COME: Test-time Adaption by Conservatively Minimizing Entropy
Authors: Qingyang Zhang
Yatao Bian
Xinke Kong
Peilin Zhao
Changqing Zhang
Venue: ICLR 2025 Poster (2025)
Keywords: Test-time adaption
Out-of-distribution generalization
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We propose an alternative to entropy minimization as a better learning principle for TTA tasks.

Abstract: Machine learning models must continuously self-adjust themselves for novel data distribution in the open world. As the predominant principle, entropy minimization (EM) has been proven to be a simple yet effective cornerstone in existing test-time adaption (TTA) methods. While unfortunately its fatal limitation (i.e., overconfidence) tends to result in model collapse. For this issue, we propose to \textbf{\texttt{Co}}nservatively \textbf{\texttt{M}}inimize the \textbf{\texttt{E}}ntropy (\texttt{COME}), which is a simple drop-in replacement of traditional EM to elegantly address the limitation. In essence, \texttt{COME} explicitly models the uncertainty by characterizing a Dirichlet prior distribution over model predictions during TTA. By doing so, \texttt{COME} naturally regularizes the model to favor conservative confidence on unreliable samples. Theoretically, we provide a preliminary analysis to reveal the ability of \texttt{COME} in enhancing the optimization stability by introducing a data-adaptive lower bound on the entropy. Empirically, our method achieves state-of-the-art performance on commonly used benchmarks, showing significant improvements in terms of classification accuracy and uncertainty estimation under various settings including standard, life-long and open-world TTA, i.e., up to $34.5\%$ improvement on accuracy and $15.1\%$ on false positive rate. Our code is available at: \href{https://github.com/BlueWhaleLab/COME}{https://github.com/BlueWhaleLab/COME}.

Published as a conference paper at ICLR 2025
COME: TEST-TIME ADAPTION BY CONSERVATIVELY MINIMIZING ENTROPY
Qingyang Zhang'; Yatao Bian?', B Xinke Kong', Peilin Zhao² and Changqing Zhang" College of Intelligence and Computing, Tianjin University! Tencent AI Lab?
ABSTRACT
Machine learning models must continuously self-adjust themselves for novel data distribution in the open world. As the predominant principle, entropy minimiza- tion (EM) has been proven be simple yet effective cornerstone in existing test-time adaption (TTA) methods. While unfortunately its fatal limitation (i.e., overconfidence) tends to result in model collapse. For this issue, we propose Conservatively Minimize the Entropy (COME), which is a simple drop-in replace- essence, COME
ment of traditional EM to elegantly address the limitation. explicitly models the uncertainty by characterizing Dirichlet prior distribution over model predictions during TTA. By doing so, COME naturally regularizes the model to favor conservative confidence unreliable samples. Theoretically, we provide a preliminary analysis to reveal the ability of COME in enhancing the optimization stability by introducing iata-adaptive lower bound the entropy. Empirically, our method achieves state-of-the-art performance on commonly used benchmarks, showing significant improvements in terms of classification accuracy and uncertainty estimation under various settings including standard, life-long and open-world world TTA, i.c., up 34.5% improvement on accuracy and 15.1% on false positive rate. Our code is available at: http://gthbbbboomBBBllllaleee.ee
INTRODUCTION
Endowing machine learning models with self-adjust ability is essential for their deployment in the open world, such as autonomous vehicle control and embodied AI systems. To this end, test-time adaption (TTA) emerges which often encounters unexpected noise or corruption (e.g.. data from rainy or snowy weather). Unsupervised losses play a crucial role in model adaptation, which can improve the accuracy of model on novel distributional test data without the need for additional labeled training data. The representative strategy entropy minimization (EM) adapts classifiers by iteratively increasing the probabilities assigned to the most likely classes, and is an integral part in the sate-oof-the-rr TTA methods (Press et al., 2024 Wang et al., 2021| /Zhang et al., 2022, Niu et al.) [022] Wang et al. 2022b Twasawa & Matsuo 2021 Niu et al. 2023 Yang et al. 2024). The initial intuition behind using entropy minimization, given by (Wang et al. 202 1) is based on the observation that models tend to be more accurate samples for which they make predictions with higher confidence. The natural extension of this observation is to encourage models to bolster the confidence on test samples.
promising strategy enhance the performance in the open world
However, this intuition may not always be true since there always exists irreducible uncertainty which arises from the natural complexity of the data or abnormal outliers. Naturally, one might expect a machine learning model to adapt itself to test data and favor higher confidence on right prediction, but of course not absolute certainty for the erroneous. This contradiction challenges the suitability of EM in TTA tasks, which greedily pursues low-entropy on all test samples. A notable example in recent research concerns that EM can be highly unstable and frequently lead to model collapse when the models encounter unreliable samples in the wild (Niu al. 2023). In this work, we hypothesize
Work done during an internship at Tencent Lab. I CCo-supervised. Correspondence to Yatao Bian cittao.bian@gmall.comm and Changqing Zhang changhhngginggejjuncdd.....

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that due to the nature of EM, previous TTA methods tend to be highly overconfident ignoring the reliability of various test samples, which further results in the unsatisfactory performance.
For the above issues, propose simple yet effeecive model-agnostic learning principle, termed Conservatively Minimizing Entropy (COME) to stabilize TTA. We first consider the model output as opinion which explicitly models the uncertainty of each sample from Theory of Evidence perspective. Then, we encourage the model to favor definitive opinions for TTA and meanwhile take the uncertainty information into consideration. This offers two-fold advantages compared to EM learning principle. First, our COME leverages subjective logic (Jsang) 2018), which is an off-the-shelf uncertainty tool in Bayesian toolbox effectively perceive the uncertainty raised upon varying test samples without altering the original model architecture or training strategy. Second, when encountering unreliable outliers, the model is regularized to favor conservative confidence and be able to explicitly express "I do not know", i.e., reject to classify them any known classes, which meets our expectation on model trustworthiness. Theoretically, our COME takes inspiration from Bayesian framework, and can be proved to correspond with a data-adaptive upper bound on the model confidence, which a desirable property for TTA where the reliability of test samples are often varying from time. The contributions of this work are summarized follows:
As a principled alternative beyond entropy minimization, we propose a simple yet effective driven strategy for test-time adaption called Conservatively Minimizing Entropy (COME) which improves previous methods by exploring and exploiting the uncertainty.
We provide theoretical analysis with insight in contrast to EM, the model confidence of our COME is provably upper bounded in a data-adaptive manner, which enables TTA methods to focus reliable samples and conservatively handle abnormal test samples.
We perform extensive experiments under various settings, including standard, open-world and lifelong TTA, where the proposed COME achieves excellent performance in terms of both classification accuracy and uncertainty quantification.
2 RELATED WORK
Test-time adaption aims to bridge the gaps between source and target domains during test-time without accessing the training-time source data. Entropy minimization performs an important rolc in test-time adaption, which has been integrated as a part of numerous TTA methods (Press et al.
2024 Wang et al.] 2021 Niu et al.] 2022; Wang et al., [022b] Iwasawa & Matsuo 2021] Yang et al. 2024] Chen et al. 2022). However, it has been observed that the performance of EM can be highly sub-optimal and unstable when encounter unreliable environments. To this end, previous works incorporate many strategies including i) Samples selection, which selectively filter out the high-entropy unreliable samples by manually setting an entropy threshold (Iwasawa & Matsuo 2021 Niu et al. 2023). ii) Constrained optimization, which heuristically enforces that the updated parameters do not diverge too much during adaption. iii) Model recovery, which lively monitor the state of the adapting model and frequently reset when detecting performance collapse (Niu) ctt al. 2023 Wang et al. 2022b). Although these strategies have shown promising performance, the underlying issues of the EM are still largely unexplored. In contrast, this work aims to handle the inherent issues of EM learning objective and validate the necessity and effectiveness in TTA settings,
Uncertainty quantification is one key aspect of the model reliability. With accurate uncertainty estimation ability, further processing can be taken to improve the performance of machine learning systems (c.g., human assistance) when the predictive uncertainty is high. This is especially useful in high-stake scenarios such as medical diagnosis (Wang et al. 2023). To obtain the uncertainty, Bayesian neural networks (BNNs) (Denker & LeCun] 19900 Mackay 1992) have been proposed to replace the deterministic weight parameters of model with distribution. Unlike BNNs, ensemble- based methods obtain the uncertainty by training multiple models and ensembling them (Rahaman) et al. 202T Abe et al. 2022). In this paper, we focus on estimating and exploiting uncertainty under the theory of subjective logic (SL, (Jsang, 2018)). Unlike BNNs ensemble, SL explicitly models the uncertainty in a single forward pass without modifying the training strategy or model architecture, which meets expectation of computational effectiveness for TTA tasks.
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3 MOTIVATION
We consider the fully test-time adaption setting in K-classification task where X' is the input space and y = {1.2,... } denotes the target space. Given classifier f X - R parameterized by which has been pretrained on training distribution ptrain, our goal is to boost f by updating its parameters l online each batch of test data drawn from test distribution ptest Note that in fully TTA setting, the training data pirain is unavailable and one can only tune 0 on unlabeled test data. This is derived from realistic concerns of privacy, bandwidth profit. Compared other closely related setting, i.e., Source-Free Domain Adaptation (SFDA), TTA focuses on online adjusting during the testing while SFDA generally perform offline. That is, TTA method aims on (unsupervised) adaption online and the inference latency matters. Entropy minimization (EM) algorithm iteratively optimizes the model minimize the predictive entropy on test sample
K
H(p(yla))
= Σ plu = k(c) ply y k|(;), k=1
where p(y]a:) is the class distribution calculated by normalizing the output logits fx) with softmax function, i.e., p(y = k//z) = Cexp f(a))
exp (an) H is the Shannon's 's entropy.
Other learning objectives. Besides EM, there also exists several TTA methods which explore
other unsupervised learning objectives. Notable examples include 1) Pseudo label (PL): LpT. -E loog(yy = /|1>) which encourages the adapted model to fit the pseudo label á predicted by the pretrained model, 2) Module adjustment (T3A) which adjusts the parameters in the last fully connected layer, and can be viewed as an implicit way to minimize entropy (Iwasawa & Matsuo E log > exp (x) which aims to minimize the free energy during adaption, and takes inspiration from energy model (Yige et al., 2024), 4) Contrastive sxep query-koy which strives to minimize the cosine distance
2021). 3) Energy minimization: LTEA- =
learning objective: LinioNCE llog Σ exp 2expquerys query-key
between the query and positive samples (key while maximizing the cosine distances between query and negative samples (Chen et al. 2022). 5) The recent advanced FOA (Niu al.] (024] which uses evolution strategy to minimize the test-training statistic discrepancy and model prediction entropy.
The overconfident issue of EM. We begin by testing EM in standard TTA setting, and put forward the following observations to detail its unsatisfying performance.
Max Softmax Probability
Accurncy per batch
per batch
0.95
Model collapsc
0.6
Overconfidence
0.85
0.7
0.4
0.80
0.6
0.75
0.5
0.3
0.70
0.4
0.3
Model collapse
0.65
0.1
100 200
300
400 500 600 700 800
100
200 300
400
500
600
700
800
100
200
300
400 500 600 700 800
Batch
Batch
Batch
No Adapt
Tent
SAR
COME
SAR+COME
Figure 1: Empirical observations of Entropy Minimization when equipped to two representative TTA methods, i.e., the seminal Tent (Wang et al., 2021] and recent advanced SAR (Niu et al.) 2023). Along the TTA process, the uncertainty of models tuned with EM quickly drops, and the false positive rate decreases temporarily for very short time horizon before quickly increasing. Along the same adaption trajectory, the model accuracy also improves for short time compared to the initial model and then quickly decreases, after which the model collapses to a trivial solution. By contrast to EM, our COME establishes stable TTA process with consistently improved classification accuracy and false positive rate. Besides, the model confidence of our COME is much more conservative, which implies fewer risks of overconfidence and more accurate uncertainty awareness.
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As shown in Figure I EM tends overconfident prediction and assign extremely high probability to one certain class. The absolutely high confidence on all test samples is obviously a rather undesired characteristic. We test ImageNet-C under snow comruption of severity level 5 typical showcase, and refer interested readers to Appendix|C... [C11 for more similar results.
4 METHODOLOGY
We propose to conservatively minimize the entropy under uncertainty modeling, a simple alternative to EM algorithm. The key idea of COME is to quantify and then regularize the uncertainty during TTA without altering the model architecture or training strategy, which avoids the overconfident nature of EM minimal cost. We first introduce uncertainty quantification by the subjective logic and then present how to regularize the uncertainty during TTA.
4.1 MODELING UNCERTAINTY BY THE SUBJECTIVE LOGIC
Given a well trained classifier f & + y, the most simple way to quantify the uncertainty of each sample is using the softmax probability as confidence in prediction. A few pioneer works propose to filter out the test samples with high-entropy predicted softmax probability for stable TTA (Niu) al. 2023 2022). However, it has been shown that softmax probability often leads to overconfident predictions, even when the predictions are wrong or the inputs are abnormal outliers (Moon et al. 2020 Van Amersfoort al., 2020). Thus this simple strategy may not be satisfied enough and highlights the necessity of better uncertainty modeling. To this end, we propose to obtain the uncertainty through subjective logie, which defines framework for obtaining the probabilities (belief masses) of different classes and the overall uncertainty (uncertainty mass) based on the evidence collected from data. Specifically, in K classification task, SL formalizes the belief assignments over a frame of discernment as a Dirichlet distribution. In contrast to softmax function that directly normalizes the logits fx) to model the class distribution p(y [z) ), SL considers the model output as evidence (denoted as e) to model Dirichlet distribution which represents the density of all possible
probability assignment f [ply 1|x), ply 2/z). . pyy K| K|x)]. That is, the predicted
categoricals fa. is also random variable itself, which yields Dirichlet distribution as follow
K
p(u[z) = Dir(wla)
1
II
Pk 0--1
a a=e11,
(2)
B(a
where Dir(u) is the Dirichlet distribution characterized by parameters a. The summation of all O% E ar is called the strength S of the Dirichlet distribution, i.e., S -x O% = Σn ex + Then SL. tries to assign a belief mass by to each class label k and an overall uncertainty mass u to the whole frame based on the collected evidence follow
Ek S
0g
K and u = S'
bg
subject to
bk
=
5
where S is the Dirichlet strength which denotes the total evidence we collected and K is the total classes number. Eq. actually describes the phenomenon where the more evidence observed for the k-thh category, the greater the belief mass assigned the k-th class. Correspondingly, the less total evidence S observed, the greater the total uncertainty u. Such assignment is so called the subjective opinion
(4) which not only describes the belief of assigning æ to each class & but also explicitly models the uncertainty due to lack-of-evidence. To ensure that > 1 which meets the requirements of valid parameters of a Dirichlet, we first deploy ReLU function on the logits and then use exponential output function for obtaining the parameters of Dirichlet distribution from the model output f()) (i.e., logits), where a = exp(ReLU(f(z)) and e = a
M(x) = b,,
... ......,,bbb be, u].
Design choices of transformation function. By definitions, the parameters of Dirichlet distribution a must be greater than and the evidence e should be non-negative. This can be achieved by applying ReLU activation function exponential function to the output logits suggested in
In Bayesian context, evidence refers the metrics collected from the input support the classification.
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Alternatively, we can get the evidence by e = ReLU(f(z)) or e = exp f - - In this paper, exponential function is chosen since we assume the model is pretrained with standard cross-entropy loss. Thus using exponential function to get the evidence can keep the training strategy unchanged. We refer interested readers to (Malinin & Gales 2018) and Gal's PhD Thesis (Gal et al. 2016) for more detailed implementation instructions and math deviations.
previous works (Han et al., 2022 Malinin & Gales 2018
Benefits of modeling uncertainty based on subjective opinion for TTA. It has been widely recog- nized that using the softmax output as confidence often leads to overconfidence phenomenon (Guo
al., 2017 Hendrycks & Gimpel 2017 Liu et al. 2020). Other advanced uncertainty measurement
methods such as MC dropout (Gal & Ghahramami 2016), Bayesian neural networks (Huang et al.
2022), deep ensemble (Lakssmminrryyanan et al. 201 Rahaman et al. 2021). calibration (Guo et al.
1017|Han et al. 2024) usually require additional computations during inference or separate valida- tion set, which are not applicable in an unsupervised TTA task. By contrast, the introduced SI. directly deduces additional uncertainty mass through one single forward pass, which is model-agnostic, light-weight, and can be seamlessly integrated in standard pretrained classifier. We defer empirical comparisons between the proposed method and other Bayesian methods or learning objectives that originate from the adaption of the unsupervised domain to Table in the Appendix.
4.2 MODEL ADAPTION BY SHARPENING THE OPINION
Vanilla EM minimizes the softmax entropy of the predicted class distribution ())] which inevitably results in assigning rather high probability to one certain class. In contrast, we propose the learning principle to minimize the entropy of opinion
K Σ
minimize H(M(z))
by log by - u log u.
(5)
Compared entropy minimization on softmax probability, which ultimately assigns all the probability to one certain class, the above learning principle offers the model with an additional option, i.e., express high overall uncertainty and reject to adapt when the observed total evidence is insufficient. In other words, by assigning all belief masses (probability) to uncertainty u, the model can now express "I do not know" as its predicted opinion. Alternatively, an additional hyperparameter can be introduced to weight the last term as - > our confidence in whether the test sample should be adapted by the model or not. We validate the effectiveness of this generalization to Table 12 in the Appendix.
k= by log - Au log u. The magnitude reflects
4.3 REGULARIZING UNCERTAINTY IN AN UNSUPERVISED MANNER
While subjective logic offers an opportunity to model uncertainty and reject to classify unreliable samples, naively minimizing the entropy of opinion for TTA may still be problematic. As shown in previous works, the model pretrained with softmax output function frequently suffers from overconfidence issue (Guo al., 2017 Nguyen al., 2015 Hendrycks et al. 2019). Therefore, the belief mass assigned to the one certain class A by the pretrained model is usually much larger than the uncertainty mass u. This results in the model tendency of increasing the belief mass during the entropy minimization process, while neglecting the uncertainty function u. Motivated by the above analysis, our next goal is to devise an effective regularization strategy for the uncertainty mass. In supervised learning tasks, previous works leverage labeled training data to constrain the uncertainty mass (Sensoy et al., 2018 Malinin Gales 2018). However, these strategies is not applicable due to the unsupervised nature of TTA task where the training data is unavailable. This motivates us to explore the uncertainty information lies the pretrained model itself for regularization without additional supervision. As one of the simplest yet effective design choices, propose to constrain the uncertainty mass predicted by the adapted model not to diverge too far from the pretrained model. This results in the following constrained optimization objective
minimize ((M(z)) subject to (u=(x) uo, (x) $ 8,
(6)
where 8, du denote the adapted and pretrained model respectively, and is the uncertainty estimated by Eq. [3] and is a threshold to prevent overly extreme model uncertainty. The magnitude of represents tolerance for uncertainty divergence.
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Rethink why we constrain on [u(x) [z(z) < 8. The uncertainty estimated by pretrained model may not be ideal. However, since in fully TTA task, we can only access unlabeled test data coming online, and the inference efficiency matters. Thus traditional methods devised for handling overconfidence like calibration, ensembling, and other Bayesian methods are not applicable. The only practically available choice is explore the uncertainty information contained within the model itself. As shown in previous works (Sensoy et al. 2018 Malinin & Gales] 2018), while the softmax probability of pretrained model tend be overconfident, subjective logic is much more reliable, which can support the proposed regularization. The most straightforward way to realize the aforementioned constraint is storage the model before and after adaption and explicitly compare the uncertainty mass. However, this may violate the efficiency requirements of TTA. Thus considering the difficulty of constrained optimization in modern neural networks, our next target to find way to convert Eq. 6 into an unconstrained form. To this end, we introduce the following Lemma.
Lemma 1. For any x E X, we have
K [lf(z))lp log K SuS
K1+1/D
llf(z))bb'
where (x) is the model output logits, K is the total class number and || -|l denotes the p-norm.
Lemma shows that the uncertainty mass subjective opinion is bounded by the norm of the total evidence collected from the model output. Thus instead of directly constraining on u(z), we can alternatively constrain on the p-norm model output logits, which is more flexible. Taking inspiration from previous work in supervised learning literature (Wei et al.] (2022), this can be achieved by factorize fz) into f(x)/|lf(x)) p flf(z) (x)]lp and then enforcing the gradient on the second term to be equal during optimization. Specifically, the final minimizing objective of COME is
f(x)
minimize H(M(z)) where f(x) [1f(z))lp llf(()) ( ppo-srad
grad.
(8)
[=] and [xf poo_red is the p-norm of (xx with zero gradient. This can be achieved by applying the
detach operation which is a common used function in modern deep learning toolbox like Py PyToc
and TensorFlow. By doing so, minimizing the entropy of opinion would not influence [I](xx) is
a hyper-parameter which controls the magnitude of recovered logits. Our COME can be implemented by modifying only a few lines of code in the original EM algorithm (shown as Algorithm [
How to set and p in practice? Noted that the tightness for the upper and lower bounds in Lemma I is determined by the choice of p. The ratio between the upper and lower bound is minimized by co. The strictness of such constraint should be selected per need by the user via trial and error: if users are extremely cautious about unreliable TTA, p should be tuned up: otherwise, if a better performance is required. In our experiments, we choose 2 and 7 1 in accordance with Occam's 's Razor. Experiments on different hypeparameters can be found in Table 13 in the Appendix.
Algorithm 1: Pscudo code of COME in a Py TyTrchh-like style.
the output logits, model: the test model
def entropy crrepp__o__ppindeee opinion (x)
belief exp (x) 1 sum (exp (x)) belief mass
uncertainty = K / sum (exp (x)) # uncertainty mass opinion = cat [[belief, uncertaintyl) | subjective opinion return -sum (opinion log (epinien)) | entropy of opinion
for data in rest_loader: + load a minibstch data x = model (data) # forward
nxrm(x, (x, p=2) nxrm(x, detaach) constraint in Eq.9 loss entropy_of_opinte (x) calculate loss
... [backwards and update the parameters]
Stability of COME, We provide preliminary theoretical understanding of the superiority of COME. As mentioned before, one notable limitation of EM is that it enforces low entropy for all test samples while ignores the instinct complexity of wild test data. Thus at the end of TTA progress, EM ultimately produces model that yields overconfident prediction. Our COME resolves this issue and

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introduces upper bound for each test sample according to its trustworthiness. This property is formalized as follows
Theorem (Model confidence upper bound), For any x E X, if u(()) uu holds, then we have
max p(y = k()) 1 (K exp
K
where max p(y k(x:) is the model confidence (class probability assigned the most likely class) k and K is the total class number. un is the shorthand of uoo (a:).
From Theorem
find that the model confidence in COME has a sample-wise upper bound according to un T) In particular, it implies that the model confidence upper bound the most can suspect that if the test model is
likely class decreases according (x)). For this reason, uncertain about some sample x (with a rather large up). it will be difficult to further increase the model confidence such x, which is desirable property for TTA in the wild.
EXPERIMENTS
We conduct experiments multiple datasets with distributional shift to answer the following questions. Q1. In the standard TTA setting, does the proposed method outperform other algorithms? Q2. How does COME perform in more realistic TTA settings, such open-world TTA or lifelong TTA? Q3. Uncertainty quantification is both the motivation behind COME and the reason for its effectiveness, does our method achieves more reliable uncertainty estimation during TTA? Q4. Ablation study what is the key factor of performance improvement in our method? Q5. Hyper- parameters how the performance of our method under different parameters?
5.1 SETUP
Following the common practice (Niu et al. 2022, 2023), we conduct experiments standard covariate-shifted distribution datasets ImageNet-CC (a large-scale benchmark with 15 types of diverse corruption), ImageNet-R and ImageNet-S. Besides, adaption setting, where the test data distribution prest is mixture of both normal data pCov and abnormal outliers pOutlier of which the true labels do not belong any known classes in ptrain Following previous work in open-set OOD generalization literature (Lee et al. 2023 Bai et al., 2023 Baek et al. 2024). pOutlier is a suit of diverse datasets introduced by (Yang) et al. 2022), including iNaturalist, Open-Image. NINCO and SSB-Hard. Compared methods. We compare our COME with board line of test-time adaption methods, including both EM-based and non-EM based TTA methods. o EM-based methods choose entropy minimization as their learning objective, including Tent (Wang al.) (02T), EATA (Niu al.. (2022). CoTTA (Wang et al.) 2022b), MEMO (Zhang et al. (022) and recent advanced SAR (Niu et al. 2023) o Non-EM methods employ other learning objectives including Pseudo Label (PL), module adjustment (Iwasawa & Matsuo 2021) (T3A) and energy minimization (Yige et al.) 2024 (TEA). Following NNu et al.) 2023). use the ViT-base architecture as our backbone and defer the results on ResNet50 to Table in the Appendix. The test batch size is 64. When equipped to previous EM-based TTA baselines, only replace the learning objective with our COME and keep all the other configures consistent to the official implementation. Tasks and Metrics. For classification performance comparison, report the accuracy (Acc) on covariate-shifted data. Besides, for open-world TTA, we report the false positive rate (FPR). The mis-classified samples and outliers considered as positive samples which should be of higher uncertainty compared to correct classification that is considered as negative. For all experiments, we run multiple times and report the average. Full results with standard deviation are deferred to the Appendix.
Datasets.
also consider open-world test-time
5.2 EXPERIMENTAL RESULTS
Performance comparison in standard TTA settings (Q1). As shown in Table 1 our COME establishes strong overall performance in terms of both classification accuracy. We highlight few essential observations. Compared to EM learning principle, our COME consistently outperforms it when equipped to the same baseline methods, including Tent (Wang al. 2021). EATA (Niu
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et al. 2022). CoTTA (Wang et al. 2022b and SAR (Nuu et al. 2023). As example of our method's improved performance, when equipped to the recent SAR, our method yields an accuracy of 64.2%, which outperforms the original implementation based on EM of 10.1%. Besides, also compare to Non-EM TTA methods, including TEA, T3A and PI.. These methods do not rely on EM learning objective, yet are less effective than EM in terms of classification performance. For more comprehensive evaluation, we conduct additional experiments on ImageNet-R and ImageNet-S, the results can be found in Table 2
Table 1: Classification accuracy comparison on ImageNet-C (level 5), Substantial (2 0.5) improve- ment and degradation compared the baseline are highlighted in blue or brown respectively. The detailed results with standard deviation are defer to Table g in Appendix C.I
Noise
Blur
Weather
Digital
Methods
COMP
Gauss.
Shot Impul.
Defoc Glass
Motion Zoom
Snow
Frost
Fog
Brit.
Contr.
Flast.
Pixel
JPRG
No Adapt
x
35.1
32.2
35.9
31.4
25.3
39.4
31.6
24.5
30.1
54.7
64.5
49.0
34.2
53.2
56.5
PL
x
49.9
48.8
50.9
48.5
41.2
52.7
42.7
24.6
42.5
63.6
73.1
65.3
44.1
63.9
62.6
T3A
X
34.6
31.5
35.5
32.7
27.5
40.7
33.5
25.6
30.8
56.5
64.9
50.8
38.0
54.3
58.4
TEA
X
44.5
39.3
45.8
37.6
35.4
46.4
31.4
8,9
46.4
60.1
72.5
59.5
45.7
62.3
58.9
LAME
x
34.8
31.9
35.5
30.9
24.4
38.9
30.7
23.4
29.5
53.3
64.2
41.0
32.7
52.8
56.0
FOA
x
46.6
43.9
48.3
47.1
40.7
49.3
43.7
53.8
52.8
64.2
76.2
63.8
48.5
62.1
64.0
X
52.5
52.1
53.4
52.8
47.4
56.7
47.4
10.5
26.4
67.2
74.3
67.3
50.4
66.5
64.6
Tent
/
53.9
53.9
55.2
55.8
51.8
59.8
52.6
58.4
61.3
71.3
78.2
68.8
57.9
70.5
68.2
Improve
A1.4
21.9
A18
2.2.9
244
/.3.2
2.5.2
/447.9
2.34
24.1
A33.9
A1.5
2.76
A4.0
4.3.6
X
56.0
56.1
57.1
54.5
54.8
59.6
58.7
61.8
60.1
71.4
75.3
68.5
62.7
69.0
66.5
EATA
56.1
56.5
57.2
58.0
57.9
62.6
59.3
65.6
63.5
72.6
78.0
69.5
66.6
72.5
70.5
Improve
2.0.1
2.0.3
2.0.1
A33.4
A.3.1
2.3.1
2.0.6
A.3.9
233.4
A.13
A.2.7
A1.
A.3.9
A3.5
2.4.0
51.9
51.7
52.8
51.5
48.9
55.5
49.5
22.2
46.9
66.2
72.9
65.8
50.9
64.0
62.8
SAR
56.4
56.6
37.4
58.3
56.9
62.9
58.3
65.3
64.5
72.7
78.5
69.6
64.0
71.9
69.7
Improve
A4.5
A449
A.46
/.6.8
2.8.1
/.7.4
2.8.8
/.43.1 1
1.7.55
2.6.5
A.5.5
A33.8
1.13.2
A.7.9
/.6.9
40.3
37.6
41.7
34.3
28.3
44.0
35.6
38.0
43.0
58.8
70.3
58.4
39.8
58.1
59.9
CoTTA
V
43.5
40.9
45.5
36.9
29.7
48.1
37.8
40.7
42.0
62.3
73.6
58.9
42.8
63.5
63.8
Improve
A3.1
23.4
43.8
22.66
414
24.0
22.
4.2.7
4/1.
4335
43.3
4..5
A3.0
A5.4
43.9
X
39.7
36.5
39.8
32.4
25.8
40.3
34.7
27.5
32.8
53.5
66.2
56.0
35.7
55.9
58.2
MEMO
/
40.6
37.5
40.6
33.4
26.7
41.2
35.4
28.7
33.7
54.7
67.1
55,9
36.6
57.2
59.3
Improve
2.088
21.0 20.8
A1.
4.0.9
41.0
4.0.
412
4.0.
41
2..
V001
2.0
A1.3
A11
Table 2: Additional results on ImageNet-R and ImageNet-S.
Dataset
COME
Tent
EATA
SAR
COTTA
X
37.73 +: 0.03
36.11 1 0.12
51.90 = 0.35
36.04 1 0.06
ImageNet-R
/
39.05 + 0.05
38.22 1 0.23
56.40
0.18
37.39 + 0.08
X
31.63 + 0.07
36.11
33.92 0.69
30.84 0.35
ImageNet-S
V
39.22 1 0.04
39.22 1 0.32
43.52 1 0.52
33.82 1 0.73
Performance comparison in open-world and lifelong TTA settings (Q2). In Table and Bndd present the results under open- world and lifelong TTA settings respectively. In open-world TTA. the test data distribution is a mixture of both normal covariate-shifted data and abnormal outliers. The mixture ratio of pCov and pOutlier 0.5 following previous work (Bai et al., 2023). i.e., ptest = 0.5 + 00.5
4 we
Such outliers arise from unknown classes that not present in training data, which should not be of high uncertainty for model trustworthiness. According to the experimental results, it is observed that our COME can consistently improve the performance of existing TTA methods.
Reliability of uncertainty estimation (Q3). We visualize the distribution of model confidence. i.c., the maximum predicted class probability in open-world world TTA setting, where the covariate-shifted samples is ImageNet-C (Gaussian noise level 3), and outliers are NINCO. As shown in Figure 2 the model confidence of our COME can effectively perceive incorrect predictions, which establishes an distinguishable margin. In contrast to the model confidence of EM which is almost identical and nearly 100% for all test samples, the model confidence of our method can provide more meaningful information with which to differentiate correct and wrong predictions.
Forr our COME, the class prohability is calculated according Eq. 5.
8

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Published as a conference paper at ICLR 2025
Table 3:
Classification and uncertainty estimation comparisons under open-world TTA settings, where plest is mixture of both covariate-shifted samples (Gaussian noise of severity level 3) and a suit of diverse abnormal outliers. Additional results with standard deviation are Appendix
None
NINCO
iNaturist
SSB-Hard
Texture
Places
Method
COME
AAct
FPR]
Acct
FPR]
Acct
FPR]
Acct
FPR]
Acct
FPR]
Acct
FPR.
No Adapt
X
64.4
63.7
64.5
69.9
64.4
69.5
64.4
72.5
64.8
65.6
64.3
56.8
PL
X
69.1
62.8
65.6
71.6
68.8
69.5
68.4
75.9
66.1
66.0
66.6
59.7
T3A
X
64.4
71.2
64.3
70.0
64.2
75.0
63.7
80.7
64.4
69.0
63.8
69.5
TEA
X
63.9
63.8
60.5
72.5
62.3
74.6
63.3
79.4
61.0
67.8
61.9
64.5
LAME
X
64.1
64.4
64.1
72.3
64.1
72.4
64.2
74.0
64.7
68.8
64.0
61.3
FOA
X
67.8
61.2
66.4
70.5
67.4
66.1
67.1
75.6
65.8
61.4
66.6
54.1
X
70.8
63.2
66.2
71.9
09.9
70.7
09.8
77.4
66.4
66.5
68.3
59.9
Tent
/
72.6
64.7
68.9
64.3
72.5
63.7
72.7
70.7
68.4
60.4
70.7
45.9
Improve
A1.7
A1.6
A2.7
77..6
A2.6
5/7.0
A.2.9
5/6.7
A2.1
<76.1
22.4
4/14.0
X
70.3
63.7
66.4
68.6
70.3
71.5
70.0
77. 4
67.3
67.1
68.8
61.9
EATA
/
73.4
62.7
70.1
60.5
73.2
63.3
73.0
70.5
70.5
55.8
72.3
45.6
Improve
A3.1
71.0
/33.7
78.2
A22.9
778.2
A.3.0
$76.9
A3.22
11..
A.3.6
116.
X
69.7
62.3
64.9
71.4
66.9
70.9
67.7
78.1
64.4
64.5
66.1
58.6
SAR
/
73.1
62.9
69.8
66.3
73.2
65.2
73.5
71..
69.5
59.4
72.3
49.8
Improve
A3.5
A.0.6
444.9
V5.1
2.6.3
V/5.6
2.5.9
5/6.3
A.5.1
</5.1
4..3
V8.8
X
67.6
63.4
65.3
69.7
70.4
69.5
70.3
76.0
65.8
66.2
66.6
59.2
COTTA
V
70.5
62.5
66.2
68.8
72.4
73.5
72.2
78.7
66.5
64.7
68.9
35.0
Improve
A2.9
<0099
A0.9
v00.9
/2.00
2:4.0
2.2.0
2.2.7
2.0.7
</1.5
42.3
V/.22
X 64.8
69.8
64.8
77.5
64.7
71.9
64.8
77.3
65.0
79.3
64.6
71.5
MEMO
V 65.2
67.8
65.9
76.4
65.2
70.8
65.3
75.0
65.4
76.7
65.3
67.6
Improve
2.0.5
41.9
A1.1
71.
2.0.5
7..
2.0.5
272.3
2.0.4
42.6
2.0.7
73.9
Table 4: Classification and uncertainty estimation comparisons under lifelong TTA settings. The model is online adapted and the parameters will never be reset, yet the test input distribution might exhibit a continual shift over time.
Noise
Blur
Weather
Digital
Methods
COMP
Gauss.
Shot
Impal.
Defoc Glass
Motion 7oom
Snow
Frost
Fog
Brit.
Contr:
Flast.
Pixel
JPFG
No Adapt
x
35.1
32.2
35.9
31.4
25.3
39.4
31.6
21.5
30.1
51.7
61.5
49.0
31.2
53.2
56.5
X
49.9
53.4
56.7
46.7
46.1
56.6
51.4
52.6
60.2
68.1
77.7
64.5
51.0
69.0
68.8
POA
x
46.5
47.0
51.1
44.8
45.5
52.3
48.1
49.7
57.9
68.0
76.4
63.6
51.7
62.2
65.3
x
52.4
56.2
58.7
50.9
51.1
57.7
52.7
54.7
60.5
68.4
77.3
64.6
53.8
69.3
68.6
Tent
/
54.7
59.0
60.3
51.2
53.7
60.6
57.4
64.1
65.8
71.0
78.6
66.9
62.7
71.8
70.5
Improve
A2..3
A2.88
A1.6
A..4
/.2.6
2.2.9
A.4.7
A.9.5
A.5.3
A.2.6
A113
A:2.3
A.9.0
A.2.5
A1.9
X
55.9
59.5
60.9
56.2
59.2
63.0
61.6
65.7
67.5
72.5
78.6
66.8
67.1
72.2
71.7
FATA
V
57.9
60.6
61.5
57.0
59.8
63.7
62.3
67.2
68.3
73.7
78.8
69.7
68.6
73.1
71.8
Improve
2.2.0
41.0
2.0.6
2.0.8
2:0.6
2.0.8
2.0.7
A.5
2.0.8
2.1.2
230.3
4:2.9
A14
2.0.9
230.1
x
52.0
54.8
56.2
50.2
52.3
56.1
52.8
50.8
26.5
0.1
3.1
0.1
0.1
0.1
0.3
SAR
V
56.0
60.0
61.1
56.4
58.1
62.9
60.4
66.1
67.4
72.2
78.7
68.0
66.0
72.6
70.9
Improve
A4.0
A5..3
A4.9
A.6.
735.8
A.6.8
A7.6
A15.3
/:41.0
2.72.1
AV75.6
/667.9
/.65.9
2.72.5
A.70.6
X
40.3
49.2
57.1
39.8
50.4
35.6
48.3
53.1
61.3
63.9
73.3
62.0
56.5
67.5
06.7
COTTA
49.7
61.7
64.2
45.7
57.0
39.0
51.1
58.2
63.1
66.0
73.4
62.9
58.0
68.9
68.2
Improve
A9.4
A12.4
A7.1
A5.9
A.6.6
A33.
A.28
A5.1
A18
A2.1
A00.1
A1.0
A1.5
A13
A1.5
Ablation study (Q4). We conduct the ablation study on different components in our COME, i.c..
with and without the uncertainty constraint in Eq. The experimental results are shown in Table where SL indicates minimizing entropy of the subjective logic opinion and UC means the uncertainty constraint described by Fq. Compared with non-constrained optimization, naively minimizing the entropy of subjective opinion can only slightly improve uncertainty estimation performance. Combin- ing with the uncertainty constraint, the average and worst-case accuracy can be both substantially improved, which indicates the optimal design of our COME.
Influence of different hyper-parameter (Q5), We conduct additional experiments to investigate the influence of different hyperparameters. The results are in Table 12 and Table [3] Our COME generally outperforms EM with moderate hyperparamters. For т, as we mentioned before, it actually controls the magnitude of uncertainty mass. If we have some prior knowledge that most test samples

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Published as a conference paper at ICLR 2025
COME (FPR=69.89)
EM (FPR=78.64)
Incorrect Right
Incorrect
Acct
FPR.
Right
SL
UC
Mcan
Worst
Mean
Worst
Density -
X
52.8
10.6
70.2
95.3
52.7
10.4
70.0
94.9
X
60.9
25.5
68.0
93.0
61.2
51.7
67.3
71.4
0.2
64
0.5
0.8
14
0.2
0.4
0.6
0.8
1.0
Figure 2: Distribution of model confidence.
Table 5: Ablation study.
should be rejected adapt during TTA, should choose relatively small and making the model confidence more conservative in this circumstance. As for (or p) which represents the tolerance of uncertainty divergence, it should be selected per need by the user via trial and error: if users are extremely cautious about unreliable TTA, 5 should be tuned down and should be tuned up: otherwise, if better performance is required. Besides, we introduce an additional hyperparameter to the learning objective. The experiments results in Table 12 shows that an additional performance improvement can be observed when [1, 100].
Table 6: Additional results with different A. We report the accuracy when our COME is equipped to Tent ImageNet-C Gaussian noise level 5 with different X. The accuracy of the original Tent using entropy minimization is 52.6.
0.10
1.00
10.0
30.0
50.0
80.0
100
150
Acc. 53.7+0.1 53.9+0.0
54.6+0.1
55.3_0.1
55.8
56.2+0.1
56.1
10.620.3
Table 7: Additional results with different hyperparameters. We report the accuracy on ImageNet-C Gaussian noise level 5 with different hyperparameters. We implement our COME with Tent. The accuracy of the original Tent using entropy minimization is 52.6.
p
p = 2
p = 3
p = 0
T = 0.5
37.8
38.5
39.1
41.1
T 1.0
53.5
53.84n.1
53.240.0
47.2+0.1
== 1.2
54.7
54.740.1
54.0+0.1
48.920.0
T 1.5
54.2+0.2
54.240.
53.2+
49.1+0.1
CONCLUSION
In this paper, we propose novel learning principle called COME to improve existing TTA methods. COME explicitly models the uncertainty raising upon unreliable test samples using the theory of evidence, and then regularizes the model to favor of conservative prediction confidence during inference time. Our method takes inspiration from Bayesian framework, and consistently outperforms previous EM-based TTA methods on commonly-used benchmarks. The simplicity of the uncertainty regularization used our implementation both an advantage and limitation. On the one hand, constraining the uncertainty mass close to the pretrain model is easy-to-deples -as-llo-eepo and meets the efficiency requirement of TTA. On the other hand, this regularization may be less effective when the pretrained model is also overconfident. We identify this as limitation of our work. Exploring more effective regularization techniques for better trade-off between the practical requirements of TTA and accurate uncertainty estimation can be promising future direction. Besides, vision foundation models such as DINOv2 are more promising to fill domain gaps. The representation learned by these foundation models are naturally expected to be much more robust than ImageNet pretrained ResNet and ViT models we used in this manuscript. Investigating TTA techniques for more advanced vision foundation models are also promising research direction.
10

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Published as a conference paper at ICLR 2025
7 ACKNOWLEDGMENT
This work was supported by the National Natural Science Foundation of China (62376193 and 61925602). The authors thank ICLR anonymous peer reviewers for their helpful suggestions.
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Jingkang Yang, Pengyun Wang, Dejian Zou, Zitang Zhou, Kunyuan Ding, Wenxuan Peng, Haoqi Wang, Guangyao Chen, Bo Li. Yiyou Sun, al. Openood: Benchmarking generalized out-of- distribution detection. Advances in Neural Information Processing Systems, 35:32598-32611, 2022.
Mouxing Yang, Yunfan Li, Changqing Zhang, Peng Hu, and Xi Peng. Test-time adaptation against multi-modal reliability bias. In The Twelfth International Conference on Learning Representations, 2024.
Yuan Yige, Xu Bingbing, Hou Liang, Sun Fei, Shen Huawei, and Cheng Xueqi. Tea: Test-time energy adaptation. In IEEE/CVF Computer Vision and Pattern Recognition Conference, 2024.
Jingyang Zhang, Jingkang Yang, Pengyun Wang, Haoqi Wang, Yueqian Lin, Haoran Zhang, Yiyou Sun, Xuefeng Du, Kaiyang Zhou, Wayne Zhang, Yixuan Li. Ziwei Liu, Yiran Chen, and Hai Li. Openood v1.5: Enhanced benchmark for out-of-distribution detection. arXiv preprint arXiv:2306.003300 09301, 2023.
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Marvin Zhang, Sergey Levine, and Chelsea Finn. Memo: Test time robustness via adaptation and augmentation. Advances in neural information processing systems, 1.38622--86442 2022.
Aurick Zhou and Sergey Levine. Bayesian adaptation for covariate shift. Advances in neural information processing systems, 34:914-927, 2021.
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APPENDICES
Proofs
15
Experimental details
17
B.1 Datasets
17
B.2 Implementation details
17
C Additional results
18
C.I Full results of accuracy with standard deviation (supplementary to Table 1)
18
C.2 Full results of open- world TTA (supplementary to Table 3)
19
C33 Comparison with source-freee domain adaption
19
C.4 Comparison with other other Bayesian methods
20
C.5 Influence of different hyperparameters
20
C.6 Comparison on ImageNet-R and ImageNet-S
20
C.7 In-Distribution performance
21
C.8 Comparison on ResNet-50
C.9 Time-consuming comparison
21
C.10 Mixed distributional shifts performance.
C.11 More visualization results (supplementary to Figure
Discussion
27
D.I Alternative Design Choice
................................
27
D.2 Limitations and Future work
30
A PROOFS
To proof Lemma and Theorem 1, we need the following lemma firstly.
Lemma 2. Let q be two real numbers. Assuming that S. then the p-norm (also called (Pnorm)) and g-norm of vector x ("1,""";0 satisfied
||0/|p S n 11/00/1/2111011
(10)
where n is the length of the vector.
Proof. Recall Hölder's inequality
Σ
aallbbl <
la
6 -
(11)
1=1
Apply this inequality to the case that lail = |20. |bal = and r = q/pp 2 1, we can derive to
Σ as bul <
(()))))))
Σ 1
x 5 1-
(12)
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Then have
|
[ pp) ) 1/p
i=1
[ng (9) ) 1- 1/p
(13)
i=1
19)))n 1
x
/ 1/ /// 11111114
n
Now we proceed to proof our main results.
Proof. Proof of Lemma Let fmax max f((x), then we have
K
exp(fmas (mxx)
exp(fi) S K exp(fmux),
(14)
Applying the logarithm the inequality, then
fuax S LSE(f) fuax + log K,
(15)
where LSE is the shorthand of LogSumExp function, i.e., LSE(x) == log Σ exp Zz. 1=1
Since assume that all the elements in logits all positive, then fmax with Lemma we can derive that
||
Thus combining
K //1l|| Illl S LSE(x)
I 11fllp + log K,
(16)
Noted that KfLSE(f), then we have
K
K1+1/p
S u <
(17)
|lfllp + log K
||fllp
Proof. Proof of Theorem 1. Assuming that the uncertainty mass u is constrained as
8 < u S 40 + 5,
(18)
then the LSE function of model output is also bounded by
K
K 40
s LSE(f(x))
(19)
uo 8
8
Noted that
max f(x) S (S)()()))
(20)
and thus
K
max f(x) Ug 8
(21)
According to Eq. 5, the model confidence is calculated by
O'k exp fmax
max (x(x)
(22)
K
Σ exp
where a; exp f((x)
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Assuming the fj is the largest element in f(z), then
1
max (x(x))
1+ Li=liig exp (fi fmax)
R
1
(23)
1 (K 1) exp (Smin fmax)
1 (K - exp (-op(
B EXPERIMENTAL DETAILS
B.I DATASETS
Covariate-shifted OOD generalization datasets. We conduct experiments ImageNet- C CHendryycks & Dietterich, 2019), which consists of 15 types of algorithmically generated corruptions from noise, blur, weather, and digital categories. Each type of corruption has 5 levels of severity, resulting in 75 distinct corruptions. Besides, we conduct additional experiments on ImageNet-R and ImageNet-S.
Abnormal outliers for open-world TTA experiments. We follow the settings of (Zhang et al., 2023), where OpenImage-O (Wang et al., 2022a), SSB-hard (Vaze al., 2022). Textures (Cimpoi et al., 2014), iNaturalist (Van Horn et al., 2018) and NINCO (Bitterwolf et al., 2023) are selected as outliers for ImageNet. Openlmage-O contains 17632 manually filtered images and is 7.8 x larger than the ImageNet dataset. SSB-hard is selected from ImageNet-21K. It consists of 49K images and 980 categories. Textures (Describable Textures Dataset, DTD) consists 5,640 images depicting natural textures. iNaturalist consists of 859000 images from over 5000 different species of plants and animals. NINCO consists with a total of 5879 samples of 64 classes which are non-overlapped with ImageNet-C. These datasets are also used in previous open-set OOD generalization works (Gao et al., 2024; Bai et al., 2023; Liang et al.; Lee et al., 2023).
B.2 IMPLEMENTATION DETAILS
Pretrained models. The pretrained ViT model is ViT-Base (Dosovitskiy et al., 2021). The model is trained on the source ImageNet- training set and the model weights are directly obtained from timm respository. Specifically, the pretrained ResNet model in C.8 is ResNet-50-BN (He et al., 2016). The model trained on the source ImageNet- IK training set and the model weights are directly obtained from torchvision library.
TEA' (Yige et al., 2024) We follow all hyperparameters that are set TEA unless it does not provide. Specifically, use SGD the update rule, with momentum of 0.9, batch size of 64 and learning rate of 0.001/0.00025 for ViT/ResNet models. The trainable parameters are all affine parameters of layer/batch normalization layers for ViT/ResNet models.
T3A6 (Iwasawa & Matsuo, 2021) We follow all hyperparameters that are set in T3A unless it does provide. Specifically, the batch size set to 64. The number of supports restore M is set to 20 for all experiments.
LAME (Boudiaf al., 2022) We follow all hyperparameters that set in LAME unless it does not provide. For fair comparison, we maintain consistent batch size of 64 for LAME, aligning it with the same batch size used by other methods in our evaluation. We use the kNN affinity matrix with the value of k 5.
httn:://hggggigggteoooooooogtttt colgeogleevvi--bss-ppacchhh
1tc
https://gihub.co//yunngggge
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FOA (Niu et al., 2024) We follow all hyperparameters that are set in FOA unless it does not provide. Specifically, the batch size is set to 64. The number of supports to restore M is set to 20 for all experiments.
Tent" (Wang et al., 2021) We follow all hyperparameters that are set in Tent unless it does not provide. Specifically, use SGD as the update rule, with momentum of 0.9, batch size of 64 and learning rate of 0.001/0.00025 for Vit/ResNet models. The trainable parameters are all affine parameters of layer/batch normalization layers for ViT/ResNet models.
EATA 10 (Niu et al., 2022) We follow all hyperparameters that are set in EATA. Specifically, the entropy constant En (for reliable sample identification) is set 0.4 10000, where 1000 is the number of task classes. The for redundant sample identification is set to 0.05. The trade-off parameter for entropy loss and regularization loss is set to 2,000. The number of pre-collected in-distribution test samples for Fisher importance calculation is 2,000. We use SGD as the update rule, with momentum of 0.9, batch size of 64 and learning rate of 0001/0.00025 for ViT/ResNet models. The trainable parameters are all affine parameters of layer/batch normalization layers for ViT/ResNet models.
SAR"! (Niu et al., 2023) We follow all hyperparameters that are set in SAR. Specifically, the entropy threshold Eg set to 0.4 x Im1000, where 1000 the number of task classes. We use SGD as the update rule, with momentum of 0.9, batch size of 64 and learning rate of 0.001/0.00025 for ViT/ResNet models. For model recovery, we follow all strategy that are set in SAR(except for the experiments of life-long).The The trainable parameters all affine parameters of layer/batch normalization layers for ViT/ResNet models.
CoTTA 12 (Wang et al., 2022b) We follow all hyperparameters that are set in CoTTA unless it does not provide. Specifically, we use SGD as the update rule, with a momentum of 0.9, batch size of 64 and learning rate of 0.001/0.01 for ViT/ResNet models. The augmentation threshold pt is set 0.1. For images below threshold, we conduct 32 augmentations including color jitter, random affine, Gaussian blur, random horizonal flip, and Gaussian noise. The restoration probability of is set to 0.01 and the EMA factor a for teacher update is set to 0.999. The trainable parameters are all affine parameters of layer/batch normalization layers for ViT/ResNet models.
MEMO 13 (Zhang et al., 2022) We follow all hyperparameters that are set in MEMO. Specifically, we use the AugMix 14 (Hendrycks et al., 2020) as set of data augmentations and the augmentation size is set to 32. We use SGD as the optimizer,with with learning rate 0.00025 and no weight decay. The trainable parameters are the entire model.
Source of standard deviation. For all the experiments, we run multiple times and report the average performance and standard deviation. The source of the standard deviation consists 1) the order in which the test mini-batches coming online and 2) the randomness of the stochastic optimization methods, e.g., SGD, Adam. Since in TTA setting, the model is initialized from the publicly available pretrained model weights (i.e., via-base- sea-base-patchloe patch16-224 from timm and resnet50 from PyTorch), there is no randomness introduced by model initialization.
C ADDITIONAL RESULTS
C.I FULL RESULTS OF ACCURACY WITH STANDARD DEVIATION (SUPPLEMENTARY TO TABLE 1)
We provide the full results with standard deviation as supplementary to Table 1 in Table 8. The results demonstrate that our COME method consistently achieves better performance than its counterparts.
12 hhar:://ghhhbb.cc///gnnngggg/oo
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Table 8: Classification accuracy comparison with standard deviation on ImageNet-C (level 5). Substantial (2 0.5) improvement and degradation compared to the baseline are highlighted in blue or brown respectively.
Noise
Blur
Weather
Digital
Methods
COME
Gauss.
Shot
Impul.
Defic
Glass
Motion
Zoom
Snow
Frost
Fog
Brit.
Contr:
Elust.
Pixel
JPDG
No Adiet
x
32Rana
20.
41.4
25.3gau
31.8gg
20.1go
14.7ana
64.5gn
34,2go
58.2g0
PL
20.0 48.5 4 41 2sea
52.7
42.7anx
42.5er
65 Bans
70.1
6553
44. aan
63.0
02 620
91.5asa
38.5201
W2-7aan 47.5usa
38.5as.
25.6as.
90.8a02
56
Mi.aasa
0800ns
54.aaa
58, 4..e
TEA
393...
45.8
97.8.04
35.4.6.7
46.4.n
S144
8.46a
46.4ana
80.1aaa
72.5
50.5an
45.7ana
623 58.9
LAME
31.9ama
36.6200
30.92u
24.2 38.
30.7an3
23.4r.
29.0amm
53.3ac.
64.220
41.02a 32.7ame
52.820 00.0aue
43.9asa
48.3.01
47.1in4
60.7.e.
40.5,oa
43.7
53.6ac.a
$2.8
64.2a3.2
76.2azx
(63.8
48.5ana
62.1 66.0,n0
152.5.o
a2..
53.4.01
52.8
47
Aue.
56.7
Aaa.
10.5aua
26.4a2x
2ac.
74.3ao.
667.8a00
50.40a
66.5
66.6.00
60.8
53
M0.
Da.Agri 61.3gn
T1.3gna
78.2an0
(888
07.9
70.5 08.22
Improve
A1.
A1.
/.2.0
.3.2
2447.9 A38.9
03.0
207.6
7:4.0
/.5.6
X
Gillana
57.1
54.5.L
54.8.1.7
39 BaL.
7....
61 Sacc.a
G0.lana
T1 Aan.a
(08.aa
B2-7ana
tiB.aao
EATA
V
56.1an
57.22m0
68.0lan
57
02.6
30.3
Bb.6ar G3.5ana
72 72.6
78.0
60..aa
72,5
70.5gn
Improve
A0.1
2003
A0.1
A34
A.3.1
A3.1
1.0.5
2.3.9
A33.
2.13
A227
A.10
2.3.
1-3.6
24.0
SAR
/
51.9m1
58,4an1 566.01
51.7am:
$7.4
52.8291
51.5am
56.9
48.9
62.9
38.3403
49.ban.2
B5.3ac.
22
64.5an
46.9a22
06 2554
T277aa
78.5aan
72.9uoa
60.8ano
64.0003
7L.9aa 69.7a03
64.0za 62.8
Improve
A455
249
44.6
A74
288.
24451
A17.5
2605
4.000
A48
2.182
479
269
10.3aug
67.6
41.7
28 Jane
40.0za1
35 35.
43.0aam
58 Sana
70.3az
58,422
58.1 59.9a
COTTA
Mi5ino AA1
46.0.04
29.7
48.1a12
42.0aa4
63.5u9 68880a
Impeove
2.3
48.8
A28
A14
44.0
2.22
2/277
433
43.
A1L5
439
C.2
FULL RESULTS OF
TTA (SUPPLEMENTARY TO TABLE 3)
We provide the full results with standard deviation as supplementary to Table 3 in Tablei, The results demonstrate that our COME method consistently achieves better performance than its counter- parts.
Table Classification and uncertainty estimation comparisons with standard deviation under open- world TTA settings, where ptest is a mixture of both covariate-shifted samples (Gaussian noise of severity level 3).
Nome
NINCO
iNaturist
SSB-Hard
Texture
Plazes
Method
COME
Acct
FPR1
Acct
FPR1
Acct
FPR.
Acct
FPRL
Acet
FPR1
Acc1
FPR
No Adapt
X
64.4guo
63. (3.7
61.6saa
70.3guu
61.4go
60.5
61.4go
72.5
66177uz
65.0gu
6615
57.
PL
69.1sa
63.0
63.6ga4
70.9a17
68.8
68.3
68.3
76.2aux
66.1am
65.7aux
67.1
644429
71.0
64.320a
4
64.1291
63.720
644492
67
64.0ao
69.6
64.1
63.3
00:2.04
72.9
6244.1
74.5.0n
63.
79.3
69.0
62:0.0
65.8
LAME
6i.
64
61.2,na
72.
611100
722
73.9
61660a
68.5
613
61-4,oa
FOA
X
67.8
62.0pa
65.9
69
69:2
67.6.0.2
66.8
67.5
76.0g03
65.8
60.6
66.7
52,8
70.9
63.6 63.61a3
66.4a0.
7L.5
70.0.a.
70.8
69.8
77.
66.7.0.
67.3
68.0
60.8
7244aa
64.9
60.0ag.1
63.920.
72.6.0.1
63.6
72:7200
70.6
68.5ga
60.3gg0
70.7200
46.3an2
Improve
A1.7
A13
A2.6
97.66
97.22
A288
996.8
A18
A2,0
914.5
x
70.2gn::
63.5
06:2g0.
68 850
70.2,0.1
71.6401
60.9gn.1
77.6
67.640.5
66.5405
68.6
62.0a0
EATA
73.310.0
63
13.1
70.1.44
60.5100
73.2,00
63.2
73.0p0
70.6
70.310a
56.8107
723,00
45.8102
Improve
2.1.1
v004
2.3.9
v8.2
2.3.0
v8.3
2331
v/6.9
4.277
v9.7
2.3.7
V/6.2
69/6.a.
62..1
65.5.0.
67 1..1
70.7
67 67.61..
77.9
65.5
65 65.8
66.2.01
50.8
73.1
62.3
7040g0.
65
65.9an2
73.
65.32an
73.5gn0
71.6am3
60-4aa
50.3
720
48.62n9
Improve
A33.5
A00.1
24.5
94.3
266.2
754
2:6.0
996.5
A33.9
76..5
A5.8
v11.5
67.7go0
63-4go1
65.3
70440a
704
08.5a11
70.1
65.7a01
65.0guu
67 02.0
888
COTTA
70.4g0
62 8a02
06
68.0ga
724
72.9
72.1
7805a2
06.402
63.8
54.540
Imgrove
428
500.6
42.4
220
2200
20.7
1/.22
v4.4
C.3 COMPARISON WITH SOURCE-FREE DOMAIN ADAPTION
There is strong connection between Source-Free Domain Adaptation (SFDA) and Test-Time Adaptation (TTA). The primary difference is that TTA focuses on online adjusting during the testing. On the other hand, SFDA approaches generally perform offline. That is, the inference deferred until the optimization is done. In contrast, our TTA method can achieve adaption and inference at the same time.
To further validate the applicability of our method, we report the classification accuracy on ImageNet- C Gaussian noise level under source-free domain adpation settings. The results are in Table 10. The baselines considered include pseudo label (PL), mutual information maximization (IM), and entropy minimization (EM) following (Liang et al., 2020).* =_" means the model accuracy collapses to random guess level.
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Table 10: Classification accuracy comparison on ImageNet-C Gaussian noise (level 5) under source- free domain adaption settings. "-" means the classification accuracy collapses random guess level. The accuracy of the original pretrained model is 35.1.
EPOCH
PL
EM
IM
TENT
EATA
SAR
COME
X
X
X
34.2
60.7
66.5
66.4
68.5 68.5.0.1
63.8
68.440.0
67.2
69.7=
65. 65.1
69. 09.2+0.0
67.8
70.2
C.4 COMPARISON WITH OTHER OTHER BAYESIAN METHODS
There exists a few Beysian inspired TTA methods closely related to our method. (Zhou & Levine, 2021) explores Bayesian model ensembling Zhou & Levine (2021) for TTA, which introduces noticeable inference latency. Since (Zhou & Levine, 2021) has not made their source code publicly available. For a fair comparison, implement the proposed COME using the same backbone (ResNet50-v2) and dataset (ImageNet-C) as (Zhou Levine, 2021) and report the average accuracy of all corruptions and levels. The results of (Zhou & Levine, 2021) are directly copied from the original paper. The results are in Table 11.
Table 11:
Classification accuracy comparison with other bayesian insipired TTA methods ImageNet-C under TTA settings averaged over all corruption types and levels. The result BACS is copied from the original paper.
No Adapt
BN
BACS
TENT X
TENT
EATA
COME
X
X
x
47.340.0
47.3+0.0
56.1
48.940.0
51.1+0.0
58.1+0.0
C.5 INFLUENCE OF DIFFERENT HYPERPARAMETHE
We conduct additional experiments to investigate the influence of different hyperparameters. The results are Table 12 and Table 13. Our COME generally outperforms EM with moderate hy- perparamters. For т, as mentioned before, it actually controls the magnitude of uncertainty mass. If we have some prior knowledge that most test samples should be rejected to adapt during TTA, we should choose relatively small т and making the model confidence more conservative in this circumstance. As for (or p) which represents the tolerance of uncertainty divergence, should be selected per need by the user via trial and unreliable TTA, 6 should be tuned down and should be tuned up: otherwise, if a better performance required. Besides, we introduce an additional hyperparameter the learning objective. The experiments results in Table 12 shows that an additional performance improvement can be observed when À E [1, 100).
if users extremely cautious about
Table 12: Additional results with different A. We report the accuracy when our COME is equipped to Tent on ImageNet-C Gaussian noise level 5 with different X. The accuracy the original Tent using entropy minimization is 52.6.
À
0.10
1.00
10.0
30.0
50.0
80.0
100
150
Acc. 53.7+0.1 53.940.0
54.6+0.1
55.3_o.1
55.8+0.1
56.2+0.1
56.1+0.1
10.6
C.6 COMPARISON ON IMAGENET-R AND IMAGENET-S
We introduce more additional datasets, i.c., ImageNet-R and ImageNet-S, to further evaluate the proposed method on commonly used OOD generalization benchmarks. The results are in Table 14 and Table 15.
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Table 13: Additional results with different hyperparameters. We report the accuracy on ImageNet-C Gaussian noise level 5 with different hyperparameters. We implement our COME with Tent. The accuracy of the original Tent using entropy minimization 52.6.
p
p = 2
p = 3
00
0.5
37.8 37.8
38.5+0.0
39.1+0.0
41.1+0.0
T = 1.0
53.5.0.0
53.8
53.2+0.0
47.2+0.1
7 1.2
54.7 10.0
54.710.1
54.040.1
48.9
== 1.5
54.210.2
54.210.1
53.210.1
49.1
Table 14: Additional results ImageNet-R
Tent
EATA
SAR
COTTA
EM
37.73 - 0.03
36.11 0.12
51.90 = 0.35
36.04 + 0,06 37.39 + 0.08
COME
39.05 0.05 38.22 0.23
56.40 = 0.18
C.7 IN-DISTRIBUTION PERFORMANCE
We compare the in in-distribution performance of proposed COME to EM- MM-based methods. As shown in Table 16, our method consistently outperforms entropy minimization.
C.8 COMPARISON ON RESNET-50
As shown in previous work (Niu et al., 2023), the TTA performance can influenced by different model architectures, especially the type of normalization layers, i.e.. batch normalization, group normalization, layer normalization, and instance normalization. To further evaluate the proposed method, we conduct additional experiments open-world TTA settings. The experimental results in Table 17 show that our COME still achieves superior performance compared entropy minimization learning principle when equipped Tent.
ResNet-50 with batch normalization layers under
C.9 TIME-CONSUMING COMPARISON
We compare the time-cost of proposed COME to EM-based methods and Non-EM based methods in Table 18. We run all the experiments one single NVIDIA 4090 GPU. Our COME does not introduce noticeably extra cost of computation.
C.10 MIXED DISTRIBUTIONAL SHIFTS PERFORMANCE.
We evaluate the proposed COME two additional settings introduced by (Niu et al., 2023). These scenarios include online imbalanced label distribution shifts, where the test data are sorted by class, and 2) mixed domain shifts, where the test data stream includes several randomly mixed domains with different distribution shifts. As shown in Table 19 and Table 20, our COME consistently outperforms entropy minimization with an exception of slightly suboptimal uncertainty estimation performance compared CoTTA
C.11 MORR VISUALIZATIONN RESULTS (SUPPLEMENTARY TO FIGURE 1)
We provide more visualization results on two representative TTA methods, i.e., the seminal Tent (Wang et al., 2021) and recent SOTA SAR (Niu et al., 2023). We observe that COME enjoys a more stable TTA progress with less risk model collapse and overconfidence across various types of corruption. We test on ImageNet-C level 5.
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Max Softmax Probability
Accurney hatch
FPR batch
0.55
0.65
0.6
0.45
0.5
0.40
0.50
0.35
0.2
100 200
300
400 500 600 700
800
100
200 300 400 500 600
700
800
0
100
200 300
400 500 600 700 800
Batch ID
Batch ID
Betch ID
No Adapt
Tent
SAR
Tem++COM
SAR+COME
Figure 3: Comparison corruption of severity level 5. By contrast to EM, our COME establishes a stable TTA process with consistently improved classification accuracy and false positive rate.
representative TTA methods ImageNet-C under Gaussian Noise
Max Softmax Probability
Accuracy batch
FPR per batch
0.8
0.55
0.75
0.70
0.5
0.40
0.35
0.60
0.30
300 400 500 600
100 200 300
100 200
300
Adapt
Tent
SAR
TeME
SARICOME
Figure 4: Comparison on two representative TTA methods on ImageNet-C under Shot Noise corruption of severity level 5. By contrast to EM, our COME establishes stable TTA process with consistently improved classification accuracy and false positive rate.
Softmax Probaiiity
Accuricy per
per
0.55
0.70
0.68
0.6
0.66
0.5
0.45
0.40
0.3
0.35
0.60
0.58
100 200 300
400 500 600 700 800
100
200 300 400 500 600
700
0
100
200
300
400 500 600 700 800
Batch ID
Batch ID
Batch ID
Adapt
Tent
SAR
Tem#+COM
SAR+COME
Figure 5: Comparison on two representative TTA methods on ImageNet-C under Impulse Noise corruption of severity level 5. By contrast to EM, our COME establishes a stable TTA process with consistently improved classification accuracy and false positive rate.
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Max Softmax Prohability
Accurncy hatch
FPR per batch
0.775
0.35
0.750
0.50
0.725
0.6
0,700
0.45
0.5
0.675
0.40
0.650
0.35
0.625
0.30
0.600
100 200 300 400 500 600
700
800
0
100
200
300 400 500 600 700 800
100
200
300
400 500 600 700 800
Batch ID
Batch ID
Batch ID
No Adapt
Teat
SAR
Tem++COE
SAR+COME
Figure 6: Comparison two representative TTA methods ImageNet-C under Defocus Blur coruuption of severity level 5. By contrast to EM, our COME establishes a stable TTA process with consistently improved classification accuracy and false positive rate.
Max Sottmax Probability
Accuracy batch
FPR per batch
0.8
0.55
0.750
0.50
0.45
0.700
0.5
0.675
0.4
0.35
0.650
0.30
0.625
0.25
100 200 300
800
Adapt
Tent
SAR
TeME
SARICOME
Figure 7: Comparison on two representative TTA methods on ImageNet-C under Glass Blur corruption of severity level 5. By contrast EM, our COME establishes stable TTA process with consistently improved classification accuracy and false positive rate.
Softmax Probubility
Accuricy per
FPR per
0.800
0.60
0.775
0.55
0.750
0.725
0.5
0.700
0.4
0.675
0.40
0.650
0.3
0.625
100 200 300 400 500 600 700 800
0
100 200 300 400 500 600 700 800
100
200 300 400 500 600 700 800
Batch ID
Batch ID
Batch ID
Adapt
Tent
SAR
Tem++CME
SAR+COME
Figure 8: Comparison on two representative TTA methods on ImageNet-C under Motion Blur corruption of severity level 5. By contrast to EM, our COME establishes a stable TTA process with consistently improved classification accuracy and false positive rate.
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Max. Softmax Probahlitt
Accurncy hatch
FPR hatch
0.55
0.80
0.7
0.50
0.6
0.70
0.4
100 200 300 400 500 600 700 800
100 200 300 400 500 600
700
800
0
100 200 300
400 500 600 700 800
Batch
Batch
Batch ID
No Adapt
Tent
SAR
Tem+COME
SAR+COME
Figure 9: Comparison on two representative TTA methods on ImageNet-C under Zoom Blur corruption of severity level By contrast to EM, our COME establishes a stable TTA process with consistently improved classification accuracy and false positive rate.
Softmax Probability
Accuracy per
FPR per
0.90
1,0
0.65
0.8
0.55
0.50
0.6
0.45
0.40
0.65
0.4
0.60
0.30
0.55
0.2
100 200 300 400 500 600 700 800
100 200 300 400 500 600
700
800
0
100
200
300
400 500 600 700 800
Batch ID
Batch ID
Batch ID
No Adapt
Tent
SAR
ToME
SARICOME
representative TTA methods ImageNet-C under Frost corruption of severity level 5. By contrast to EM, our COME establishes stable TTA process with consistently improved classification accuracy and false positive rate. Although the SAR method recover the model when it collapses to trivial solution, its performance remains poor. Our COME method addresses the issue of overconfidence that leads model collapse.
Figure 10: Comparison
Max Softmax Prebability
Accuracy per batch
FPR per batch
0.65
0.6
0.60
0.5
0.55
0.4
100 200 300 400 500 600 700
100 200 300 400 500 Batch
100 200 300
800
Batch
Batch ID
Adapt
TeME
SAR+COME
Figure 11: Comparison on two representative TTA methods on ImageNet-C: under Fog corruption of severity level 5. By contrast to EM, our COME establishes stable TTA process with consistently improved classification accuracy and false positive rate.
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Max Softmax Prohability
Accurney hatch
FPR per batch
0.775
0.66
0.8
0.750
0.64
0.62
0.6
0.58
0.650
0.5
0.625
100 200 300 400 500 600
700 800
0
100
200 300 400 500 600 700
800
100 200 300 400 500 600 700 s0n
Batch ID
Batch ID
Batch ID
No Adapt
Tent
SAR
Tem+CCME
SAR+COME
Figure 12: Comparison on two representative TTA methods ImageNet-C under Brightness corruption of severity level 5. By contrast to EM, our COME establishes a stable TTA process with consistently improved classification accuracy and false positive rate.
Max Sottmax Probability
Accuracy batch
FPR per batch
0.70
0.800
0.775
0.750
0.6
0.5
0.55
0.700
0.4
0.675
0.3
0.50
0.2
0.625
100 200 300
Adapt
Tent
SAR
TeME
SARICOME
Figure 13: Comparison on two representative TTA methods on ImageNet-C under Contrast cor- ruption of severity level By contrast to EM, COME. establishes stable TTA process with consistently improved classification accuracy and false positive rate.
Softmax Probability
Accuricy per
per
0.60
0.7
0.75
0.55
0.50
0.70
0.45
0.4
0.65
100 200 300 400 500 600 700 800
100 200 300 400 500 600
800
0
100
200 300 400 500 600 700 800
Batch ID
Batch ID
Batch ID
Adapt
Tent
SAR
Tem++CME
SAR+COME
Figure 14: Comparison on two representative TTA methods on ImageNet-C under Elastic Transform corruption of severity level 5. By contrast to EM, our COME establishes a stable TTA process with consistently improved classification accuracy and false positive rate.
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Max Softmax Probability
Accuracy batch
FPR per batch
0.74
0.70
0.8
0.72
0.7
0.65
0.70
0.68
0.60
0.66
0.55
0.64
0.62
0.4
0.50
0.60
Batch
Batch
Adapt
Tent
SAR
TeML+COME
SAR+COME
Figure 15: Comparison on two representative TTA methods on ImageNet-C under Pixelate corruption of severity level 5. By contrast to EM, COME establishes stable TTA process with consistently improved classification accuracy and false positive rate.
Max Softmax Probability
Accuracy batch
FPR batch
0.70
0.74
0.66
0.7
0.72
0.64
0.6
0.62
0.5
0.60
0.4
0.56
0.64
Batch
Batch
Adapt
Tent
SAR
Ten+COME
SAR+COME
Figure 16: Comparison on two representative TTA methods on ImageNet-C under Jpeg Compression corruption of severity level 5. By contrast to EM, our COME establishes stable TTA process with consistently improved classification accuracy and false positive rate.
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Published as a conference paper at ICLR 2025
Table 15: Additional results on ImageNet-S.
Tent
EATA
SAR
COTTA 30.84 + 0.35
EM
31.63 + 0.07
36.11 + 0.19
33.92 = 0.69
COME
39.22 + 0.04
39.22 - 0.32 43.52 = 0.52
33.82 + 0.73
Table 16: Comparison w.r.t. in-distribution performance, i.e., on clean/original ImageNet validation set, with VIT as the base model. Substantial (2 0.5) improvement and degradation compared to the baseline are highlighted in blue or red respectively.
TENT
SAR
EATA
CoTTA
MEMO
Avg.
COME
Acct
Acct
Acc
Acc f
Acct
Acct
X
81.4
80.7
81.3
82.1
80.3
81.2
83.1
83.1
83.1
82.8
80.6
82.6
Improve
A1.7
222
A1.8
2.0.7
A1.4
D DISCUSSION
D.1 ALTERNATIVE DESIGN CHOICE
Choices of transformation function to obtain the opinion. By definitions, the parameters of Dirichlet distribution a must be greater than and the evidence should be non-negative. This can be achieved by applying ReLU activation function or exponential function to the output logits as suggested in previous works (Han et al., 2022; Malinin & Gales, 2018). That is, we can get the evidence by
ReLU(((x))
(24)
or
exp f(x) 1.
(25)
In this paper, we choose the exponential function. Since we assume the pretrain model is trained with standard cross-entropy loss, using exponential function to get the evidence can keep the training strategy unchanged. Besides, based on our early empirical findings, using exponential function can achieve better classification performance compared ReLU.
We refer interested readers to (Malinin & Gales, 2018) and Gal's PhD Thesis (Gal et al., 2016) for more detailed implementation instructions and math deviations.
Choices of uncertainty constraint. In Lemma 1, we prove that by constraining on the model output logits, we can control the uncertainty mass not diverge too far from the pretrained model. Previous work (Wei et al., 2022) proposes to mitigate the overconfidence issue by normalizing the logits during pretrain progress in supervised learning tasks. Following their implementation, we propose to optimize that the optimization progress is not related to the magnitude of f(x), i.c., its norm. from Wei et al. (2022), recover the magnitude by multiplying the direction vector with its norm (detached), rather than a constant to avoid an additional hyperparameter. However, the uncertainty estimated by pretrain model may not be ideal. However, please kindly remind that in fully TTA task, we can only access unlabeled test data coming online and the inference efficiency matters. Thus traditional methods devised for handling overconfidence like calibration (Guo et al., 2017), ensembling (Zhou & Levine, 2021). BNNs (Huang et al., 2022) and other Bayesian methods like dropout (Gal & Ghahramani, 2016) are not applicable. The only practically available choice is to explore the uncertainty information contained within the model itself. As shown in previous works, while the softmax probability of pretrained model tend to overconfident, subjective logic much more reliable (Sensoy et al., 2018: Malinin & Gales, 2018), which can support the proposed regularization. Exploring more effective and efficient regularization is an interesting future research direction.
expect the direction vector of f(x). i.e., f(x)/Ilf(x) Ip. and thus Different
Choices of p-norm. The tightness for the upper and lower bounds in Lemma is determined by the choice of p. By considering the simple model where f()) outputs the same logits for all classes, the ratio between the upper and lower bound is minimized by 20. A larger p can lead to a more strict constraint ju u_0l < 8. We conduct additional experiments varying p. When using
27

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Published as a conference paper at ICLR 2025
Table 17: Classification and uncertainty estimation comparisons under open-world TTA settings with ResNet-50-BN, where prest = 0.5 + (epsee noise of severity level 3) and a suit of diverse abnormal outliers as same with Table Substantial (2 0.5) improvement and degradation compared to the baseline are highlighted in blue red respectively.
None
NINCO
iNaturist
SSB Hard
Texture
Places
Method
COME
Acct
FPR]
Acct
FPR|
Acct
FPR]
Acct
FPR..
Acc1
FPR]
Acc|
FPR.
Acct
FPR
No Adupt
X
3.0
81.6
3.0
91.2
3.0
89.4
3.0
90.2
3.0
88.3
2.8
90.7
3.0
88.6
PL
X
26.9
71.2
16.1
84.3
12.9
88.4
15.9
87.8
18.1
86.1
16.9
82.9
17.8
83.5
TEA
28.5
73.5
84.0
84.5
16.8
82.9
17.4
82.9
52.5
67.5
43.7
79.6
52.1
78.3
51.9
82.1
44.4
78.6
48.7
73.2
48.9
76.6
Tent
/
55.0
67.6
46.3
75.5
54.3
75.1
54.4
81.9
45.8
75.6
50.8
64.0
51.1
73.3
Improve
42.6
4.2.6
1/1.2
2.2
1/3.2
2.2.5
A.14
1/3.1
2...2
1/9.2
2..
1/3.3
55.9
68.2
80.8
78.4
52.2
48.7
49.3
74.5
51.2
76.5
EATA
/
58.0
66.2
52.9
74.8
57.6
73.2
57.4
81.3
52.5
70.7
55.4
62.9
55.6
71.5
Improve
A2.0
<72.0
/.5.1
$76.0
214.5
4/5.
2.5.2
$70.7
/33.9
174.6
2.6.0
1711.6
244.5
$75.0
x
51.8
64.6
42.4
78.3
47.6
81.3
48.1
84.4
42.7
79.1
46.0
76.7
46.4
77.4
SAR
/
56.3
64.0
46.7
77.9
55.3
77.1
55.1
81.6
46.4
77.5
52.5
68.1
52.0
74.4
Improve
/44.5
<70.6
A4.3
$70.3
AT.7
<74.2
A.6.9
472.8
/33.7
171.6
A.6.5
778.6
A5.6
$73.0
X
22.6
70.7
14.4
87.4
21.1
78.6
19.7
84.1
15.5
87.3
15.8
82.0
18.2
81.7
COTTA
81.9
Improve
A1.8
1..
1/..
2.2.9
2:2.0
2.0.5
1/1.6
2.0.6
2.0.7
2.0.5
X
8.0
83.6
7.5
89.0
7.9
87.9
7.9
89.8
7.8
88.6
7.7
88.4
7.8
X7.9
77.9
90.2
87.3
9.1
9.1
8.7
88.8
86.8
Improve
A11
775.7
A1.2
/1.
A1.2
470.7
A1.2
770.6
A1.3
71.
A1.1
A1.2
V1.1
Table 18: Comparisons w.r.t. computation complexity. Accuracy (%) and FPR (%) are average results on ImageNet-C (level 5) with ViT-Base. The Wall-Clock Time (seconds) and Memory Usage (MB) are measured for processing 50,000 images of ImageNet-C on a single RTX 4090 GPU.
Method
COME
Acc t
FPR
Memory
Run Time
No Adapt
X
39.8
67.5
853
59
LAME
x
38.7
69.7
T3A
67.7
179
PL.
51.3
6393
128
FOA
53.7
63.6
869
1687
TEA
X
46.9
68.3
17266
2865
70.1
6393
Tent
66.5
6393
130
Improve
28.4
V3.6
X
62.1
65.1
6394
135
EATA
64.5
63.8
6394
134
Improve
V1.3
X
54.2
66.7
6393
253
SAR
64.2
63.8
6393
254
Improve
A10.1
V22.
X
46.1
67.9
19612
738
COTTA
49.1
67.5
19611
739
Improve
A3..
v0.3
X
42.3
72.1
5392
20576
MEMO
/
43.2
70.8
5392
20530
Improve
2..9
7...
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Table 19: Comparison w.r.t. imbalanced label shifts performance. Results obtained on ViT and ImageNet-C (level 5) under imbalanced label shifts TTA setting, where the imbalance ratio is co. Substantial (2 0.5) improvement and degradation compared to the baseline are highlighted blue or red respectively.
Noise
Blur
Weather
Digital
Avg.
Methods
CURE
Gauss.
Shot
Impul.
Defoc
Glass
Motion
Zoom
Snow
Frost
Fog
Bril.
Contr:
Elast.
Pixel
JPEG
Acc1
No Adapt
x
35.1
32.2
35.9
31.4
25.3
39.4
31.6
24.5
30.1
54.7
64,5
49.0
34,2
53.2
56.5
39.8
X
49.7
48.6
50.9
49.8
41.5
53.0
41.9
26.6
49.0
64.3
73.6
65.6
45.2
63.9
52.4
T3A
X
33.4
30.3
34.2
31.3
26.8
38.7
32.1
25.1
29.3
54.5
62.8
48.8
37.4
51.9
56.2
39,5
44.9
35.2
12.1
72.7
LAME
X
47.0
43.3
48.2
39.8
31.8
50.3
39.4
30.5
37.1
66.0
75.4
63.5
42.0
65.1
68.1
49.8
X
52.4
51.9
53.3
53.8
48.1
$7.0
46.2
10.3
53.5
67.9
74.2
67.1
52.3
66.5
64.9
54.6
62.9
78.1
Improve
2.2.5
2.3.2
2/2.9
233.4
2.6.5
2.4.6
233.1
A.52
2.10.6
244.4
2.40
A2.2
A10.4
2.1.9
4.4.1
A.7.9
X
51.8
51.7
52.7
51.9
48.2
55.6
47.8
20.3
52.9
66.8
73.2
66.0
52.2
64.1
62.8
54.5
SAR
/
56,0
56.0
57.2
58.0
56.3
62,3
54,1
64,0
64.3
72.4
78.3
69.6
64.0
71.5
69.1
63.5
Improve
/.42
A66.6
/663
//43.8
/.5.7
/.5.1
/33.
A.9.0
52.0
53.6
53.9
49.3
49.5
54.4
55.6
58.1
56.9
69.6
74.9
63.6
61.1
68.0
64.2
59.0
EATA
54.9
56.4
54,7
56.5
56.3
62.1
59.0
67.0
65.4
73.4
78.4
73.0
64.2
Improve
A2.8
A2.8
A.08
AV.33
A6.8
A7.7
/33.4
/V8.9
7.8.5
/33.
A.3.5
/.4.4
/.6.8
/55.0
/.6.
/.5.3
42.9
40.00
44.6
36.0
29.7
44.8
37.2
42.3
46.4
60.7
72,9
65.0
45.4
61.6
48.8
COTTA
/
51.6
49.0
52.9
41.7
37.0
51.6
43.8
46,7
53.2
65.9
74.4
65.6
52.8
66.7
65.9
54.6
Improve
A8.6
A7.4
A668
/.6.6
/.4.4
A1.5
/.5.1
A5.8
x
39.7
36.5
39.8
32.4
25.8
40.3
34.7
27.5
32.8
53.5
66.2
56.0
35.7
55.9
58.2
42,3
26.7
41.2
35.4
28.7
54.7
67.1
Improve
20.8
2.10
2.0.8
1.0
2.0.9
1.10
2.077
A1.
2.0.9
412
2.0.8
2.0.9
4.1
4.1
4.0.9
Table 20: Comparison w.r.t. mixed shifts performance. Results obtained on ViT and ImageNet-C (level 5) under mixed shifts TTA setting, the performance evaluated on single data stream consisting of 15 mixed corruptions. Substantial (2 0.5) improvement and degradation compared to the baseline are highlighted in blue or red respectively.
TENT
SAR
EATA
CoTTA
Avg.
COME
Acct
FPR.
Acct
FPR]
Acct
FPR]
Acct
FPR/
Acct
FPR]
X
58.0
72.3
53.6
68.2
58.8
71.3
62.0
69.7
58.1
70.4
/
61.2
67.9
62.3
66.9
61.8
67.0
65.1
70.7
62.6
68.1
Improve
/.3.2
44.4
A8.6
V 1.3
A3.0
74.3
A3.1
/.0.9
A445
72.3
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infinity norm, suboptimal classification accuracy is observed. We suppose this is because overly strict constraint can be harmful to TTA. Since on reliable test samples, we still expect to reduce the uncertainty (in conservative manner).
D.2 LIMITATIONS AND FUTURE WORK
Many stte-of-the--nd TTA methods are equipped with entropy minimization learning principle; but the potential pitfalls lie in this optimization objective empirical analysis towards understanding the failure mode. These findings motivate to further explore the connection between uncertainty learning and reliable TTA progress, which further implies principle to design novel learning principle as alternative entropy minimization. Finally, we perform extensive experiments on multiple benchmarks to support our findings. In the work, simple yet effective regularization
well understood. this paper, we provide
the uncertainty mass is devised, and other regularization techniques could be explored. Another interesting direction further explore the relationship between overconfidence issue and model collapse theoretically.
30


Reviews:
[{"review_id": "Vp0vz2Rkif", "reviewer": "Reviewer_saue", "summary": "This paper addresses the model collapse of the popular Entropy Minimization algorithm for Test-Time Adaptation. Motivated by the observation that the amplification of model over-confidence causes model collapse, this paper proposes to minimize entropy with respect to an augmented output distribution that includes the probability to reject a sample, which is an uncertainty estimation technique known as subjective logic. Moreover, a logit normalization is designed in order to avoid degenerated solutions. Theoretical analysis reveals that the resulting approach upper bounds model confidence during adaptation based on the sample-specific confidence of the initial model. The resulting algorithm, COME, can be easily embedded into general EM-based TTA methods with a few lines of code revision. Experiments across TTA, open-world and life long TTA settings demonstrate a significant and consistent improvement upon current EM baselines.", "questions": "1. The tightness for the upper and lower bounds in Lemma 1 is determined by the choice of p. By considering the simple model where f(x) outputs the same logit for all classes, the ratio between the upper and lower bound is minimized by $p=\\infty$. Is is a better choice to consider $\\| f(x) \\|_\\infty = \\max_k | f_k(x) |$?\n\n2. The usage of exp transformation of logits in the softmax function appears pivotal to the proof of Theorem 1. And if we take b(x) as a general non-negative function of f(x), the upper bound may reduce to 1. And minimizing the entropy of opinion in equation 6 can also lead to the Dirac distribution, which is over-confident.  Is there a characterization for the class of functions that could be used to form a reasonable belief b(x)?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "This paper accurately spots the paradox of EM's learning objective: minimization of entropy leads to over-confidence. And the paper proposes a simple yet effective solution to minimize entropy with respect to a probability distribution that faithfully estimates the uncertainty without over-confidence. It is a very reasonable idea to differentiate between the statistics used for prediction and for uncertainty estimation, which has long been considered the same in the TTA literature. Therefore, the algorithm enjoys the feature that the entropy minimization can be tailored to samples with different uncertainty, which is also supported by the monotonicity result in Theorem 1. The introduction of SL for uncertainty estimation is natural and perfectly compatible with softmax functions used for training models in most cases. As a result, the implementation is light-weight, model-agnostic, and extremely easy to embed into any TTA algorithms based on the EM objective. The experiments are convincing by covering both standard TTA tasks and more challenging settings of open-world TTA. A surprisingly significant 34.5% improvement on accuracy is reported on the model of SAR. And the algorithm has further addressed uncertainty estimation under continual distribution shift as a side product, which itself is also an important problem.", "weaknesses": "These are not necessarily weaknesses but rather some questions that I would like to confirm with the author.\n1. How does the algorithm ensure that $b_k$ is non-negative for the computation of entropy, since $b_k$ is implemented as \n$(e^{f_k(x)}-1)/ \\sum_{k'} e^{f_k'(x)}$ which could be negative?\n\n2. Why does the algorithm keep $u$ close to $u_0$? Does it imply that the uncertainty estimation for the pretrained model is trusted? What if the pretrained model is over-confident? What about the alternative constraint $u \\geq u_0$ which seems to be more conservative as is the objective of COME?\n\n3. Average false positive rate is used in experiments to assess uncertainty estimation. However, FPR measures the correctness of uncertainty estimation with such a binary perspective: for the samples we predict 1, what is the actual proportion of 0. Uncertainty estimation considers a more sophisticated question: for the samples we predict with a probability 0.7, is the actual proportion of 1 exactly 0.7? Expected calibration error (ECE) is a better metric in this sense.\n\n4. Are there standard errors of the reported results?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Xb6E6bj26O", "reviewer": "Reviewer_4L2X", "summary": "The authors propose Conservatively Minimizing Entropy, a method for test-time adaptation (TTA) that improves test-time adaption by managing prediction uncertainty. Unlike traditional entropy minimization, which can lead to overconfidence, COME uses a Dirichlet distribution to model uncertainty, allowing the model to avoid forced classification on unreliable data.", "questions": "Please refer to weaknesses.", "limitations": "", "rating": 5, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- This paper is well-motivated, and the story makes sense. \n- Extensive experiments have been done to support the proposed method.", "weaknesses": "My major concerns include:\n- For the proposed method: Why Dirichlet distribution is used? How is the Dirichlet distribution related to the final algorithm in Algorithm 1. In addition, what is the role of delta in Algorithm 1? It seems that the authors tell a long story about their algorithm, but the algorithm itself is rather simple.\n- For the theoretical analysis: Could the authors provide a more detailed (theoretical) comparison between the proposed method and traditional EM? What is the benefit?\n- For baselines: Some baselines are missing, for example,  [1] and [2].\n- For the datasets: I'm curious why the authors follow literatures on outliers detection.\n- For Theory 1: What is the exact benefit of the upper bound of model confidence? I think it will also hurt the performance on some \"confident\" samples. \n\n[1] Nado, Z., Padhy, S., Sculley, D., D'Amour, A., Lakshminarayanan, B., & Snoek, J. (2020). Evaluating prediction-time batch normalization for robustness under covariate shift. arXiv preprint arXiv:2006.10963.\n[2]Zhou, A., & Levine, S. (2021). Bayesian adaptation for covariate shift. Advances in neural information processing systems, 34, 914-927.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "QR0pnFmEJs", "reviewer": "Reviewer_WHFP", "summary": "This paper investigates the issue of model collapse in entropy minimization algorithms for test-time adaptation. The authors propose a novel entropy minimization approach that models prediction uncertainty by defining a Dirichlet prior distribution over model predictions. This method regularizes the model to favor conservative confidence for unreliable samples. Experiments on benchmark datasets demonstrate the effectiveness of the proposed algorithm.", "questions": "see weakness", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The proposed algorithm introduces a rejection mechanism for unreliable samples in the TTA process, preventing the model from learning from potentially noisy labeled data. It is simple to integrate into existing TTA frameworks, and the experimental results indicate satisfactory performance.", "weaknesses": "The proposed method and its theoretical analysis rely heavily on existing techniques, which limits its technical novelty.\n\nThe core concept shares some similarity with research on learning with rejection. It is recommended to discuss how the proposed loss function compares with the loss functions used in learning with rejection, as outlined in [1].\n\nThere is a lack of experiments involving real-world applications with distribution shifts, as exemplified in [2]. Testing the proposed algorithm on real-world data streams in dynamic environments is suggested to validate its robustness.\n\nReferences:\n\n[1] Cortes, Corinna, Giulia DeSalvo, and Mehryar Mohri. \"Learning with rejection.\" Algorithmic Learning Theory (2016).\n\n[2] Yao, Huaxiu, et al. \"Wild-time: A benchmark of in-the-wild distribution shift over time.\" Advances in Neural Information Processing Systems 35 (2022): 10309-10324.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "L2xGirkAIb", "reviewer": "Reviewer_5e1E", "summary": "The paper proposes a Bayesian inference technique to address the overconfidence problem in test-time domain adaptation. Experiments demonstrate its effectiveness, achieving a SOTA performance.", "questions": "- (Eq. (6)) What if a hyperparameter $\\lambda \\in \\mathbb{R}$ is introduced as $-\\sum_{k=1}^K b_k \\log b_k - \\lambda u \\log u$? This seems to be a straightforward generalization of Eq. (6).\n\n- Is there any quantitative correspondence between $\\tau$ and $\\delta$ (Eq. (9) & (7))?\n\n- How can we set $p$ and $\\tau$ (or $\\delta$) in practice?\n\n- To my understanding, the regularizer (Eq. (9)) effectively prevents spurious training that would drive the second term in Eq. (6) to zero by enforcing $e \\rightarrow 0$. Is this correct?\n\n- To me, the proposed method seems to be a simple combination of known techniques (to clarify, I do *not* claim that simplicity  alone is grounds for rejection at all). Could you clarify the differences of the proposed technique from other Bayesian approaches, confidence calibration algorithms, semi-supervised learning, and entropy regularization techniques? A quantitative and objective discussion would be preferable, which would significantly enhance the paper's contribution and clarity.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The motivation of the proposed method is clear. \n\n- The idea of transforming the optimization problem with a constraint (Eq. (7)) into a simpler form (Eq. (9)) is interesting and effective, which significantly simplifies the problem.\n\n- Several theoretical results, as well as empirical results, are provided. Theorem 1 helps quantitative understanding of the proposed method.\n\n- Code is available.", "weaknesses": "- Experiments on the dependence of the results on $p$ and $\\tau$ are lacking. I would like to see the results and discussions when $p \\neq 2$ and $\\tau \\neq 1$.\n\n- (Major) The paper applies a Bayesian inference (Eq. (5)) with a regularization to the overconfidence problem inherent in TTA. The \"EM\" (entropy minimization) mentioned in the paper is, in a more general context, the unsupervised learning using soft pseudo-label, which has a wide variety of applications beyond TTA.\nWhile the proposed method is technically sound, it would benefit from discussion in a broader context—such as unsupervised learning with a pretrained model, unsupervised domain adaptation, source-free domain adaptation, and semi-supervised learning—to emphasize its wide applicability as a key contribution.\n\n- (Major) Error bars are missing. Could you provide error bars because several performance gains are marginal?\n\n- The performance metric \"Avg.\" in the tables are nonsense, while it is actually a bad convention in the field. Obviously, a \"1% gain\" is quite different in iNaturalist and SSB-Hard, for example.\n\n- (Minor) A large part of the paper is dedicated to reviewing previously known works, such as the overconfidence problem, which hinders readability.\n\n- (Minor) Much of the paper, particularly the description of the proposed method, is redundant. A more direct tone is generally preferable in academic writing.\n\n- Overall, while the paper is well-written and the proposed method is interesting and effective, the paper would benefit from a more refined articulation of its contributions and focus. Additionally, the reproducibility issue should be addressed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)