = 1 , equivalently) if and only if lim t → τ ϵ t = p 1 .

Proof. Under Assumption 1, we have E P t [ X t ] = p 0 µ 0 +(1 -p 0 ) µ 1 . Also note that:

where the second equality follows Eqs. 9-10. Therefore:

Eq. 15 shows positive correlation between ˆ p 0 ,t and ϵ t . Given lim t → τ ϵ t = p 1 , taking the limit introduces:

Similarly, having lim t → τ ˆ p 0 ,t = 1 , the false negative rate ϵ t when t → τ is:

Since ˆ p 0 ,t + ˆ p 1 ,t = 1 , lim t → τ ˆ p 1 ,t = 0 , equivalently. Towards the collapsing point, the model tends to predict a single label (class 0 in the current setup). In addition, the FNR of the model ϵ t also raises correspondingly.

B.2 Proof of Lemma 2.

Lemma 2 ( ϵ -GMMC After Collapsing ) . For a binary ϵ -GMMC model, with Assumption 1, if lim t → τ ˆ p 1 ,t = 0 (collapsing), the cluster 0 in GMMC converges in distribution to a single-cluster GMMC with parameters:

Proof. From Eqs. 9-10, under the increasing type II collapse of ϵ -GMMCsetting, the perturbation does not affect the approximation of µ 1 . Meanwhile, when ϵ t increases, one can expect that ˆ µ 0 ,t

moves further away from µ 0 toward µ 1 . Frist, the mean teacher model of GMMC (Eq. 4, main text) gives:

By defining u y,t = E P t [ˆ µ y,t ] , we obtain the following recurrence relation between u 0 ,t and u 0 ,t -1 :

Given lim t → τ ˆ p 0 ,t = 1 , it follows that lim t → τ ϵ 0 ,t = p 1 by Lemma 1. From this point:

Taking the limit t →∞ :

The second equation is obtained by solving the recurrence relation. When lim t → τ ˆ p 0 ,t = 1 , { ˆ µ y,t } y ∈{ 0 , 1 } becomes a deterministic values. Hence, giving u y,t = E P t [ˆ µ y,t ] = ˆ µ 0 ,t ( ∀ t > τ ) and

Repeating the steps above with Eqs. 11-12 in place of Eqs. 9-10, we obtain a similar result for σ 2 0 ,t :

By Lévy's continuity theorem (p. 302, [42]), from Eqs. 17-18, when t → ∞ , the estimated distribution of the first cluster N ( x ; ˆ µ 0 ,t ˆ σ 2 0 ,t ) converges to the whole data distribution P t ( x ) (Eq. 13) when collapsing.

B.3 Proof of Theorem 1 and Corollary 1.

Theorem 1 ( Convergence of ϵ -GMMC ) . For a binary ϵ -GMMC model, with Assumption 1, let the distance from ˆ µ 0 ,t toward µ 1 is d 0 → 1 t = | E P t [ˆ µ 0 ,t ] -µ 1 | , then:

Proof. Substituting Eq. 15 into ˆ p 0 ,t of Eq. 16 gives:

Hence, we have the distance from u 0 ,t toward µ 1 :

The last inequality holds due to the triangle inequality. Equivalently,

Let d 0 → 1 t = | E P t [ˆ µ 0 ,t ] -µ 1 | , we conclude that:

Corollary 1 ( ACondition for ϵ -GMMCCollapse ) . With fixed p 0 , α, µ 0 , µ 1 , ϵ -GMMCiscollapsed if there exists a sequence of { ϵ t } τ τ -∆ τ ( τ ≥ ∆ τ > 0 ) such that:

Proof. Initialized at µ 0 , ϵ -GMMC is collapsing when ˆ µ 0 ,t converges to the mid-point p 0 µ 0 + p 1 µ 1 (Lemma 2), i.e., moving closer to µ 1 . From Thm. 1, the distance towards µ 1 d 0 → 1 t < d 0 → 1 t -1 if

When there exists this sequence { ϵ t } τ τ -∆ τ ( τ ≥ ∆ τ > 0) it follows that d 0 → 1 t < d 0 → 1 t -1 and ϵ t > ϵ t -1 is guaranteed ∀ t ∈ [ τ -∆ τ , τ ] . Hence, lim t → τ ϵ t = p 1 (model collapsed, by Lemma 1).

C Further Justifications on Gaussian Mixture Model Classifier

One may notice that in ϵ -GMMC (Sec. 4.2), the classifier is defined f t ( x ) = argmax y ∈Y Pr( x | y ; θ t ) (maximum likelihood estimation) while in general, f t ( x ) = argmax y ∈Y Pr( y | x ; θ t ) (maximum a posterior estimation), parameterized by a neural network. In this case, since the equal prior (i.e., Pr( y ; θ t ) = Pr( y ′ ; θ t ) , ∀ y, y ′ ∈ C ) is enforced in ϵ -GMMC, the two definitions are equivalent .

Proof. Having:

We conclude that the two definitions are equivalent. In fact, it is well-known that maximum likelihood estimation is a special case of maximum a posterior estimation when the prior is uniform.

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 environments (or corruptions) as in [59, 40], the distribution of class labels is also temporally correlated (non-i.i.d.) as suggested by [15, 61] to 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 to 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, 61]. 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 over-simplicity 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 a 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 our setup is a simple yet sufficient scenario to demonstrate the vulnerability of existing continual TTA methods when facing the issue of model collapse (compared to CCC [45], a 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 P 1 → P 2 → ··· → P D → P 1 → P 2 → ··· → P D . 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.

Experiment Setup. We refer to the setting same-order as using one order of image corruptions in [61] for all recurrences (specifically, on CIFAR-10/100-C and ImageNet-C: motion → snow → fog → shot → defocus → contrast → zoom → brightness → frost → elastic → glass → gaussian → pixelated → jpeg → 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 is fixed after generation. Without updating its parameters, the performance of the source model is trivially independent of the order of corruptions.

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, a 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 we promoted in recurring TTA .

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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 of a TTA algorithm, enhancing the regular mean-teacher update step . 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, as a

Algorithm 1 Persistent TTA (PeTTA)

shorthand notation, ϕ θ t -1 ( X t | ˆ Y t = y ) denotes the empirical mean of all feature vectors of X ( i ) t (extracted by ϕ θ t -1 ( X ( i ) t ) ) if ˆ Y ( i ) t = y, i = 1 , · · · , N t in the current testing batch.

E.2 Anchor Loss

KL Divergence Minimization-based Interpretation of Anchor Loss . In Sec. 4, we claimed that minimizing the anchor loss L AL is equivalent to minimizing the relative entropy (or KL divergence) between the output probability of two models parameterized by θ 0 and θ .

Proof. Having:

Hence,

Intuitively, a desirable TTA solution should be able to adapt to novel testing distributions on the one hand, but it should not significantly diverge from the initial model. L AL fits this purpose, constraining the KL 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 ( L AL ) and the Regularizer term ( R ( θ ) ) is 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 1 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 to 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 category-balanced 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 is 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, a 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 a 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 a 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 is not universally applied in every real-world scenario, we believe that this is a reasonable assumption, and commonly adopted in TTA practices. Nevertheless, exploring alternative ways for reducing the memory size (e.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 µ y 0 and Σ y 0 in Practice. One may notice that in PeTTA, computing γ y t requires the pre-computed empirical mean ( µ y 0 ) and covariance ( Σ y 0 ) 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:

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.

With only the source pre-trained model available, assume we can sample a 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 pseudo 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 a Fisher matrix (for anti-forgetting regularization purposes). Our work - PeTTA follows the same second setup as 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 our experiments.

We believe that having the empirical mean and covariant matrix pre-computed on a portion of the source distribution in PeTTA is a 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 a 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-10, 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 of images is 10 , 000 in CIFAR-10/A00, 50 , 000 in ImageNet, and 69 , 622 in DomainNet. We randomly sample 25% , 50% , 75% , and 100% of the images in this set to run PeTTA for 20 rounds of recurring. The result is provided in Tab. 6 below.

The default choice of PeTTA is using 100% samples of the validation set 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 an 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, we propose a harmonic way of combining those elements adaptively to achieve a persistent TTA process.

The design of PeTTA 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 on the tasks CIFAR-100 → CIFAR-100-C, and real → clipart, painting, sketch of DomainNet are provided in Tab. 7 and Tab. 8. To prevent model collapse, the adaptability of PeTTA is more constrained. As a 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 to the space constraint, the corresponding standard deviation values are now reported in Tab. 9. Generally, the average standard deviation across runs

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77.3

77.2

77.1

76.8

76.9

76.5

76.4

76.4

76.3

76.4

76.2

76.2

76.1

76.4

76.1

76.0

75.8

76.5

MECTA [22]

94.6

98.4

98.6

98.8

99.1

99.0

99.0

99.0

99.0

99.0

99.0

99.0

99.0

99.0

99.0

99.0

99.0

99.0

99.0

99.0

98.7

RoTTA [61]

44.3

43.8

44.7

46.7

48.7

50.8

52.7

55.0

57.1

59.7

62.7

65.1

68.0

70.3

72.7

75.2

77.2

79.6

82.6

85.3

62.1

RDumb [45]

44.3

44.4

44.3

44.5

44.2

44.2

44.3

44.5

44.4

44.2

44.3

44.3

44.3

44.3

44.5

44.3

44.2

44.3

44.4

44.3

44.3

PeTTA

(ours)

(

∗

)

43.8

42.6

42.3

42.3

42.6

42.8

42.8

43.0

42.9

42.9

43.1

43.0

42.9

43.0

43.0

43.1

43.0

42.8

42.9

42.9

42.9

stays within ± 0 . 1% for small datasets (CIFAR-10-C, CIFAR-100-C) and ± 0 . 5% for larger datasets (ImageNet-C, DomainNet).

F.2 An Inspection of PeTTA

In Fig. 7, we showcase an inspection of our PeTTA on the task CIFAR-10 → CIFAR-10-C [19] in a typical recurring TTA with 20 visits. Specifically, the visualizations of PeTTA parameters ( ¯ γ t , λ t , and α t ), adaptation losses ( L CLS , L AL ) and regularization term ( R ( θ ) ) are provided. Here, we observe the values of adaptive parameters λ t and α t 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 rate persists as a result. A similar pattern is observed on other datasets (CIFAR-100-C [19] and DomainNet [44]).

F.3 Does Model Reset Help?

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 is named RDumb [45]. We thoroughly conducted experiments to compare the performance of RDumb with PeTTA. The implementation of RDumb in this setting is as follows. We employ RoTTA [61] as the base test-time adaptor due to the characteristics of the practical TTA [61] stream. The model (including model

parameters, the optimizer state, and the memory bank) is reset after adapting itself to T images. 1 For each dataset, three values of this hyper-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 (CIFAR-10/100-C), T = 5 , 000 (ImageNet-C) and T = 24 , 237 (DomainNet). 2 This value is equal to the number of samples in the test set of a single corruption type , i.e., the model is reset exactly after visiting each P i '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-10/100-C), T = 75 , 000 (ImageNet-C) and T = 72 , 712 (DomainNet). This number is equal to the number of samples in one recurrence of our recurring TTA, i.e., the model is reset exactly after visiting P 1 →··· → P D . Here, D = 15 - types of corruptions [19] for CIFAR-10/100-C and ImageNet-C and D = 3 for DomainNet ( clipart, painting, sketch ). For example, the model is reset 20 times within a recurring TTA 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 an oracle model with a 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.

Discussions. Across datasets and reset frequencies, our 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 or recurrence. Importantly, this oracle information is typically unavailable in practice.

Noteworthy, it is clear that the performance of RDumb varies when changing the choice of the reset frequency. For a given choice of T , the better performance on one dataset does not guarantee the same performance on other datasets. For example, T = 1 , 000 - 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 of 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 in the TTA setting where a 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. 8, we provide the evaluation results of PeTTA on recurring with 40 recurrences.

F.5 The Sensitivity of Hyper-parameter Choices in PeTTA

There are two hyper-parameters in PeTTA: α 0 and λ 0 . The initial learning rate of α 0 = 1 e -3 is used for all experiments. We do not tune this hyper-parameter, and the choice of α 0 is universal across all datasets, following the previous works/compared methods (e.g., RoTTA [61], CoTTA [59]).

Since λ 0 is more specific to PeTTA, we included a sensitive analysis with different choices of λ 0 on PeTTA, evaluated with images from CIFAR-10/100-C and ImageNet-C in Tab. 15. Overall, the choice of λ 0 is not extremely sensitive, and while the best value is 1 e 1 on most datasets, other choices such as 5 e 0 or 5 e 1 also produce roughly similar performance. Selecting λ 0 is intuitive, the larger value of λ 0 stronger prevents the model from collapsing but also limits its adaptability as a trade-off.

In action, λ 0 is 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, this hyperparameter can be tuned similarly to the hyper-parameters of other TTA approaches, via an additional validation set, or some accuracy prediction algorithm [29] when labeled data is unavailable.

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. 9 presents an additional examination of the ablation study conducted on the task CIFAR-100 → CIFAR-100-C [19] for our PeTTA approach. We plot the classification error (top) and the value of ¯ γ t (bottom) for various PeTTA variations. As the model diverges from the initial state, the value of ¯ γ t increases. Unable to adjust α t or constraint the probability space via L AL limits the ability of PeTTA to prevent model collapse. In all variations with the model collapse in ablation studies, the rapid saturation of ¯ γ t is all observed. Therefore, incorporating all components in PeTTA is necessary.

F.7 More Confusion Matrices in Recurring TTA Setting

For the task CIFAR-10 → CIFAR-10-C [19] in recurring TTA setting (with 20 visits), we additionally showcase the confusion matrix of RoTTA [61] (Fig. 10) and our proposed PeTTA (Fig. 11) at each visit. Our PeTTA 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 . 80 GHz i7-10700K 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 are i.i.d. distributed. The adaptation configuration of PeTTA follows the same settings as used on ImageNet-C, while the same setting introduced in Sec. F.3, with T = 1000 is used for RDumb [45].

G.3 Test-time Adaptation Methods

Pre-trained Model on 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 on source distributions. Specifically, for ImageNet-C and DomainNet experiments, a ResNet50 model [17] pre-trained on ImageNet V2 (specifically, checkpoint ResNet50_Weights.IMAGENET1K_V2 of torchvision ) is used. From RobustBench , the model with checkpoint Standard and Hendrycks2020AugMix_ResNeXt [20] are adopted for CIFAR10-C and CIFAR-100-C experiments, respectively. Lastly, experiments on DomainNet dataset utilize the checkpoint ( best_real_2020 ) provided in AdaContrast [8] study. 4

Optimizer. Without specifically stated, Adam [26] optimizer with learning rate equal 1 e -3 , and β = (0 . 9 , 0 . 999) is selected as a universal choice for all experiments.

More Details on PeTTA. Since designing the batch normalization layers, and the memory bank is not the key focus of PeTTA, we conveniently adopt the implementation of the Robust Batch Norm layer and the Category-balanced Sampling strategy using a memory bank introduced in RoTTA [61].

G.4 The Use of Existing Assets

Many components of PeTTA is utilized from the official repository of RoTTA [61] 5 and RMT [12]. 6 These two assets are released under MIT license. All the datasets, including CIFAR-10-C, CIFAR100-C and ImageNet-C [19] are publicly available online, released under Apache-2.0 license. 7 DomainNet dataset [44] (cleaned version) is also released for research purposes. 8

0

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AL

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CIFAR-10-C

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ImageNetC

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PeTTA -

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AL

0: airplane

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6: frog

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0.79 0.01 0.04 0.03 0.02 0.01 0.01 0.02 0.05 0.02

0.02 0.82 0.01 0.01

0

0.01 0.01 0.01 0.01 0.09

0.06

0

0.68 0.07 0.04 0.03 0.06 0.03 0.01 0.01

0.02 0.01 0.04 0.66 0.04 0.08 0.07 0.05 0.01 0.02

0.03

0

0.04 0.06 0.68 0.02 0.06 0.09 0.01 0.01

0.03

0

0.05 0.15 0.03 0.61 0.03 0.07 0.01 0.01

0.02 0.01 0.03 0.07 0.02 0.02

0.8

0.02

0

0.01

0.01

0

0.02 0.03 0.03 0.02 0.01 0.87

0

0.01

0.09 0.02 0.02 0.02 0.01

0

0.02 0.01 0.77 0.04

0.03 0.03 0.01 0.01

0

0

0.01 0.01 0.03 0.85

0: airplane

1: auto

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3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

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0.76 0.01 0.03 0.03 0.01

0

0.03 0.02 0.07 0.03

0.02 0.76

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0.01

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0

0.03 0.01 0.02 0.16

0.07

0

0.63 0.08 0.06 0.02 0.08 0.04 0.01 0.01

0.02

0

0.04

0.7

0.04 0.04 0.09 0.05 0.01 0.02

0.03

0

0.03 0.05 0.73 0.01 0.06 0.08 0.01 0.01

0.01

0

0.03 0.23 0.04 0.53 0.06 0.08 0.01 0.01

0.02

0

0.02

0.1

0.02 0.01 0.81 0.01

0

0.01

0.01

0

0.01 0.05 0.03 0.01 0.01 0.87

0

0.01

0.08 0.01 0.01 0.02 0.01

0

0.02 0.01

0.8

0.04

0.03 0.02 0.01 0.02

0

0

0.02 0.01 0.02 0.87

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

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0.01 0.03 0.04 0.02

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0.03 0.03 0.09 0.06

0.01 0.72

0

0.01

0

0

0.04

0

0.01

0.2

0.07

0

0.56

0.1

0.08 0.02 0.09 0.04 0.01 0.02

0.01

0

0.03

0.7

0.05 0.02 0.13 0.04

0

0.02

0.04

0

0.03 0.07 0.69

0

0.08 0.07 0.01 0.01

0.01

0

0.04 0.26 0.05 0.42 0.13 0.07

0

0.01

0.01

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0.02 0.11 0.03

0

0.8

0.01

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0.02 0.06 0.05 0.01 0.04

0.8

0

0.01

0.07 0.01 0.01 0.03 0.01

0

0.03 0.01 0.78 0.05

0.02 0.01 0.01 0.02 0.01

0

0.04 0.01 0.02 0.86

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

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0.62 0.01 0.03 0.06 0.03

0

0.05 0.04 0.09 0.08

0.01 0.66

0

0.02 0.01

0

0.04

0

0.02 0.25

0.07

0

0.48 0.13

0.1

0.02 0.13 0.03 0.01 0.02

0.01

0

0.02 0.68 0.05 0.02 0.17 0.03

0

0.02

0.03

0

0.02 0.07 0.67

0

0.12 0.07 0.01 0.01

0.01

0

0.02 0.29 0.07 0.39 0.14 0.06

0

0.01

0.01

0

0.01 0.11 0.04

0

0.8

0.01

0

0.01

0.01

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0.02 0.08 0.06 0.01 0.06 0.75

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0.05 0.01 0.01 0.04 0.02

0

0.05 0.01 0.74 0.07

0.01 0.01

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0.05 0.01 0.02 0.86

True label

1 st

visit

2 nd

visit

3 rd

visit

4 th

visit

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

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0.56

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0.03 0.07 0.04

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0.01 0.01

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0.07

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0.02 0.26

0.08

0

0.42 0.13 0.13 0.02 0.15 0.03 0.01 0.02

0.02

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0.01 0.62 0.06 0.02 0.21 0.03

0

0.02

0.03

0

0.02 0.06 0.66

0

0.16 0.06 0.01 0.01

0.01

0

0.02

0.3

0.08 0.34 0.17 0.06

0

0.02

0.01

0

0.01 0.12 0.07

0

0.76 0.01

0

0.02

0.01

0

0.02

0.1

0.08 0.01 0.08 0.69

0

0.02

0.05 0.01 0.01 0.05 0.02

0

0.09 0.01 0.68 0.09

0.01 0.01

0

0.03 0.02

0

0.09 0.01 0.02 0.83

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

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0.51

0

0.02 0.07 0.04

0

0.09 0.03

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0.14

0.01 0.56

0

0.01 0.02

0

0.09

0

0.02 0.29

0.08

0

0.35 0.15 0.16 0.02 0.18 0.03 0.01 0.03

0.02

0

0.01 0.57 0.07 0.02 0.27 0.02

0

0.03

0.04

0

0.01 0.08 0.62

0

0.18 0.05 0.01 0.01

0.01

0

0.01 0.29 0.09

0.3

0.21 0.05

0

0.02

0.01

0

0.01 0.12 0.09

0

0.75

0

0

0.01

0.02

0

0.01 0.11 0.12 0.01

0.1

0.6

0

0.03

0.06 0.01

0

0.04 0.02

0

0.09

0

0.66 0.11

0.01 0.01

0

0.02 0.03

0

0.11

0

0.02

0.8

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

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0.48

0

0.02 0.08 0.04

0

0.11 0.03 0.11 0.13

0.01 0.54

0

0.01 0.02

0

0.11

0

0.02 0.28

0.09

0

0.3

0.16 0.16 0.02 0.21 0.02 0.01 0.03

0.02

0

0.01 0.51 0.08 0.01 0.33 0.01 0.01 0.02

0.03

0

0.01 0.05 0.65

0

0.21 0.03 0.01 0.01

0.02

0

0.01 0.27 0.11 0.25 0.28 0.03

0

0.02

0.01

0

0.01 0.12

0.1

0

0.75

0

0

0.01

0.02

0

0.01 0.11 0.13 0.01 0.13 0.56

0

0.03

0.06

0

0

0.06 0.03

0

0.13

0

0.6

0.11

0.02 0.01

0

0.03 0.04

0

0.15

0

0.02 0.73

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

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9

0.46

0

0.01 0.07 0.06

0

0.13 0.01 0.09 0.15

0.01 0.48

0

0.01 0.04

0

0.16

0

0.01 0.28

0.09

0

0.27 0.15 0.19 0.01 0.23 0.01 0.01 0.03

0.02

0

0.01 0.44 0.12 0.01 0.37 0.01 0.01 0.02

0.04

0

0.01 0.05 0.63

0

0.23 0.02 0.01 0.01

0.02

0

0.01 0.25 0.13 0.22 0.33 0.02

0

0.01

0.01

0

0

0.11 0.15

0

0.71

0

0

0.01

0.02

0

0.01 0.09 0.22

0

0.15 0.47

0

0.02

0.08

0

0

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0.24 0.12 0.22 0.01 0.24 0.01 0.01 0.03

0.03

0

0

0.4

0.12 0.01 0.39

0

0.01 0.02

0.05

0

0.01 0.06 0.61

0

0.23 0.02 0.01 0.01

0.03

0

0.01 0.22 0.15

0.2

0.35 0.02

0

0.02

0.01

0

0

0.11 0.15

0

0.7

0

0.01 0.01

0.03

0

0.01 0.08 0.25

0

0.15 0.44 0.01 0.03

0.09

0

0

0.04 0.07

0

0.14

0

0.55

0.1

0.02 0.01

0

0.03 0.05

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0.16

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0.02

0.7

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

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9: truck

0

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0.46

0

0.01 0.05 0.07

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0.14 0.01

0.1

0.16

0.04 0.43

0

0.02 0.07

0

0.14

0

0.03 0.27

0.11

0

0.22 0.11 0.23 0.01 0.26 0.01 0.02 0.03

0.04

0

0

0.33 0.16 0.01 0.43

0

0.01 0.02

0.05

0

0

0.03 0.66

0

0.23 0.01 0.01 0.01

0.04

0

0.01 0.22 0.15 0.18 0.37 0.01 0.01 0.02

0.01

0

0

0.1

0.19

0

0.68

0

0.01 0.01

0.03

0

0.01 0.08 0.28 0.01 0.16 0.41 0.01 0.03

0.11

0

0

0.04 0.05

0

0.14

0

0.56 0.09

0.04 0.01

0

0.02 0.08

0

0.18

0

0.02 0.65

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.47

0

0.01 0.04 0.07

0

0.14

0

0.1

0.16

0.04 0.42

0

0.01 0.07

0

0.15

0

0.05 0.26

0.11

0

0.21

0.1

0.26 0.01 0.26

0

0.02 0.03

0.05

0

0

0.31 0.18 0.01 0.42

0

0.01 0.02

0.06

0

0

0.04 0.65

0

0.21 0.01 0.01 0.02

0.04

0

0.01 0.17 0.21 0.15 0.39 0.01 0.01 0.02

0.01

0

0

0.1

0.24

0

0.64

0

0.01 0.01

0.04

0

0.01 0.09 0.28

0

0.16 0.39

0

0.03

0.14

0

0

0.03 0.07

0

0.14

0

0.52 0.09

0.05 0.01

0

0.03

0.1

0

0.18

0

0.03 0.61

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.49

0

0.01 0.03 0.06

0

0.14

0

0.11 0.17

0.07

0.4

0

0.01 0.07

0

0.12

0

0.07 0.27

0.13

0

0.19 0.08 0.27 0.01 0.25

0

0.02 0.03

0.07

0

0

0.27 0.19

0

0.43

0

0.02 0.03

0.07

0

0

0.02 0.64

0

0.23 0.01 0.01 0.01

0.06

0

0.01 0.19 0.18 0.13 0.39 0.01 0.01 0.02

0.02

0

0

0.09 0.22

0

0.65

0

0.01 0.01

0.05

0

0

0.07 0.32

0

0.15 0.36 0.01 0.04

0.17

0

0

0.03 0.07

0

0.12

0

0.53 0.08

0.06 0.01

0

0.01 0.13

0

0.17

0

0.03 0.59

True label

9 th

visit

10 th

visit

11 th

visit

12 th

visit

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.5

0

0

0.02 0.08

0

0.13

0

0.1

0.15

0.09 0.37

0

0.01 0.11

0

0.11

0

0.08 0.24

0.15

0

0.18 0.07 0.31 0.01 0.24

0

0.03 0.02

0.09

0

0

0.24 0.17

0

0.44

0

0.02 0.03

0.08

0

0

0.02 0.66

0

0.19 0.01 0.02 0.02

0.08

0

0.01 0.15 0.23 0.11 0.38 0.01 0.01 0.02

0.02

0

0

0.08 0.31

0

0.55

0

0.02 0.01

0.05

0

0

0.05 0.37

0

0.14 0.34 0.01 0.04

0.2

0

0

0.03 0.06

0

0.12

0

0.52 0.08

0.08 0.01

0

0.01 0.11

0

0.15

0

0.04 0.59

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.54

0

0

0.02 0.06

0

0.11

0

0.12 0.15

0.13 0.35

0

0.01

0.1

0

0.09

0

0.12 0.21

0.16

0

0.18 0.07 0.29 0.01 0.24

0

0.03 0.02

0.11

0

0

0.22 0.19

0

0.42

0

0.03 0.03

0.08

0

0

0.03 0.65

0

0.2

0.01 0.02 0.01

0.09

0

0.01 0.12 0.29 0.08 0.37

0

0.02 0.02

0.02

0

0

0.09 0.29

0

0.56

0

0.02 0.01

0.06

0

0

0.05 0.39

0

0.13 0.32 0.01 0.04

0.23

0

0

0.02 0.07

0

0.1

0

0.51 0.07

0.12 0.01

0

0.01 0.11

0

0.13

0

0.05 0.57

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.56

0

0

0.02 0.08

0

0.1

0

0.12 0.12

0.18 0.32

0

0

0.11

0

0.08

0

0.13 0.19

0.18

0

0.15 0.05 0.34

0

0.2

0

0.04 0.02

0.12

0

0

0.19 0.27

0

0.36

0

0.04 0.02

0.09

0

0

0.02 0.69

0

0.15 0.01 0.02 0.02

0.11

0

0

0.1

0.33 0.07 0.33

0

0.03 0.01

0.03

0

0

0.09 0.35

0

0.5

0

0.02 0.01

0.08

0

0

0.04 0.43

0

0.1

0.29 0.01 0.04

0.26

0

0

0.02 0.08

0

0.08

0

0.51 0.06

0.15 0.01

0

0.01 0.12

0

0.1

0

0.07 0.55

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.58

0

0

0.01 0.07

0

0.09

0

0.13 0.11

0.16 0.32

0

0

0.11

0

0.07

0

0.16 0.18

0.18

0

0.15 0.05 0.36

0

0.19

0

0.04 0.02

0.14

0

0

0.18 0.26

0

0.35

0

0.05 0.02

0.1

0

0

0.01 0.69

0

0.15 0.01 0.03 0.01

0.11

0

0

0.1

0.36 0.05 0.32

0

0.04 0.01

0.03

0

0

0.08 0.38

0

0.46

0

0.03 0.01

0.09

0

0

0.04 0.43

0

0.09 0.29 0.02 0.04

0.29

0

0

0.02 0.09

0

0.08

0

0.47 0.06

0.18 0.01

0

0.01 0.11

0

0.08

0

0.1

0.5

True label

13 th

visit

14 th

visit

15 th

visit

16 th

visit

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.6

0

0

0.01 0.08

0

0.08

0

0.13

0.1

0.2

0.28

0

0

0.1

0

0.06

0

0.19 0.17

0.2

0

0.14 0.05 0.36

0

0.18

0

0.05 0.02

0.17

0

0

0.16 0.28

0

0.29

0

0.08 0.02

0.1

0

0

0.01 0.71

0

0.11 0.01 0.04 0.02

0.13

0

0

0.1

0.4

0.04 0.27

0

0.05 0.01

0.04

0

0

0.09

0.4

0

0.41

0

0.04 0.01

0.1

0

0

0.04 0.45

0

0.07 0.27 0.03 0.04

0.34

0

0

0.01 0.08

0

0.05

0

0.47 0.05

0.22 0.01

0

0.01 0.13

0

0.06

0

0.12 0.44

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.62

0

0

0.01 0.09

0

0.08

0

0.13 0.08

0.24 0.26

0

0

0.1

0

0.05

0

0.19 0.15

0.2

0

0.13 0.04 0.41

0

0.16

0

0.05 0.02

0.16

0

0

0.14

0.3

0

0.29

0

0.09 0.02

0.11

0

0

0.01

0.7

0

0.1

0.01 0.05 0.02

0.14

0

0

0.09 0.42 0.02 0.27

0

0.05 0.01

0.03

0

0

0.09 0.44

0

0.39

0

0.04 0.01

0.12

0

0

0.04 0.43

0

0.06 0.28 0.03 0.04

0.35

0

0

0.01 0.07

0

0.06

0

0.46 0.04

0.26 0.01

0

0.01 0.13

0

0.06

0

0.13 0.41

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.67

0

0

0

0.1

0

0.05

0

0.11 0.06

0.3

0.21

0

0

0.1

0

0.03

0

0.21 0.13

0.26

0

0.11 0.04

0.4

0

0.11

0

0.06 0.02

0.2

0

0

0.13 0.32

0

0.21

0

0.12 0.02

0.13

0

0

0.01 0.72

0

0.07 0.01 0.05 0.01

0.2

0

0

0.09 0.42 0.01 0.19

0

0.08 0.01

0.04

0

0

0.08 0.49

0

0.3

0

0.07 0.01

0.16

0

0

0.03 0.45

0

0.04 0.24 0.05 0.03

0.42

0

0

0.01 0.06

0

0.04

0

0.43 0.04

0.33 0.01

0

0

0.11

0

0.03

0

0.15 0.36

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.69

0

0

0

0.08

0

0.04

0

0.13 0.05

0.34 0.17

0

0

0.1

0

0.03

0

0.23 0.13

0.24

0

0.1

0.03 0.44

0

0.11

0

0.07 0.01

0.21

0

0

0.11 0.32

0

0.21

0

0.14 0.01

0.14

0

0

0.01 0.71

0

0.06 0.01 0.06 0.01

0.21

0

0

0.07 0.44

0

0.16

0

0.1

0.01

0.05

0

0

0.08 0.51

0

0.25

0

0.1

0.01

0.17

0

0

0.03 0.46

0

0.04 0.21 0.06 0.04

0.46

0

0

0.01 0.06

0

0.03

0

0.41 0.03

0.34

0

0

0

0.12

0

0.03

0

0.18 0.32

Predicted label

Predicted label

Predicted label

Predicted label

True label

17 th

visit

18 th

visit

19 th

visit

20 th

visit

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.77 0.01 0.04 0.03 0.03 0.01 0.02 0.02 0.05 0.02

0.02 0.84 0.01 0.02

0

0.01 0.02 0.01 0.02 0.06

0.04

0

0.69 0.07 0.05 0.05 0.05 0.02 0.01 0.01

0.04 0.01 0.05 0.62 0.05

0.1

0.06 0.04 0.01 0.02

0.03

0

0.06 0.07 0.68 0.05 0.04 0.05 0.01 0.01

0.01

0

0.04 0.14 0.03

0.7

0.03 0.04 0.01 0.01

0.01 0.01 0.04 0.06 0.03 0.03 0.78 0.01 0.01 0.01

0.03

0

0.03 0.04 0.04 0.04 0.01 0.79 0.01 0.01

0.08 0.02 0.02 0.02 0.01 0.01 0.02 0.01

0.8

0.03

0.03 0.05 0.02 0.02 0.01 0.01 0.01 0.01 0.03 0.82

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.77 0.01 0.04 0.03 0.02 0.01 0.03 0.01 0.06 0.02

0.01 0.87 0.01 0.01

0

0.01 0.01

0

0.02 0.05

0.04

0

0.7

0.09 0.05 0.03 0.06 0.02 0.01 0.01

0.03 0.01 0.06 0.64 0.05 0.08 0.06 0.04 0.01 0.02

0.02

0

0.05 0.06 0.74 0.03 0.05 0.04 0.01 0.01

0.01

0

0.05 0.15 0.04 0.66 0.04 0.04 0.01 0.01

0.02 0.01 0.04 0.06 0.02 0.02 0.78 0.01 0.01 0.03

0.02

0

0.03 0.05 0.05 0.03 0.01 0.81

0

0.01

0.05 0.02 0.01 0.02 0.01

0

0.02 0.01 0.83 0.03

0.02 0.05 0.01 0.02 0.01 0.01 0.01 0.01 0.03 0.83

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.74 0.01 0.05 0.03 0.02

0

0.03 0.01 0.06 0.02

0.02 0.87 0.01 0.02

0

0

0.01

0

0.02 0.05

0.05

0

0.7

0.07 0.05 0.03 0.06 0.02 0.01 0.01

0.02 0.01 0.05 0.68 0.05 0.07 0.07 0.03 0.01 0.02

0.02

0

0.05 0.06 0.77 0.02 0.04 0.03

0

0

0.01

0

0.07 0.15 0.04 0.65 0.04 0.03 0.01 0.01

0.01

0

0.03 0.07 0.03 0.02 0.83 0.01

0

0.01

0.01

0

0.03 0.04 0.04 0.02 0.01 0.82

0

0.01

0.06 0.02 0.01 0.02 0.01

0

0.02

0

0.85 0.02

0.02 0.05 0.01 0.02 0.01

0

0.01 0.01 0.03 0.85

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.76 0.01 0.05 0.04 0.02

0

0.02 0.01 0.07 0.03

0.01 0.87 0.01 0.01

0

0

0.01

0

0.02 0.05

0.04

0

0.73 0.06 0.05 0.03 0.06 0.01 0.01 0.01

0.01 0.01 0.05 0.71 0.05 0.06 0.06 0.03 0.01 0.01

0.02

0

0.04 0.05 0.78 0.02 0.04 0.03 0.01

0

0.01

0

0.06 0.17 0.04 0.64 0.04 0.03 0.01 0.01

0.01

0

0.03 0.06 0.03 0.01 0.85 0.01

0

0.01

0.01

0

0.04 0.04 0.05 0.02 0.01 0.81 0.01 0.01

0.05 0.02 0.01 0.02 0.01

0

0.02

0

0.84 0.02

0.02 0.05 0.01 0.02 0.01

0

0.02 0.01 0.04 0.83

True label

1 st

visit

2 nd

visit

3 rd

visit

4 th

visit

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.76 0.02 0.04 0.04 0.02

0

0.02 0.01 0.08 0.02

0.02 0.86 0.01 0.02

0

0

0.01

0

0.02 0.05

0.04

0

0.73 0.07 0.05 0.03 0.05 0.01 0.01 0.01

0.01 0.01 0.06 0.69 0.05 0.06 0.07 0.02 0.01 0.01

0.02

0

0.05 0.07 0.76 0.02 0.04 0.03 0.01

0

0.01

0

0.07 0.17 0.04 0.64 0.03 0.03 0.01 0.01

0.01

0

0.03 0.07 0.02 0.01 0.84 0.01 0.01 0.01

0.01

0

0.04 0.04 0.06 0.02 0.01 0.81 0.01 0.01

0.04 0.02 0.02 0.02 0.01

0

0.02

0

0.86 0.02

0.02 0.05 0.02 0.02 0.01

0

0.02 0.01 0.03 0.82

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.74 0.01 0.05 0.04 0.02

0

0.03 0.01 0.07 0.02

0.01 0.88 0.01 0.01

0

0

0.01

0

0.02 0.05

0.05

0

0.74 0.07 0.05 0.02 0.05 0.01 0.01

0

0.01

0

0.05

0.7

0.06 0.06 0.07 0.02 0.01 0.01

0.01

0

0.04 0.06 0.79 0.02 0.04 0.02 0.01

0

0.01

0

0.06 0.17 0.04 0.65 0.04 0.03 0.01 0.01

0.01

0

0.03 0.07 0.02 0.01 0.85

0

0

0.01

0.01

0

0.04 0.04 0.06 0.02 0.01

0.8

0.01 0.01

0.04 0.02 0.01 0.02 0.01

0

0.02

0

0.87 0.02

0.02 0.05 0.02 0.02

0

0

0.02 0.01 0.04 0.83

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.76 0.01 0.04 0.04 0.02

0

0.03 0.01 0.07 0.02

0.01 0.88 0.01 0.01

0

0

0.01

0

0.02 0.05

0.04

0

0.74 0.06 0.05 0.02 0.05 0.01 0.01 0.01

0.01 0.01 0.06 0.68 0.05 0.06 0.08 0.02 0.01 0.01

0.01

0

0.05 0.06 0.79 0.01 0.04 0.02

0

0

0.01

0

0.07 0.18 0.05 0.61 0.04 0.03 0.01 0.01

0

0

0.03 0.06 0.02 0.01 0.86

0

0

0

0.01

0

0.04 0.04 0.06 0.02 0.01

0.8

0

0.01

0.06 0.02 0.02 0.02 0.01

0

0.02

0

0.84 0.02

0.02 0.05 0.01 0.03 0.01

0

0.02 0.01 0.04 0.81

0: airplane

1: auto

2: bird

3: cat

4: deer

5: dog

6: frog

7: horse

8: ship

9: truck

0

1

2

3

4

5

6

7

8

9

0.75 0.01 0.04 0.04 0.02

0

0.03 0.01 0.08 0.02

0.01 0.87 0.01 0.01

0

0

0.02

0

0.02 0.06

0.04

0

0.73 0.08 0.05 0.02 0.05 0.01 0.01 0.01

0.01

0

0.05 0.73 0.05 0.05 0.07 0.02 0.01 0.01

0.01

0

0.05 0.06 0.79 0.01 0.05 0.02 0.01

0

0.01

0

0.06 0.18 0.04 0.63 0.04 0.02 0.01 0.01

0

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1. Claims

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Justification: We have discussed the limitations and potential future work of our study in Sec. 6. 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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Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

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Justification: Due to the limited computing resources, we only extensively evaluate the performance of our proposed method (PeTTA) across 5 independent runs, with different random seeds. Specifically, the mean values in 5 runs are reported in Tab. 1, Tab. 2, Tab. 7, and Tab. 8. The corresponding standard deviation values are provided in Appdx. F.1.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS:\n(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.\n\nWEAKNESSES:\n(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.\n\nQUESTIONS:\nPlease see the weaknesses.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\n1. 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.\n\nWEAKNESSES:\n1. 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.\n\nQUESTIONS:\nSee 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.\n\nLIMITATIONS:\nThere 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.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS:\n- 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.\n\nWEAKNESSES:\n- 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.\n\nQUESTIONS:\nThe 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?\n\nLIMITATIONS:\n- 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.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}

comments: 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.

STRENGTHS:
1. The proposed recurring TTA scenario reflects the challenging and practical situation well.
2. 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.
2. 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.

QUESTIONS:
1. The authors said EATA is the baseline. However, I couldn't find any results of EATA.
2. It seems that the images on the left and right in Figure 4(c) are switched.
3. How about applying only $\mathcal{L}\_\mathrm{AL}$ without $\lambda$ in Table 3?

LIMITATIONS:
Yes
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-14