Decoupled Entropy Minimization
Authors: Jing Ma
Hanlin Li
Xiang Xiang
Venue: NeurIPS 2025 poster ()
Keywords: Entropy Minimization
Test-Time Adaptation
Self-Supervised Learning
Primary area: general_machine_learning

Abstract: Entropy Minimization (EM) is beneficial to reducing class overlap, bridging domain gap, and restricting uncertainty for various tasks in machine learning, yet its potential is limited. To study the internal mechanism of EM, we reformulate and decouple the classical EM into two parts with opposite effects: cluster aggregation driving factor (CADF) rewards dominant classes and prompts a peaked output distribution, while gradient mitigation calibrator (GMC) penalizes high-confidence classes based on predicted probabilities. Furthermore, we reveal the limitations of classical EM caused by its coupled formulation: 1) reward collapse impedes the contribution of high-certainty samples in the learning process, and 2) easy-class bias induces misalignment between output distribution and label distribution. To address these issues, we propose Adaptive Decoupled Entropy Minimization (AdaDEM), which normalizes the reward brought from CADF and employs a marginal entropy calibrator (MEC) to replace GMC. AdaDEM outperforms DEM*, an upper-bound variant of classical EM, and achieves superior performance across various imperfectly supervised learning tasks in noisy and dynamic environments.

Decoupled Entropy Minimization
Jing Ma Hanlin Li Xiang Xiang School of Computer Science and Tech, Huazhong University of Science and Tech, China School of Al and Automation, Huazhong University of Science and Technology, China
Abstract
Entropy Minimization (EM) is beneficial to reducing class overlap, bridging domain gap, and restricting uncertainty for various tasks machine learning, yet its potential is limited. To study the internal mechanism of EM, reformulate and decouple the classical EM into two parts with opposite effects: cluster aggregation driving factor (CADF) rewards dominant classes and prompts peaked output distribution, while gradient mitigation calibrator (GMC) penalizes high-confidence classes based on predicted probabilities. Furthermore, we reveal the limitations of classical EM caused by its coupled formulation: 1) reward collapse impedes the contribution of high-certainty samples in the learning process, and 2) easy- class bias induces misalignment between output distribution and label distribution. To address these issues, we propose Adaptive Decoupled Entropy Minimization (AdaDEM), which normalizes the reward brought from CADF and employs a marginal entropy calibrator (MEC) to replace GMC. AdaDEM outperforms DEM*, upper-bound variant of classical EM, and achieves superior performance across various imperfectly supervised learning tasks in noisy and dynamic environments.
Introduction
Entropy Minimization (EM) is common self-supervised optimization method, which minimizes the conditional entropy of model predictions to reduce the class overlap []] It facilitates low-density separation between classes and enhances confident predictions. EM has been shown to be useful for clustering, sumi-supervised, and unsupervised learning 2 3 4 Conditional entropy is also widely validated calibration tool for Deep Neural Networks (DNNs), since it measures the prediction error and distribution shifts to some extent [5]. EM helps to bridge the domain gap and push the model's decision boundary towards the low-density regions of the target distribution. Therefore, EM is widely
used in active learning, domain adaptation, and online learning as well 6 7 2
Although Entropy Minimization applied to a variety tasks due to its simplicity and generality, previous literature indicates that the potential of classical EM is limited [T) causing unsat- isfactory performance. To investigate the internal mechanisms by which EM effectively optimizes model parameters in unsupervised manner, attempt to reformulate and decouple the EM. We divide the conditional entropy in EM into two independent parts with entirely opposite effects: a positive-acting factor termed Cluster Aggregation Driving Factor (CADF) and negative-acting factor called Gradient Mitigation Calibrator (GMC). Minimizing CADF rewards dominant classes and promotes a peaked output distribution of the model, while minimizing GMC (log-sum-exp of the logits) penalizes the maximum logit and high-confidence classes based on predicted probabilities and calibrates the optimization direction, serving regularization term. This reformulation enables systematic analysis of these two parts and refining the classical EM, as shown in Fig.
Importantly, we demonstrate that the highly coupled formulation of conditional entropy inherently constrains the effectiveness of classical EMM First, samples with lower confidence exert a more pro-
"quual contribution, co-first author; Correspondence xex@hust. edu, (also with Peng Cheng Lab)
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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nounced impact in the learning process, while the contribution of high-certainty samples diminishes significantly as their predicted probabilities approach 1.0, as shown Fig. (left) This phenomenon is named reward collapse, which is not preferable because samples with higher certainty can provide more reliable and informative signals for self-supervised learning. Second, classical EM tends to exhibit systematic bias toward dominant and easy classes, assign most samples to single cluster, resulting in severe misalignment between the model's output distribution and the ground-truth label distribution, as shown in Fig. (right). Easy-class bias undermines the adaptability of classical EM in handling noisy or imbalanced tasks, particularly in dynamic or non-stationary environments.
Through the investigation in Sec. 3.2 we find that replacing the classical EM with minimizing CADF alone sig- nificantly enhances EM performance, yet it exhibits weaker robustness to
CEM
L pi logg
{
(pi
target data distribution shifts. How- ever, introducing a temperature r to reshape the reward curve (by raising the value of 7 when model's average predicted probability on target data in- creases) effectively improves CADF's robustness. Mcanwhile, minimizing GMC helps reject erroneous highly confident predictions from poorly cal- ibrated models [13]. Consequently, a weight a is employed to scale and control GMC's influence, mitigating model overfitting in noisy tasks and dynamic environments. Decoupled Entropy Minimization* (DEM*) inte-
Model
Softmax
f
PDEM =
Σ Epize Prizi + alug Σ e
CADP
GMC
Logits
Probabilities
CAMADEM
(pr P
Forward
Backward
Figure 1: EM is decoupled into two parts with opposite effects: CADF and GMC. DEM* softens the model's pre- diction via temperature and scales GMC via weight a, searching for optimal (T*, a**) to maximize classical EM's performance. AdaDEM normalizes CADF reward by 5 (LI-norm of the gradients) to prevent reward collapse, and replaces GMC with Marginal Entropy Calibrator (MEC, i.e. p reduce easy-class bias.
grates CADF and GMC with searched optimal hyperparameters (r*,a") in target data distributions. DEM* represents the performance upper bound of classical EM on the target task to some extent, but incurs additional computational overhead for identifying optimal hyperparameters.
To address these issues, we propose Adaptive Decoupled Entropy Minimization (AdaDEM). To enhance the relative contribution of high-certainty samples in the learning process, AdaDEM employs the Ll-norm of rewards brought from CADF to normalize the conditional entropy. Additionally, replace the GMC, which counteracts the
Marginal Entropy Calibrator (MEC) is proposed overwhelming influence of dominant and easy classes by maximizing the estimated marginal entropy. Unlike existing methods that assume uniform label distributions 12. MEC climinates the need for label priors and instead leverages dynamic estimation during learning. In this way, AdaDEM, without requiring tunable hyperparameters, achieves comparable or even superior performance to DEM
We provide an insightful view to study Entropy Minimization by reformulating and decoupling the classical EM into CADF and GMC with opposite effects. We investigate the roles of these two parts and reveal the inherent limitations of classical EM caused by its highly coupled formulation, i.e., reward collapse and easy-class bias phenomena. 2) We propose DEM* explore the performance upper bound of classical EM, and propose AdaDEM to mitigate reward collapse and easy-class bias while eliminating hyperparameter tuning requirements, thereby unleashing EM's potential. AdaDEM is validated to outperform DEM* 3) Extensive experiments] across various tasks, including supervisd and unsupervised learning. domain adaptation, and reinforcement learning, demonstrate AdaDEM's superior performance. Additional evaluations on noisy/imbalanced benchmarks and dynamic/non--sttionnn environments further validate the effectiveness of AdaDEM.
In summary: 1)
2 Related Work
Shannon entropy [4] is a fundamental concept in information theory, which is used to measure the degree of uncertainty of random variable. The higher the entropy value, the greater the uncertainty. For discrete random variable x with n possible values [2,, xr and corresponding prob-
Source code is available

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Classical Entropy Minimization
Predicted Class Distribution by VIT-8/16
0.8
classical EM (Tent)
20
0.7-
ADaDEM
L1-Nern of Gradlent
2.3
0.6-
Proportion (86
0.5
2.0
0.4-
2.5
0.5-
0.5
0.0
0.2 0.4 0.5 0.8
1.0
200
000
1000
Maximum Predicted Probability
Classes
Figure 2: (Left: Reward Collapse) We compare the gradient magnitudes of the classical EM for samples with different predicted probabilities, which collapse to 0.0 when the maximum probabilities approach 1.0. (Right: Easy-Class Bias) The output distribution of ViT-B/16 after test-time adaptation using classical EM and our AdaDEM on class-balanced Gaussian- weeeeoooiiiecoeoeee corrupeed ImageNet-C benchmark, with all classes sorted by their predicted proportions in descending order.
abilities {PI.P2. Pa.....ph where ps 2 and E Pi = 1, the Shannon entropy (zx) is defined as
Hxx== = - log
(I)
In machine learning, Shannon entropy is generally used as an information-theoreti measure of uncertainty or impurity. represents the amount of information needed to encode a distribution [15]
Given an input distribution X, a target distribution Y, and the sampled examples {xr, y,y. We build a model f that maps X to Y that parameterized by 0. The conditional output of f is denoted as
p(y| 8). Substituting it into Eq. (.). we can obtain the uncertainty of the model's ' output, which is defined as the expected value of the information carried by a sample from the output distribution:
X y
H(YIX)= = N
Σ
plule 0) logp(ulz,
The conditional entropy HY| is measure of class overlap (I. Intuitively, if want p(ylær 0) to be highly peaked, Le., the model's prediction for x is certain, we want H({(X) to be low. On the other hand, if we want p(y ar. 0) to be flat predictions to be uncertain, XX)) which, in the limit, will lead to a uniform conditional distribution over classes [16].
maximize the entropy
By minimizing the conditional entropy (Y|XX) for samples, the overlap of model's output dis- tribution can be reduced, leading the density of data points to get lower at the decision boundary [17. It facilitates low-density separation between classes, commonly prior assumption for semi- supervised learning [88. A number of studics show that unlabeled data can be more informative there is less overlap between classes [19]. Thus, EM has been demonstrated to be effective for clus-
tering (avoiding trivial solutions where most instances concentrate in a single cluster) 2 12: 20 21),
semi-supervised learning (enhancing model prediction accuracy per data point) 1
3
10
22 23 24
and unsupervised learning (yielding peaked conditional class distributions) 4 16
20:
25|
26).
Deep Neural Networks (DNNs) suffer from distribution shifts. DNNs trained on the source domain tend to produce over-confident (low-entropy) predictions for in-distribution data and under-confident (high-entropy) predictions for out-of-distribution data (27). Conditional entropy HY|)X) is widely validated calibration tool for DNNs, since it measures the prediction error and distribution shifts to some extent. More confident predictions are all-in-all more correct. More severe shifts result in higher entropy [3]. One possible way to bridge the domain gap is to push the model's s decision boundary towards the low-density regions of target distributions. On this basis, EM is widely used in
active learning (reducing the number of possible hypotheses as rapidly as possible) 6 15). domain
adaptation (aligning the target data distribution with the source data distribution)
7 27 and
online learning (connecting entropy to error and shift) [ g
30.
Entropy Maximization plays role in encouraging exploration in Reinforcement Learning
31 32 33
Adding the entropy of policies to the objective function prevents premature convergence to sub- optimal deterministic policies and is particularly beneficial for tasks that require hierarchical behavior
3

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B1J. It is worth noting that maximizing the conditional entropy [X] leads to uniform output distribution obtain uncertain predictions, while maximizing the marginal entropy, i.e.
Y H(Y) = Σ plu, 0) log p(y. 0)
(3)
over all data points, where p(y, 0) N Ei= p(y 0). encourages the cluster sizes of classes be uniform [12.. The combination of minimizing XX) and maximizing H (Y) makes the model's output distribution individually certain and globally diverse [6).
Unlike Cross- Enssooy and KL-Divergence, which employ external supervision in the learning process, Entropy Minimization is self-supervised learning paradigm. Therefore, addressing the potential limitations of classical EM can promote the development of imperfectly supervised machine learning. EM exhibits similar concepts to Self-Training [34] and Pseudo-Labeling [17], which leverage the
model's own predictions for learning. Reward collapse and easy-class bias are also prevalent in these methods, and AdaDEM can be plug-and-play. While Early-Learning Regularization [35] mitigates label noise by using an exponential moving average of predictions as self-supervised signals, MEC in AdaDEM employs dynamic estimation of predictions to penalize dominant and easy classes.
3 Rethinking Entropy Minimization
3.1 Reformulating Conditional Entropy
Notations. For an example X, the output of model f(0) is z [= RC_ 2, is the logit of the i-th class predicted by the model, and C is the number of classes. We apply the Softmax function (-) to convert the real-valuee vector into probability vector p [p:]/= RC, i.e.,
P: ea1 Eラ=1 e where 0 S P S 1, Vi E {1, ..., C).
For simplicity, define the conditional entropy in Eq. (2)
H(z) = Σ logp
(4)
The objective of EM is to minimize (z) as the sole loss function or regularization term, namely,
X
x
arg min Σ H(z|,, 0) = arg min ΣΣ pi(zl,, log
(5)
1=1
where 8* is defined as the optimal solution for model f(0).
To analyze the contributions of the two decoupled parts with opposite effects from classical EM to the model's output, we define the terms "reward" and "penalty." We define "reward" as the negative partial derivative of the loss function L = (z)) with respect to the logit 2,, i.e., 02.02,, which aligns with the gradient descent direction for solving Eq. (5). We define "penalty" as the opposite of "reward", i.e., 0C/Oz, We subsequently demonstrate that minimizing CADF rewards the model's output logits, while minimizing GMC penalizes them.
Reformulation. To investigate the internal mechanisms by which EM effectively optimizes model parameters in an unsupervised manner, we attempt to reformulate and decouple the conditional entropy H (z) in EM, which be rewritten as
C
H(z) = Σ pi(z) log 1=1
Pi(z) z4 + log
e
(6)
1=1
CADF
GMC
As shown in Eq. (6. the conditional entropy is reformulated into two independent parts with entirely
opposite effects. To analyze the impact of these two parts on EM, we derive the partial derivatives of
-i=1 the first term T(z) Σ pz(z)2, and the second term Q(z) log D e with respect to 2,
0T(z) = (T +2+ R₂ Oz,
aQ(z)
e
RQ
Oz,
Σラ es =
=Pi
(7)

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Reward Logits
0.8
(m=1.)
(a=1.0)
1.5
Vin
23-36]
[10
Datta
mid
0.6
2-20 la=1.))
[E0 4g)
Shat
0.4
alue
1.3
5me
1.2
0.2
(34-77)
(3-40)
0.0
1.0
Predicted Probability
0.2 D.6
0.1 0.2 0.3 0.5 0.6 0.7
Averaged Probability
C10 100 533
142 391
Reedust
Srale
Figure 3: (Left) Reward curves DEM with varying T values for a 10-way classification task. (Center) The best value positively correlates with the average predicted probability of source models on target data. (Right) Detailed TTA results using the optimal T across 15 target domains, with exact values sourced from F (Center) "NoAdapt" denotes the baseline using fixed source model parameters without adaptation, hence its performance remains consistent for both single-domain and continual TTA tasks. Values in brackets indicate the corresponding axis range.
Mil
Deta
Reward for Logits
Accuracy for Single
30 W
Accuracy Conti.
mpu
0.8
40
0.6
0-1.5
Shot
Snw
0.2
26
Per
-0.2
lgg
0.4
p1.50
0.0
0.2
0.4
0.6
0.8
1.0
1.122414 Value a
1.8.0
Predicted Probability
(56
[17-27)
Cord T7%.
Single TTA
Figure 4: (Left) Reward curves of DEM with varying a values for a 10-way classification task. (Center) Effects of different a values static and dynamic target data distribution shifts of single- domain and continual tasks. (Right) Detailed TTA results across 15 target domains, using the optimal = 1.0 for single-domain TTA tasks and 1.3 for continual TTA tasks. These values are selected based on the ablation results in Fig. (Center). The definitions "NoAdapt" and values in brackets are consistent with Fig. (Right).
Ri denotes the reward obtained by minimizing (z)) for logits 2,, which leads to an positive increase in logit values. The reward curve of Ry w.r.f. the probability pt is shown in Fig. (left) (a 0.0 & = 1.0). Minimizing ('zz) results in higher rewards for logits with elevated predicted probabilities, exhibiting an approximately linear trend. However, when reflected in probabilities through the Softmax function, this trend becomes amplified into exponential growth. Consequently, minimizing T(z) causes the model to favor predicting classes with higher probabilities, thereby granting more rewards to dominant and easy classes. Conversely. RQ -P- < O consistently imposes penalties on logits based on predicted probabilities. Minimizing (zz) causes higher-confidence classes to receive greater penalties, which drives the model's 's output distribution toward uniformity. So we name T(z) the Cluster Aggregation Driving Factor (CADF) and (z)) the Gradient Mitigation Calibrator
(GMC). According to Eq. (6) and 7 we introduce Σ p;č; that shares identical partial derivatives w.r.t logits as (z)), where p; a constant excluded from the computation graph
Therefore, the conditional entropy can also be rewritten as
(z))
2p. 2-)
(8)
i=1
Next, we investigate the effects of CADF and GMC on entropy minimization and propose enhance- ments to address these limitations.
3.2 Decoupled Entropy Minimization
Role of CADF and GMC. To investigate the effects CADF and GMC on EM, we conduct ablation studies test-time adaptation (TTA) tasks using Tent (5), Tent serves as appropriate study subject
In PyTorch, the "detach()" method can he used to obtain
5

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Table 1: (Left) Ablation studies in single-domain and continual TTA tasks. DEM* searches optimal hyperparameters (T*,0*) on subset of target data. A denotes the performance improvement relative
to the baseline. (Right) The sensitivity testing of learning rates demonstrates that AdaDEM has expanded 10x tolerance range compared to classical EM.
Single-Domain Domain TTA
Methods
Single
Contl.
60
Nobdast
ResNer50
EMtRenG
Top-1 Accuracy
55
Asa DEM
NoAdapt
31,5.000
31.5+000
EM (Tent)
40.0anos
+0.0
31.2s0111
+0.0
50
CADF
41.7anos
+1.7
36,
+4.9
-5
DEM*
41 8ancs
+1.8
39.0zn02
+7.8
AdaDFM-Noree (w/o MFC)
43. 13.7
+3.7
37.5.a0s
+6.3
AdaDEM-MER (w/o Norm)
44.4s
+4.4
37.5
+6.3
10
10-2
10-2
10
AdaDEM (w/ MEC & Norm)
44.8
+4.8
37.7aa0s
+6.5
Learning Rate
Continual TTA
VIT-B/16
65
NoAdast
60
EM TRenG
NoAdapt
38 8ancoo
38. S20.00
Top-1 Accuracy
Asa DEM
EM (Tent)
53. Janos
+0.0
58 6enoy
+0.0
555
CADF
48 68.62
-4.5
63.0sao1
+4.4
DEM
56 06..22
+2.9
64.1
+5.5
AdaDEM-N Norm (w/o MEC)
54. Oan4a
+0.9
59.3a004
+0.7
-5
AdaDEM- MEC (w/o Norm)
58. Janzs
+5.0
62, 14
+3.8
40
AdaDEM (w/ MEC & Norm)
61.5.n.2
+8.4
13.2
+4.6
10-
10-3
10-2
10
Learning Rate
since it solely employs the classical EM as the loss function (where conditional entropy is measured from streaming data) to mitigate DNNs' performance degradation during online testing. Target data distribution shifts are categorized into static single-domain tasks and dynamic continual tasks. We adopt ResNet50 (36) and ViT-B/16 (37). two widely used DNNs pre-trained on ImageNet-11 (38), as source models for TTA starting points. Other implementation details are provided in Sec. 4.1 The experimental results in Table [ demonstrate that replacing the classical EM with CADF alone significantly enhances EM performance, yet it suffers from poor robustness, as exemplified by the single-done Ainle-domain TTA case with ViT-B/16. To address these issues, we propose our improved solution.
Reshaping the Reward Curve. We introduce temperature in CADF to soften the probability ps. thereby reshaping the reward curve. This operation is defined in Eq. (9). with its partial derivative w.r. logit z, given in Eq. (17) (refer to Appendix A..)) Based Eq. (17). the Fig. (left) plots EM's reward curve across different values, which degenerates to classical EM when r = 1.0. Notably, the reward collapses to 0 when predicted probability approaches 1.0 or 1 1/C. Introduction of 7 ensures high-probubility predictions remain within the high-reward interval, thus mitigating reward collapse.
ea/T
(2)2) = Er((z)e
Pri(z)=
Σ 65/T
We search the optimal maximize performance on target tasks, and empirically demonstrate that the best value exhibits positive correlation with the average predicted probability of source models on target data, NV E maxigisc Pij. as shown in Fig. (center). Detailed results in Fig. (right).
Scaling the Influence GMC. Note that Q(z) = log Ei= e takes the form of the log-sum-exp of the logits 2, which provides smooth approximation to the maximum of logits. Therefore, introduce a weight a to control the penalties imposed by minimizing GMC on the maximum logit, preventing model overfitting caused by rapid growth in logit magnitudes, which is defined
Qa(z) = a log
(10)
thereby scaling the reward of Qa (z) to Rq. ap, We plot the reward curve of EM with varying values, as shown Fig. (left). When a 1.0, EM with (z) penalizes logits corresponding to predicted probabilities near .0, helping to reject erroneous highly confident predictions from poorly calibrated models 13). The results in Fig. 4 (center) demonstrate that slightly increasing the value of effectively mitigates model overfitting in noisy tasks and dynamic environments.

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In summary, the Decoupled Entropy Minimization (DEM) is formulated as
= Qa(z)= = Pr.(z)) + a log Eea.
(11)
To explore the performance upper bound of classical EM, we search for the optimal combination within the search space defined by and a using a fast TPE algorithm [39]. In Proposition A.1 we theoretically prove that valid values of in DEM satisfy 0 < 7 S 2/a where >> 0. DEM* searches the optimal hyperparameters (r", ,a) on a subset of target data. Implementations in Sec. 4.1
3.3 Adaptive Decoupled Entropy Minimization
Classical EM suffers from reward collapse and easy-class bias phenomena, demonstrated in Fig. DEM* enhances the contribution of high-certainty samples in the learning process by improving the CADF, while refining the GMC to penalize dominant and easy classes. However, DEM* cannot fundamentally resolve these limitations, as the dynamic interplay between model parameters updates and shifting data distributions causes the model's output to evolve over time. While continuous optimization or designing evolution strategies for hyperparameters T and a remains feasible, such approaches incur substantial computational overhead, which is drawback effectively addressed by the next proposed Adaptive Decoupled Entropy Minimization (AdaDEM).
We employ the LI-norm of the reward brought from CADF, i.e., ò = | --0T(z)z 0)/Oz|11 to quantify the extent of changes in the model's output logits z = [2:]-= before and after EM optimization, which is caused by updates to the model parameters 0 induced by sample x. We adopt 1/ /5 to normalize rewards across different deterministic samples, providing more fundamental and direct approach compared to using the reciprocal of conditional entropy as the normalization factor [I. We have further verified that must be computed from the gradients of CADF, while using the gradients of the overall conditional entropy (z) would introduce penalties that compromise the accurate assessment of rewards brought by learning samples. Refer to Appendix f.6 details.
Most existing methods for addressing easy-class bias employ a label prior to guide EM, typically by maximizing the marginal entropy align the model's output distribution with a uniform label prior distribution 77: 12). In contrast, we eliminate the label prior assumptions and propose a Marginal Entropy Calibrator (MEC) to replace GMC. The MEC dynamically estimates the marginal entropy during learning via an exponential moving average of 1 1/Nk N= pk where Pk E RC is probability vector of the k-th class. The estimated probability is dynamically updated as ! p = 0.9 Pk O.1/NN Σ No ph where denotes the iteration index and Ps = [1/C] E RCC
---
In summary, the AdaDEM takes the form of
C (pi(z) - k == arg m.).
Hzz) -
Σ
(12)
5 1=1
AdaDEM requires no tunable hyperparameters and achieves comparable or even superior performance DEM*, as demonstrated in Table [ Notably, AdaDEM also reduces the sensitivity of classical EM
to learning rates, as shown in Table (right). Detailed analyses are provided in Appendix D
4 Experiments
4.1 Setups
Benchmarks. We conduct experiments on four benchmarks, i.e., Test- Time Adaptation (TTA), Semi- Supervised Learning (SSL), Unsupervised Domain Adaptation (UDA), and Reinforcement Learning types of image corruptions, as well 42). -R. (3), and -S. (44] that represent natural distribution shifts. For SSL., we consider CIFAR- 10, CIFAR-100 45). STL-10 [46), EuroSat [47], TisSueMNIST 48) and Semi-Aves 44) Regarding synthetic-to-red UDA, mainly experiment with the semantic segmentation task using GTAS dataset (50) as the source domain and Cityscapes dataset (5]) the target domain. We employ Minigrid [52]. series of discrete environments, to conduct RL experiments. We also conduct experiments on class-imbalanced benchmarks of
(RL) tasks. For TTA, we adopt ImageNet-C C 40) containing 15
as ImageNet 38 and its variants: -A [4 V2.
CIFAR- (p = 100) and CIFAR-IO0-LT (p=10) 45 Refer to Appendix [.I for detailed setups.
7

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Table 2: Experiments on single-domain & continual TTA tasks (left) and the test-time prompt tuning task (right). Top-1 classification accuracy (%) is reported. We highlight the highest accuracy in bold and the second best underline. A denotes the performance improvement relative to the baselines.
Methods
Single- DoggeeDomain Mean
A
Continual Mean
Methods
-1K
ImageNet -V2
-S.
Avg.
A
NoAdapt
38.8
38,
18.8-00
CLIP RN50
Tent
53. Lece
+0.0
58 58.6
+0.0
Zero-Shot
58.2scm
Satooo
51.4am
56.2wm
33.4
44.2ammm
DEM
56.0uess
+2.9
La,la
+5.5
Ensemble
59.8.om
23.2.000
52,9um
60.7
35.5
46.4
+2.2
+ AdaDEM
20
+8.4
63.
+4.6
TPT
60.7
26. 10000
14.6
58.9
35.2-00
47.1
+2.9
Tent
52.7.c
+0.0
48. S
+0.0
DEM
61.3gtm
25.5a
10.0a0.1
59.7s0.2
35.62003
47.4g0n
43.2
DEM
55, Jaca1
+2.4
64.5am
+1600
AdaDEM
60.7a0m
29. 29.2
54.8
58.8
35.4
47.8amur: +3.6
AdaDEM
26.2.
+13.5
64.4am02
+15.9
CoOp
63. 63.3gt00
23. 1
55.4a000
56.62000
34.72000
46.62000
+2.4
ETA
65. 1st.10
+0.0
64.2am04
+..0
TPT (CoOp)
65.4:00
28.9.n
58.2+n
59.0
36.3-015
49 19.6.amm
+5.4
DEM
66.3-2
+1.2
65.7-00s
+1.5
AdaDEM
68.6ram5
31.3
58.5an
39.3
36.3
50.2
+6.0
AdaDEM
66.8.crc
+1.7
66. Laua
+1.9
CLIP-VTT-/16
62.2
+0.0
64.9aus
+0.0
DEM
64.422 to
2:2
2-00
+1.3
Zero-Shot
66.7atm
47.9 47.9
60.9am0
74.0aam
46.12to0
59.1anum
+0.0
65. AdaDEM
+3.1
66.4
+1.5
Ensemble
68.3emm
49,9
60.9a000
77.7
48.2s000
61.2samm
+2.1
DeYO
62.6 6ue32
+0.0
$7.6ucs
+0.0
TPT
69.02394
54.5a009
63 43.4an.13
77.0zum
48.02013
62,420u5
+3.3
DEM
+3.0
65.4an12
+7.8
DEM
68.9
54,8
63.5wmm
77.1aum
47.9uum
62.5anmm
AdaDEM
62. 1o
+0.0
59 S0.0s
+14
AdaDEM
69.4iom
58.8-n1x
64.0
77.6
48.6wn0s
63.74ams
+4.6
SAR
+0.0
57.0
40.0
CoOp
71.5m
49.7amo
24,2amto
75.2
48.0am00
61.7amm
+2.6
DEM
57.9 57.9.20
+3.7
62 Amnas
+5.4
TPT (CoOp)
73.6
57.9-a13
66.9
77.2
49.2-0m
64.9
+5.8
AdaDEM
46.0
AdaDEM
73.7
60.3
66. 66.92m.3
77.9
49.3
65.6uauz
+6.5
Table 3: Experiments on semi- supervipe learning tasks. We report the average Top-1 I classification accuracy (%) under different numbers of labeled samples. We highlight the highest accuracy in bold and the second best as underline. A denotes the performance improvement relative to the baselines.
CIFAR-I0
CIFAR-100
STL-10
EuroSat
TissieMNIST Mean
Semi-Aves
Methods
Mean
Mean
Mean
Mean
Mean
Ein. Min.
96.4.19
+0.0
72.6as
+0.0
82.4as
+0.0
76.5xs6
+0.0
47.3x16
+0.0
59.9w77
+0.0
* AdaDEM
97.2-a>
+0.8
75.8-01
+3.2
84.8-03
+2.4
83.7ao
47.2
49.3.13
+2.0
61.0
+1.1
Var Ent. Min.)
97.4-3n
+0.0
77.5.at
+0.0
85.4wa
+0.0
86.6.5
+0.0
45.2
+0.0
61.0ma
+0.0
+ AdaDEM
98.5
+1.1
78.8
+1.3
86.9a
+1.5
91,2609
+4.6
49.4696
44.2
61 8500
+0.8
MixMatch AdaDEM
98.5443
+0.0
73.8-04
+0.0
82.9
+0.0
79.34:3
40.0
48.0w17
40.0
62 6i00
+0.0
98.3.6
0.2
74.8
+1.0
85.2mm
+2.3
83,9
+4.6
50.0.39
+2.0
62.5aa
40.1
FixMatch
98.4-77
+0.0
79.3:06
+0.0
88.5a12
+0.0
91 9.36
40.0
46.753
40.0
68.2
+0.0
+ AdaDEM
98.7443
+0.3
79.5
87.9
96.9u0
+5.0
49.9.13
+3.2
67.9kk
FrecMatch
98.9.a
+0.0
Geni
+0.0
89.7ats
+0.0
95.8s10
+0.0
45.6
+0.0
+0.0
* AddDEMM
99.0-s1
+0.1
83.1-a1
+0.5
9166
+1.9
95.9ka
40.1
47.9-a4
+2.3
67. 3.00
+0.3
Implementations. We consider the proposed DEM* and AdaDEM as alternatives to the clas- sical EMM Therefore, we replace EM in the ob- jective function of existing methods with our implementations of DEM* or AdaDEM. For in- stance, substitute the loss function of Tent
Table 4: Experiments on unsupervised domain adaptation task of semantic segmentation. Stan- dard mloU (%) is reported.
Methods
MinFar
AdaDEM
AdvFnt
DDM
AdaDEM
mioU
41.604:
42.7am
43.6au
41.6.132
44.9.m
[3] with Eq. (T] or 12 DEM* employs a fast TPE algorithm [39 search for the best hyper- parameters (r",a"). Proposition A.. heepps hhepss to reduce the search scope. We use subset comprising 20% of test data with ground-truth labels for TPE, which is only applied for DEM* explore the upper bound performance of classical FMM. Note that AdaDEM requires neither test data labels nor hyperparameter tuning, strictly adhering to the unsupervised learning paradigm. We mainly utilize ResNet [36), ViT (37], CLIP Model [[53]. Deeplab-V2 (54), MLP and other DNNs as the research subjects. Meanwhile, for TTA, we adopt Tent (5), ETA &. EATA B. De M. SAR (2), TPT 30 for SSL, we employ Ent. Min. []. VAT B, MixMatch (24, FixMatch 23 FreeMatch 22 for UDA, we use MinEnt & AdvEnt (77: and for RL, we take PPO
+0.0
-11
+0.0
+1.0
+1.3
B2 as baselines. We
follow the implementations and hyperparameter setups of original methods unless otherwise specified.
Uniformly, we report Top-1 accuracy for classification tasks, mloU (55) for semantic segmentation tasks, and average return for RL tasks. All experiments are running in 3 seeds (e.g., {1, 2, ))) which control the orders of training/testing samples and the initial conditions of algorithms. Performance metrics are reported as "mean + std". Refer to Appendix B.2 for detailed implementations.
8

---

CIFAR-1O-LT (p=100)
CIFAR-100-LT (p=10)
Classical EM (Avg. F1: 0.67)
Clasiccal EM (Avg. Fl: 0.49)
1.0
Our AdaDEM (Pgg. F1: 0.87)
Our MMGGEM lNeg. r1: 0.53)
Per-Class F1 Score
Per- Class F1 Score
0.5
0.8
9.6
0.4
0.2
0
1
20
6.
Classes
Classes
Figure 5: Experiments on class- imbalance benchmarks. p denotes the sample ratio between the
most and least populous classes. Both per-class F1 scores and the average F1 score are reported.
RedBlueDoors 6x6
KeyCoridorS3R1
DoorKey-Sx5
Unlock
Fetch-Sx5-W2 N2
0.8
0.0
Average Retum
0.8
9.8
5 0.6
9.5
0.5
0.7
0.5
Averoge
0.6
6..4
0.3
0.3
6.2
0.0
A.CAAMEEM
0.1
1ROHAMSEN
0.4
2.0 500
5. 2/2 9'4 o's
0.s 1.5
2
5.0
02
04
0.6
33
1.0
0.0
0.2
0,5 Timestaps
0.4
0.5
0.0
0.4
12
1.0
Timestaps
Timesteps
Timestaps
Timesteps
Figure 6: Experimental results on Reinforcement Learning tasks in Minigrid environments.
4.2 Main Results
In the following, we present the experimental results in various tasks. Details in Appendix C
Test-Time Adaptation. As shown in Table 2 ETA, EATA, SAR, and DeYO are based on sample selection and assign larger weights to higher certainty test samples for mitigating the reward collapse. The application of DEM* and AdaDEM improves the original performance by addressing the potential limitations of the classical EM, which also enhances the prior-based EATA and SAR. For test-time prompt tuning task based on episodic TTA, DEM and AdaDEM provide a solution suitable for the aggressive optimization strategy with large learning rate and multiple adaptation steps, which brings a significant improvement in unsupervised prediction to CLIP Models and TPT.
Semi-Supert Supi-vuper Learning. Entropy Minimization is widely used in SSL to facilitate low-density separation between classes. We set different numbers of available labeled samples, i.e., Ni per class, for the training of DNNs. Specifically, we set N; 4/25/ 400 for CIFAR-10. N₂ 2/ 2/4/25 for CIF,R-100, N₂ = 4/10 for STL-10, N₁- = 2/4 for EuroSat, N; = 10/50 for TissueMNIST, and N; = 15 53 for Semi-Aves. In Table 3 we report the average classification accuracy under different settings. We replace the classical EM in Ent. Min. and VAT (w/ Ent. Min.) with AdaDEM, and introduce AdaDEM as part of the loss functions in MixMatch, FixMatch. and FreeMatch, which do not use EM. We also validate on class-imbalanced benchmarks for SSL, as shown in Fig. 5 and provide an imbalance study on highly skewed data with varying degrees of imbalance severity, refer to Appendix DD Experimental results show that AdaDEM outperforms the classical EM. Meanwhile, incorporating AdaDEM into SSL methods effectively improves the performance of original algorithms.
Unsupervised Domain Adaptation in Semantic Segmentation. Beyond image classification, we also verify the effectiveness of DEM* and AdaDEM in the semantic segmentation task. As shown in Table compared with the classical EM, AdaDEM can better bridge the domain gap between the source and target distributions, thus achieving a higher mIoU on the target data. For detailed results, refer to Table 10 DEM* and AdaDEM can improve the accuracy of tail classes and prevent the model from being inclined to predict dominant classes. We also visualize the segmentation results and pixel entropy in Fig. 7 AdaDEM reduces the prediction entropy of pixels more effectively, resulting in clearer object boundaries of dit-of--istribution data.
Learning, Entropy Maximization is beneficial for encouraging agents' exploration in RL tasks. To demonstrate that our AdaDEM is not confined to the minimization strategy, we compare the effects of applying the classical EM and AdaDEM to PPO while keeping entropy maximization strategy unchanged. The experimental results, as depicted Fig. 6 indicate that AdaDEM can achieve higher or comparable average return.
Reinforcement

---

4.3 Ablation Studies
We report the optimal hyperparameters (T", xx*) searched for DEM* in different methods and tasks in Table I
For additional discussions, refer to Appendix Dior for details.
Effect of Temperature 7 in DEM* As shown in Fig. increasing the value of T promotes DNNs to learn high-certainty samples by reshaping the reward curve. The best T for a target data distribution
positively correlated with the average prediction probability of DNNs for target samples.
Effect of Weight a in DEM As shown in Fig. (left), GMC employs a to scale the penalty applied to logits, reducing the interference of over-confident samples for poorly calibrated DNNs. Decreasing the value of a is suitable for aggressive optimization strategies, while increasing is suitable for alleviating the catastrophic forgetting caused by model overfitting, as shown in Fig. (center).
Sensitivity of AdaDEM to Optimizers and Learning Rates. We compare the impacts of different optimizers and learning rates classical EM and AdaDEM. As shown in Table I AdaDEM effec- tively reduces EM's sensitivity to the learning rate. Meanwhile, Table demonstrates performance gains when applying AdaDEM to various optimizers including vanilla SGD (Tent)), Momentum (ETA, EATA, YO),) Adam (Tent), and SAM (SAR). The results indicate that AdaDEM maintains compatibility with SGD optimizers utilizing first/seecodd-rrde momentum estimation.
Robustness of Ad.DEM. As shown Table 2 and Fig. source models' predictions on target data are significant noisy due to severe distribution shifts. Continual TTA tasks are evaluated under dynamically changing target data distributions. The results across these tasks demonstrate that AdaDEM exhibits robustness to noisylimbalanced tasks and dynamic/non-ssationarry environments.
Conclusion
This paper provides an insightful view to study EM by reformulating and decoupling into CADF and GMC with opposite effects. We reveal the limitations of classical EM caused by its highly coupled formulation: reward collapse and easy-class bias phenomena. We propose AdaDEM to overcome these limitations, which outperforms DEM*, an upper-bound variant of classical EM, and achieves superior performance across various machine learning tasks.
Limitations. DEM and AdaDEM overcome the limitations of classical EM only from the perspective of objective functions. Although improving performance. they cannot completely solve all the problems of self-supervised learning. Additional targeted techniques and designs are required.
Acknowledgements
This work supported by the HUST Interdisciplinary Research Support Program (2025JCYJ077), the project of Peng Cheng Lab (PCL22255S2214) 14), the 2026 Optics-Valle Valley Excellence Project funded by the National Graduate College for Elite Engineers of HUST, the Natural Science Fund of Hubei Province (2022CFB823). and the School of Computer Science and Technology, School of Artificial Intelligence and Automation, Hopreroft Center for Computing Science, and Al Institute, all at HUST.
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13

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APPENDIX
Contents
A More Design Details of DEM and AdaDEM
15
Pseudo Code
15
A.2 Theory and Proof
16
More Implementation Details
18
B. I Experimental Protocols
18
B.2 Method Implementations
19
More Experimental Results
20
Experiments on Single-Domen Continual Test- Time Adaptation Tasks
20
Experiments on Test- Time Prompt Tuning Task
21
Experiments Semi-Supervised Learning
22
Experiments on Unsupervised Domain Adaptation
23
C.5
Experiments on Reinforcement Learning
23
D
Additional Discussions
24
D.I Effect of Temperature 7 in DEM
24
D.2
Effect of Weight in D
25
................................
D.3
Optimal Hyperparameters
for DEM
................................
25
D.4
Ablation Studies DEM and AdaDEM
................................
25
D.5
Sensitivity of AdaDEM to Optimizers and Learning Rates
................................
26
D.6
Ablation Study the Role of ò in AdaDEM
27
D.7
Impact of Adding Scale MEC AdaDEM
27
D.8
Impact of Different Norm Types on AdaDEM
28
D.9 Impact of Different on AdaDEM
28
D. Ablation Study Imbalanced Benchmarks
29
Broader Impacts
29
14

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A More Design Details of DEM and AdaDEM
A.1 Pseudo Code
Algorithm Pseudo code of DEM and AdaDEM in the PyTorch-like style
logits: the output logita the model, in the shape of (N, C)
for CADF alpha: hyper-parameted for GMC in DEX
tau:
# avg-pred: the averaged predicted probabilities for C classes, initialized as None reset: flag for resetting the MEC in AdaDEM, in bool type
import torch
import torch.me mn. functional as F
Decoupled Entropy Minimization (DEM)
p_tau F.softmar tau, dim=1)
# CADF
cadf = (p_tau logits) sum (dim=11)
GMC
gmc alpha torch, (ogsumexp(logits,, dim=1)
Total loss of DEM
dem_ loss cadf gmc
# Adaptive Decoupled Entropy Minimization (AdaDEM)
p t.softmax (logits, dim=1) pseudo label p. a.ggmax
Initialize avg-pred to (C, C) tensor of 1/C
if reset is True or avg-pred is None:
C = torch. te size (1), devicesp.device avg_pred forch.ones((C) C), devicesp.devcce) 1.0 / c
with torch. no_grad():
Update avg-pred for MEC for label unique (pseudo_label)
avg_pred[label] 0.9 [[abbl] +
0.1 p[pseudo_label == label] .ean www.mmaannmmmnn (dim=0)
Calculate the L-1 norm of the rewards of CADF T (p * logits) suu sdu(imm,,, (dim=1, keepdin=True)
grad (logits + T + 1) delta grad. abs sum (dim=1, keepdin-True)
Original
# ....atcch()
# AdaDEM-Norm
# p (p p.detach()) / delta
AdaDEM-MBC
p = avg_pred [pseudo____d
# AdaDEM
P
(p avg. org.pred wws_preedpseud__aaaae
delta
Total loss AdaDEM
adadem_lose = (p * logits) sum www.ssummmma.... (dim=1)
15

---

A.2 Theory and Proof
Firstly, we present the detailed derivation process of reformulating and decoupling the conditional entropy for Eq. O Given the logit vector z = [21 22.....2 .... 20 E IC predicted by the model f()) for sample and the probability vector p [pI. ..... Pt. ..... PC) E RC obtained by applying the
Softmax function o.). The conditional entropy can be rewritten as
((2) = Σ
pj log =
Σ 1=1
Pi
log
e
Σ/=1 E
c
C
P:z Σ
log
e
1=1
i=1
C
C
C
Σ
Pizz
log
E
X
Pi
(13)
i=1
/=l
1=1
C
C
Σ
Pid log
Σ
e
i=1
1=1
Piz: + log
i=1
CADF
GMC
where Pi -E e. and E 1. Note that log Σ Dj=1 e is independent of i.
Next, we derive the partial derivatives of the two independent parts in Decoupled Entropy Minimiza- tion (DEM) with respect to the logit z; for Eq. 7 respectively. According to the Softmax function, the partial derivatives of with respect to 2, and 2y are
0p: 02, (5 E3
j== e 22
0p: e
P and
PPP5--
(14)
Øzj
e
Let T = Ei= Pize denote the Cluster Aggregation Driving Factor (CADF), and let Q log DC e denote the Gradient Mitigation Calibrator (GMC). The partial derivatives of and Q with respect to are calculated follows.
0T
Ozipi
OziPi 0z:
Zi
jfi
Opi
Opi
Σ
Pi 74 Zz
2.
Ozi
jti
D: 2p
-pî) + Σ
23(
(o--PPPp)
(15)
P: + p:2
(pf +
Σ
PsP}2z)
C
P: + P:a
PiP321
2g
1=1
C
Pi + Pi Ps Σ
pj2j
j=1
--x(TT + 24 + 1),
16

---

aQ
ôlog 2j=1 6
0 8E e
e
Oz,
dz.
2j=1 e*
Ōz,
Σ E==1
e²
Pt-
(16)
Note that T and Q treat 2,, Vi (1.,,..,C) equally.
Proposition A.I. The valid value of temperature in Decoupled Entropy Minimization is 0 < S 2/a where a >
Proof. Considering the introduction of a temperature T in CADF to reshape the reward curve,
obtain T, -E Pri24, where Pri = e² Σi=1 ets/T, Following the derivation process in Eq. 15
calculate the partial derivative of T. with respect to 2;
OT, 1 Pri (T+ + TT)
(17)
02;
Therefore, obtain the partial derivative of the conditional entropy used by DEM with respect to 2 as
(z ØT, 0z
aQa
1 PPi + 24 + 7).
Ōz:
Ōz:
ap;
7
(18)
Let F T + app:/Pri. and we have AH (2)/ -F x pri/T. We take the derivative of F respect to 2; in order derive the second derivative of H (z) with respect to zg:
OF, 1-11--r(111
+ T uT (T- Pi + 5
Pi 7
(19)
0z:
TPri
Pri
Under the boundary condition of 2; 2g k. V i + j. we have p. Pri 1/C, Vi E [1, CJ. T. E:= Priž; = h Σ.
1=1 Pri = k, and = 7(1 - -a).. Then, we can obtain the second
derivative of Hz) with respect to 2
8 Hz)
Pri 0F
F 0p
02,
T 02,
T 02:
1
1
1 7 - ) 1 + =) (1 a1
1
(1
TO
(20)
7C
C
7
C
1 a
2 T
(1 C (
For the logits z predicted for a sample we assume that > 2j, Vj * m. It means that the probability of the m-th class greater than others. We hope that the reward for zm is greater than zero, and the rewards for other 2j, Vjj # are less than zero, i.e., -HH (z)/Oam -0H(z) (z)/Ozz < 0j & m, in order to achieve low-density separation between classes and reduce the class overlap. Equivalently, we hope that 0² H (z))0zz S under the boundary condition of = 2y. vi * Because C > 1 and is set to be > 0, thus
0 and
Hz)
2
2
<0
(21)
0z,
TC
When > 0, we have T s 2/a. The above inequality does not hold when 7 < 0.
Considering the limit conditions + 0+ and + O=, have
8 Hz)) 00, and T-0-
Hz)
lim ++
lim
+ +00.
(22)
Oz:
0z:
Overall, the valid value of 7 in Decoupled Entropy Minimization is 0 < S 2/a, where a > 0.
17

---

B More Implementation Details
B.1 Experimental Protocols
B.1.1 Single-Domain Test-Time Adaptation
ImageNet-C 40 is used to construct the single-domain TTA task. ImageNet-C contains 15 different versions of corruptions applied to 50, 000 images from the validation set of ImageNet-IK [88 These corruptions Gaussian noise (Gauss), Shot noise (Shot), Impulse noise (Impul), Defoeus blur (Defcs), Frosted Glass blur (Gls), Motion blur (Mtn), Zoom blur (Zm), Snow (Snw), Frost (Frst), Fog (Fg), Brightness (Brt), Contrast (Cnt), Elastic (Els), Pixelation (Px), and JPEG (Jpg). Each type of corruption consists of 5 levels, with the 5-th level indicating the highest degree of damage. Following previous TTA methods [5: g [ 29], we uniformly use the corruption at the 5-th level as the test datasets. We measure the Top-1 classification accuracy of TTA algorithms on each corruption subset and calculate the average accuracy as the metric for the single-domane TTA task. We employ 3 random seeds, to run experiments on ImageNet-C. These seeds determine the order in which test samples are seen by TTA algorithms and models over time. Since TTA is an online learning task, the arrival order of test samples can significantly impact the performance of TTA algorithms. We report the average accuracy and standard deviation across the random seeds as metrics to measure the performance and robustness.
B.1.2 Continual Test-Time Adaptation
Unlike the single-domain task tested separately on each corruption, continual TTA constructs a continuously changing testing environment using ImageNet-C 40) Specifically, it concatenates the 15 corruptions in the 5-th level in the order of Gaussian noise - Shot noise - Impulse noise + Defocus blur + Frosted Glass blur + Motion blur + Zoom blur - Snow - Frost - Fog + Brightness + Contrast - Elastic + Pixelation + JPEG. This setup requires the TTA algorithms to have the ability adapt aggressively in local static distribution shifts and maintain stability in the face of dynamically changing environment without suffering from catastrophic forgetting. Similarly, we employ the Top-1 classification accuracy as the metric, conduct experiments with 3 random seeds, and report the average accuracy and standard deviation.
B.1.3 Test-Time Prompt Tuning
We mainly use ImageNet-1K (38) and ImageNet variants, including ImageNet-A [41]. ImageNet-V2 42]. ImageNet-R [43), and -keeechh 44] as the test datasets. They used to measure the robustness of TTA algorithms and models against natural distribution shifts. ImageNet-A consists
of 7, 500 natural adversarial samples that are misclassified by the standard ResNet-50 (36 and contains 200 ImageNet categories. ImageNet-V2 includes 10, 000 images and 1, 000 ImageNet categories collected from the source different from ImageNet- ImageNet-R collects 30, 000 artistic renditions from 200 ImageNet categories. ImageNet-Skeet and-white sketches, independently collected from the original ImageNet validation set, covering 1. 000 ImageNet categories. A main research object of test-time prompt tuning is the CLIP models [53] a series foundational vision-language models. uses test samples to update the input learnable prompts of the text encoder in CLIP models. Test-time prompt tuning focuses on performing episodic online learning on an individual test sample, that is, only one sample is used for prompt tuning at time. The updated prompt is applied to make predictions on the current sample again, and then the learnable prompt is reset and waits for the next test sample to arrive. Similar to other TTA tasks, employ the Top-1 classification accuracy as the metric and calculate the average accuracy and standard deviation over 3 random seeds.
is composed of 50, 000 black-
1.1.4 Semi-Supervised Learning
We use CIFAR-10 -10 45
CIFAR-100 45
STL-10 46). EuroSat 477) TisSueMNIST 48 and
Semi-Aves (49 as the datasets for evaluating SSL algorithms. CIFAR- 10 and CIFAR- 100 contain 50, 000 32 x 32 training images and 10, 000 test images, covering 10 and 100 categories respectively. STL-10 consists of 13. 000 96 x 96 RGB images, with 5, 000 labeled images and 100, 000 unlabeled images for training, and 8, 000 for testing. The AuroSAT dataset focuses on land use and cover classification, which includes 10 categories, 16, 200 training images, and 5, 400 test images. The
18

---

TisSUeMNIST dataset is related to biomedical images, mainly containing 8 biomedical scenarios and tissue types, with 165, 466 28 28 training images and 47, 280 test images. The Semi-Aves dataset is a bird dataset for semi-supervised image classification, extracted from the iNaturalist-201 0 dataset, containing 200 bird species, 3, 959 labeled images and 26, 640 unlabeled images for training. and 4, 000 for testing, with an average of 15 ~ 53 labeled images per class. We strictly follow the configuration of various comparison methods and datasets in USE" [56), a unified semi-supervised learning benchmark. We report the average Top-1 classification accuracy and standard deviation from running experiments on random seeds.
B.1.5 Unsupervised Domain Adaptation in Semantic Segmentation
We focus the synthetic-to-real unsupervised domain adaptation task, in which the model is trained on fully annotated synthetic images and validated on real-world data. Several unlabeled real images accessible during training. We use the GTA5 dataset [50] as the source domain data, which contains 24, 966 synthetic frames and pixel-level semantic annotations of 33 categories captured from video game. The Cityscapes dataset [51] is used as the target domain data, and its 2, 975 unlabeled images are utilized for training. The 19 common categories of GTA5 and Cityscapes are selected. inttersetioon-vvr--UUee metric [55], evaluated on 500 validation images. We also report the average mIoU and standard deviation over random seeds.
We measure the segmentation performance with the standard
(mloU)
B.1.6 Reinforcement Learning
We mainly consider RL tasks in discrete environments and utilize the Minigrid environment [52]. Minigrid offers series of grid environments with different layouts and tasks, enabling agents to move within these grids. The agents' actions are typically discrete, such as basic actions like moving up. down, left, and right. We conduct experiments in environments, including DoorKey-5x5. -apply-Random-5x5. Fetch-5x5-N2, FourRooms, GoToooor-5x3. KeyCorridor3R1,, PutNear-6x N2, RedBlue and Unlock. We employ the RI. Baselines3 Zod a training framework for Stable Baselines3 RI. agents, to run our experiments and strictly follow the configurations of RL algorithms implemented therein. We report the mean and standard deviation of the average return over 10 episodes, which is measured on 6 random seeds.
B.2 Method Implementations
We primarily run experiments on one NVIDIA GeForce RTX 4090 GPU with 24 GB of memory.
DEM In Sec. 3.2 we propose DEM and introduce two hyperparameters, 7 and a, to improve the classical EM. When 1.0 and 1.0, DEM is completely equivalent classical EM. To find the most suitable hyperparameter configuration, we construct a search space where T and a range from 0.0 to 2.0 (including 0.0 and 2.0), sampled at intervals of 0.1. Through TPE [39]. fast hyperparameter search algorithm integrated in NN we can search for the optimal hyperparameter configuration (r", a²)). We call this method DEM However, when applying DEM* replace or add to the loss functions of existing TTA and SSL algorithms, it requires a significant amount of additional computational overhead search the suitable hyperparameters. Although we do not recommend this, if necessary, the way of introducing and searching for (r" a²)) in DEM# can also be applied to AdaDEM, which discussed in Appendix D We share our practical experience in tuning and a for quick hyperparameter searching. For DEM, changes in the value of a have greater impact on the original algorithm than changes the value of т. This means that we can fix 7 1.0 and search for the optimal a", then fix a and search for the optimal r", thus obtaining a sub-optimal hyperparameter combination. We also notice that the valid value of T in DEM is < T S 2/o, which is considered to help determine the search scope of r.
AdaDEM. To avoid the complications brought by hyperparameter search, we do not introduce and a in AdaDEM, even though it is feasible, as discussed in Appendix D We uniformly use the
loss function in Eq. (12) to replace or add to existing TTA and SSL algorithms. Specifically, if the original algorithmss employ the Entropy Minimization loss as a part of the objective function, we
httt::///ggccabbb.cm////croros/s www.sssseeeeeeeeeeee learning
http:://ggttabb..commmmcrootttt
19

---

Table 5: Detailed experimental results on single-domain TTA task. T accuracy (%) is reported. We highlight the highest accuracy in bold and the second best as underline. denotes the performance improvement relative to the baselines.
SingleDommin Domain TTA
Methods
Naise Shol
Blur Gls
Weather
Digital
Mcaan
A
Gauss
Impul
Defes
Min
Zm
Snw
Fist
Fg
Brt
Cut
Els
Ps
Jpg
NoAdapt
29.1
31.2
39.3
31.5
24.6
64.5
48.4
34.2
52.5
55.1
38.8-oco
Tent"
54.2
54.8
55.4
56.6
54.0
60.8
34.2
3.8
9.6
70.9
76.5
69.4
59.5
70.0
67.2
53. 33.1ames
+0.0
DEM
51.7
51.9
53.3
54.8
51.2
58.7
50.8
14.7
44.3
69.9
17.0
68.5
57.3
69.0
66.8
56.0am3
+2.9
AdaDEM
57.2
58.2
58.1
58.9
60.2
65.5
63.3
2.0
66.0
73.7
78.0
68.7
69.2
73.3
70.4
+8.4
Tent
51.6
52.0
53.1
52.3
47.6
56.6
46.7
10.2
29.7
67.2
74.2
67.2
50.9
66.5
64.3
52. 12.7
+0.0
DEM*
50.1
50.1
51.7
52.1
46.6
55.6
46.3
30.0
52.2
67.5
15.8
67.0
50.4
66.1
64.2
55. Jani1
+2.4
AdaDEM
57.5
58.5
58.3
60.3
66.3
65.3
63.8
73.6
78.5
67.8
70.8
74.2
71.7
66.2
+13.5
ETA
57.1
57.3
58.3
63.9
61.2
66.6
77.6
70.0
67.2
72.4
70.1
+0.0
DEM*
57.3
58.2
58.3
59.7
60.0
65.4
62.8
68.2
67.0
74.2
79.1
20.7
69.0
73.9
71.4
666.3
+1.2
AdaDEM
58.7
60.8
66.6
65.4
68.0
71.3
74.1
71.6
66.8
EATA
55.2
55.9
56.5
54.0
54.9
61.9
58.8
61.9
60.3
71.6
75.4
68.6
63.0
69.3
66.3
62 2a0
+0.0
DEM
56.2
56.7
56.9
55.6
58.5
63.6
63.1
67.1
64.9
71.3
76.7
65.6
68.2
71.7
69.3
64 4400
+2.2
AdaDEM
36.9
57.9
58.0
57.5
58.8
63.8
63.8
68.0
72.7
77.0
66.6
69.4
72.6
70.0
3
+3.1
DeYO
53.3
54.4
54.3
55.1
55.1
61.9
55.0
64.3
63.1
71.7
77.2
67.2
65.8
71.5
68.4
62 6..61132
+0.0
DEM
56.4
57.2
57.4
59.1
58.8
64.3
59.8
67.3
66.8
74.1
79.3
70.7
67.9
73.8
70.9
Ganen
+3.0
AdaDEM
53.6
54.7
54.7
54.5
56.2
62.1
54.4
63.2
63.5
71.7
77.1
67.5
65.2
71.7
68.9
62.6wn
+0.0
SAR
51.0
51.6
52.4
52.4
49.5
56.2
49.2
21.7
43.9
66.6
73.2
66.5
5144
64.4
63.3
54 2.
+.0
DEM
51.1
51.1
52.8
53.4
48.5
56.4
48.9
54.8
56.7
67.8
75.7
67.4
52.9
66.3
64.0
57.9
+3.7
AdaDEM
56.6
57.8
57.6
58.5
60.0
64.1
62.2
68.4
66.5
73.3
78.9
66.9
69.0
74.1
71.5
63.7anev
+11.5
replace the classical EM with AdaDEM, such as Tent, ETA, EATA, De, SAR, TPT, Ent. Min., VAT, MinEnt, and etc.. Otherwise, we add AdaDEM, calculated using only unlabeled samples, to the original objective function, such as MixMatch, FixMatch, FreeMatch, AdvEnt, and etc.. Since Entropy Maximization applied in RL to encourage exploration, we replace the implementation of conditional entropy in PPO with our AdaDEM while keeping the maximization strategy unchanged.
Test-Time Adaptation Methods. We compare with several TTA methods, including Tent [3). ETA 8. EATA 8. YO [T SAR (29), and TPT 30 We use ResNer50 and ViT-B/16 pre-trained on for test-time prompt wning. We follow the original methods' hyperparameter settings unless otherwise specified.
ImageNet- IK the source models, except for using CLIP-RN50 and
Semi-Supervised Learning Methods. We adopt VAT (3] MixMatch [24], FixMatch [23], and FreeMatch [22] implemented in USB (56) as the base algorithms. Meanwhile, we implement Ent. Min. as a baseline. Ent. Min. calculates the cross-entropy loss for labeled samples and the entropy minimization loss for unlabeled samples based on the work [.. where EM is weighted by 0.3. ViT-S/2-32px is applied for CIFAR and EuroSat, ViT-S/16-2240 2244x for Semi-Aves, ViT-B/16-96p for STL-10, and VIT-T/2-322xx for TISST. We follow the hyperparameter settings of these methods unless otherwise specified.
Unsupervised Domain Adaptation Methods. We mainly follow the work 27 to implement unsupervised domain adaptation in semantic segmentation. We use Deeplab-V2 (54] as the backbone.
We employ MinEnt (27] and AdvEnt (27] as baselines and follow the hyperparameter settings of the original methods unless otherwise specified.
Reinforcement Learning Methods. We adopt Proximal Policy Optimization (PPO) 32 as the baseline. In PPO, Entropy Maximization is used as part of the objective function, which is weighted by 0.001 by default. We replace the calculation method of conditional entropy PPO with our AdaDEM while keeping the intropy-maximizaii strategy unchanged. We follow the hyperparameter setup implemented RI. Baselines3 Zoo unless otherwise specified.
C More Experimental Results
C.1 Experiments on Single-Domain & Continual Test- Adaptation Tasks
For single-domain and continual TTA tasks, we compare with previous state-of-the-art (SOTA) TTA methods, including Tent, ETA, EATA, DeYO, and SAR. We replace the classical EM loss function in these methods with our proposed DEM* and AdaDEM. As shown in Tables 5 and we employ the ViT-B/16 pretrained on ImageNet-IK IK as the source model. Compared with the official Tent which
20

---

Table 6: Detailed experimental results continual TTA task. Top-1 accuracy (%) is reported. We highlight the highest accuracy in bold and the second best as underline. denotes the performance improvement relative to the baselines.
Continual ITA
Time
Methods
Noise Shot
Blur
Weather
Digital
Mcan
Gauss
Impul
Defes
Gis
Min
7m
Snw
Frst
Fg
Bri
Cnt
Els
Px
Jpg
NoAdape
29.1
29.6
31.6
31.2
39.3
31.5
24.6
64.5
48.4
34.2
52.5
55.1
38.8-000
Tent
49.4
54.2
56.3
46.9
56.7
$2.0
55.7
77.2
64.5
52.8
68.1
67.5
58.6mms
+0.0
DEM
50.8
57.1
59.3
56.4
58.7
62.6
62.0
65.9
66.2
72.0
76.6
66.6
66.5
71.4
69.1
1400s
+5.5
AdaDEM
60.1
58.6
64.3
61.7
71.3
63.2ams
Tent
51.6
56.2
57.5
49.2
51.2
57.5
53.2
56.0
61.2
68.2
77.2
64.8
20,9
1.6
1.8
48, 58.5:
+0.0
DEM
53.1
58.7
59.5
56.2
58.2
63.1
63.5
66.1
65.9
72.1
76.4
66.6
67.3
71.7
69.1
64.52m04
16.0
AdaDEM
54.7
59.7
60.4
55.9
58.1
62.6
61.5
65.5
66.4
72.3
77.1
66.5
64,9
71.7
69.5
64.4amsz
15.9
ETA
56.2
60.0
60.6
55.5
58.8
61.9
60.7
65.1
65.8
70.6
78.0
61.9
66.5
72.0
69.4
64.22m04
+0.0
DEM
57.0
60.8
61.0
57.0
63.6
62.3
66.5
78.5
66.9
68.0
72.9
70.7
65. 65.7an84
+1.5
AdaDEM
57.0
61.0
60.8
57.0
60.0
64.3
65.1
67.7
67.4
73.2
77.3
66.8
69.8
73.2
70.6
66.1
+1.9
EATA
55.2
58.9
59.8
56.7
58.8
63.1
61.4
65.9
67.5
72.6
78.6
65.5
66.5
72.2
71.2
64.9
+0.0
DEM
56.3
60.3
60.7
$7.4
60.0
64.2
62.9
67.6
68.6
73.8
78.9
68.5
68
73.3
71.6
66 25005
+1.3
AdaDEM
60.7
60.3
57.2
65.4
69.3
68.5
63.7
70.7
73.6
66.4.0s
DeYO
53.3
56.6
56.5
44.9
53.0
56.2
22.9
59.0
60.2
67.4
76.2
57.5
62.9
69.9
67.1
57 Ganas
+0.0
DEM
56.2
60.3
61.1
57.3
60.0
63.8
56.4
66.2
67.4
73.3
79.0
67.7
67.9
73.4
70.9
65 450
+7.8
AdaDEM
51.5
54.5
55.3
50.7
53.2
57.0
50.9
57.9
60.8
64.5
75.6
60.5
59.2
67.6
66.3
59.0wess
+1.4
SAR
51.0
54.2
55.2
50.5
52.5
56.5
52.8
50.8
41.4
66.7
75.5
64.2
52.9
65.1
65.6
57 07n0s
+0.0
DEM
57.1
58.9
53.6
60.1
56.3
62.3
65.0
67.2
61.1
70.6
70.1
4.03
+5.4
AdaDEM
54.4
58.5
58.6
53.4
57.5
61.2
59.0
62.9
64.0
69.8
77.8
63.1
64.4
71.3
68.8
63.0auss
+6.0
Table 7: Detailed experimental results on test-time prompt tuning task. Top-1 accuracy (%) is
reported. OOD Avg. is average accuracies on ImageNet variants, including -A, -V2., -R., and -S..
Methods
ImageNet-IK ImageNet-A ImageNet-V2.
ImageNet-R.
ImageNNttSS
Average
A 00D Avg.
A
CLIP RN50
Zem-Shot
58.2-0c0
21 8-ndo
$1.4
56.24200
33,4
44.24000
+0.0
40.74000
+0.0
Ensemble
59,8anc
23. 2.000
$2.92om0
60.7anc0
35.5mmm
46.4100
+2.2
43. 13no
+2.4
TPT
60.7
54.62002
58.9200s
35,2amm
47.1a00s
+2.9
43.7am05
+3.0
DEM*
61.3ane9
25 Sanav
55. Dasuso
$9.7aa.1
35.6mus
47.4a004
+3.2
43.92nc
+3.2
AdaDDM
60.7an04
29 200
54.8
58. Bancs
35.4
47.8
+3.6
44.52000
+3.8
CoOp
63.3ancn
23. Lasoo
55 Aacooo
56.6as00
34.7mmm
46.61000
+2.4
42. 42.4anoo
+1.7
TPT (CoOp)
Asnoo
28. 920014
240.1
59. Oeaes
36.3
49. 6.000
+5.4
Geno77
+4.9
AdaDEM
65.6
31.3
58.5
59.32au
36.3
50.2a000
+6.0
46,4
+5.7
CLIP VIT W/16
Zero-Shot
66.7=nco
47.9 9e0dd
60. 92000
74.0a200
46. 15000
+0,0
57. 27.22000
40.0
Ensemble
68.3
49.9 9unoo
61.5 9.oo
77.7.000
48.2iam
61,2100s
+2.1
59.4.no0
+2.2
TPT
54.5.000
A2aus
77.0xaos
48. 48.0an.1
62,4a00s
+3.3
60.7a0e6
+3.5
DEM*
68.9.sss
54. 8aoov
63. Saous
47.9umm
62.5up0s
+3.4
60.8incs
+3.6
AddaDEM
49.42
58 Becus
64.02oot
77.62221
48.620u
63.7aaus
+4.6
62. 2ance
+5.0
CoOp
71 Sancn
49.7.000
64. 2,00o
75. 25.20
48.0amm
61.710m
+2.6
59, Janoo
+2.1
TPT (CoOp)
23. Ganes
57. 9..921
66 9c0ns
77.24000
49.2ssum
64,9 9200s
+5.8
62.8am0
+5.6
+ AdaDEM
73.7
60.3amc
66.92o11
77.9
49.3
65.610m2
+6.5
63,6u004
+6.4
applies SGD with momentum, our implemented Tent utilizes vanilla SGD as the optimizer and achieves better performance on both tasks. However, SGD with momentum has higher potential, and applying DEM* and AdaDEM can bring higher performance gains. ETA, FATA, SAR, and De adopt sample-selection- -aased methods to screen high-confident test samples for model optimization. These methods assign higher weight to high-certainty test samples. This re-weighting approach partially alleviates the reward collapse phenomenon in the classical EM. Nevertheless, there is still room for improvement, which can be observed from the performance gains brought by DEM* EATA and SAR use the Fisher information of in-distribution samples from the source model and the sharpness aware optimizer respectively to stabilize the model's 's optimization in the dynamically changing environment. Our DEM* and AdaDEM can be directly applied to these prior-based methods and achieve significant performance improvements.
C.2 Experiments on Test Prompt Tuning Task
Test-time prompt tuning is an episodic TTA task. It focuses on optimizing the learnable text prompts of the source model using single test sample and making predictions for the current batch with
21

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Table 8: Detailed experimental results (a) on semi-supervised learning. Top-1 aceuracy (%) is reported. We highlight the highest accuracy in bold and the second best as underline. A denotes the performance improvement relative to the baselines.
Mothods
25
CIFAR-10
400
Mean
A
CIFAR-100
25
Mean
A
Semi-Aves 15-53
A
Ent. Min.
93.2.37
97.5un
98.5um
96.4uL
+0.0
58.84za
74.6a1.1
84.4aa
72. 22.6
+0.0
59.9wa7
+0.0
AdaDEM
95.744x
97.4.6
98.5wm
97.2wa3
:0.8
66.2416
76.3ana
85.1aa
75.8.63
+3.2
61.0wn
+1.1
Vat (w/ Eut, Min.)
94.7=ae
98 6an
98.9 Ston
97.4
+0.0
68 Ass.
78.3
85 8es4
77.5
+0.0
61 Dana
+0.0
AdaDEM
97.8ua
98.7am1
98,8ana
98.5
+1.1
70.8a8
79.2
86.3
78.8
+13
61.
+0.8
MixMatch
1-47
98. 1
98. 5.43
62.8-67
73.6w0
84.9
73.8
40.0
62.6wa
+0.0
AdaDEM
98. 6
98 9011
98. Jate
-0.2
64. Tater
74.6
85.7
74. 8:na
+1.0
62.5pu
-0.1
FisMatch
9u.5
98.8 98.8ae1
98. 4ua7
69.9.17
80.6
87 2..2
79.3 29.3am0
+0.0
+0.0
AdaDFRM
98.4-4 4
98.74mm
99.1.0
98 7-43
40.3
71.344
80.1 1
87.0+a
79.5-69
401.
67.9ml
-0.3
FreeMasch
98.7 7-4
99.0-a1
99.1-na
98 9
+0.0
76.5-13
83.6-10
87 3-3
82.6-a1
40.0
67.0wa
+0.0
AdaDEM
98.8.a2
99.0uco
99,1ana
99.0.a
+0.1
77.5.c0
84.3aa
87.7.a
83.
+0.5
67.3wn2
+0.3
Table 9: Detailed experimental results (b) semi-supervised learning. Top-1 accuracy (%) is reported. We highlight the highest accuracy in bold and the second best as underline. A denotes the performance improvement relative to the baselines.
Methods
10
STI-10
Mean
2
FaroSar
Mcan
A
10
TisseMNNIST
50
Mcan
A
Ent. Min.
76.3as
38.5
82.4
66.6.51
86.4kc
76.5as
44.6x37
50.0.s
+0.0
a AdallEM
80.447
89.2ml
84.8-6x
+2.4
76.4.
91.046
83.7-ax
47.2
46.8-19
S1 8-88
49.3-13
+2.0
(w/ Min)
81.6118
85.4
81.9ss
86.624
+0.0
41.6219
45.2as
+0.0
+ AdaDEM
83.8.e3
89.9ws
86.9uo
+1.5
90.3.:
92.1
91.2
+4.6
48.0.13
50.8
49.4
+4.2
MixMatch
76.7au.
K2.9
72.0s5s
86.611
79. 3
+0.0
44.7at3
3:13
48.0ai:
+0.0
+ AdaDEM
80.2am
90.2vm
85 2am.
+2.3
76.4as1
91/4w
83.9uz4
+1.6
46.8417
53.2
50.0wm
+2.0
FixMatch
84.0-36
1315
88.5.13
+0.0
87.7
96.2461
91.941s
+0.0
44.6418
48 8.31
46.7435
+0.0
+ AdaDEM
83 8ans
Iata
87 9anx
96.0cus
97.8ma
96.9mm
+5.0
46.3a17
53.4
49.9t
+3.2
FreeMasch
$6.9.1
92.5
89.7.16
+0.0
95.2
96.4ans
95.8a11
+0.0
42.8A48
48 3us0
45.6ass
+0.0
AAdaDEM
89.6-0
93.7mi
91.6-03
95.6-11
98.94a
46.3wm
49.6-ax
47.9wn
+2.3
the updated parameters. It emphasizes the ability to learn rapidly and adapt extremely well to a single test sample, often employing aggressive optimization strategies such as more complex image augmentation functions, larger learning rates, and more test-time adaptation steps. We adopt CLIP- RN50 and VIT-BV11 as the source models. As shown in the Table. 7 CLIP models' zero-shot prediction and the experimental results based on a prompt ensemble method [33 are employed as baselines. We compare with TPT, widely used method for test-time prompt tuning, and CoOp, a prompt-tuning method for supervised few-shot learning. We modify the loss function of TPT to our proposed DEM* and AdaDEM. We find similar conclusions in other TTA tasks, that is, DEM* and AdaDEM improve the optimization performance by addressing the potential limitations in the classical EM. They enable the source model to adapt in aggressive optimization strategies with large learning rates and multiple steps for TTA.
C.3 Experiments Semi-Supervised Learning
Entropy Minimization is widely used in semi-supervised learning (SSL) tasks. A common prior assumption for SSL is the low-density separation between classes. EM reduces the overlap of models' output probability distribution by decreasing the conditional entropy of the data, causing the density of data points to be lower at the decision boundary. To verify the effectiveness of our proposed AdaDEM in SSL tasks, we conduct experiments on six common SSL benchmarks. The results are presented in Table 8 and | For each benchmark, we set different numbers of available labeled samples N; per class for training Specifically, we set N/ = 4/25/ 400 for CIFAR- 10, N₁ 2/4/25 for CIFAR-100. Ni = 4/10 for STL-100 N₁ = 2/4 for EuroSat, N₂ 10/50 for TissueMNIST, and N; 15 ~ 53 for Semi-Aves. We report the average top-1 classification accuracy under different numbers of labeled samples. We replace the classical EM in Ent. Min. [ and VAT (w/ Ent. Min.) 3 with AdaDEM, and introduce AdaDEM as part of the loss functions in MixMatch (24), FixMatch 23 and FreeMatch (22) which do not use Entropy Minimization. Through experiments, we demonstrate that AdaDEM outperforms the classical FM. Meanwhile, incorporating AdaDEM into existing SSL methods can effectively improve the performance of the original algorithms.
22

---

Table 10: Detail experimental results unsupervised domain adaptation in semantic segmentation. The standard mloU is reported. We highlight the highest accuracy in bold and the second best as underline. A denotes the performance improvement relative to the baselines.
Sidewalt
boilding
mbhle
rrad
bike
Methods
mloU
MinEnt
84.4
24.3
77.4
22.1
21.3
26.5
33.1
19.2
82.7
30.9
764
58.0
26.6
75.4
31A
38.0
2.0
25.5
35.5
41.6s94
+0.0
AdaaDEM
85.5
22.6
78.8
21A
24.6
27.2
34,4
19.6
82.5
28.8
58.5
28.5
80.1
34.4
41.2
1.8
24.5
38.1
42.7-am
+1.
AdvEnt
89.1
25.4
8144
26.7
25.6
29
32.8
22.5
83.8
35.1
77.5
$7.9
28.8
84.3
30.3
41.8
0.9
29.3
26.0
43.6.a.1
+0.0
DEM
89.0
32.2
80.9
28.3
35.6
58.8
32.6
44.6-33
+1.0
AdaDEM
88.9
30.4
81.2
27.9
27.0
28.7
34.0
21.0
84.1
35.4
78.6
39.1
29.8
84.4
36.4
42.6
166
28.0
33.4
44.9
+1.3
Original Image
Pixel Prediction
Pixel Entropy
MinEnt
+ AdaDEM
AdvEnt
+ AdaDEM
Figure 7: Visualization of the pixel prediction results and pixel entropy of baselines and AdaDEM.
C.4 Experiments on Unsupervised Domain Adaptation
Another role of Entropy Minimization is bridge the domain gap between the source training distribution and the target test distribution. The classical EM pushes the decision boundaries of DNNs towards the low-density regions of the target distribution. On the other hand, conditional entropy is also widely validated calibration tool for DNNs, which measures the prediction error and distribution shifts some extent. Vu et al. vu2019advent demonstrate that the semantic segmentation model trained in the source distribution has high prediction entropy and blurred boundaries for objects images. We verify the effectiveness DEM and AdaDEM in the semantic segmentation task, and use MinEnt and AdvEnt 27 as baselines for comparison. The experimental results are shown in Table TO.
also visualize the segmentation results of the baselines and AdaDEM, as shown in Fig. The experimental results show that compared with the classical EM, AdaDEM can effectively improve the accuracy of semantic segmentation, especially for tail classes, and avoid DNNs' tendency predict dominant classes. At the same time, AdaDEM can effectively reduce the prediction entropy of DNNs for image pixels, resulting in clearer object boundaries and thus bridging the domain gap between the source and target distributions.
C.5 Experiments Reinforcement Learning
For reinforcement learning, Entropy Minimization could increase the certainty of policies, thereby limiting the exploration of agents. Therefore, reinforcement learning methods commonly use the
23

---

MiniGrid-Unlln
MiniGrri Min&Grid.Doarkey -xc5
MiniGrid Empty Random 5xx5
0.9
Retur F
Average Retum
0.960
0.905
0.4
0.3
0.
0.945
0.2
0.3
PPO + EM
PPO EM
PPO AdaDEM
0.2
PPO AdwDEM
0.915
AduDEN
0.1
0.8
0.2
0.4
0.6 Timestops
0.8
1.
Timesteps
Timesteps
MiniGrd-fedBlueDDors-sss
MiniGnd--Fechh-5S----
MiniGr MinGGr--eyyorrddo55531
PP0 FM
Average Reeurn
0.8
Average Return
0.8-
AdaDEM
Average Return
0.8
0.60
0.7
0.6
0.4
0.5-
0.2
PPO
0.2
PPO EM
PPO
AdoDEM
PPO DEM
0.00 0.25 0.50
0.75
1.00 135
1.50
1.55
2,50
Timesteps
X10
Timesteps
X10
Timesteps
X12
MiniGrid- GoToDoor-55 5x5
MinniriddPPuNNea-Bae
MiniGrid- Fooiiddoou
0.200
0.6
Average Return
0.150
PPD MMDEM
Average Return
0.5
sat
0.125
0.4
0.5
&
0.100
0.3
0.4
0.075
0.2
0.050
0.1
0.1
0.00
0.25 0.50 0,75 1.00
Timestops
PPO + AdaDEM
1.50 1.75 2,00
0.30
150 LiS Timesteps
1.00
xtpt
0.0
0.00
0.25 0.0 0.75 1.00 1 1:50 1.75 2:00 Timesteps
PPO AdaDEM
Figure 8: The curves of average return with respect to timesteps in 9 environments of Minigrid for Reinforcement Learning.
Entropy Maximization strategy to prevent premature convergence and encourage exploration. We aim prove that AdaDEM is not limited to the Entropy Minimization strategy and is also applicable the Entropy Maximization strategy in reinforcement learning. We adopt Proximal Policy Optimization (PPO) as the baseline and compare the experimental results of adding the classical EM and our AdaDEM to PPO, as shown in Fig, [8] We keep the Entropy Maximization strategy unchanged. The experimental results demonstrate that replacing the classical EM with AdaDEM can effectively improve or maintain a comparable performance of reinforcement learning methods.
D Additional Discussions
D.1 Effect of Temperature in DEM*
As stated in Sec. 3.2 T controls the contribution of samples with different certainties to model optimization by reshaping the reward curve. As shown in Fig. 3 (left), the larger the value of is, the higher the reward given to high-confident samples than to low-confident ones, which makes the model more inclined to learn high-confident samples. On the contrary, the lower the value of T is, the more the model tends to learn low-confident samples. When DNNs classify all samples accurately, a lower value of T promotes the model to rapidly improve the prediction probability of samples and accelerate the model optimization. However, poorly calibrated DNNs often have very low accuracy on out-of-distribution samples. In this case, low value of leads to incorrect model optimization. Therefore, setting to a value slightly larger than 1.0 enables DNNs to stably and effectively learn from data distributions with low certainty (distributions that deviate significantly from the source training distribution). Additionally, as shown in Fig. (center), we find that the best value of is positively correlated with the average prediction probability of DNNs for samples in a specific distribution. That is, when the model has higher average prediction accuracy for a certain data distribution, increasing the value of maximizes the efficiency of DNNs in learning this data distribution within limited optimization time.
24

---

Table 11: Optimal hyperparameters a²) of DEM" applied to different methods on various tasks, and the resulting performance improvement brought by DEM*
Methods
Toal (RN)
DEM
Teut' (ViT) + +DMM
ETA
DEM
EATA
DEM
DeYO
+ +DMM
SAR
DEM
Single Domain Test Time Adaptation
1.0
1.1
1.0
1.3
1.0
1.2
1.0
1.3
1,0
1.1
1,0
1.0
1.0
1.8
1.2
1.0
1.0
1.6
Acc.
40.0,am
41.8ane
53,1
56.0wu
65.1 +0.0
66.3
62.2au4 +0.0
64.4003
62.6x002
65 6..u
54,2
57.9ka14
+0.0
+0.0
+2.9
+1.2
+0.0
+3.0
+3.7
Continual Test-Time Adaptation
T'
1.0
1.4
1.0
0.8
1,0
1.1
1.0
1.3
1.0
0.9
1.0
1,0
1.0
1.4
1.0
1.0
1.0
1.2
1.0
1.2
1.0
1.4
1.0
1.8
Acc.
31
39.0suu +7.8
58 Gana
64, Lauos
64.2:aM4 +0.0
65.7am
64.9tus
6602 +1.3
57.6
45-4
57.0cmms
62.4ca0
A
+0.0
+0.0
+5.5
+1.
+0.0
+0.0
+7.8
+0.0
+5.4
Methods
CLIP-RN50
TPT
DEM
CLI--VT--/16
TVT
+ DEM*
Test: Time Prompi Tuning
1.0
1.0
0.8
o²
1.0
0.6
1.0
1.0
Acc. 44.2-am
47.1-4
47.4.am
--.1
62.4
62.5wam
A
+0.0
+2,9
+3.2
+0.0
+3.3
+3.4
D.2 Effect of Weight a in DEM*
In GMC, suppresses the overfitting of DNNs over-confident samples by scaling the penalty on logits. As shown in Fig. 4 (left), increasing the value of a causes DNNs to impose greater penalties on samples with prediction probabilities approaching 1.0. For poorly calibrated DNNs, increasing the value of a reduces the disruption of over-confident samples to DNNs optimization. For data distributions with high prediction accuracy. decreasing the value of a promotes rapid model optimization and enhances the learning efficiency of nigh-confident samples. Therefore, lower value of a is suitable for the aggressive optimization strategy, while a higher value of suitable for alleviating the catastrophic forgetting problem caused by DNNs overfitting. The ablation study in Fig. 4 (center) validates this point.
D.3 Optimal Hyperparameters (r".a") for DEM
In Table 11 obtain the optimal hyperparameters (r", for different methods on various tasks through hyperparameter search by the fast TPE algorithm. For test-time prompt tuning, we employ well-calibrated CLIP models as the source models. It is suitable for leveraging an aggressive optimization strategy to enable CLIP models to quickly learn the target data distribution. Therefore, lower value of 7 or a optimal. We mainly use VIT-B/16 for TTA. Hence, for single-domain TTA, a lower and higher 7 are suitable for the rapid optimization of ViT-B/16 and enhance the contribution of high- high-confident samples to model optimization. For continual TTA, higher values of t and a prevent ViT-B/16 from overfitting to Single static distribution and improve the model's s adaptability in dynamically changing environments and its ability to resist catastrophic forgetting.
D.4 Ablation Studies of DEM and AdaDEM
we report the detailed ablation experiment results of DEM and AdaDEM, respectively. Different from the official Tent (5), which uses SGD with Momentum as the optimizer, we employ vanilla SGD to verify the learning efficiency of TTA algorithms on real-time incoming test samples, while avoiding the influence of historical gradients on the learning of current samples. This implementation leads to better accuracies than the official Tent. In some cases, applying CADF alone improves the performance of DNNs on the single-domain TTA task. However, CADF suffers from catastrophic forgetting in dynamically changing environments. Therefore, DEM* searches the optimal hyperparameters (T" aa)) to stabilize the model optimization process and enhance the model's learning efficiency for the target distribution. AdaDEM DaaDM-ooomm solves the reward collapse phenomenon in the classical EM by introducing 8, the Ll-norm of the reward brought from CADF. It improves the performance of Tent. Meanwhile, AdaDEM-MEC uses the estimated prior label distribution in the marginal entropy calibrator to alleviate the model's overfitting to dominant classes, resulting in performance improvement on continual tasks. AdaDEM comprehensively utilizes AdaDEM- Norm and AdaDEM-MER as alternatives to the classical EM and DEM. It avoids the additional
In Tables 12 and 13
25

---

Table 12: Detailed ablation studies of DEM and AdaDEM in the single-domain TTA task. We highlight the highest accuracy in bold and the second best as underline. denotes the performance improvement relative to the baselines.
Single-Domain TTA
Methods
Noise Shot
Blur Gls
Weather Frst
Digital
Mean
A
Gauss
Impul
Defes
Mtn
Zm
Snw
Fg
Brt
Cot
Els
Px
Jpg
ResNer50
NoAdapt
15.3
15.9
15.8
15.1
15.2
26.4
38.9
34.3
33.1
48.1
65.2
16.8
44.1
48.8
39.6
31.5-e
EM (Teni)
24.8
26.3
25.8
24.4
23.7
43.9
39,4
55.6
66.8
26.7
52.1
56.5
49.7
40.0ac
CADF
28.3
29.2
29.3
27.2
26.5
40.6
48.5
46.8
40.9
56.7
66.0
22.8
54.1
57.6
51.5
41,7-am
+1.7
28.3
29.2
27.1
26.5
46.8
41.0
56.8
66.0
23.2
54.2
57.6
51.5
AdaDEM-Norm
30.5
32.7
32.0
29.6
28.5
44.0
50.7
49.2
41.0
58.6
67.6
22.1
56.4
59.7
53.6
43.7at.0
+3.7
AdaDEM-MEC
31.7
34,2
33.1
30.3
29.6
44.8
50.0
49,5
41,8
57.9
66.0
30.0
55.6
58.6
53.0
44.4
AdaDEM
31.8
33.9
33.0
30.6
30.0
44.0
50.3
49.3
42.8
58.0
66.2
34.2
55.7
58.7
53.0
44.8-c
+4.8
VIT-B/16
NoAdapt
29.1
29.6
31.6
31.2
25.1
39.3
31.5
24.6
30.2
54.3
64.5
48.4
34.2
52.5
55.1
38.8st
EM (Tent)
54.2
54.8
55.4
56.6
54.0
60.8
34.2
3.8
9.6
70.9
76.5
69.4
59.5
70.0
67.2
+0.0
CADF
50.1
50.6
51.6
49.0
44.3
58.0
46.
6.6
15.0
64.2
72.8
65.3
33.0
63.4
61.6
48.6wc
4.5
DEM
51.7
51.9
53.3
54.8
51.2
58.7
50.8
14.7
44.3
69.9
27.0
68.5
57.3
69.0
66.8
56.0ae32
+2.9
AdaDEM-Nomn
54.3
55.2
55.6
56.5
54.2
61.1
41.0
3.7
13.4
71.2
76.4
69.3
60.5
70.3
67.4
54,0a245
+0.9
AdaDEM-MEC
56.0
57.1
57.0
41.4
59.1
64.1
61.2
2.7
62.5
73.4
27.0
50.2
67.9
71.9
69.7
+5.0
AdaDEM
57.2
58.2
58.1
58.9
60.2
65.5
63.3
2.0
66.0
73.7
78.0
68.7
69.2
73,3
70,4
61.5pt
+8.4
Table 13: Detailed ablation studies of DEM and AdaDEM in the continual TTA task. We highlight the highest accuracy in bold and the second best as underline. A denotes the performance improvement relative to the baselines.
Continual TIA
Time
Metbods
Weather
Digital
Mean
Gauss
Shot
Impul
Defes
Gls
Min
Za
Snw
Frst
Fg
Brt
Chl
Els
Px
Jps
ResNer50
NoAdaat
15.3
15.9
15.8
15.11
15.2
26.4
38.9
34.3
33.1
65.2
16.8
44.1
48.8
39.6
31.54am
EM (Teni)
24.8
32,9
32.7
24.2
25.9
30.2
37.7
30,2
28.3
36.4
49.2
17.7
32.5
35.3
30.0
31,2ac11
+0.0
CADF
18.1
24.2
26.6
22.6
36.9
36.4
47.2
59.9
26.3
49.3
43.8
+4.9
DEM
18.7
26.2
28.9
23.9
26.9
36.5
45.6
39.8
39.5
49.8
62.8
32.8
49.6
54.0
49.3
39.0smm:
+7.8
AdaDEM-Norm
18.9
25.2
22.6
38.0
37.9
28.4
49.3
52.8
47.4
37.5asm
AdaDEM-MEC
20.5
28.0
29.5 5
23.5
27.0
34.2
43.8
36.9
37.0
47.2
60.5
27.8
48.5
51.5
46.5
37.5
+6.3
AdaDEM
20.5
28.3
29.9
23.8
27.5
34.5
44.0
36.9
36.9
47.4
60.3
28.7
48.6
51.5
46.7
37.7-4m
B/T
NoAdapt
29.1
29.6
31.6
31.2
25.1
39.3
31.5
24.6
30.2
54.3
64.5
48.4
34.2
52.5
55.1
38.
EM (Ten))
49.4
54.2
56.3
46.9
48.1
56.7
52.0
55.7
61.0
68.2
77.2
64.5
52.8
68.1
67.5
58.6atee
+0.0
CADF
50.1
54.8
56.8
53.2
57.4
61.9
60.1
64.9
66.4
71.5
77.0
66.7
64.0
70.7
68.9
63.0acm +4.4
DEM*
50.8
57.1
54.3
56.4
58.7
62.6
62,0
65.9
66.2
72,0
76.6
66.6
66.5
71.4
69.1
64.
+5.5
AdaDEM-Norm
48.9
54.4
56.5
47.7
49.5
57.4
53.1
58.1
62.0
68.5
77.3
65.1
55.2
68.4
67.9
59,3
+0.7
AdaDEM-MEC
54.4
58.0
58.7
52.7
55.5
60.6
57.2
62.4
64.1
70.1
77.7
65,2
59.9
70.7
68.8
62.4-e
+3.8
AdaDEM
54.9
59.2
60.1
53.3
56.9
58.6
63.3
65.3
70.9
27.6
64.3
61.7
21.3
69.3
63.2-4
computational overhead caused by hyperparameter search and achieves the best performance on both single-domain and continual tasks.
D.5 Sensitivity of AdaDEM to Optimizers and Learning Rates
Setting different learning rates for the optimizer leads to different learning efficiencies the model on unlabeled samples. A larger learning rate corresponds to an aggressive optimization strategy, which can effectively promote the model's learning of limited online samples, improving the speed at which DNNs adapt to the target distributions. But it may also cause the model to easily overfit to a single data distribution, resulting in the catastrophic forgetting problem. A smaller learning rate adopts more conservative optimization approach, accelerating the model's adaptation to the target distribution while ensuring that the performance of DNNs does not degrade significantly. As shown in Fig, 9 we compare the sensitivities of the classical EM, AdaDEM-Norm AdaDEM-MEC and AdaDEM to different learning rates. Existing TTA algorithms, especially Tent, which simply uses the classical EM to optimize DNNs, are susceptible the settings of the optimizer and learning rate, making it necessary to carefully adjust the optimization strategy. However, we demonstrate that AdaDEM normalizes the contributions of samples with different certainties to model optimization and introduces the estimated prior label distribution as regularizer. It alleviates the model's 's tendency to optimize low-confident samples and the overwhelming influence of dominate classes. Therefore, the sensitivity of AdaDEM to the learning rate is significantly lower than that of the classical E.M.
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Single-Domain TTA
Continual
65
NoAdapt
NoAdapt
(Tent]
AdaDEM.Nomm
Ado Der.NNom
Top-1 Accuracy (%%
AdeDEM-MEC
Top-1 Accuracy (%)
AdaDEM
Ads
55
50
40
10-4
10
10-4
10
10
10a
Learning Rate
Learning Rate
Figure 9: Comparison the sensitivity to the learning rate among the classical EM, AdaDEM-Nome AdaDEM-MEC, and AdaDEM for single-domain and continual TTA tasks.
Table Ablation study on the role We compare two methods for computing: 8 derived from the reward brought from CADF, and 6, derived from the reward brought from overall conditional entropy. AdaDEM and AdaDEM-V denote the approaches using and 6, as reweighting factors.
Single-Domain TTA
Continual TTA
LR
LR
Classical EM AdaDEM
AdaDEM-V
Classical EM
AdaDEM
AdaDEM-V
0.0001
40.5
39.7
39.9
0.0001
44.7
42.6
42.7
0.00025
42.2
40.8
41.0
0.00025
48.2
43.8
31.2
0.0005
44.1
0.0005
$1.0
26.0
0.001
46.5
44.1
0.001
53.8
51.7
10.8
0.0025
49.8
47.6
29.2
0.0025
57.0
55.7
3.8
0.005
51.5
50.9
14.6
0.005
58.6
58.0
1.5
0,01
52.5
53.8
6.3
0.01
49.1
60.1
0.7
0.025
53.1
57.2
2.2
0,025
6.2
62.5
0,3
0.05
48.3
59.5
1.2
0.05
2.3
63.2
0.2
0.1
41.9
61.5
0.7
0.1
1.1
41.6
0.2
D.6
Ablation Study on the Role of AdaDEM
In Sec. 3.3 the parameter is introduced to reweight the conditional entropy Hzz) in AdaDEM.
d is defied as the Ll-norm of the reward brought from CADF (i.e., 8 which quantifies the extent of changes in the model's s output logits before and after EM optimization. Another common reweighting method uses the L.I-norm of the reward from the overall loss function as the reweighting factor, specifically employing the partial derivative of the conditional entropy (z) with respect to the logits z, i.e., 5, for constructing AdaDEM using 5 and 6,,, respectively. Experimental results in Table [4] demonstrate that employing 5, leads to degraded performance and robustness, whereas 8 must be computed from CADF gradients. Since d, incorporates gradients from the overall loss function, it risks introducing penalties from components such as GMC or MEC into the normalization process. This may undermine the targeted mitigation of reward collapse by diluting CADF-specific rewards and altering optimization dynamics, such as misaligning sample contributions or impairing robustness in noisy dynamic environments.
0T 22, 0) 9)/Oz|] 1).
ÖH (z) æ. 0)/2zll1. We compare these two methods
D.7 Impact of Adding to Scale MEC in AdaDEM
Similar to introducing the GMC of DEM to scale the penalty logits as mentioned in Sec. 3.2 we can also introduce an in AdaDEM to scale the effect of MEC. We conduct an ablation study on adding a to AdaDEM single and continual TTA tasks, as shown in Fig. T0 The results indicate that AdaDEM-Nomn is less sensitive to the value of than AdaDEM-MEC. Meanwhile, changing the value of also improves performance AdaDEM, additional computational overhead required to determine appropriate value for
27

---

Single-Domain Test-Time Adaptation
Accuracy
55
NoAdapt
50
AdaDEM-N
AdaDEM, MEC
The value a
Continual Test-Time Adaptation
60
Accuracy
50
40-
30
NoAdapt
AdaDEM-Norm
AdaDEM
0.0
0.2
0.3
0.4
0.5
0.7
0.8
1.0
111
1.2
13
2.0 1.4
15
16
1.7
1.8
19
value
Figure 10: Ablation study of adding different a to scale MEC AdaDEM.
Single Damain TTA
Continual TTA
66
66
AdDDEM
AdaDEM
AdaaEMM-Nom
AdaDEMM-Nom
Accuracy (%6
Top.1 Accuracy (96
tee
52
50
L1-norm
L2-norm Type
Linf-norm
50
L1-norm
L2-norm
Linf-norm
Norm
Figure 11: Ablation study of different Ln-norm types to normalize the reward brought from CADE.
D.8
Impact of Different Norm Types on AdaDEM
we default to using the Li-norm of the reward brought from CADF, i.e., 5 ||āT/0z|l. to normalize the loss of an individual sample. In Fig. I1 we compare applying Ll-norm, L2-norm, and Loo-norm as criteria for measuring the reward of CADF on logits, which are then used to normalize the loss function. The results show that the impact of different norm types on AdaDEM-Nome compared to the other two methods. The reason is that the L.-norrm directly measures the Manhattan distance of the logits predicted by the model for sample before and after EM optimization.
As stated in Sec. 3.3
negligible. However, the I.1-norm achieves better performance for AdaDEM
D.9 Impact of Different on AdaDEM
As stated in Sec. 3.3 we use the exponential moving average of the marginal entropy predicted by the model for each class as the estimated prior label distribution. Further, we can change the momentum
the moving average equation, i.e., P = x) p + т Р&. to alter the follow-up ability of the estimated label distribution to the model's prediction probability. Specifically, we set क = 1.0, 0.1, and 0.01 for the ablation study, as shown in Fig. 12 When π 1.0, MEC uses marginal entropy estimated for the samples in the current batch the regularizer. When approaches 0.0, the update of the estimated label distribution becomes slow. The experimental results show that in the single-domain task, 0.1 is suitable for AdaDEM and AdaDEM-MEC. However, for
28

---

66-
AdaDEM AGADEMMEC
AdaDEM
64
AdADFM-MEC
Top Arcuracy (%)
62-
Top 1 Accuracy 196
62-
60
60
58
0.01
0.1
1.0
0.01
0.1
1.0
Momentum
Momentum
Figure 12: Ablation study of different π to estimate prior label distribution for MEC.
Table 15: Imbalance study on Classical EM and AdaDEM. Top-1 classification accuracy (%) is reported. / denotes the sample ratio between the most and least populous classes. A represents the accuracy improvement of AdaDEM over Classical EM.
CIFAR-I0-LT
CIFAR-I00-H
Methods
Methods
100
500
1000
10
100
500
Classical
82.1
Classical EM
58.6
AdaDEM
91.4 +6.0
88.6
AdaDEM A
66.2
64.4
A
+6.8
+6.5
+6,2
+5.8
EuroSat-LT
TisSucMIIST-LI
Methods
Methods
= 100
= 500
1000
100
p 500
1000
Classical EM
67.1
65.7
63.4
Classical EM
47.8
47.4
46.8
AdaDEM
76.6
74.8
73.6
AdaDEM A
49.6
48.7
48.4
A
+9.5
+9.1
+10.2
+1.8
+1.3
+1.6
continual tasks, a smaller value of i improves the performance the algorithms for both AdaDEM and AdaDEM-MEC.
D.10
Ablation Study on Imbalanced Benchmarks
Table 13 presents experimental results CIFAR-IO-LT, CIFAR-100-LT, EuroSat-LT, and TNsSTE. In addition to CIFAR-LT, we employ EuroSat-LT (focusing land use and land cover classification) and TissueMNIST-LL (targeting biomedical scenarios and tissue type classification). We vary the configuration of p (the sample ratio between the most and least populous classes). larger / indicates greater skewness in class sample sizes and higher imbalance severity. Results in Table along with Fig. 5 in the manuscript, demonstrate that AdaDEM more effectively mitigates easy-class bias and further improves accuracy compared the classical EM across tasks with varying degrees of imbalance.
E
Broader Impacts
This paper aims to improve the widely-used classical Entropy Minimization method in the machine learning community. Our proposed method achieves eredible performance across multiple machine learning tasks, including semi-supervised and unsupervised learning, domain adaptation, and rein- forcement learning, thereby directly facilitating real-world applications of self- supervueee entropy minimization strategies. Our work involves no human subjects and complies with legal requirements, with anticipated harmful consequences. Through this study, we seek enhance research and societal awareness regarding imperfectly supervised machine learning and deep learning systems.
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Reviews:
[{"review_id": "L36sS4fiAg", "reviewer": "Reviewer_TNtq", "summary": "The paper proposes a novel way to decouple entropy minimization, which is beneficial to semi-supervised/unsupervised learning, domain adaptation, and other exploration-related problems in machine learning. The paper proposes Decoupled Entropy Minimization* (DEM*), which reformulates the entropy minimization into Cluster Aggregation Driving Factor (CADF) and Gradient Mitigation Calibrator (GMC). Then, the paper proposes an improvement over DEM, replacing GMC with Marginal Entropy Calibrator (MEC), naming the method Adaptive Decoupled Entropy Minimization (AdaDEM). The paper then performs various evaluations to validate the methods.", "questions": "1. How does the proposed method compare with other ways of uncertainty quantification and minimization, such as evidential learning and Bayesian approaches, in principle or in practice?\n\n2. What is the significance of $\\mathbf{p}_i$? Does it break the normalization of predictive probabilities?\n\n3. Does the subset calibration for DEM* mean that it is not a fair comparison in those tasks?", "limitations": "Yes", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n1. The paper is clearly written and all the results in figures and tables are clearly presented.\n\n2. The paper provides a fresh perspective for uncertainty quantification and entropy minimization. The proposed method is also applicable in various settings, providing a general solution for multiple scenarios. The experiments covering different tasks also support the proposed method well.\n\nWeaknesses: \n\n1. There’s some concern regarding the motivation for the proposed method. The reformulation (section 3.1) converts the entropy minimization problem into expressions concerning the logits. However, this could be overcomplicating the problem. While the standard entropy measure is a first-order uncertainty quantification (w.r.t. predicted probabilities), there are existing uncertainty quantification methods that analyze and study higher-order uncertainty measures.  For example, in evidential deep learning (Sensoy, Murat, Lance Kaplan, and Melih Kandemir. \"Evidential deep learning to quantify classification uncertainty.\" Advances in neural information processing systems 31 (2018).). There are also many works that introduce Bayesian flavors into deep learning to achieve the goal of uncertainty decomposition. It is unclear why we should complicate a first-order uncertainty measure instead of exploring better measures.\n\n2. Some experimental results are not very convincing. In the TTA and semi-supervised tasks, the advantage of the proposed method is not very obvious.", "was_revised": true, "final_justification": "I appreciate the rebuttal and revisions to the paper made by the authors. However, I am still not convinced that reformulating entropy minimization is necessary, especially since the improvement in TTA tasks is marginal. The proposed method evaluates entropy from the logits perspective, while I believe that one could directly study the problem from the logits side without getting into probability-based entropy metrics.", "rebuttal": "We sincerely appreciate the time and effort put into reviewing our paper and providing valuable feedback. We would like to address your questions below.\n\n> **R1Q1**: What's the motivation for reformulating Entropy Minimization (EM) in terms of logits?\n\nThe reformulation of EM into logit-based terms (CADF and GMC) is motivated by the need to dissect EM's internal mechanisms and address its inherent limitations, such as reward collapse and easy-class bias, which hinder performance in noisy and dynamic environments. While higher-order uncertainty methods like evidential deep learning offer valuable alternatives, our approach specifically targets enhancing EM's efficacy within its established paradigm—revealing coupled formulation issues and enabling improvements (AdaDEM) that outperform even the upper-bound EM variant (DEM*) across diverse tasks, without requiring complex new uncertainty frameworks or hyperparameter tuning. This provides a focused, practical advancement for EM-dependent applications.\n\n> **R1Q2**: Some experimental results are not convincing.\n\nThe experimental results demonstrate clear advantages of AdaDEM. In TTA tasks (Table 2), AdaDEM consistently outperforms baselines (e.g., +8.4% over Tent for ViT-B/16 in single-domain adaptation). For semi-supervised learning (Table 3), AdaDEM improves multiple methods (e.g., +5.0% for VAT on EuroSat). These gains are statistically significant (reported with standard deviations) and observed across diverse benchmarks including class-imbalanced settings (Fig. 5). The improvements stem from resolving fundamental limitations (reward collapse and easy-class bias) as validated in ablation studies (Table 1).\n\n> **R1Q3**: How does the proposed method compare with other ways of uncertainty quantification and minimization?\n\nAdaDEM fundamentally differs from Bayesian approaches (which rely on probabilistic priors and sampling, incurring high computational costs) and evidential learning (which explicitly models evidence or uncertainty distributions) by decoupling entropy minimization into CADF and GMC components for self-supervised optimization. AdaDEM adaptively normalizes rewards and replaces GMC with a marginal entropy calibrator to avoid hyperparameter tuning, enabling superior efficiency and robustness in noisy, dynamic environments (e.g., test-time adaptation and semi-supervised learning) without requiring explicit uncertainty modeling or external supervision, as empirically validated across diverse benchmarks.\n\n> **R1Q4**: What is the significance of $\\mathbf{p}_i$? Does it break the normalization of predictive probabilities?\n\n$\\mathbf{p}_i$ in Eq. 8 represents the predicted probability for class $i$ from the model's softmax output, which is a constant excluded from the computation graph. It does not break normalization, as $\\mathbf{p}_i$ is inherently normalized via softmax ($\\sum \\mathbf{p}_i = 1$), and its use in AdaDEM's loss function (e.g., Eq. 12) for optimization preserves this normalization, as the loss modifies gradients without altering the probabilistic output directly.\n\n> **R1Q5**: Does the subset calibration for DEM* mean that it is not a fair comparison in those tasks?\n\nThe subset calibration for DEM* uses ∼20% of test data with ground-truth labels to search for optimal hyperparameters. However, DEM* is explicitly framed as an upper-bound exploration of classical EM's potential, while AdaDEM serves as the fair, hyperparameter-free method in evaluations."}, {"review_id": "253OIu7xYj", "reviewer": "Reviewer_LD3y", "summary": "The authors revisit classical entropy minimization (EM) and show that its conditional-entropy objective can be rewritten as two antagonistic terms: a cluster aggregation driving factor (CADF), which amplifies high-confidence predictions, and a gradient mitigation calibrator (GMC), which penalizes them . They argue that the tight coupling of these terms causes reward collapse (vanishing gradients for very confident samples) and **easy-class bias** (over-assignment to dominant classes) .\n\nTo expose the upper performance bound of standard EM they introduce DEM, which separately scales CADF with a temperature τ and GMC with a weight α, choosing (τ\\*, α\\*) through hyper-parameter search . They then propose AdaDEM, an adaptive variant that (i) normalises CADF by the L1 norm of its own gradient and (ii) replaces GMC with a marginal entropy calibrator (MEC) that tracks class-wise probabilities online, eliminating the need for label priors or tuning . Across test-time adaptation, semi-supervised learning, unsupervised domain adaptation and reinforcement learning, AdaDEM consistently outperforms both classical EM and DEM\\*, while remaining hyper-parameter free .", "questions": "See the weaknesses", "limitations": "See the weaknesses", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# Strengths\n\n* **Clear decomposition** of entropy into two interpretable forces (CADF and GMC).\n* **Practical algorithm**: AdaDEM matches or exceeds DEM\\* without any tuning.\n* **Broad empirical evidence** across four benchmarks and many architectures .\n\n\n# Weaknesses\n\n1.  Beyond the gradient expressions in Eq. 7, there is no convergence or stability analysis. For instance, how the adaptive normalisation in AdaDEM affects optimisation dynamics is unexplored; the paper only provides empirical plots. Also, the claim that DEM\\* gives an upper bound on EM performance is heuristic; it relies on searching τ and α on held-out labelled data, which breaks the unsupervised assumption and may over-fit that validation split.\n\n2.  DEM\\* demands a two-dimensional hyper-parameter search (τ, α); even the authors note “significant additional computational overhead” when integrating it into existing pipelines . Although AdaDEM removes these tunables, it still needs per-batch gradient-norm computation and a running class-wise EMA, increasing memory bandwidth compared with vanilla EM.\n\n3.  Many headline gains come from tasks that already use EM (e.g., Tent); for methods that do not depend on EM such as MixMatch or FixMatch, improvements are marginal and sometimes negative . Reported gains on semi-supervised benchmarks often fall within one standard deviation; statistical significance is not tested. Reinforcement-learning results are shown only on MiniGrid; gains are small and confidence intervals large .\n\n\n5.  AdaDEM still rewards CADF after normalisation; in highly skewed data the L1 scaling may be insufficient to avoid collapse toward a few confident classes. No synthetic imbalance study is provided beyond standard long-tail CIFAR-LT.\n\n6.  The paper focuses purely on objective-level fixes; it concedes that many problems of self-supervised learning persist and must be handled by separate mechanisms . No discussion is given on how AdaDEM interacts with modern calibration techniques such as temperature scaling or focal loss; compatibility with these is uncertain.", "was_revised": true, "final_justification": "I believe this paper in general has a marginal contribution. Also as I said in my original comment, some extra experiments are needed to justify the performance of the method. The authors promised to provide some experiments in the camera ready, yet they did not provide any results during the rebuttal. SO it is not clear if they can get a good and reasonable results for the new setup. If they had provided the experiments I asked them, I would be more willing to flag this paper as \"accepted\". But now, still I would say it is a borderline paper.", "rebuttal": "We sincerely appreciate the time and effort put into reviewing our paper and providing valuable feedback. We would like to address your questions below.\n\n> **R2Q1**: There is no convergence or stability analysis and the claim that DEM* gives an upper bound on EM performance is heuristic\n\nThis paper primarily provides an empirical analysis and improvement of classical EM. Its optimization dynamics and benefits (e.g., reduced learning rate sensitivity and enhanced robustness) are empirically validated across tasks (Table 1, Fig. 9). DEM* serves to explore EM's empirical limits, and AdaDEM addresses this by eliminating hyperparameter tuning while matching or exceeding DEM* performance in unsupervised settings (Sec. 3.3, Table 1).\n\n> **R2Q2**: Questions about the computational overhead of DEM*'s hyperparameter search and the increased memory bandwidth in AdaDEM compared to vanilla EM.\n\nThese additions are designed for efficiency—$\\delta$ leverages existing gradients, and EMA updates are $O(C)$ per batch—resulting in negligible memory bandwidth increase relative to classical EM, while enabling superior performance and robustness without costly hyperparameter optimization, as validated in Sec. 4.2 and Table 1.\n\n> **R2Q3**: Questions about the significance of AdaDEM's improvements.\n\nFor EM-based tasks like Tent, AdaDEM's direct optimization of entropy minimization drives substantial gains by resolving coupled limitations (e.g., reward collapse and easy-class bias). In non-EM methods such as MixMatch or FixMatch, AdaDEM acts as a lightweight regularizer, yielding consistent average improvements (e.g., +0.2% to +5.0% in Table 3) by enhancing robustness to noise and imbalance, though occasional decreases stem from dataset-specific variances. Semi-supervised gains, while often within one standard deviation, reflect consistent positive trends across six benchmarks (Table 3, 8, 9). Reinforcement learning on MiniGrid shows reliable average improvements (e.g., higher returns in 7/9 environments, Fig. 6, 8), with variability due to task stochasticity, validating the effectiveness of AdaDEM.\n\n> **R2Q4**:  Whether AdaDEM's L1-normalized CADF reward sufficiently prevents collapse to dominant classes in severe data imbalance.\n\nWe acknowledge this valid concern. While AdaDEM’s marginal entropy calibrator (MEC) inherently counters class skew by dynamically penalizing over-confident predictions (Sec. 3.3), we agree that synthetic imbalance studies beyond CIFAR-LT (e.g., higher skew ratios ρ>>100) would strengthen validation. We will add experiments simulating extreme imbalance (e.g., ρ=500) in other tasks to rigorously test AdaDEM’s robustness. The L1 normalization mitigates reward collapse, while MEC aligns outputs with the estimated label distribution—jointly reducing bias even when $\\delta$ alone may not fully offset severe skew.\n\n> **R2Q5**: Questions about AdaDEM's compatibility with modern calibration techniques.\n\nThe paper focuses on revealing the limitations of classical EM and enhancing EM’s efficacy through AdaDEM via adjustments at the objective level. We explicitly validated AdaDEM’s compatibility with modern calibration techniques, such as EATA and DeYO for test-time adaptation tasks, and MixMatch and FreeMatch for semi-supervised learning tasks. Compatibility with temperature scaling or focal loss remains an area for future work."}, {"review_id": "Rnf9U1XgXy", "reviewer": "Reviewer_pbbx", "summary": "This paper explores the reformulation and decouple of classical entropy minimisation into two parts.\n\nOne part is namely as cluster aggregation driving factor rewards and another one is called gradient mitigation calibrator.\n\nThey identify the weaknesses of classical EM, and propose adaptive decoupled entropy minimisation.\n\nThe evaluations and analysis are performed on several benchmarks, covering different learning objectives, and show consistent improvements compared to the use of classical EM.", "questions": "Please refer to weaknesses for questions.", "limitations": "Limitations have been discussed in conclusion section.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 1, "significance": 2, "originality": 3, "strengths_and_weaknesses": "strengths:\n\n+ The idea of decouple classical EM into two parts where each part serves a different aim is interesting and new.\n\n+ The comprehensive experiments, analysis, evaluations and discussions somehow show the effectiveness of the proposed model.\n\n+ The paper is overall well written, somehow clean and clear.\n\nweaknesses:\n\nMajor:\n\n- In figure 1, some terms are not properly explained and unclear to reviewer. For example, the introduction of $\\alpha$, $\\tau$, $\\delta$, as well as some unknown operations appear to be too early. This figure does not present much insights or novel elements for readers. In addition, as the first figure in the paper, it does not seem to be attractive to readers at all. Also is it necessary to have 3 unclear equations and terms presented in the figure, and are there better ways to present them in a clear, interesting, and easy-to-understand manner?\n\n- This paper appears to have several redundancies. For example, Line 107-108 is the same as Line 19-20; Line 114-116 is the same as Line 23-24. These sentences look almost identical and just the references differ. This is inappropriate.\n\n- It would be better to be clear eg what are scalars, vectors, and matrices. Some equations appear to be a bit strange. For example, eq 4, left side input is $z$, but the right side there is no $z$, although $p$ and $z$ have connections, as a research paper, it would be better to make it clear and rigorous, rather than a technical report with some levels of inconsistencies.\n\n- Fig 3 and 4 right subfigures, what are the optimal values used? the right most two subfigures are not being properly explained. For example, what are those numbers mean inside brackets, shouldn't the readers need to know those concepts? Regarding NoAdapt, shouldn't the plot (yellow lines) the same in both plots? These concepts are not being properly explained and made clearer enough to readers. \n\n- Line 177-180 is unclear, for example, this part refers to figure 3 left and eq 7, but eq 7 does not contain either $\\alpha$ and $\\tau$ at all. the motivation for $\\alpha$ and $\\tau$ is unclear. What motivate the use of $\\alpha$ and $\\tau$?\n\n- Some of the operations are not being clearly explained, for example, Line 216, and eq 12 regarding $-t$ and its roles are unclear.\n\n- Line 248, \"on multiple random seeds\", how many runs exactly? The reviewer thinks that as a research paper, it is necessary to be as clear as possible, also what are the random seeds, are random seeds in random manner when calculating mean and std?\n\n- Line 296-299, table 1 and 2, especially 2, seem does not reflect on the discussions made in this part? \n\nMinor:\n\n- Line 142, what is that $\\mathcal{L}$, this is not being properly introduced / explained before using it. Also in Line 142-144, it is unclear why such operation is performed. It would be better to provide justifications.\n\n- apart from redundancies in texts, figures also have some levels of redundancies. For example, fig 3 left, and figure 4 left, why $\\alpha=1.0$ appears 3 times, and $\\tau=1.0$ appears 3 times in both legends.\n\n- Using `\\eqref` for referencing equations, instead of using `\\ref`. \n\n- table 1, DEM* should be explained in caption.\n\n- The reviewer also reads the appendix. Page 22, figure 7 top row, texts and figures a bit cluttered.", "was_revised": true, "final_justification": "The authors have addressed the concerns I raised in a clear and thorough manner.\n\nI have also carefully reviewed the comments from the other reviewers, along with the authors’ responses. \n\nI believe this work presents some meaningful insights. However, I also acknowledge and share some of the reservations expressed by reviewers who maintain a borderline assessment.\n\nThat said, based on the improvements in the revised version, I am satisfied with the authors’ efforts and am happy to raise my score from a borderline reject to a borderline accept, aligning with the direction suggested by other reviewers (Borderline).", "rebuttal": "We sincerely appreciate the time and effort put into reviewing our paper and providing valuable feedback. We would like to address your questions below.\n\n> **R3Q1**: Concerns about Fig. 1.\n\nWe revise Fig. 1 to explicitly label $\\alpha$, $\\tau$, and $\\delta$ using annotations. Core novelty is emphasized with intuitive visual cues. The figure is reorganized as a conceptual flowchart comparing classical EM, DEM*, and AdaDEM, using succinct text and highlighting information flow.\n\n> **R3Q2**: Concerns about redundancies in the manuscript.\n\nThese repetitions are fully eliminated in the revised version to ensure conciseness and adherence to publication standards. Thank you for highlighting this issue.\n\n> **R3Q3**: Concerns about clearer definitions in equations.\n\nWe thank the reviewer for the constructive feedback on enhancing mathematical rigor. We agree that explicitly denoting $p$ as a function of $z$ would improve clarity, and we revise all equations to rigorously define scalars, vectors, and matrices, ensuring consistency throughout the paper.\n\n> **R3Q4**: Questions about Fig. 3 and Fig. 4.\n\nWe acknowledge the need for clearer explanations. In Fig. 3 (right), the optimal values of $\\tau$ are represent in Fig. 3 (center), and the optimal values of $\\alpha$ in Fig. 4 (right) are $\\alpha=0.0$ for single-domain TTA and $\\alpha=1.3$ for Continual TTA. NoAdapt (yellow) represents the consistent source-model performance without adaptation, so that the results are the same for both single-domain and continual TTA tasks. We clarify brackets as \"mean±std\" and add captions to define NoAdapt uniformly.\n\n> **R3Q5**: Questions about the motivation for introducing $\\alpha$ and $\\tau$.\n\nThe reference to Eq. 7 in lines 177-180 serves to contextualize the gradient behavior before introducing $\\alpha$ and $\\tau$. $\\tau$ in Eq. 9 reshapes the reward curve in CADF to enhance contributions from high-confidence samples (Fig. 3 left), while $\\alpha$ in Eq. 10 scales GMC penalties to prevent overfitting in noisy/dynamic environments (Fig. 4 center), both optimizing robustness and performance upper bounds in DEM*.\n\n> **R3Q6**: Concerns about Eq. 12.\n\nIn Eq. 12 and the surrounding text (e.g., line 216), $t$ denotes the iteration index for the exponential moving average in the Marginal Entropy Calibrator (MEC), where $\\overline{\\mathbf{p}}\\_{k}^t$ represents the estimated probability vector for class $k$ at step $t$, dynamically updated as $\\overline{\\mathbf{p}}\\_{k}^{t} = 0.9 \\cdot \\overline{\\mathbf{p}}\\_{k}^{t-1} + 0.1 \\cdot \\mathbf{p}\\_{k}$ using the current prediction $\\mathbf{p}\\_{k}$, thereby replacing GMC to mitigate easy-class bias without label priors.\n\n> **R3Q7**: Concerns about random seeds.\n\nIn the paper, we use 3 random seeds for all experiments, specifically $\\{1, 2, 3\\}$. The random seeds are chosen to determine test sample order or initial conditions, ensuring reproducibility, and the mean and standard deviation are computed across these runs to account for variability, as specified in Sec. 4.1 and Appendix B.1.\n\n> **R3Q8**: Whether the discussions in lines 296-299 and Tables 1-2 are adequately reflected in the paper.\n\nThe paper explicitly addresses the results in Table 1 and Table 2 within the main text: Table 1's ablation findings on CADF and robustness are discussed in Sec. 3.2, while Table 2's TTA performance gains with DEM*/AdaDEM are analyzed in Sec. 4.2. Lines 296-299 directly reference Table 2 to explain how sample selection methods like EATA/SAR are enhanced, mitigating reward collapse and easy-class bias, which aligns with the tables' data on accuracy improvements and $\\Delta$ values). This ensures coherence between results and discussion.\n\n> **R3Q9**: What is the $\\mathcal{L}$?\n\n$\\mathcal{L}$ refers to the entropy minimization loss $H(z)$ (Eq. 5), which is the conditional entropy minimized in EM.\n\n> **R3Q10**: Concerns about the redundant labeling in Fig.3 and Fig. 4.\n\nWe acknowledge the redundancy in the legends of Fig. 3 (left) and Fig. 4 (left), where $\\alpha=1.0$ and $\\tau=1.0$ are repeated for clarity in the original plot. This will be rectified in the final version by retaining only unique labels (e.g., one instance of $\\alpha=1.0$ and $\\tau=1.0$) to eliminate redundancy while preserving interpretability.\n\n> **R3Q11**: Suggestions of using \\eqref to replace \\ref, explaining DEM* in Table 1's caption, and address the cluttered presentation in Fig. 7.\n\nWe acknowledge the reviewer's feedback and optimize the manuscript to enhance clarity by: (1) Using \\eqref for equation references, (2) Adding an explanation of DEM* to Table 1's caption, and (3) Modifying Fig. 7's layout to improve visual spacing."}, {"review_id": "tC2wKFTnEI", "reviewer": "Reviewer_xd1y", "summary": "In this paper, the authors propose decomposing the entropy minimization objective into two distinct components, each responsible for a different task, one pushing for a peaked output distribution and one penalizing high-confidence classes. Next, they introduce a temperature factor to control reward collapse for highly certain samples, and a separate weighting factor to control the regularization effect of the second component. To avoid the need for manual hyperparameter search, they propose an adaptive variant that automatically adjusts the weight of the loss based on the norm of gradients. The authors conduct extensive experiments on a wide range of datasets, demonstrating improved performance.", "questions": "- Could the authors clarify the exact connection between DEM and AdaDEM? Specifically, is AdaDEM intended to be a direct reformulation of DEM with adaptive scaling, or does it represent a fundamentally different objective? How the hyper-pararameters are related to the gradient-based rescaling of the objective? \n\n- Given that $\\delta$ δ is computed as the norm of the gradients of $T$, have the authors explored or could provide discussion on the effect of using the norm of the full loss gradients instead? Would this lead to different behavior, and if so, how?\n\n- Eq. (8) appears to be very close to standard conditional entropy with a sample-based normalization term. Could the authors clarify how AdaDEM differs in practice from standard conditional entropy minimization under this view? What about comparisons with existing sample based reweighting approaches? Should the method be compared against them given the high similarity with such approaches?\n\n-  Regarding the term introduced in lines 161–162 and its role in the derivation of Eq. 8, could the authors provide further justification for this simplification?  Why is this particular simplification needed or desirable?", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "The paper is overall well written and is supplemented with extensive experiments that demonstrate the effectiveness of the proposed approach. The idea of the proposed decomposition of entropy is interesting and can provide an alternative view on these entropy objectives, which is positive. However, the ideas presented in the paper are loosely connected and not well motivated. Please find some important comments below:\n\n- The connection between DEM and AdaDEM is somewhat vague and not very clear. If my understanding is correct, AdaDEM does not use the temperature factor or the $\\alpha$ parameter. So, AdaDEM does not remove these hyperparameters by providing adaptive estimations for them, but rather by using a different formulation. In fact, AdaDEM is Eq. 8 with the added  $\\frac{1}{\\delta}$  factor. $\\delta$ is calculated as the norm of gradients of T. Therefore, even though AdaDEM is presented as an attempt to remove the hyperparameters of DEM, it seems to me that AdaDEM can be directly derived from DEM without introducing any hyperparameters. This is also supported by the signficantly better performance of AdaDEM compared to DEM.\n\n- This also brings out another quite important concern. If (8) is equivalent to (4), then AdaDEM is actually the regular conditional entropy with just a sample-based normalization term. Another question that naturally arises here is: what if we do not use the gradients of  $T$ to calculate $\\delta$, but rather the whole loss? Appropriate supporting discussion and/or ablation studies are needed here to support this. Furthermore, this approach is very similar to prior works on reweighting samples based on gradient directions, e.g., Ren, Mengye, et al. \"Learning to reweight examples for robust deep learning.\" International Conference on Machine Learning. 2018, as well as gradient norm approaches, e.g., Chen, Zhao, et al. \"GradNorm: Gradient normalization for adaptive loss balancing in deep multitask networks.\" International Conference on Machine Learning. PMLR, 2018, and similar extensions.\n\n- The motivation for introducing the term proposed in lines 161–162 should be better justified by the authors. It is not directly evident how  p_i  should be computed or why we need to perform this simplification that leads to Eq. 8. This is a minor concern.\n\nOverall, the paper provides strong performance on benchmark datasets and extensive comparisons, demonstrating the effectiveness of the proposed approach. However, I find that there are significant gaps in the presentation and motivation of the proposed method, so I am reluctant to provide an acceptance decision at this stage. I appreciate the insights provided by the discussion on entropy decomposition, but it seems that the proposed method essentially amounts to a relatively simple sample-based reweighting scheme, a concept that has already been explored in various forms in prior work.", "was_revised": true, "final_justification": "After considering the authors’ rebuttal, I am increasing my rating to borderline accept. However, I still have concerns regarding the incremental contribution and the novelty over existing formulations.", "rebuttal": "We sincerely appreciate the time and effort put into reviewing our paper and providing valuable feedback. We would like to address your questions below.\n\n> **R4Q1**: Questions about the connection between DEM and AdaDEM, specifically how AdaDEM eliminates hyperparameters without adaptive estimation and whether it is directly derivable from DEM.\n\nAdaDEM is derived from the decoupled framework of classical EM (as in DEM) but fundamentally differs by replacing the hyperparameter-dependent components: it normalizes the Cluster Aggregation Driving Factor (CADF) reward using the L1-norm of its gradient ($δ = || -\\partial T / \\partial z ||\\_1$) and substitutes the Gradient Mitigation Calibrator (GMC) with a Marginal Entropy Calibrator (MEC) for dynamic prior estimation. This reformulation, expressed in Eq. 12, avoids $\\tau$ and $\\alpha$ by design, not through adaptive hyperparameter tuning, and directly addresses DEM's limitations (e.g., reward collapse and easy-class bias), leading to superior performance without introducing new hyperparameters.\n\n> **R4Q2**: Whether AdaDEM is equivalent to normalized conditional entropy and requests clarification on using gradients of the entire loss versus CADF for $\\delta$.\n\nAdaDEM is not equivalent to simply normalizing classical entropy. Eq. 8 reformulates entropy via decoupled components (CADF/GMC), but AdaDEM replaces GMC with MEC and normalizes using $\\delta$ derived exclusively from CADF’s gradients (Eq. 12). Ablations in Table A validate that $\\delta$ must be computed from CADF gradients—using the whole loss degrades performance. Such a substitution of using the full loss would likely introduce penalties from components like GMC or MEC into the normalization, potentially disrupting the targeted mitigation of reward collapse by diluting CADF-specific rewards and altering optimization dynamics, such as misaligning sample contributions or affecting robustness in noisy/dynamic environments.\n\nTable A: Ablation study of $\\delta$. AdaDEM-V means that using $\\delta$ computed from the whole loss.\n\n| Single-Domain TTA |      |        |          | Continual TTA |      |        |          |\n| ----------------- | ---- | ------ | -------- | ------------- | ---- | ------ | -------- |\n| LR                | EM   | AdaDEM | AdaDEM-V | LR            | EM   | AdaDEM | AdaDEM-V |\n| 0.0001            | 40.5 | 39.7   | 39.9     | 0.0001        | 44.7 | 42.6   | **42.7**     |\n| 0.00025           | 42.2 | 40.8   | **41.0**     | 0.00025       | 48.2 | 43.8   | 31.2     |\n| 0.0005            | 44.1 | 42.1   | 39.4     | 0.0005        | 51.0 | 48.3   | 26.0     |\n| 0.001             | 46.5 | 44.1   | 37.7     | 0.001         | 53.8 | 51.7   | 10.8     |\n| 0.0025            | 49.8 | 47.6   | 29.2     | 0.0025        | 57.0 | 55.7   | 3.8      |\n| 0.005             | 51.5 | 50.9   | 14.6     | 0.005         | **58.6** | 58.0   | 1.5      |\n| 0.01              | 52.5 | 53.8   | 6.3      | 0.01          | 49.1 | 60.1   | 0.7      |\n| 0.025             | **53.1** | 57.2   | 2.2      | 0.025         | 6.2  | 62.5   | 0.3      |\n| 0.05              | 48.3 | 59.5   | 1.2      | 0.05          | 2.3  | **63.2**   | 0.2      |\n| 0.1               | 41.9 | **61.5**   | 0.7      | 0.1           | 1.1  | 41.6   | 0.2      |\n\n> **R4Q3**: Discussion of relations to prior gradient-based reweighting methods.\n\nThe design of AdaDEM addresses EM-specific limitations (reward collapse/easy-class bias), distinct from general sample reweighting schemes. Unlike supervised reweighting (Ren et al.) or multitask balancing (GradNorm), AdaDEM’s $\\delta$ is unsupervised, dynamically adjusted per-sample via the L1-norm of CADF’s reward, and integrated with MEC for distribution calibration without label priors. The approach is tailored for EM’s decoupled dynamics, as evidenced by gains over baselines (Tables 1–4).\n\n> **R4Q4**: Why introducing the term in lines 161–162 and clarifying the motivation for the simplification leading to the $\\mathbf{p}_i$ in Eq. 8.\n\nWe clarify that $\\mathbf{p}_i$ is introduced as a constant (excluded from the computation graph) to reformulate the conditional entropy equivalently to Eq. 6–7, enabling identical partial derivatives as GMC while simplifying gradient computation by avoiding dependencies on variable probabilities. This rewrite (Eq. 8) facilitates practical implementation and analysis of decoupled effects without altering the optimization objective, as $\\mathbf{p}_i$ requires no explicit computation beyond its role as a fixed scaling term."}]

Author rebuttal:

Meta-review: This paper revisits entropy minimization by decomposing it into interpretable components, diagnosing key weaknesses, and proposing AdaDEM, a hyperparameter-free adaptive variant. The method is simple yet broadly applicable, and demonstrates consistent improvements across test-time adaptation, semi-supervised learning, domain adaptation, and reinforcement learning benchmarks. While some reviewers initially questioned the incremental novelty and modest improvements, the rebuttal was highly constructive, providing new imbalance experiments, additional clarifications on the mathematical formulation, and significantly improved presentation. These efforts directly addressed reviewers’ concerns, leading to multiple reviewers raising their scores and shifting the overall balance in favor of acceptance. The final consensus is that the combination of a novel perspective, practical algorithm, and thorough rebuttal-driven clarifications make the contribution both useful and impactful for the community.
Decision: Accept (poster)