Dependency-aware Maximum Likelihood Estimation for Active Learning
Authors: Beyza Kalkanli
Tales Imbiriba
Stratis Ioannidis
Deniz Erdogmus
Jennifer Dy
Venue: Accepted by TMLR ()

Abstract: Active learning aims to efficiently build a labeled training set by strategically selecting samples to query labels from annotators.
In this sequential process, each sample acquisition influences subsequent selections, causing dependencies among samples in the labeled set. However, these dependencies are overlooked during the model parameter estimation stage when updating the model using Maximum Likelihood Estimation (MLE), a conventional method that assumes independent and identically distributed (i.i.d.) data. We propose Dependency-aware MLE (DMLE), which corrects MLE within the active learning framework by addressing sample dependencies typically neglected due to the i.i.d. assumption, ensuring consistency with active learning principles in the model parameter estimation process. This improved method achieves superior performance across multiple benchmark datasets, reaching higher performance in earlier cycles compared to conventional MLE. Specifically, we observe average accuracy improvements of 6%, 8.6%, and 10.5% for k=1, k=5, and k=10 respectively, after collecting the first 100 samples, where entropy is the acquisition function and k is the query batch size acquired at every active learning cycle.

Published in Transactions on Machine Learning Research (09/2025)
Dependency-aware Maximum Likelihood Estimation for Active Learning
Beyza Kalkanlı
kalkanli@ece.neu.cmm
Department of Electrical and Computer Engineering Northeastern University
Tales Imbiriba
cales.imbirihaohamm edu
Department Computer Science
University of Massachusetts Boston
Stratis Ioannidis
ioannidis@cce.neen edu
Department of Electrical and Computer Engineering Northeastern University
Deniz Erdogmus
endogmaus@ece.rer edu
Department Electrical and Computer Engineering Northeastern University
Jennifer Dy
Etyy@ece.meu.cce
Department of Electrical and Computer Engineering Northeastern University
Reviewed OpenReview: https:// oppnnrvveeeenrvveee. net/ ptr/ffrmm? id= gDVDSXXGK
Abstract
Active learning aims to efficiently build a labeled training set by strategically selecting sam- ples to query labels from annotators. In this sequential process, each sample acquisition inffuences subsequent selections, causing dependencies among samples in the labeled set. However, these dependencies are overlooked during the model parameter estimation stage when updating the model using Maximum Likelihood Estimation (MLE), conventional method that assumes independent and identically distributed (i.i.d)) data. We propose Dependency-aware MLE (DMLE), which corrects MLE within the active learning framework by addressing sample dependencies typically neglected due to the i.i.d. assumption, ensur- ing consistency with active learning principles in the model parameter estimation process. This improved method achioves superior performance across multiple benchmark datasots, reaching higher performance in earlier cycles compared conventional MLEE. Specifically, we observe average accuracy improvements of 6%, 8.6%, and 10.5% for k = 1, k 5, and k: = 10 respectively, after collecting the first 100 samples, where entropy the acquisition function and & is the query batch size acquired at every active learning cycle.
1 Introduction
Supervised learning methods rely the availability of abundant labeled data (Alzubaidi et al. 2023 The significance of large labeled sets becomes more acute with ver-parameterized deep learning models that can be prone to overfitting (Brigato & Tocchi 2021). Resource raists in acquiring labeled data present a practical challenge and must be balanced against the benefits having more labeled data (Alzubaidi et al. 2023 Bansal et al. 2022). In particular, the costs associated with labeling motivate the need to use labeling resources as effectively as possible. Data augmentation (Shorten & Khoshgoftaar 2019) and transfer learning methods (Tan et al. 2018 Yu et al. 2022 have been adopted methods to deal with this (labeled) data

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0
Chec
Cass
Use
Clas
Dew
Passive learning with IMLE (700 sam- plas)
Active learning with IMLE and entropy as an acquisition function (140 samples)
Active learning with DMLE and entropy as an acquisition function (140 samples)
Figure 1: Comparison of decision boundaries and selected samples across different learning scenarios: On the left, all 700 samples are used for model training without active learning. In the active learning experiments shown in the middle and right, the entropy acquisition function is used for sample selection, with one sample acquired at each cycle, resulting in a total of 140 samples. The middle setup employs IMLE for model parameter updates, while the right setup uses DMLE. The shaded regions represent the decision boundaries, and the yellow crosses in the active learning setups highlight the selected informative samples. Both the model trained with all samples using IMLE and the active learning model updated with DMLE achieve 99.5% accuracy, while the active learning model using IMLE for parameter updates achieves only 92% accuracy.
scarcity challenge. In contrast to these methods, active learning aims to intentionally select which samples to acquire labels to request based on the latest available model, attempting to achieve high return on performance improvement per data acquisition or labeling resource spent (Ren et al. 2021).
Active learning is an iterative process that alternates between two steps: (1) querying samples/ samples/labels /labels and (2) adapting or retraining the model with the updated training set (Settles 2009). Each complete iteration of these steps is referred batch of samples during the query sampling step. So, in an active learning sctup, proviously selected data have control over the selection of new samples at each cycle through the updated model parameters. This sequential nature of data collection in active learning inherently introduces dependencies between samples across cycles in the labeled set. However, conventional Maximum Likelihood Estimation (MLE) assumes samples are ii.d., overlooking dependencies during model parameter estimation; hence, we refer to it as Independent MLE (IMLE)..
as a cycle. Tn each cycle, the model might either aequire one sample or a
Sample dependency has been studied in the literature by revisiting the sample acquisition stage Sener & Savarese 2018 Ash et al. 2019a) or re-weighting the objective function (Farquhar et al. 2021 Beygelz- imer al. 2008;; however, these approaches are insufficient to capture the dependency within the objective function, as they still rely on the i..d. assumption of MLE. Neglecting sample dependency during model parameter estimation is fundamentally incompatible with the nature of active learning. Ignoring this de- pendency during training leads to a vicious cycle of suboptimal model parameter estimation and, in turn, poor sample acquisition decisions in upcoming cycles as illustrated in Figure [ The model updated with DMLE, using the same nunber of samples, outperforms the one updated with IMLE, creating a decision boundary similar to the one achieved by using all data samples with passive learning. The diverse and well-separated sample selections made by the DMLF-updated model show that better model leads to more effective sample selection and better results with fewer samples. This motivates to explore ways to model this dependency in the training objective. To the best of our knowledge, this is the first work that attempts to correct for dependencies among samples across active learning cycles during model parameter estimation by removing the incompatible i.i.d. data assumption from the training objective. Our main contributions are as follows:
We propose Dependency-aware Maximum Likelihood Estimation (DMLE), novel approach that corrects MLE by eliminating the i.i.d. data assumption and aligning with active learning principles through explicit modeling of sample dependencies during the parameter estimation stage.
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We theoretically derive the dependency term in MLE that arises from the inffuence of previously selected samples on subsequent sample selections within the active learning framework.
Our empirical results and hypothesis tests indicate that, in an active learning process, DMLE out- performs the conventional independent MLE (IMLE) approach in various benchmark datasets and several sample selection strategics.
The remainder of this paper is organized as follows. The related works are discussed in Section - Section provides the necessary definitions and background in active learning. We present the proposed methodology of DMLE to use MLE within active learning in Section J Experiments and results are presented in Section 5 while final remarks are made in Section
2 Related Work
In machine learning, the dataset used for training plays a significant role in resulting model performance (Shui
ct al.] 2020 Gal et al.] 2017). However, labeling large datasets not always convenient and straight.or-
ward, especially in domains such as medicine (Hoi al. 2006 Smailagic et al., 2018 [udd et al.) 2021),
hyperspectral imaging (Cao et al. 2020 Lei et al. 2022 Thoreau et al.) 2022). bioinformatics (Mohamed]
et al. 2010 De Angeli et al.) 2021). In the active learning paradigm, the model is allowed to choose the data from which it learns, aiming to enhance labeling efficiency (Settles 2009). Thus, using active learning becomes crucial to create the best training set under limited resources. For the oplimization of this sample selection procedure, extensive research in active learning has been focused on developing various acquisition
stratogies (Schohn & Cohn 2000: Zhong & Padmanabhan 2002 Sourati ct al. 2017 Ash et al., 2021 Kim
et al. 2023) aiming to make better selections at each cycle.
Initial work in active learning aimed to combine this paradigm with simple machine learning models, such as Support Vector Machine (SVM), decision tree classifier (Lewis & Catlett
1994a).
nearest neighbor clas-
sifier (Lindenbaum et al., 2004], and logistic regression model (Schein & Ungar 2007
Zhang & Oles 2000).
With neural networks' increasing popularity, the deep active learning research focused on combining neural
networks with active learning has become accelerated (Wang et al. 2021 Ban et al. 2023 Despite the mer-
its of deep active learning, [Settles] (2011) draws attention to the sample correlation problem that emerges in batch sample acquisition. While batch selection may reduce the overall computational expenses of train- ing after each cycle, it can lead to redundant labeling if the tradc-off between uncertainty and diversity of samples is not well-balanced, potentially hindering the objectives of active learning (Settles) 2011 Krishnan et al., 2021b. To solve this issue, various corrective approaches have been proposed mainly attempting to balance inference uncertainty and sample diversity during batch sample selection (Sener & Savarese 2018
Ash et al. 2019a; Kirsch et al. 20196 Ash et al., 2019b Bryık et al., 2019 Citovsky et al. 2021 Kirsch et al, 2019a). Kirsch el al. (2021) highlights computationn
expenses of methods aiming to combine uncertainly and diversity of samples and proposes a stochastic batch sampling strategy with less additional costs during the sample selection. Although sample correlations are comprchensively rescarched, the focus has far been on the dependency between the batch of samples selected in a cycle, primarily addressing the sample dependency within the sample acquisition stage. Our aim differs from the earlier works as we emphasize sample dependency across cycles and address it during the model parameter estimation stage.
Achieving good results at earlier cycles even when starting from a few or zero labeled samples is highly preferable in active learning (Jin et al.] 2022 Yuan et al.) 2020: Houlsby et al. 2014). Challenges include dealing with biased data (Gudovskiy et al. 2020 Singh et al. 2023), the creation of imbalanced training data
across the cycles (Kottke et al. 2021 Y Yang& & Loog] 2022 Szúcs & Papp 2022). the selection of outliers (Chen et al. 2022 Beygelzimer et al. 2008 Prabhu et al. 2019 Dasgupta & Hsu 2008), or sample dependency in data collection leading to a possible discrepancy between the acquired training data distribution and the actual population distribution (Dasgupla 2011). Among the listed challenges, the one most aligned with our focus is the emphasis on sample dependency. Previous work on this issue has proposed using re-weighting
the objective function (Farquhar et al.) 2021 Beygelzimer et al.] 2008) or clustering techniques (Dasgupta & Hsu 2008) to address sample dependency, with focus on achieving a representative approximation of the true distribution. While Dasgupta & Hsu (2008) proposes a clustering-basel sample selection process
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aimed at ensuring good representation of the population with the selected sample set, their approach is limited to optimizing the sample acquisition stage with similar goal to that of batch sample selection methods. [Beygelzimer et al. (2008) on the other hand, proposes an importance-weighter active learning method to address the sample dependency relationship by applying importance-samplig corrections in a dasea active learning scenario, which limited simple binary classifier models. Although [Farquhar] et al. (2021) also focuses on neural networks and addresses an unlabeled pool scenario similar to ours, their method is limited re-weighting within the objective aiming come up with an unbiased risk estimator. They conclude that applying this methodology with neural networks leads to performance deficiencies due to the loss of generalizability. However, the sample dependency issue we highlight arises specifically during the model parameter estimation stage due to the ii.d. data assumption inherent in MLE. The proposed methods canmol capture this issue, as fixing it by re-weighting the loss term for each sample during training will still rely on the ii.d. assumption and neglect the dependency term (See Appendix D for experimental comparison on MNIST datasct).
3 Background
We consider active learning setup (Settles) 2009 Takezoe et al. 2023 Joshi et al. 2009), in which large number of unlabeled samples are available for an evaluation in terms of an acquisition score and labeling by an oracle (e.g,, an expert) on demand. An active learning algorithm proceeds in stages, collecting a batch of & labels from the labeling oracle, training a model from the labeled data, and selecting which batch to collect next. At cycle t, we define the unlabeled dataset as U2 {ar and the labeled set No.
()) where x2 E X' C Rdz the features, Y/ E E {1, (1,.... K} categorical labels. We also denote by Nt. the number of unlabeled samples and Ný, the number of labeled samples at cycle t. We assume that, conditioned
D. ((r),,)))
parameters 0 and sample features H, label y is sampled from a conditional distribution P(- | x; 0) parameterized by 0 E Rdd, For example, in a deep network (DNN) setting, where the DNN model is represented by function fo :o:20 + {), 1] the probability would be given by the softmax function, i.e., P(y x. 0) e(f(())yy) Ey elolt)yy for y. y E E. For clarity, a glossary of the terminology is included in Appendix
Overall, the above active learning algorithm is formally defined by two components, illustrated in Figure query sampling, which includes acquisition function and sample selection strategy, and model parameter eslimation method.
U
6.
MLE
D
Model
5 5)
3
3
Model
Parameter
D. 1)
Estimation
E..1 7)
Model Predictions
Query Sampling
((x,y), S..1)
Active
Sample
Learning
-
Selection
S
Oracle
Acquisition
Function
Function
Figure 2: Components of active learning procedure. In active learning, the model selects the samples to be acquired where the model parameters are cyclically updated with the updated labeled set. First, uncertainty scores for the samples in the unlabeled sample pool Ur are calculated using the current model with the acquisition function a(x, Next, samples S441 selected based on the sample selection strategy are labeled by an oracle and included in the labeled set Dr, resulting in D++1 D, U {(x, y). E S After the query sampling step, the updated labeled set is used for the model update with model parameter estimation method, concluding one cycle.
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3.1 Acquisition Function
2009) model the value of acquiring a label for an unlabeled sample towards the quality of the trained model. Formally, the acquisition function X X + R takes as arguments an a Uz, current labeled set D,, model parameters 0, and returns the value of the sample given the current parameters.
Acquisition functions (Lewis & Catlett, 1994b Sett les
This often modeled through the uncertainty of the label prediction of the classifier trained so far at an unlabeled sample. Examples include the entropy [Shannon 1948), Bayesian Active Learning by Disagreement (BALD) (Houlsby et al., 2011), margin sampling (Roth & Small 2006), and least confidence (Settles 2009). The entropy acquisition function quantifies the uncertainty of sample using the entropy of label follows:
axx, Dr. 0) = Hly 8; 6; D -- Σ
Ply 2; Dn D),
(1)
yEY
The BALD acquisition function evaluates the uncertainty of a sample as the mutual information bet label predictions and model posterior:
alx, Dr,0) Hlyle, D., 0] ( 0]]
(2)
The least confidence acquisition function quantifies the uncertainty of sample a(x, Dr, 0) max Ply D)
(3)
Some acquisition methods extend beyond uncertainty- selection by incorporating sample diversity, ensuring that selected samples cover a broader representation of the data. This approach is considered deterministic, as demonstrated by methods such as Core-set (Sener Savarese 2018). BADGE (Ash et al. 2019b), and Cluster Margin (Citovsky scores can be designed to align with the selection procedure. We provide acquisition score for the Core- set method below, where C' is the set of cluster centers derived from De, and the feature representations extracted by the model with paramcters A are denoted as Φ(x,0):
al. 2021]. Depending on the acquisition method, acquisition
alx, 0) min [, 0) - 6(2',0)11
(4)
.'EC
3.2 Sample Selection Strategy
The selection strategy determines the samples from the unlabeled set be labeled in the next cycle relying the acquisition scores. Two of the most widely employed approaches for sample selection are Top-k selection (Gal et al. 2017 Kirsch et al. 2019a) and stochastic batch sampling: proposed recently. The latter was shown to outperform Top-k in selecting informative samples (Kirsch et al. (021) with less computational expenses which makes it preferable for sample selection during cycles. Independently of the selection strategy, having selected S4,1, the corresponding labels are collected from the labeler, yielding a dataset.
0D = {(xx v) a St11).
and sets D. and U, are adjusted accordingly, i.e., D = D. 0D+41 and U++1 = U. \ Se+1.
Top-k Selection. In Top-k selection procedure, the selected batch of size given by:
S
arg Ea(z (i,
(5)
SCU. DES
where Dr is the labeled set at cycle and U, the unlabeled set. As the objective function equation submodular, is maximized by the items of the highest acquisition value (Kirsch et al. 2021
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Table 1: Possible sample selection probability distributions spanned by the sampling method of Kirsch al.) h021h. Here, a(x, D,0) represents the acquisition score, is the descending ranking of the acquisition scores (xx, D, 0), with the smallest rank as 1, and denotes the coldness parameter.
Distribution
P(x]D;;0)
Softmax
(,,((,,,.0)) aa Dr, 0y3
Power
Soft-rank
x xr
Stochastic Batch Selection. For stochastic batch selection, Kirsch et al. (0221) propose three probabilistic sampling schemes: Softmax, Power, and Soft-Rank Acquisition, as listed in Table L Stochastic batch selec-
tion methods differ from the conventiona ional Top-k selection method by perturbing the acquisition scores/ raanks of the samples in the unlabeled set with Gumbel noise prior to the Top-k acquisition, formulated as follows:
S arg max SCU,|Si=k WES
Σ [a(x, 02, +c..
where is a modified acquisition function and 6 is Gumbel noise. This perturbation introduces proba- bilistic selection into the sample selection procedure. Kirsch et al. (2021) show that, given an acquisition function a(-), the sample selection distributions P(z[D,;;) presented on Table can be implemented via equation through an appropriate choice of ā(-) and parameters of the Gumbel noise. Details are provided in Appendix in the supplement.
There are two ways to incorporate clustering- active learning methods acquisition functions with the selection methods outlined here. The first approach involves using measurement based on the cluster- ing/diversity method as the acquisition score, which can be used for stochastic softmax sampling, stochastic power sampling, or top-k sampling. The second approach entails developing a ranking of the samples in the unlabeled set to be used with stochastic soft-rank sampling. As long as acquisition scores or ranks can be computed, sample dependency can be integrated into model training within any active learning framework.
3.3 Model Parameter Estimation
In the model adaptation//re-training step, typically, Maximum Likelihood parameter Estimation (MLE) is preferred:
1+1 In P(S, Dr-1:0)
MME arg max in P(D arg max ++
Σ
In P(y 2:; 0)
(7)
0
(
The foundational assumption of i.i.d. data in MLE BBshhop 2006 causes the In P(S, D+-1;0) term to be constant and leads to the following conventional objective function:
MLE arg max
In P(y x;
(8)
(a.a) De+1
which we refer to as Independent Maximum Likelihood Estimation (IMLE) in the rest of the paper to highlight this sample independence assumption (see Appendix for detailed derivation).
4 Dependency-aware Maximum Likelihood Estimation
In the itcrative framework active learning, the sample acquisition followed by the of the model. Conventional approaches often employ MLE given in equation as a way of doing model pa- rameter estimation during this stage. Conventional MLE, referred to here IML.F, introduces fundamental

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incompatibility in active learning: its assumption of sample independence conflicts with the sequential nature of sample acquisition. The i.i.d. data assumption in IMLF, eliminates the contribution of sample dependen- cies during estimation (see Appendix BJ. However, labels in Di-1 are in fact not independent, as the data samples in the labeled set are not iid:: their sclection is determined via the active learning algorithm itself, which lurn depends on the past collected labels. As a result, IMLE is an approximation of reality and overlooks sample dependencies the estimate of Our major contribution is to account for these sample dependencies during model parameter estimation.
Ideally, we aim to preserve the valuable information embedded in sample dependencies within the likelihood P(D++1:0), which is typically disregarded under the i.i.d. assumption. In equation 7 if the second term is not eliminated due to the i.i.d assumption, it gives rise to the dependency term in equation [1] This dependency appears as a conditional probability term of the form P(S, D)-1:8). This accounts for the dependence of batches on previous cycles and may enable conditional chain computation of the likelihood P(D++1;0) f) as follows:
1+1
k
OOMLE arg max
Σ
Σ
P(y Tr,6,) + In P(S, Dr-1;
(9)
H+1
= arg max
Σ
In P(y (x;)) +
In P(S+ D+-1:0)
(10)
ED+4)
arg max
Σ
In P(y x;
Σ
Σ In P(r aES,
(11)
(
IMLE
Dependency
We propose taking into account these dependencies as evident in equation during the model parameter estimation. We refer to this approach as Dependency-aware Maximum Likelihood Estimation (DMLE), in which model parameters are updated while incorporating the sample dependencies introduced during active learning cycles.
However, this poses challenge, P(S, D+-1;0) distribution is difficult to express in a closed form: it depends the labels in D4+1 through the generation of an estimate of which affects the acquisition function and in turn the sample selection. Fortunately, when utilizing the stochastic batch sampling approaches introduced by Kirsch et al. (2021) and discussed in Section 3.2 an appropriate corresponding probability distribution P(S, D)-1:6) is associated with them (see Table This allows us to introduce an appropriate
correction term for equation []] When using the Top-k sampling Sinpling& discussed in Section 3.2 we approximate
P(S, D+-1; 0) by the Softmax distribution given in Table choice provided in Appendix [
Bishop 2006), with empirical support for this
Unlike the formulations presented by Raginsky & Rakhlin 2011 (equation 11) and discussed in Chap. 15 of Lattimore & Szepesváii (2019), our approach explicitly models both the data likelihood and the sample selection distribution as functions of a single underlying parameter 0. This choice reflects the interactive nature active learning, where the evolving model directly influences which samples are selected, and the acquired labels in turn refine the model. By updating the selection distribution at each cycle using the most recent parameter estimates, our formulation ensures that past acquisitions are re-cvaluated under the current model, capturing the feedback loop between selection and learning. This stands in contrast to approaches that treat the selection mechanism as fixed or independent of the model dynamics, which neglect the dependencies inherent to the sequential data generation process in active learning.
The comparison of DMLE in equation 1 with IMLE in equation g highlights their discrepancy in the latter term. We interpret this term as accounting for the sample dependencies resulting from active learning in MLE. Indeod, this term incorporates the likelihood of observed sample features (rather than labels alone) under a given 8, as induced by sampling. Thus, the selection of the sampling strategy results in varying terms in the objective function, which are proportionate to the summation of acquisition scores of samples in
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the labeled set, can be deduced from Table (sce Appendix for the corresponding objective functions). This likelihood function directly influences the Fisher Information matrix by incorporating the dependencies between selected samples, leading to stronger information measure. As result, the estimator benefits from lower variance guarantees. In Appendix C.1 we prove:
Theorem Let (0) and (DM be the Fisher Information malrices for IMLE and DMLE, respec- tively. As DMLE accounts for sample dependencies, 1 (0) = (1)) By the Cramér-Bod Rao bound, this implies Cov(DDMLL) Coo()IMLE), ensuring that DMLE provides lower variance eslimates compored to IMLE.
Thus, the incorporation of sample dependencies in DMLE strengthens the estimator, leading to improved parameter precision. To better understand the impact of training with DMLE, it is helpful to examine the expected log-likelihhook over samples drawn from ptree which is the true data distribution. This provides a theoretical justification for DMLE, as it reveals how the decomposition of the log-likelihood guides the alignment of model distributions with the true data distribution. In Appendix
C.2 wc show:
Theorem Let ptrse be the true data distribution, (x', ptrve, and P(y d' D.:0) and P(x' D2; be the distributions estimated from actively selected training data Dr. The expected lokeiikldhood on this data decomposes as:
(x y') ptrrue (n = DDKI(P( 12') II P(y' Dr, DxL(P(z') II P(d D,;
When the training data actively selected, DMLE, which includes both terms in the log-likelihood decom- position, minimizes the sum of two KL divergences, ensuring better alignment with the true distribution compared to IMLE, which ignores the dependency term. The dependency term, DKL (P(x') P(x' ()):
measures the discrepancy between the true distribution P(x') and the model's estimate P(x") Dz; learned from the actively selected data. Maximizing the expected log-likelihood is equivalent minimizing the of two KT. divergences, ensuring that both the predictive and data distributions of the model remain close to the true distributions. By addressing this discrepancy, DMLE reduces distributional mismatch and enhances the model's approximation of the true data distribution. In contrast, IMLE, by neglecting this term, risks a greater distributional gap. The dependency term ensures that the model's input distribution remains aligned with the true data distribution. The overall algorithm with DMLE in the active learning framework is provided Alg.
Algorithm Dependency-aware Maximum Likelihood Estimation for Active Learning
Input: Unlabeled dataset U, = No Labeled dataset D. = { 31)
NI ) Model fe, Sample selection
size k
for 1,.,.., do
Acqusition Function: Compute acquisition scores a(x, D,, 0) for all x E Ur with fo
Sample Selection Strategy: Select batch S,+1 C U, of size k
Oracle Labeling
Request labels from Oracle for samples in S,+1
Obtain labeled samples ((r,, ma)e,
Update Labeled Dataset D,+1 D, U ((x, w) aa 8
Update Unlabeled Dataset U++1 + U, S841
Model Parameter Estimation
Compute acquisition scores a(x, 0) for all E Dr+1
Update model parameters 8,41 with DMLE (equation 11 on D
end for

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5 Experiments
5.1 Experiment Setting
Datasets and Models. We evaluate the impact of including the dependency term during model parameter estimation for the active learning process on the model's prediction performance by testing it on datasets consisting of images, texts, or fcatures; to have diverse experiments, we used datascts with different sizes
and complexities. We use eight classification datasets: Iris (Fisher 1988). Mnist (Deng) 2012), Fashion- Mnist (FMuist) (Xiao et al. (017)) SVHN (Netzer et al. 2011)) Reuters-21511 21578 (Reuters) (Lewis 1997),
Connist-Letters (Bmnist) (Cohen et al., 2017). Cifar-10 (Krizhevsky et al., 2009), and Tiny ImageNet (Le & Yang 2015). We randomly sample portion of each datasct and assume that only the subset was available during training. We used subset of 30000 samples for Mnist and FMnist, 110 samples for Tris, 14651
samples for SVHN, 8083 samples for Reuters, 4440 samples for Emnist, 3000 samples for Cifar-10, and 1500 samples for Tiny ImageNet. Depending on the dataset, we adjust the complexity of the model. We used LeNet (Lecun et al. 1998) for Mnist, FMnist, SVHN, and Emnist, a two-layer MLP for Iris, a neural network with embedding and convolutional layers for Reuters, ResNet-50 (He et al. 2015) for Cifar-10, and a visual transformer (ViT-B/16) with 200 classes fine-tuned from ImageNet weights (Dssovitskiy et al. 2021 for Tiny ImageNet.
Comparing methods. In this paper, we address the i.i.d. data assumption in model parameter estima- tion, emphasizing its fundamental incompatibility with active learning. We propose the Dependency-awarr Maximum Likelihood Estimation (DMLE) method, which accounts for dependencies between data points, a problem that has not been tackled in the literature before, and applies to any setup irrespective of the em- ployed sample selection strategy or acquisition function. Thus, we compare our method with the conventional model parameter estimation technique in active learning setups, which we refer to as Independent Maxi- mum Likelihood Estimation (IMLE). With the experiments, we assess the performance disparity between using DMLE and IMLE while utilizing different combinations of acquisition functions and sample selection strategies to show the impact of considering sample dependency during the model parameter estimation.
We use Keras for the implementation of the neural networks CChollet et al. 2015 Aiming a fair comparison, at each active learning cycle, for all combinations, we use the same number of epochs, the same hyperpa- rameter combinations, and the same acquisition functions. We present results with 3 = 1 for the stochastic batch selection as practiced and suggested by Kirsch et al. (2021) which can be important for the model and acquisition performance. Different values for B might be explored further for better results. For all models, we use Adam optimizer (Kingma & Ba 2017 with learning rale of 0.001.
At each cycle, we select a fixed number of new samples and acquire the labels corresponding to them. Each experiment starts with the selection of one sample randomly and continues with the selection of the remaining samples with an acquisition function through active learning cycles. For each combination, we repeat the experiments eight times with different seeds.
Hardware. We utilized two different computing resources during the experiments. For the experiments with smaller models and datasets like Tris, we used an Intel(R) Core(TM) 19-10900KF processor paired with RTX 3090 GPU. For larger datasets and more complex models, which constitute the rest, we used an internal cluster with an Nvidia Tesla K80 GPU.
Evaluation metrics. We report the test accuracy on separate test sets for each dataset. where the test sets for Mnist and FMnist consist of 384 samples while Tris has 30 samples, SVHN has 510 samples, Reuters has 450 samples, Emnist has 336 samples, Ci has 386 samples, and Tiny ImageNet has 296 samples. Additionally, we have separate validation sets of similar size to the test sets of each dataset.
Time complexity. The time complexity analysis consists of two parts: sample acquisition and model parameter estimation. Sample acquisition is identical for IMLE and DMLE, as they differ only in the model parameter estimation process. The complexity of the sample selection process is ONy + c)) for both Top-k and Stochastic Batch Selection methods, where Ny is the number of samples in the unlabeled pool, k is the size of the batch of samples selected in each cycle, and is the cost of making an acquisition call per sample.
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Table 2: Comparison of mean classification test accuracies =1 standard deviation for different sample selec- tion sizes (k) during cycles and different sample selection strategics where number of samples in the labeled set Nr,- 100. We use entropy for the acquisition function and SSMS (Stochastic Softmax Sampling), SPS (Stochastic Power Sampling). SSRS (Stochastic Soft-rank Sampling), and Top-k sampling for the sample selection strategy. The bold text highlights the higher mean accuracy and lower standard deviation when
comparing DMLE and IMLE under various sampling schemes.
SSMS
SPS
SSRS
Top-k
Dataset
DMLE
IMLE
DMLE
IMLE
DMLE
IMLE
DMLE
IMLE
0.81 1 0.03
0.79 0.02
0.83 1 0.03
0.82 1 0.01
0.78 1 0.04
0.75 1 0.06
0.72 1 0.07
0.69 1 0.04
MNIST
0.82 1 0.02
0.75 0.03
0.82 1 0,02
0.79 1: 0.03
0.75 1: 0,04
0.74 1: 0.04
0.68 1 0.05
0.61 +: 0,07
10
0.80
0.02
0.76 0.03
0.79 + 0.03
0.77 + 0.05
0.73 0.05
0.71 + 0.07
0.64 + 0.08
0.48 + 0.06
0.39 0.03
0.37 0.02
0.39 + 0.02
0.36 + 0.03
0.35 = 0.04
0.33 = 0.03
0.32 + 0.03
0.30 + 0.03
EMNIST
0.39 +
0.35 + 0.02
0.40 0.04
0.36 + 0.01
0.36 = 0.03
0.33 = 0.02
0.31 = 0.04
0.31 = 0.03
10
0.40
0.37 0.002
0.40 1: 0.02
0.38 0.03
0.35 1 0.02
0.34 1 0.04
0.28 1 0.03
0.24 1 0.03
0.55
0.47
0.54 0.01
0.50 + 0.02
0.48 + 0.05
0.46 = 0.04
0.45 + 0.06
0.40 + 0.06
REUTERS
0.54
0.51 + 0.02
0.52 + 0.02
0.51 + 0.03
0.47 + 0.07
0.46 + 0.04
0.47 + 0.04
0.34 + 0.06
0.00 10
0,49 +: 0.02
0.50 1: 0.02
0,48 + 0.02
0.47 + 0.04
0.46 = 0.03
0.41 + 0.04
0.32 + 0.11
0.02
0.27 0.03
0.30 0.02
0.28 0.03
0.25 0.03
0.03
0.22 0.01
0.17 0.03
SVHN
0.27 0.05
0.23
0.25 0.04
0.24 0.03
0.22 0.03
0.04
0.22 0.02
0.02
10
0.26 0.03
0,22
0.24 0.04
0,22 + 0.01
0.21 0.03
0.21 = 0.03
0.21 + 0.01
0.20 + 0.03
0.74 0.02
0.72 + 0.03
0.75 + 0.02
0.71 + 0.02
0.70 + 0.04
0.71 + 0.04
0.52 + 0.07
0.53 + 0.06
FMNIST
0.02
0.71 0.02 0.02
0.73 0.02
0.68 0.06
0.66 + 0.03
0.49 1 0.08
0.49 - 0.08
10
0.03
0.72 0.02
0.72 0.01
0.72 0.04
0.69 0.04
0.70 = 0.04
0.48 0.11
0.43 + 0.05
0.24 0.02
0.21 0.02
0.23 1: 0.02
0.22 1: 0.02
0.22 1: 0.03
0.21 + 0.02
0.18 1 0.03
0.19 1 0.02
CIFAR-10
0.22 0.03
0.19
0.22 + 0.03
0.19 0.02
0.21 0,03
0.19 = 0.03
0.17 = 0.03
0.17 = 0.03
10
0.20 0.04
0.02
0.20 0.02
0.16 0.02
0.19 0.02
0.19 = 0.02
0.17 + 0.03
0.15 + 0.02
0.96
0.97 0.01
0.96:1:.00
0.96 - 0.01
0.97 1 0.01
0.97 1 0.01
0.99 1 0.01
0.94 1: 0.02
TRTS
0.96 + 0.03
0.87 + 0.02
0.96 + 0.03
0.87 + 0.02
0.93 + 0.03
0.86 + 0.04
0.92 + 0.05
0.81 + 0.06
10
0.93 + 0.03
0.79
0.93 + 0.03
0.79 + 0.03
0.85 + 0.03
0.81 + 0.02
0.85 + 0.08
0.77 + 0.05
The complexity of model parameter estimation differs between IMLE and DMLE due to the dependency term in the DMLE objective function, which requires additional calculations per sample. We perform the complexity analysis per sample in an epoch, assuming the use of Stochastic Gradient Descent for gradient updates and disregarding its associated complexity. Under these conditions, the time complexity of IMLE is O(Nz), while that of DMLE is O(NNr., where Nr. represents the number of labeled samples. Experimental timing results are provided in Appendix
5.2 Results
Accuracy results: With the experiments, we seek to highlight the impact of not eliminating the depen- dency term during the model parameter estimation. Thus, the experiments present the test performance discrepancy between using DMLE vs. IMLE for multiple benchmark datasets with various sample selection schemes and acquisition functions. We provide the experiment results with the entropy and Corc-set acqui- sition functions in this section. A similar case study for BALD and least. confidence acquisition functions is presented Appendix
Table 2 demonstrates the test accuracy results of using DMLE vs. IMLE after 100 collected samples across seven datasets, employing entropy the acquisition function, with four different sample selection schemes-Stochastic Softmax Sampling (SSMS), Stochastic Power Sampling (SPS), Stochastic Soft-rank Sampling (SSRS), and Top-k sampling along with batch sizes k E (1, 5, 10). When SSMS or SPS are used for the sample selection, DMLE leads to better test accuracy results for all datasets and batch sizes except for the Iris dataset with = 1. In the case of SSRS as the sample selection strategy, test accuracics are either improved when DMLE is employed as the method of model parameter estimation or competitive with IMLE,, except for FMisst with k = 1 or k = 10. For Top-k selection, DMLLE either outperforms leads to
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comparable results with IMLE in all cases except Emnist with l: 5, FMnist with & 1, and Cifar-10 with k = 1. The average test accuracy improvements of 9.6%, 7%, 3.9%, and 11.8% are observed for SSMS, SPS, SSRS, and Top-k sampling strategies respectively, which underline the significance of sample dependencies in model parameter estimation. Using different values for the sample selection size yields average test accuracy improvements of 6%, 8.6%, and 10.5% for = 1, = 5, and k 10 respectively. We observe performance degradation as the sample selection size k increases for both DMLE and IMLE. We conjecture that this drop occurs due to the reduced diversity in the collected samples, where similar samples might have been collected in active learning cycle. Even though we observe trend for the test accuracy to drop when k is increased, the reported average accuracy improvements for different & values suggest. that DMLE leads to better results than IMLE. The experiments in Appendix F) further underline the importance of using DMLE in active learning for model parameter estimation and its impact on model performance.
SPLL
DNLA
Actio
IRIS-SSMS
TRIS-SPS
TRIS-SSRS
IRIS-Top-
MLL
BILL
SVHN-SSMS
SVHN-SPS
SVHN-SSRS
SVHN-Top-k
Actind Leantrn Cydee
Aust Leerney Ortn
Actied Learney Cyde
REUTERS-SSMS
REUTERS-SPS
REUTERS-SSRE
REUTERS-TopH
OPLE
DNLL
Acve Laming Tiny Imagenet-SSMS
Adha Leining
Adh Leaning
Acteeaating Opta
Tiny Imagenet-SPg
Tiny Imagenet-SSRi
Tiny Imagenet-Topp.l
Figure The average test accuracy comparison with #1 standard deviation for DMLE and IMLE over cycles for Iris, SVHN, Reuters, and Tiny ImageNet datasets for different sample selection strategies, namely Stochastie Softmax Sampling(SSMS), Stochastic Power Sampling(SPS), Stochastie Soft-rank Sam- pling(SSRS), and Top-k Sampling where sample selection size k = 1 for all except Tiny ImageNet and k 5 for Tiny ImageNet.
To analyze the accuracy evolution over the active learning cycles we plot the mean test accuracy +1 standard deviation for DMLE and IMLE using four sample selection strategies with entropy acquisition function. Figure X shows such curves for the Iris, SVHN, Reuters, and Tiny ImageNet datasots where the sample selection size k for the first three and i 5 for Tiny ImageNet, with the remaining four datasets (MNIST, Fashion-Mnist, Emnist, and Cifar-10) provided in Appendix F In the first row of Figure 3 the test
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accuracy performance for the Iris dataset after 100 cycles of collecting one sample at each cycle is shown for different sample selection schemes. The initial unlabeled set size for the Tris dataset is 110 which means after 100 cycles, we collected almost all of the samples in the unlabeled set. Thus, it is observed that DMLE and IMLE exhibit similar performance in the later cycles, while DMLE yiclds better test accuracy in the earlier cycles, demonstrating its advantage within the active learning framework. In the second and third rows of Figure g
the average test accuracy plots for 500 cycles on SVHN and Reuters datasets are given. The improvement gained by accounting for sample dependencies during model parameter estimation with DMLE is evident in all plots, regardless of the sample selection strategy employed. In the last row of Figure & the performance difference between DMLE and IMLE is observable across all selection strategies for the Tiny ImageNet dataset, with DMLE showing a particularly greater improvement when Top-k selection is used.
k=5
OMLE
PALL
DNIC
Actve Learing Cydea MNIST-SSMS
Actve teernng
Adha Learbng Sycke
Actine Laarnng Eydle
MNIST-SPS
MNIST-SSRS
MNIST-Top--
k-10
Actos Lauring Ori
Riss Leurny Cyea MNIST-SPS
Levske Cyck
Lestin Syda
MNIST-SSMS
MNIST-SSRS
MNIST-Top-k
The experiments comparing DMLE and IMLE using the clustering- based Coreset approach with different sample selection Prrategies-SSMS. SPS, SSRS, and Top-k Sampling-aro presented. The average test accuracy plots with 1 standard deviation over cycles are shown, with sample selection size k = 5 for the first row and = 10 for the second row. The plots demonstrate that DMLE outperforms IMLE, particularly in the earlier cycles, consistent with our observations using uncertainty-based acquisition functions.
Figure 4:
In Figure provide the accuracy plots for the Core-set acquisition score, which is diversity-baset clustering approach for sample selection where k 5, 10. It can be observed that, across all sample selection strategies combined with this acquisition function, DMLE consistently outperforms IMLE from the carlier cycles onward. This demonstrates that taking into account the dependency term during model parameter estimation an important problem regardless of the acquisition function and can be applicable to any active learning setup as long as acquisition scores/ranks can be obtained.
Hypothesis testing: We used the Wilcoxon signed-rank test (Conover 1971] compare the performance of DMLE and IMLE, based on 8 random seeds per experiment for the experiments shown in Table 2 Table 4 and Table As this non-parametric test. does not, assume normality, it is suitable for cases where the differences between paired observations may not follow a normal distribution, such as accuracy values in active learning experiments with varying random initializations. Our results show that DMLE outperforms IMLE in 92% of these experiments, with statistical significance (p < 0.05). This suggests that DMLE outperforms IMLE significantly, providing strong evidence of its effectiveness improving model accuracy by accounting for sample dependencies.
6 Conclusion and Future Works
In this paper, we introduced Dependency-aware Maximum Likelihood Estimation (DMLE) for the active learning framework, novel approach that highlights the incompatibility of the i.i.d. data assumption in
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conventional parameter estimation methods with the active learning paradigm. Specifically, DMLE preserves the sample dependency that would otherwise be overlooked in conventional independent MLE (TMLE), ensuring they are properly taken into account in the estimation. We examined the impact of this dependency term across various acquisition functions, sample selection strategics, and sample sizes, evaluating its effect on multūple benchmark datasets with varying model complexities. Our experiments revealed & significant improvement average test accuracy when using DMLE compared IMLE, underlining the critical role of accounting for sample dependencies in model parameter estimation.
A promising direction for future work is the theoretical characterization of the dependency term's distribu- tion under Top-k selection strategies, beyond the current empirical analysis we provide. This would lead to deeper insights into the approximation behavior of DMLE and help identify scenarios where its benefits may diminish. Additionally, examining DMLE's performance across varying uncertainty and diversity conditions represents a valuable line of research, with potential to inform improvements the acquisition process. Another promising direction is exploring connections between DMLE and submodular optimization frame- works for large-batch selection. Prior work has shown that submodularity can e guide the selection of informative and diverse subsets under efficiency constraints (Golovin & Krause 2017). Adapting DMLE within such combinatorial framework may help scale its benefits to larger selection sizes while maintaining tractable optimization.
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Appendix
A Stochastic Batch Selection
Recently, Kirsch al. [2021) have proposed Stochastic Batch Acquisition probabilistic sampling strategy in active learning, which is known to outperform conventional Top-k sampling while avoiding additional com- putational expenses. Their approach includes three different stochastic batch acquisition methods depending on the perturbation method they use: Softmax, Power, and Soft-rank acquisition where each corresponds to sampling from the corresponding probability distribution given in Table [ We generate the sample set S.+1 by assuming we sample from a probability distribution without replacement when we make batch selection, Under this assumption, the probability that sample the sequence of elements St+11 this order depends the stochastic sampling method and defined as follows:
[31, 22,--. .... , in
Softmax acquisition
P(s++1 Dc:0)= Zi
II
Dolz..D.))) where Z1
Σ
Da(2,D.,9),
1=1
ZEU
Power acquisition
P(s++1 D2:0)
1 II
a Dy, where Z₁
Σ
alx, Dy,
Z,
==1
zEU, (it
Soft-rank acquisition
k
P(S Dr:0)
I1
r where Ze, =
Σ
B-
Ze
xEUi MFI....,;;
1
where for all acquisition methods the normalization constant Z, = II Ze,i r is the descending ranking
of the acquisition scores a(z, Dr,0), with the smallest rank being 1, and B temperature parameter: the higher it is, the closer the selection
Top-k.
Note that this probability characterizes the sequence of elements sample, from (unordered) set S.11- We also denote by
8 to distinguish it
d [[x1, v1), (22, (IKK, wk)]
the corresponding sequence of collected tuples of data points and labels. Morcover
P(de+1 D); P(S D2:0)
II P( xz0)), ==1
(12)
where Do =
We couple our method DMLE with these sample selection procedures. We explicitly model sample de- pendency and implement it via sampling from the probability distributions given in Table [ to model P(S, D7-1;0) term which leads to the objective functions for the following stochastic sampling strate- gies:
Softmax acquisition
++1 &
Bata, D. 11
DMLE arg max
P(y 2),4;0) + In
(13)
Zr.i
++
arg max
Σ
In P(y x: 0)
In
(14)
Z
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Power acquisition
t-1
a(FT,i, 17-1, 0)
@DMLE [+1
arg max
In P(Ur,
[r,s; + In
(15)
Zr
T=1
k
a 0)
arg max #
Σ (2.0)EDi41
In P(y x; +
In
(16)
Zz
t=1
Soft-rank acquisition
++1
Tr,
= max
In P(ur.i 0) + In
(17)
ZT,
2m
arg E (a.))D
In P x; +
In
r 7,
(18)
Zz,6
Thus, depending on the sampling strategy follow in active learning, the dependency term that ap- pears in the model parameter estimation objective varies. For the active learning framework that uses the Stochastic Softmax acquisition function, equation 14 utilized for the model parameter estimation objective, while equation 16 is used for the Stochastic Power acquisition function, and equation 18 for the Stochastic Soft-rank acquisition function.
Evaluation of the normalization constant: To analyze the relevance of the Z term in equation [4 equation 16 and equation 18 experimented with the MNIST dataset where the trend for the term over the active learning cycles is observed. Figure shows the evolution of this term for different sample selection strategies. Notably, the resulting curves are nearly constant for all. These results suggest a simplified optimization problem where the normalization constant is neglected, which is followed for the experiments in this manuscript.
DMLE/SSMS
DMLE/SPS
DMLE/SSRS
SMap/igp--Samplig
Active Learning Cycles
Figure 5: The change of the term Et= In(2,) in the model parameter estimation objective function through the active learning cycles for the MNIST dataset. One can note that the term changes marginally over the cycles which motivales the elimination of this term from the model parameter estimation while taking into account the computational expenses it introduces into the process.
B Independent Maximum Likelihood Estimation (IMLE) Derivation in Active Learning
Due to the sequential collection of the labeled training data, we can write the likeliioo formula with the cycle notation follows:
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@MLE ++1
arg max 0
In P(eD, Dr-1:0)
(19)
arg
In P( Mr,ks..... 2 Nr.1 Dr 15
(20)
With the application of product rule:
++1
arg max
In P.y Urr TT,k.... ,r.k,,..,,,,,111 (r-1,, + in P( , D- 1:01
(21)
0
By relying the modeling assumption of fo y::
++1
MLE ++1
arg max
Σ In P(rr 2r.; 0) + In P( Fr, 1)-1;0)
(22)
As S is the samples selected in cycle for labeling, ( 27,11 can be written follows:
Z+I Σ
MML = arg
Σ
In P(yra 27,1;0) + In P(Sr Dr-1i0)
(23)
6
The i.i.d. sample assumption in conventional Maximum Likelihood Estimation causes In P(S, term to be a constant during the estimation which leads the climination of that term as follows:
D)-1:0)
MLE 4+1
arg max
Σ In P(yra 27, 0)
(24)
arg max
Σ
In P(y 12:
(25)
C Proofs
C.1 Proof of Theorem 1.
The Fisher Information matrix quantifies the precision of parameter estimates, where variance is inversely related to Fisher Information. For simplicity in the proof, we consider sample selection size of & = 1 at each cycle, though the result can be extended to larger k.
For IMLE:
I IMLE (8) = Σ In P(yt x);0),
(26)
where are labels, and are inputs, and 0 is the parameters.
For DMLE::
DMLB = Σ In P(yr x);0) + In P(xr Dr-1;0)),
(27)
T=1
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where P(z+ D)-1:0) accounts for sample dependency.
The individual Fisher Information terms are given by:
0²
(0) E
In P(u+ 2r:00]
(28)
0000T
0²
(0) E
In P(x+ 1r)
(29)
d0a0r
The Fisher Information matricos for IMLE and DMLE are:
T(QINLE) Σ (0),
(30)
I(ODMLE) Σ
1:00 + (0)
(31)
7=1
Matching corresponding terms, we see:
[(0DMLE) = I(@IMLE) + (0).
(32)
Since (0) is positive semi-definile:
T(@DMLE) I(BIMLE).
(33)
The Cramér-Rao Lower Bound (CRLB) indicates:
2 Cov(0)=1(0))1
(34)
Since T(@DMLE) = T(@IMLE):
Cov(@DMLE) ((MLE)
(35)
Thus, DMLE yields lower variance guarantees and more precise estimates compared to IMLE.
C.2 Proof of Theorem 2.
The expected logelikelihood of the learned model over the true data distribution pirue is given by:
y/) peroue In L(0)] = (o' (orr)))))m) In P(w 0z':0) + In P(x' D.:00)
(x''
(36)
where 0 = arg max In P(D++1;0) arg max E(E,)) In P(y 1;;0) + E+t Exes, In P(r|
Expanding the expectation:
Σ p(d) ((d)EPG P(y 2x') In P(y x²;0) + In P(x') 0r:
(37)
Rewriting as separate summations:
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Σ P(d') Σ P(y' | P(y |x:;0) In P(a DG:0).
(38)
Using KL divergence and entropy identities:
(PuL(P(y x') P( 1z);6) + HPP (2)) - DKL(P(a') || P(d' D;:0)) + H((())))
(39)
Since H(P(y' xx')) and H(P(x') are constants, they can be omitted:
(f pi In [(i) = - DKI. (PW x²) II P(y/ (z'00) DKI(P(a') || P(d' D2;)).
(40)
This result shows that maximizing the expected log-likelihood is equivalent to minimizing the sum two KL divergences, ensuring that both the predictive and data distributions of the model remain close to the true distributions.
Comparison with Farquhar et al. (2021)
We conducted experiments using the MNIST dataset with entropy as the acquisition function and various sample selection methods. For model parameter estimation, we utilized IMLE, DMLE, and the statistical bias mitigation approach proposed by Farquhar et al. (2021). The results are consistent with the findings of Farquhar et al. (2021), which show that mitigating bias with their proposed method leads to performance degradation with neural networks compared to conventional MLE (IMLE) in active learning. In contrast, our Dependency-awar MLE (DMLE) enhances model performance.
DPLL
DPLL
DALL
DALL
Actiee Louning Ore
Letese Oce
Leart Crck
Actve Acive Cycki
MNIST-SSMS
MNIST-SPS
MNIST-SSRS
MNIST-ToP-R
Figure 6: The experiments comparing DMLE and IMLE with statistical bias mitigation method proposed by Farquhar et al. (2021) where different sample selection strategies StrategiesSSSMS SPS, SSRS, and Top-k Sam- pling are employed. The average test. accuracy plots with +1 standard deviation over cycles are shown for sample selection size k = 1. The plots demonstrate that DMLE outperforms the bias mitigation method proposed by Farquhar et al. (0221)
E Experimental Timing Results
To demonstrate that DMLE does not impose significant computational burden during training, measured clapsed times using timer. Table 3 presents the mean clapsed times for DMLE versus IMLE, combined with various sample selection functions and entropy as the acquisition function, with selection sizes of k=1, k=5, or k=10 on the MNIST dataset. We selected MNIST due to its large unlabeled pool of 30,000 samples. In the table, the additional time required to include dependency with DMLEE is negligible compared to the overall elapsed times. Additionally, the mean extra time due DMLE is 1.88 minutes across all combinations, indicating that the time complexity remains largely unaffected.
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Table 3: Comparison of mean elapsed times for DMLE and IMLE with different sample selection sizes k and number of cycles T on the MNIST dataset (30000 samples in the unlabeled pool at the beginning). The table shows the mean elapsed times over 8 experiments per combination for A = 1, k 5, and A: = 10. On average, DMLE causes an additional 1.88 minutes of elapsed time compared to IMLE for Stochastic Softmax Sampling (SSMS), Stochastic Power Sampling (SPS), and Stochastic Soft-rank Sampling (SSRS).
k
T
Sampling Strategy
DMLE Time
IMLE Time
SSMS
465:30.31 min
462:50.19 min
1
2500
SPS
473:30.78 min
471:19.08 min
SSRS
477:17.79 min
467:51.34 min
SSMS
92:58.25 min
92:28.56 min
5
500
SPS
92:54.06 min
91:12.24 min
SSRS
92:28.00 min
92:48.54 min
SSMS
46:03.89 min
45:57.16 min
10
250
SPS
48:00.51 min
48:52.36 min
SSRS
49:03.53 min
49:18.87 min
As the difference between DMLE and IMLE lies in the objective function, we also plot the loss processing times per cycle better observe the computational expense introduced by the dependency term in DMLE, shown in Figure 7 This experiment, conducted on the MNIST dataset collecting 500 samples with sample selection size &- 1, shows that the loss processing time increases for both methods across cycles as more samples are labeled. Although the difference between DMLE and IMLE grows over time, it remains relatively small practice.
0.07
Loss Processing
0.05
0.04
0.03
DMLE
IMLE
0.01
100
200
300
400
500
Active Learning Cycles
Figure 7: Per-cycle loss processing time for DMLE and IMLE during active learning on MNIST (500 cycles, sample selection size = 1). This plot illustrates how the computational cost evolves across cycles, em- phasizing the impact of DMLE'ss dependency-aware loss formulation. While both methods exhibit incrcased processing times as the labeled grows, DMLE shows steeper rise due the additional overhead of mod- eling dependencies. Nonetheless, within the context of active learning, where the objective is to achieve high performance with a limited number of labeled samples, the overall runtime remains practical, demonstrating that the additional computational cost of DMLE is manageable in typical active learning scenarios.
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F
Additional Results
To further analyze the importance of taking the sample dependencies into account during model parameter estimation in active learning, we plot the test accuracy plots with #1 standard deviation for four more datasots, c.g., Mnist, Fashion-Mnist (FMnist), Emnist, and Cifar-10, where for each we utilize four different sample selection stralegies explained in Section 3.2 In Figure g we observe the performance improvement with DMLE when compared with LMLE in most cases. For the cases where DMLE doesn't outperform IMLE, we believe further hyperparameter tuning for 3 might lead to better performance. For all experiments here,
use B 1 suggested by Kirsch et al. 22011)
DKL 1ML6
OPLE MEL
DNL
OMLL
X0
23 MNIST-SSRS
MNIST-SSMS
MNIST-SPS
MNIST-Top-.
1P11
IDE
16
18
46
24
16
CIFARIO-SSMS
CIEAR1O-SPS
CIFAR1O-SSRS
CIFARIO-Top-k
Paa A00LEEC
19.6
Cyees
(1) FMNIST-SSMS
FMNIST-SPS
(k) FMNIST-SSRS
(I) FMNIST-Top-E
Loaniniinn Syone
Adle Loaring Cyoes
Aclessentin Kyons
(m) EMNIST-SSMS
(n) EMNIST-SPS
(0) EMNIST-SSRS
(p) EMNIST -
Figure 8: The average Lest accuracy comparison with 11 standard deviation for DMLE and IMLE over cycles for Mnist, Cifar-10, FMnist, and Emnist datasets for different sample selection strategies, namely Stochastic Softmax Sampling(SSMS), Stochastic Power Sampling(PSS), Stochastic Soft-rank Sampling(SSRS), and Top- k Sampling where sample selection size is k = 1. In the earlier cycles, DMLE was observed to outperform IMLE in all cases except for FMnist with SSRS and Top-k which is highly advantageous in active learning.
In addition to providing average test accuracy results in Table for multiple datasets with different & values used for sampling al each cycle until 100 samples are collected, we also plot. the accuracy improvement over cycles for the Iris dataset to better observe the importance of DMLE. In Figure g we observe that when k 1, the curves for DMLE and IMLE tend to converge to the same accuracy. However, with and k 10, using DMLE becomes significantly more important, as the resulting accuracy values for DMLE and
IML.E diverge significantly.
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OPIL
OPIL
Actve Leaming Gycha
170
Adive Leuring Oyc
idha Laaminggyc
Active Leariig Oycaa
IRIS-SSMS (k=1)
IRIS-SPS (k=1)
IRIS-SSRS (k=1)
TRIS-Top-k (k=1)
DRLA
Aetvo Laanin Sycles
Adiv Liurnng Cycat IRIS-SPS (k=5)
Activ Leaving Sycle IRIS-Topk-k (k=5)
IRIS-SSMS (k=5)
IRIS-SSRS (k=5)
Main OPLL
DARL
BMLE
Actire Oycla IRIS-SSMS (k=
Aths Leaniig Cyca
Laurnng Oytkr
Actie Learing Cycks IRIS-Top-k (k=10)
IRIS-SPS (k=10)
IRIS-SSRS (k=10)
The average test accuracy comparison with +1 standard deviation for DMLE and IMLE over cycles for Iris for different sample selection strategies, namely Stochastic Softmax Sampling(SSMS) Stochastic Power Sampling(SPS). Stochastic Soft-rank Samplig(SSRS), and Top-k Sampling where sample selection sizes k = 1, k 5, and h 10. As the sample selection size increases, the significance of using DMLE over IMLE becomes more acute.
Figure 9:
Table A is based on the same experimental sctup as Table 2 where the only difference is the acquisition function used for the experiments. For these results, we use the BALD acquisition function. The average test accuracy improvements resulting from using DMLE over IMLE for different sample selection schemes are 3.5%, 4.1%, 3.8%, and 5.5% for SSMS, SPS, SSRS, and Top-k respectively. Moreover, for k == 1, k: 5, and k- 10, the average test accuracy improvements of 7.6%, 5.2%, and 3.8% are attained when DMLE is employed. Both with different sample selection strategies and sample selection sizes, it is notable that taking the sample dependency into account via DMLE during model parameter estimation elicits better average test accuracy performance when compared with IMLE.
In Table 5 we use the same experimental setup as Table 2 with Least Confidence as the acquisition function. The average test accuracy improvements obtained when DMLE used instead of IMLE for different sample selection strategies are 4.6%, 3.7%, 2.3%, and 3.8% for SSMS, SPS, SSRS, and Top-k respectively. When experiment with different sample selection sizes for sample acquisition at each cycle, we average test accuracy improvements of 1.6%, 4.9%, and 5% for k = 1, k = 5, and k = 10 respectively. Both experiments with different sample selection strategies and sample selection sizes demonstrate the importance of accounting for sample dependencies during model parameter estimation in active learning.
G Impact Statement
The method we propose may impact fields where acquiring labeled data is scarce, dangerous, or costly by allowing better performance with less data. This is the case in many problems in the medical, bioinformatics, remote sensing, and materials fields, to name but few.
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Table 4: Comparison of mean classification test accuracies =1 standard deviation for different sample selec- tion sizes (k) during cycles and different sample selection strategics where number of samples in the labeled set Nr.- = 100. We use BALD for the acquisition function and SSMS (Stochastic Softmax Sampling), SPS (Stochastic Power Sampling). SSRS (Stochastic Soft-rank Sampling), and Top-k sampling for the sample selection strategy. The bold text highlights the higher mean accuracy and lower standard deviation when
comparing DMLE and IMLE under various sampling schemes.
SSMS
SPS
SSRS
Top-k
Dataset
DMLE
IMLE
DMLE
IMLE
DMLE
IMLE
DMLE
IMLE
0.77 T 1.02
0.77 1 0.03
0.77 0.03
0.78 1 0.03
0.82 1 0.02
0.80 1 0.03
0.80 1 0.05
0.80 1 0,04
MNIST
0.76 0.01
0.75 0.02
0.76 1 0,03
0.73 1: 0.03
0.79 1: 0,03
0.78 :: 0.02
0.80 1 0.03
0.79 1: 0.02
10
0.76 + 0.03
0.76 + 0.03
0.76 + 0.04
0.75 + 0.02
0.79 0.03
0.76 + 0.02
0.79 + 0.03
0.78 + 0.02
0.36 0.02
0.36 +: 0.03
0.37 +: 0.02
0.35 + 0.04
0.36 = 0.03
0.36 = 0.03
0.36 + 0.03
0.34 0.01
EMNIST
0.38 +.0
0.38 + 0.02
0.36 + 0.02
0.37 = 0.02
0.35 = 0.02
0.35 = 0.02
0.36 = 0.03
0.37 = 0.04
10
0.36 0.03
0.36
0.37 1 0.03
0.35 1 0.03
0.36 1 0.03
0.38 1 0.02
0.38 1 0.03
0.36 1 0.04
0.51 +:1:000 0.02
0.49
+
0.48 + 0.02
0.51 0.02
0.50 = 0.03
0.50 + 0.01
0.48 + 0.04
REUTERS
0.51 0.03
0.51 + 0.03
0.51 0.01
0.51 + 0.01
0.49 0.03
0.48 + 0.01
0.48 + 0.01
0.47 + 0.01
0.48 0.04 10
0,47 + 0.04
0.48 : 0.02
0.47 + 0.01
0.46 = 0.01
0.45 = 0.03
0.44 + 0.01
0.42 = 0.04
0.03
0.26
0.03
0.25 0.02
0.30 0.02
0.28
0.05
0.04
0.22 0.04
SVHN
0.24 0.02
0.003
0.04
0.23 0.02
0.25 0.03
0.25
0.23
0.23 0.02
10
0.22 0.03
0.22
0.24 0.03
0,23 0.03
0.23 0.05
0.23 = 0.04
0.22 + 0.03
0.21 + 0.03
0.73 +73++00 0.03
0.02
0.70 + 0.03
0.69 + 0.02
0.72 + 0.04
0.73 + 0.02
0.70 + 0.04
0.72 + 0.04
FMNIST
0.70 0.00 0.03
0.70 + 0.03
0.73 0.02
0.73 + 0.02
0.72 0.02
0.69 0.05
0.00
0.00
0.00
0.71 + 0.03
0.72 0.02
0.73 0.02
0.72 0.04
0.69 0.05
0.24 0.02
0.23
0.25 1: 0.03
0.24 +: 0.02
0.24 1: 0.04
0.22 1 0.02
0.23 1 0.03
0.21 1: 0.03
CIFAR-10
0.23 0.04
0.19
0
0.19 0.02
0.21 0,02
0.18 = 0.02
0.22 = 0.02
0.19 0.04
10
0.20 0.02
0.18
0.19 0.03
0.04
0.20 0.03
0.16 = 0.03
0.20 + 0.03
0.17 + 0.02
0.95 0.002
0.95 0.02
0.95 1: 0.02
0.96 1 0.02
0.95 1 0.02
0.95 1 0.02
0.97 1 0.01
0.96 + 0.01
TRIS
0.94 + 0.02
0.86 0.03
0.90 + 0.02
0.83 + 0.05
0.94 + 0.02
0.86 + 0.02
0.94 + 0.03
0.86 + 0.06
10
0.85 + 0.03
0.80 0.03
0.80 + 0.03
0.78 + 0.02
0.84 + 0.01
0.78 + 0.03
0.80 + 0.04
0.78 + 0.03
H
Dataset/Model Licenses and Sources
All datasets we use are publicly available. Our experiments include Mnist. and SVHN datasets both dis- tributed under the GNU General Public License. Reuters-21578 collection resides with Reuters Ltd. where Reuters Ltd. and Carnegie Group, Inc. allows free distribution of the dataset for research purposes only, Additionally, we utilize Emnist, Fashion-Mnist, Cifar- 10, and Iris all with MIT License. We download Mnist, Reuters-21578, Fashion-Mnist, Cifar-10, and Tris from Keras datasets repository; Emnist from Tensorflow datasets repository, and SVHN from cls/ncural networks we use, is distributed under MIT License whereas ResNet-50 under Apache License 2.0.
hhousnumberss//// htt.ssannnrdded///oosnuumaess/// LeNet, one of the mod-
Ablation Study for Dependency Approximation under Top-k Sampling
The primary motivation for choosing the Stochastic Softmax approximation is its computational simplicity. As shown in equation [4] this approximation enables the reuse of already computed acquisition scores without requiring additional operations. In contrast, the Stochastic Power method (equation 16) involves computing the logarithm of the scores, while the Stochastic Soft-rank method (equation 18) requires both sorting and logarithmic computation. These additional steps introduce extra computational overhead during query selection.
To assess the practical impact of this choice, we conducted empirical comparisons using these distributions to model the dependency under Top-k selection. In Figure 10 we present results on the MNIST, Iris, Reuters, and Ting-Im datasets, which were selected to reflect varying data modalities, complexities,
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Table 5: Comparison of mean classification test accuracies =1 standard deviation for different sample selec- tion sizes (k) during cycles and different sample selection strategics where number of samples in the labeled set Nr.- 100. We use Least Confidence for the acquisition function and SSMS (Stochastic Softmax Sam- pling), SPS (Stochastic Power Sampling), SSRS (Stochastic Soft-rank Sampling), and Top- sampling for the sample selection strategy. The bold text highlights the higher mean accuracy and lower standard deviation
when comparing DMLE and IMLE under varions sampling schemes.
SSMS
SPS
SSRS
Top-k
Dataset
DMLE
IMLE
DMLE
IMLE
DMLE
IMLE
DMLE
IMLE
0.80 1 0.05
0.77 1 0.03
0.78 1 0.03
0.78 1 0.03
0.78 1 0.03
0.79 1 0.03
0.78 1 0.04
0.72 1 0.04
MNIST
0.79 1 0.03
0.77 +: 0.02
0.77 0,04
0.76 4: 0.03
0.77 1: 0,02
0.74 1: 0.04
0,68 1 0.06
0.66 +: 0,05
10
0.77 + 0.03
0.76 + 0.03
0.76 + 0.02
0.77 + 0.03
0.75 0.04
0.73 + 0.05
0.62 + 0.08
0.54 + 0.07
0.35 0.00
0.35 + 0.03
0.37 + 0.03
0.37 0.01
0.37 = 0.02
0.35 = 0.04
0.31 + 0.02
0.33 + 0.02
EMNIST
0.37 +
0.36 + 0.03
0.35 0.03
0.34 = 0.04
0.36 = 0.04
0.35 = 0.02
0.30 = 0.08
0.30 = 0.03
0.36 0.00 10
0.37 0.03
0.36 0.02
0.37 1 0.03
0.35 1 0.03
0.34 1 0.04
0.27 1 0.02
0.27 1 0.02
0.50 0.03
+.49
0.49 0.49 0.02
0.48 + 0.03
0.48 + 0.03
0.49 - 0.02
0.40 + 0.05
0.40 0.02
REUTERS
0.53 + 0.02
0.51 + 0.03
+.52+0000
0.52 + 0.01
0.48 0.03
0.46 + 0.04
0.41 + 0.07
0.39 0.08
10
0.51 +: 0.03
0,47 + 0.04
0.50 0.02
0,48 + 0.02
D.44 0.07
0.46 = 0.04
0.33 : 0.10
0.30 + 0.09
0.04
0.28 + 0.02
0.29 0.02
0.03
0.04
0.21 0.03
0.23
SVHN
0.02
0.23 0.04
0.03
0.23 0.03
0.04
0.22
0.04
0.02
0.04
0.23 0.03
0.23
0.23 0.03
0.03
0.03
0.21 0.04
0.20 - 0.02
0.20 - 0.02
0.20 0.02
0.72 + 0.01
0.72 0.03
0.71 + 0.02
0.73 + 0.01
0.70 + 0.04
0.71 + 0.03
0.63 + 0.04
0.62 + 0.06
FMNIST
0.71 0.04
0.02 0.00
0.72 0.01
0.71 0.03
0.73 0.04
0
0.53 0.06
0.04
0.70
0.03
0.71 + 0.03
0.69 0.04
0.70 - 0.03
0.52 + 0.06
0.51 + 0.08
0.23 1: 0.02
0.23
0.24 1: 0.03
0.23 +: 0.02
0.22 +: 0.02
0.21 1: 0.02
0.20 1 0.05
0.20 1 0.03
CIFAR-10
0.22 0.04
0.23 0.02
0,20 0.03
0.20 0,02
0.19 = 0.03
= 0.19
0.18 = 0.02
10
0.21 + 0.02
0.08
0.21 +
0.18 0.02
0.17 0.01
0.17 = 0.03
0.18 = 0.03
0.17 + 0.03
0.95 0.002
0.96
0.96 0.02
0,95 + 0.02
0.97 1 0.01
0.97 1 0.01
0.96 1 0.01
0.95 + 0.02
TRIS
0.94 + 0.01
0.87 0.02
0.94 0.01
0.86 + 0.03
0.94 + 0.01
0.88 + 0.04
0.91 + 0.07
0.81 + 0.06
10
0.91 + 0.03
0.80
0.95 + 0.02
0.80 + 0.03
0.85 + 0.03
0.79 + 0.02
0.87 + 0.06
0.76 + 0.06
and sample selection sizes. The results show that the Stochastic Softmax consistently outperforms the other approximations in terms of average test accuracy under Top-k sampling.
This empirical evidence supports the choice of softmax-based approximation, highlighting its efficiency and effectiveness.
5585
5005
Acter
Lyoningtycet
teaning
MNIST
IRIS
REUTERS
TINY IMAGENET
Figure 10: Comparison of average test accuracy (#1 standard deviation) using different distribution ap- proximations for the dependency term with Top-k selection: Stochastic Softmax (SSMS), Stochastic Power (SPS), and Stochastic Soft-rauk (SSRS). These methods were evaluated during model parameter estimation on MNIST, Tris, Reuters, and Tiny ImageNet datasots. The results demonstrate the effect of each approx- imation across diverse data modalities and complexities, with sample selection sizes of k for the first three datasets and l = 5 for Tiny ImageNet.
28

---

Published in Transactions on Machine Learning Research (09/2025)
J Glossary
This scction presents a glossary of variables and notations used throughout the paper, provided in Table 6
Term / Symbol
Definition
X C Rd
Input feature space of dimension dg-
y (I...., K
Label space with possible classes. Input sample from the feature space X.
Corresponding label of sample x, belonging to y
U. (x))
Unlabeled dataset at cycle containing N2 samples.
D. (1))))
No
Labeled dataset at cycle I, containing NÊ labeled samples.
N
Number of unlabeled samples at cycle t. Number of labelod samples cycle t.
NN
Se+1
Batch of samples selected for labeling at cycle t.
E Redo
Model parameters of dimension dg-
a(x,
Acquisition function assigning score an unlabeled sample based labeled data D, model parameters 0.
fo : X + 10,
Prediction function (e.g., deep neural network) parameterized by 0, outputs scores for K classes.
Current cycle.
Table 6: Glossary of terms and symbols used in the active learning framework.
29


Reviews:
[{"review_id": "FdnE38BXJf", "reviewer": "Reviewer_b9XG", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "S1. Well-formulated problem from a specific MLE-aware perspective of active learning\nS2. Well-established theorems to provide deep insight on the problem and the author's solution (pls refer to the summary of contributions)\nS3. Extensive evaluation to understand the effectiveness of proposed algorithms under various controllable simulation with real-world datasets.\n\nMy major concern is the relevance of this work to the lanscape of current machine learning research. The concern regarding the applicability of batch-based active learning with the reported experimental setup to the frontiers of the domain is quite pertinent.\n\nW1. The experiments, while covering a variety of data types, primarily utilize datasets (e.g., MNIST, Iris, Cifar-10) and network architectures (e.g., LeNet, small MLPs) that, while standard for benchmarking certain algorithms, are modest in scale compared to the massive datasets and extremely large models that characterize much of cutting-edge AI research today. The ResNet-50 and ViT-B/16 are more contemporary, but the overall experimental scope might not fully capture the dynamics at the current AI frontier.\n\nW2. The query batch sizes ($k=1, 5, 10$) are relatively small. In large-scale applications, the cost of retraining or fine-tuning enormous models might necessitate much larger batch queries to be practical, or different paradigms altogether.\n\nW3. While the paper analyzes time complexity and reports manageable additional costs for DMLE on the tested scales, the computational implications of the dependency term, especially the calculation or approximation of $P(x|D_{\\tau-1};\\theta)$ and the additional summation in the loss, could become more pronounced with vastly larger datasets and more complex models. The authors do simplify the normalization constant $Z_{\\tau,i}$ based on empirical observations on MNIST to manage computational expenses.\n\nI am considering leveraging a pre-trained VLM combined with prompt engineering as a method to efficiently generate high-quality annotations for all unlabeled images. This approach could significantly streamline the annotation process while maintaining accuracy. Additionally, models like CLIP could be utilized to assign pseudo-labels to the unlabeled data. These pseudo-labels can serve as valuable inputs for supervised or semi-supervised machine learning tasks, potentially enhancing overall model performance and achieving better outcomes in downstream applications.\n\nAfter all, the concept of active learning was originally founded on the premise that high-quality data annotation is both time-consuming and expensive, making it essential to carefully select an optimal subset of data for annotation. In the current machine learning landscape, however, this challenge has likely evolved to a more complex level. Relying on traditional batch-based sampling methods applied to relatively small and straightforward classification datasets may no longer be sufficient to address the demands of modern, large-scale, and diverse datasets. You should consider something that the pre-trained models (which are almost free) cannot do.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces a thoughtful approach to a subtle yet important aspect of the active learning process. Active learning aims to build an efficient labeled training set by strategically selecting samples for annotation. This paper focuses on the model parameter estimation stage within this iterative process.\n\nThe primary technical contribution of this work is the introduction of \\emph{Dependency-aware Maximum Likelihood Estimation (DMLE)}, a novel method for model parameter estimation within the active learning framework.The authors identify a fundamental incompatibility: Active learning is an iterative process where each queried sample influences subsequent selections, thereby creating \\emph{dependencies} among the samples in the labeled set. However, conventional MLE estimator, a standard for model updates, assumes that data samples are \\textbf{independent and identically distributed (i.i.d.)}. This assumption is violated in the active learning context.\n\nDMLE aims to rectify this by explicitly accounting for these dependencies. The standard MLE objective (which the paper refers to as Independent MLE or IMLE when the i.i.d. assumption is applied) is typically:\n$\\hat{\\theta}_{t+1}^{\\mathrm{IMLE}} = \\argmax_{\\theta} \\sum_{(x,y) \\in D_{t+1}} \\log P(y|x;\\theta) $\n\n\nThis formulation arises from a more complete likelihood expression that includes a term for the probability of selecting the samples themselves:\n$ \\hat{\\theta}_{t+1}^{\\mathrm{MLE}} = \\argmax_{\\theta} \\left( \\sum_{(x,y) \\in D_{t+1}} \\ln P(y|x;\\theta) + \\sum_{\\tau=1}^{t+1} \\ln P(S_{\\tau}|D_{\\tau-1};\\theta) \\right) $\n\nThe term $P(S_{\\tau}|D_{\\tau-1};\\theta)$ represents the probability of selecting the batch of samples $S_{\\tau}$ given the previously labeled data $D_{\\tau-1}$ and model parameters $\\theta$. Under the i.i.d. assumption, this term is considered constant and thus dropped.\n\nThe technical advancement of DMLE lies in retaining and modeling this dependency term. The resulting DMLE objective function is:\n$$ \\hat{\\theta}_{t+1}^{\\mathrm{DMLE}} = \\argmax{\\theta} \\left( \\sum_{(x,y) \\in D_{t+1}} \\ln P(y|x;\\theta) + \\sum_{\\tau=1}^{t+1} \\sum_{x \\in S_{\\tau}} \\ln P(x|D_{\\tau-1};\\theta) \\right) $$\n\nA key challenge is that the distribution $P(S_{\\tau}|D_{\\tau-1};\\theta)$ can be difficult to express in a closed form. The authors cleverly address this by leveraging stochastic batch sampling strategies (like Softmax, Power, and Soft-Rank acquisitions as proposed by Kirsch et al., 2021), which have defined probability distributions for sample selection. For the common Top-k selection strategy, DMLE approximates this probability using the Softmax distribution.\n\nThe authors provide two theorems as theoretical support for DMLE. The first one posits that DMLE achieves a Fisher Information matrix $I^{\\text{DMLE}}(\\theta)$ that is greater than or equal to that of IMLE ($I^{\\text{IMLE}}(\\theta)$). According to the Cramér-Rao bound, this implies that DMLE can provide parameter estimates with lower variance ($\\text{Cov}(\\hat{\\theta}^{\\text{DMLE}}) \\le \\text{Cov}(\\hat{\\theta}^{\\text{IMLE}})$), leading to enhanced precision. The second one demonstrates that maximizing the expected log-likelihood with DMLE is equivalent to minimizing the sum of two Kullback-Leibler (KL) divergences. One KL divergence measures the difference between the true conditional label distribution and the model's estimate, while the second (the dependency term often ignored by IMLE) measures the discrepancy between the true data distribution $P(x')$ and the model's estimate of it based on actively selected data $P(x'|D_t; \\hat{\\theta})$. This ensures the model aligns better with both the predictive and the underlying data distributions.\n\nThe paper empirically evaluates DMLE against IMLE across several benchmark datasets, including image (MNIST, Fashion-MNIST, SVHN, EMNIST, Cifar-10, Tiny ImageNet), text (Reuters), and feature-based (Iris) datasets. Various acquisition functions (entropy, Core-set, BALD, least confidence) and sample selection strategies (Stochastic Softmax, Stochastic Power, Stochastic Soft-rank, Top-k) are employed.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please address the weakness I mentioned above.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "QLFKGpeX8S", "reviewer": "Reviewer_hMbu", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n\n1. **Novel Problem Formulation**  \n   The paper addresses a fundamental oversight in existing active learning literature: the incompatibility between the i.i.d. assumption of conventional Maximum Likelihood Estimation (MLE) and the sequential, dependent nature of data acquisition in active learning. This is a well-motivated and original contribution that fills a clear gap in methodology.\n\n2. **Principled Theoretical Framework**  \n   The proposed Dependency-aware Maximum Likelihood Estimation (DMLE) is theoretically grounded. The authors derive the dependency term \n   $$\n   \\sum_{\\tau=1}^{t+1} \\sum_{x \\in S_\\tau} \\log P(x | D_{\\tau-1}; \\theta)\n   $$\n   and prove that incorporating it results in a Fisher Information gain, leading to lower-variance estimators according to the Cramér-Rao bound:\n   $$\n   \\text{Cov}(\\hat{\\theta}_{\\text{DMLE}}) \\preceq \\text{Cov}(\\hat{\\theta}_{\\text{IMLE}})\n   $$\n\n3. **Generality and Practicality**  \n   DMLE can be applied with a variety of acquisition functions (entropy, BALD, Core-set) and selection strategies (Top-k, stochastic softmax, soft-rank, power sampling), making it widely usable in real-world active learning setups.\n\n4. **Strong Empirical Results**  \n   The method demonstrates consistent and significant improvements across 8 diverse benchmark datasets, with test accuracy gains of up to 10.5% compared to IMLE. The experiments are thorough, covering multiple acquisition functions, sampling strategies, and batch sizes.\n\n5. **Statistical Rigor**  \n   Results are backed by Wilcoxon signed-rank tests over multiple seeds, showing that DMLE significantly outperforms IMLE in 92% of settings, with p-values under 0.05.\n\n6. **Reproducible and Clear Experimental Protocol**  \n   The authors detail their implementation setup, dataset splits, model choices, and training hyperparameters clearly, supporting reproducibility and transparency.\n\n### Weaknesses\n\n1. **Computational Overhead**  \n   Although DMLE adds meaningful improvements, the additional computation required to model the dependency term can increase training time. The paper could benefit from deeper discussion or mitigation strategies (e.g., approximations) for large-scale settings.\n\n2. **Approximation of the Dependency Term**  \n   For Top-k sampling, the dependency term \n   $$\n   \\log P(S_\\tau | D_{\\tau-1}; \\theta)\n   $$\n   is approximated using a softmax distribution. While this is practical, the paper does not deeply analyze how this approximation affects theoretical guarantees or empirical results. A sensitivity analysis or ablation would strengthen the claims.\n\n3. **Limited Discussion on Limitations**  \n   The paper could include a more critical reflection on scenarios where DMLE may not perform well—e.g., when the sample selection strategy already leads to highly diverse data, or when the model is overparameterized and less sensitive to data redundancy.\n\n4. **No Real-world Application Case Study**  \n   While the synthetic and benchmark datasets demonstrate DMLE's benefits, applying it to a real-world active learning task (e.g., medical annotation or industrial settings) could enhance the practical relevance and highlight potential deployment considerations.\n\n5. **Notation and Readability**  \n   Some formulas (particularly in Section 4) and variable definitions are dense and could benefit from clearer explanations or visual aids (e.g., a schematic of dependency modeling).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This submission introduces a novel approach, Dependency-aware Maximum Likelihood Estimation (DMLE), to address a fundamental mismatch between active learning and conventional model training techniques. Standard Maximum Likelihood Estimation (MLE) operates under the assumption that training samples are independently and identically distributed (i.i.d.), an assumption that fails in active learning where sample acquisition is sequential and each selection affects subsequent ones.\n\nThe key contributions are:\n\n1. Identifying the MLE-i.i.d. mismatch in Active Learning  \n   The paper shows that conventional Independent MLE (IMLE) fails to account for dependencies introduced by sequential querying in active learning. This mismatch leads to inaccurate model updates and suboptimal performance in future acquisition cycles.\n\n2. Proposing Dependency-aware MLE (DMLE)  \n   The authors propose DMLE, which corrects the training objective of MLE by explicitly incorporating the dependency structure of labeled data points across active learning cycles. The improved objective includes a correction term:\n\n   $$\n   \\hat{\\theta}_{\\text{DMLE}} = \\arg\\max_\\theta \\sum_{(x,y) \\in D_{t+1}} \\log P(y \\mid x; \\theta) + \\sum_{\\tau=1}^{t+1} \\sum_{x \\in S_\\tau} \\log P(x \\mid D_{\\tau-1}; \\theta)\n   $$\n\n   This formulation captures both label likelihood and feature-level sample selection probability, correcting the i.i.d. bias in IMLE.\n\n3. Theoretical Analysis and Fisher Information Gain  \n   The authors show that DMLE leads to lower variance estimators than IMLE by proving:\n\n   $$\n   \\mathcal{I}_{\\text{DMLE}}(\\theta) \\succeq \\mathcal{I}_{\\text{IMLE}}(\\theta) \\Rightarrow \\text{Cov}(\\hat{\\theta}_{\\text{DMLE}}) \\preceq \\text{Cov}(\\hat{\\theta}_{\\text{IMLE}})\n   $$\n\n   via the Cramér-Rao lower bound. This formalizes that DMLE yields more reliable parameter estimates.\n\n4. General Applicability Across Acquisition Functions  \n   DMLE is shown to be agnostic to acquisition functions and sampling strategies, and can be plugged into any active learning pipeline as long as acquisition scores or ranks can be computed. The paper discusses entropy, BALD, Core-set, and least confidence as examples.\n\n5. Empirical Validation Across Diverse Benchmarks  \n   Extensive experiments on 8 datasets (MNIST, FMNIST, SVHN, CIFAR-10, Reuters, Tiny ImageNet, Emnist, Iris) show that DMLE consistently outperforms IMLE across multiple acquisition functions (Entropy, BALD, Core-set) and sampling strategies (Top-k, Stochastic Softmax, Power, Soft-Rank Sampling). The gains are especially notable in early cycles, for example, average accuracy improvements of 6%, 8.6%, and 10.5% for batch sizes \\( k = 1, 5, 10 \\) respectively, after 100 acquired samples.\n\n6. Statistical Significance and Robustness  \n   Using Wilcoxon signed-rank tests over multiple random seeds, the authors show that DMLE outperforms IMLE in 92% of the experiments, confirming the robustness and significance of the improvement.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "### Requested Changes\n\n#### Critical\n\n1. **Clarify the impact of the softmax approximation for Top-k sampling**  \n   The dependency term \\( \\log P(S_\\tau \\mid D_{\\tau-1}; \\theta) \\) is approximated using a softmax-based distribution. However, the paper lacks analysis of how this approximation may affect theoretical guarantees or empirical results.  \n   **Action**: Include either a justification or empirical ablation (e.g., sensitivity to the approximation) to demonstrate the robustness of this assumption.\n\n2. **Expand discussion on computational overhead**  \n   While DMLE introduces a theoretically grounded correction, it also incurs additional computational cost, especially in large-scale scenarios. The current discussion does not quantify the trade-off clearly.  \n   **Action**: Provide additional experimental details or complexity scaling plots that show the runtime/memory cost of DMLE versus IMLE under different settings.\n\n3. **Improve notation clarity in Section 4**  \n   The presentation of the DMLE objective (particularly equations 9–11) can be dense for readers unfamiliar with active learning internals.  \n   **Action**: Define all variables precisely (e.g., clarify what each subscripted term like \\( S_\\tau \\) and \\( D_{\\tau-1} \\) refers to), and consider adding a small example or schematic.\n\n#### Optional (but recommended to strengthen the paper)\n\n4. **Include an ablation on acquisition functions and batch sizes**  \n   The paper reports performance across several acquisition functions and \\( k \\)-values, but could benefit from a focused comparison showing how DMLE performs relative to IMLE in particularly diverse or uncertain data regimes.  \n   **Suggestion**: Add a supplementary ablation plot or table showing accuracy improvements stratified by acquisition function difficulty.\n\n5. **Discuss limitations and future directions more explicitly**  \n   The conclusion hints at future work, but a more structured discussion of DMLE's limitations (e.g., approximation assumptions, scalability, or diminishing returns in late cycles) would improve transparency.  \n   **Suggestion**: Add a short paragraph on when DMLE may not provide benefits, and how future work could address that.\n\n6. **Real-world case study**  \n   The paper is well-validated on benchmarks, but its practical relevance could be enhanced with an example application in a high-stakes domain (e.g., medical, annotation-efficient language modeling).  \n   **Suggestion**: If space permits, add a discussion or a short real-world example, even if preliminary.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "SV3P0izzJt", "reviewer": "Reviewer_Vmqk", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Strengths\n1. The problem studied in this paper is important and interesting in the active learning field.\n2. The proposed approach is effective.\n3. The toy experiment in Figure 1 is convincing.\n\n# Weaknesses\n1. In addition to the example dependency mentioned in this article, there is also a deep dependency between the selected example and the learned model. The selected samples are determined by the uncertainty of the model, and these samples further affect the updating of the model. Can you discuss these more?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper studies an important issue in active learning, i.e., how to handle the example dependencies in the iteration process. Specifically, each sample acquisition influences subsequent selections, causing dependencies among samples in the labeled set. Existing methods ignore this issue. The author proposes DMLE to address sample dependencies and ensure consistency with active learning principles in the model parameter estimation process. The experimental results validate the effectiveness of the proposed approach. \n\nOverall, I think this is a good article and tends to be accepted.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "N/A", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)