A High-Dimensional Statistical Method for Optimizing Transfer Quantities in Multi-Source Transfer Learning
Authors: Qingyue Zhang
Haohao Fu
Guanbo Huang
Yaoyuan Liang
Chang Chu
Tianren Peng
Yanru Wu
Qi Li
Yang Li
Shao-Lun Huang
Venue: NeurIPS 2025 poster ()
Keywords: multi-source transfer learning
K-L divergence
high-dimensional statistics
Primary area: general_machine_learning
TL;DR: We propose a theoretical framework based on asymptotic analysis to determine optimal sample transfer quantities in multi-source transfer learning, yielding an efficient algorithm (OTQMS) that enhances accuracy and data efficiency.

Abstract: Multi-source transfer learning provides an effective solution to data scarcity in real-world supervised learning scenarios by leveraging multiple source tasks. In this field, existing works typically use all available samples from sources in training, which constrains their training efficiency and may lead to suboptimal results. To address this, we propose a theoretical framework that answers the question: what is the optimal quantity of source samples needed from each source task to jointly train the target model? Specifically, we introduce a generalization error measure based on K-L divergence, and minimize it based on high-dimensional statistical analysis to determine the optimal transfer quantity for each source task. Additionally, we develop an architecture-agnostic and data-efficient algorithm OTQMS to implement our theoretical results for target model training in multi-source transfer learning.  Experimental studies on diverse architectures and two real-world benchmark datasets show that our proposed algorithm significantly outperforms state-of-the-art approaches in both accuracy and data efficiency. The code is available at https://github.com/zqy0126/OTQMS.

A High-Dimensional Statistical Method for Optimizing Transfer Quantities in Multi-Source Transfer Learning
Qingyue Zhang'; Haohao Fu'; Guanbo Huang'; Yaoyuan Liang', Chang Chu', Tianren Peng', Yanru Wu', Qi Yang Li Shao-Lun Huang"
TTsinhuaa Shenzhen International Graduate School, Tsinghua University
Abstract
Multi-source transfer learning provides an effective solution to data scarcity real- world supervised learning scenarios by leveraging multiple source tasks. In this field, existing works typically use all available samples from sources in training, which constrains their training efficiency and may lead suboptimal results. To address this, we propose a theoretical framework that answers the question: what is the optimal quantity of source samples needed from each source task jointly train the target model? Specifically, we introduce a generalization error measure based on K-L divergence, and minimize it based on high-dimensional statistical analysis to determine the optimal transfer quantity for each source task. Additionally, develop an architecture-agnostie and data-efficient algorithm OTQMS to implement our theoretical results for target model training in multi- source transfer learning. Experimental studies on diverse architectures and two real-world benchmark datasets show that our proposed algorithm significantly outperforms state-of of-the-art approaches in both accuracy and data efficiency. The code is available at wttttt::/// : gi gighhuu..... com 2a2p/12/00120MM
Introduction
0.5
Nowadays, various machine learning algorithms have achieved remarkable success by leveraging large-scale labeled training data. However, in many practical scenarios, the limited availability of labeled data presents significant challenge, where transfer learning emerges as an effective solution 33 Transfer learning aims to leverage knowledge from tasks with abundant data or be- ing well-trained, known as the source tasks, to improve the performance of a new learning task, known the target task. Given its numerous applications, transfer learning has gained wide popularity and seen success variety of fields, such as computer vision (28), natural language processing (24). recommendation systems [ and anomaly detection [26). Traditionally, trans- fer learning has focused on the transfer between single source task and target task. However, there growing emphasis on multi-source transfer learning, which leverages multiple source tasks enhance the training of the target task 23
Accuncy
AllSources Target
0.6
Parget-Only
0
73
100
150
200
250 300 350
400
Target Training Samples Class
Figure 1: More source samples does not al- ways mean better performance. Incorporating all source samples may bring negative impact, which is illustrated by the comparison of two strategies, using target task samples with source sam- ples (blue) and using target task samples only (red), evaluated on the equally divided 5-task CI- FAR10 dataset. Theoretically, although incorporat- ing more source samples reduces model variance by expanding the training data, the discrepancy between the source and target tasks introduces ad- ditional bias.
"Equal contribution.
i Corresponding authors: Yang Li (yangl@sx.tsngghua.cdd...) Shao-Lun Huang(twn2golde gmail.com)
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Table I: Comparison across matching-baser transfer learning method, based on whether they
tailored to multi-sources,, have task generality, have shot generality, and require target labels. The represents obtaining the corresponding aspects, while X the opposite. Task generality denotes the ability to handle various target task types, and shot generality denotes the ability to avoid negative transfer in different target sample quantity settings including few-shot and non-few-shot.
Method
Multi-Source
Task Generality
Shot Generality
Target Label
MCW
Supervised
Leep
Supervised
Tong
Supervised
DATE
Unsupervised
H-ensemble 31
x
X
Supervised
OTQMS (Ours)
Supervised
In multi-source transfer learning, traditional methods usually jointly train the target model using all available samples from sources without selection [32 22 [1]. This evidently poses severe limitation to training efficiency, considering the vast number of available samples from various potential source
tasks in real-world scenarios [19]. Moreover, directly assuming the use of all available samples seriously constrains their solution space, which possibly leads to suboptimal results, as illustrated in Figure Therefore, it is critical establish theoretical framework to answer the question: what is the optimal transfer quantity of samples from each source task needed in training the target task?
In this work, we formulate sample- based multi- transfer learning problem estimation problem. By employing high sigh-dimensional statistical methods to analyze it, we establish a theoretical framework to determine the optimal transfer quantity for each source task. Specifically, we introduce the expectation of Kullback-Leibe target task samples and the distribution learned from training samples as a measure of generalization error. This measure is then minimized in the asymptotic regime to derive the optimal transfer quantity of each source task. Building on this, we propose practical algorithm, named Optimal Transfer Quantities for Multi-Source Transfer learning (OTQMS), to implement our theoretical results for training multi-source transfer learning models. Notably, OTQMS is data-efficient and compatible with various model architectures, including Vision Transformer (ViT) and Low-Rank Adaptation (LoRA). It also demonstrates advantages in task generality and shot generality (as illustrated in Table Tb since we establish theoretical framework without restricting it to a specific target task type or limiting the range of target sample quantity. summary, our main contributions are as follows:
parameter
(K-L) divergence between the true distribution of
a) We use high nigh-dimensional statistics to analyze the parameter estimation problem formu- lated by the sample- based multi-source transfer learning problem. Based on this, novel theoretical framework that optimizes transfer quantities is introduced.
b) Based on the framework, we propose OTQMS, an architecture-agnostie and data-efficient algorithm for target model training in multi-source transfer learning. In particular, we propose dynamic strategy in OTQMS to alleviate the estimation error of transfer quantities caused by the scarcity of target task samples.
c) Experimental studies on few-shot multi-source transfer learning tasks, two real-world datasets and various model architectures demonstrate that OTQMS achieves significant improvements in both accuracy and data efficiency. In terms of accuracy, OTQMS out- performs state-of-the thetthearttt approaches by an average of 1.5% on DomainNett and 1.0% on Office-Hoof Home.. In terms of data efficiency. OTQMS reduces the average training time by 35.19% and the average sample usage by 47.85% on DomainNet. Furthermore, extensive supplementary experiments demonstrate that OTQMS facilitate both full model training and parameter-effiiient training, and OTQMS is also applicable to multi-task learning tasks.
2 Related Work
This work is related to two main lines of research. The first is transfer learning theory, where we propose measure based on K-L divergence as novel measure of generalization error, different from

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previously adopted measures 100 25 I B1I. The second is multi-source transfer learning, where most existing studies either utilize all samples from all source tasks adopt task-level selection strategies [
22 25 31 In contrast, our framework explicitly optimizes the transfer quantity from each individual source task. Other closely related work is discussed further in Appendix B
3 Problem Formulation
Consider the transfer learning setting with one target task T, and K source tasks {Si...., SS,,ss The target task T is not restricted specific downstream task category. Generally, we formulate it parameter estimation problem under a distribution model Px.g- For example, when T is a supervised classification task, Px:0 corresponds the joint distribution model of input features Z and output labels Y, i..., = (Z, Y). Our objective is to estimate the true value of f. which corresponds to optimizing the neural network parameters for target task T. Here, 0 denotes I-dimensional parameter, and @ denotes high-dimensional parameter. Furthermore, we assume that the source tasks and the target task follow the same parametric model and share the same input space X. Without loss of generality, we assume X' to be discrete primarily for clarity of writing and to avoid redundancy, and our results can readily extended to continuous spaces. The target task T has No training samples No = {Ii.....22 No i.i.d. generated from some underlying joint distribution Px:e, where the parameter in Rd. Similarly, the source task S; has N; training samples X.N. (=t....., ,xx; i..d. generated from some underlying joint distribution Px.g, where E [1, K], and the parameter O, E 1Rd, In this work, we use the Maximum Likelihood Estimator (MLE) to estimate the true target task parameter Og. Moreover, the following lemma characterizes the asymptotic behavior of MLE.
Lemma 1. (Asymptotic Normality of the MI.E) (29 samples to estimate O, i.e.
When we use MLE only based on target task
MMLE arg max
Σ
log Px:e(z),
0 No
zEXNo
under appropriate regularity conditions, the following holds:
VNo (IMLE - 2o) S N (0, ((B))
(2)
where " "-1" denotes the matrix inverse and the (() the Fisher information matrix defined as:
((
2 30
(
i (0)
log Px.e)) )
E log lxxe)
When we transfer m1..... 7g samples from the (S Sx). where n, E [0, N.). we denote these training sample sequences as X",,"", "x.""...,,""" X". learning, is the parameter value that maximizes the empirical mean of the likelihood of samples from source tasks and target task, i.e.
Then, the MLE for multi-source transfer
K
arg max 0
Σ log (x. + NO
Σ Σ
log Px.e (2x)
(4)
No + Ex
1=1 2EX"
In this work, our goal derive the optimal transfer quantities ni...., nk of source tasks St....,Sp Sk to minimize certain divergence measure Div between the true distribution of target task Px,g, and the distribution Px.e learned from training samples, i.e.
ni ni = arg min Div Px: p).
(5)
Besides, we provide the notations table in Appendix A
4 Theoretical Analysis and Algorithm
In this section, we will first introduce a new K-L divergence based measure for the optimization problem in (5. Then, we will analyze it based on high-dimensional statistics to derive the optimal transfer quantities for both single-source and multi-source scenarios. Finally, we will develop a practical algorithm based on the theoretical framework.
3

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Definition 2.
(The K-L divergence) 14 The K-L divergence D (PIIQ) measures the difference
between two probability distributions P P(X) and ((X) over the same probability space. defined as:
( = Σ p(x) log p(z) MEX
q(x))
In this work, we apply the expectation of the K-L divergence between the true distribution model of the target task and the distribution model learned from training samples measure the generalization error. Compared to other measures, the K-L divergence exhibits a closer correspondence with the generalization enror as measured by the cross-entropy loss
Finally, our generalization error measure is defined as:
Div(Pxee Px.e) = E ( (xx2)
(6)
4.1 Single-Source Transfer Learning
The direct computation of the proposed K-L divergence based measure is challenging, Fortunately, we can show that the proposed measure is computable in the asymptotic regime using high-dimensional statistical analysis. To be specific, we can prove that the proposed measure directly depends on the Mean Squared Error (MSE) in the asymptotic regime, and the MSE can be calculated by an extension of Lemma Therefore, we perform an asymptotic analysis of this measure.
In this section, we begin by presenting the theoretical results in Lemma 3 and Theorem where the parameter is 1-dimensional, and subsequently extend them the high-dimensional parameter setting in P 6 To begin, we consider the setting with target task T with No samples generated from model with I-dimensional parameter.
Lemma 3. (proved in Appendix C.Ty Given a target task T with No ii.d samples generated from a I I-dimensional underlying model Px:,,. where On E IR, and denoting ô as the MLE ① based on the No samples, then the proposed measure (6) be expressed
E
[ (Px
P. Px00
2No
No
The result of Lemma demonstrates that when there is only one target task without any source task, the proposed measure is inversely proportional to the number of training samples. Next, we consider the transfer learning scenario where have target task T with No training samples and one source task S with N, training samples. this case, we aim to determine the optimal transfer quantity ni [1, N. To facilitate our mathematical derivations,
assume No and N, asymptotically comparable, and the distance between the parameters of the target task and source task is sufficiently small (i.e. 10, 01 ( (,r)), )) Considering the similarity of low-level features
among tasks of the same type, this assumption made without loss of generality and supported by related studies 200. Furthermore, demonstrated in subsequent analysis, our conclusions remain valid even extreme cases where the distance between parameters is large.
Theorem 4. (proved in Appendix C.2) Gäven a target task TT with No i.i.d. samples generated from I-dimensional underlying model Px:0a- and source task S1 with N₃ i.i.d. samples generated from I-dimensional underlying model Px:,,, where 0, O1 E TR and 16 0ı1 = O( /Noo ). Ô is denoted as the MLE (4) based on the No samples from T and n samples from S,, where nı E [0,N.], then the proposed measure ( can be expressed as:
1
No + -
1
+
(8)
No + mh
NNo + n
variance term
bias term
where
t A J(O) - -0
(9)
Moreover, the optimal transfer quantity ni is
N1.
No t 0.5
(10)
min
NN,, 2Not-1
No 1> 0.5"

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From 8 we observe that the proposed measure decreases No increases, which aligns with
our intuition that utillizing all available target samples is beneficial. In addition, the trend of (8) with respect to is more complex. We plot (8) as function of under two different regimes, determined by the value of No L, as shown in Figure 2 Our goal is to explore the optimal value of ni to minimize (8)
Case (No -t S 0.5): The proposed measure monotonically decreases Obviously, the optimal point is n Nj. This indicates that when the source task and the target task are highly similar, L.c, when i is small, an increase in the transfer quantity will positively impact the results.
increases.
Case (No t > 0.5): The proposed measure first decreases and then increases as n1 No increases. It attaining its minimum n 2Not-T- should be noted that when is large enough, the point ri approaches 0. This aligns with the intuition that when the discrepancy between the source and target tasks is substantial, avoiding transfer yields better results. Furthermore, if n exceeds N,, we should utilize all N, samples, so nj
No min [N1. 2Not-1
As the dimensionality of the model parameter increases to higher dimensions, we derive the following propositions.
(8) y t
Proposition 5. (proved in Appendix C.3 In the
case where the parameter dimension is d, i.e. Lo Rd, with all other settings remaining the the proposed measure (6)
11 =
same as the Lemma 3 can be expressed
n1 (a) No ts 0.5
n1
(b) No t 0.5
Figure 2: The function curve figures of (8) under
d
(11)
different regimes determined by the value of Nv- t (blue). The vertical axis denotes the value of proposed measure (8). while the horizontal axis denotes the variable n2.
2No
No
C.4 In the case where the parameter dimension d, i.e. on 0, E Rdd, with all other settings remaining the same as the Theorem 4 the proposed measure
Proposition 6. (proved in Appendix
6
can be expressed as
n
2
No + TA
+
(No n m1)3
+o
No mi
(12)
where we denote
t 0 TT) J(O,) (0, -0n)
(13)
d
In addition, is a sealar. J(0,) d matrix, and (0, 0,) d dimensional vector, which is the element-wise subtraction of two d-dimensional vectors e and f.-
Compared Theorem Proposition exhibits similar mathematical form, allowing us to derive the optimal transfer quantity through similar approach. Furthermore, observe that the parameter dimension increases, the K-L error measure (12) increases. This suggests that for a complex model, knowledge transfer across tasks becomes more challenging, which is consistent with the findings in 25
4.2 Multi-Source Transfer Learning
Consider the multi-source transfer learning scenario with K source task (Si,..., and one target task T. We aim to derive the optimal transfer quantity ne of each source.
Theorem 7. (proved in Appendix C.5) Given a target task T with Ng i.i.d. samples generated from the underlying model Px:e,- and K source tasks SI....,Sx with NI....,NN Nu i.i.d. samples generated from the underlying model Px.0,,..., Px:0ge where Qg ........,... ,0k E RRd is denoted the MLE 4 based on the No samples from T and ni...., n. samples from S,,...,K, where
,,,
5

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K
as the proportion of different source tasks, then the proposed measure (6) be expressed as:
n E [0,N.]. Denoting 8 ==
Σ n4 as the total transfer quantity, and ag = 2s
:=1
d 1
x² + (No + 8)2 No +
(14)
2
+o
+
8
In 14 is scalar denoted
OF J(L,)
(15)
where @= .......... ak] is a K-dimensional vector, and Odxk 10, 2o................... ........ ........... 2x ool-
According to Theorem 7 we can derive the optimal transfer quantities ni.... ................................ " by minimizing (14). Equivalently, we need to find the optimal total transfer quantity 8* and the optimal proportion vector a² which minimize (14). The analytical solutions of 8* and a* are difficult to acquire, and we provide method to get their numerical solutions in Appendix E Eventually, we can get the optimal transfer quantity of each source through n. = s" a
4.3 Practical Algorithm
Along with our theoretical framework, we propose a practical algorithm, OTQMS, which is applicable sample-based multi-source transfer learning tasks, presented in Algorithm In OTQMS, when computing the optimal transfer quantities based on Theorem 7 we use the parameters of pretrained source models to replace 2..... Ωk- Considering that the source model can be trained using sufficient labeled data, it is reasonable to use the learned parameters as good approximation of the true underlying parameters. In contrast, the number of target data in transfer learning often insufficient, so it is difficult to accurately estimate the true parameter l using only the target data. Therefore, as shown lines 5-14 of Algorithm 1, we adopt a dynamic strategy. Specifically, in the first epoch, we train a 6, using only the target data. This fo is then used, along with Theorem 7 to determine the optimal transfer quantity from each source task, and we use random sampling to form new resampled training dataset. Finally, we continue training 9, on this new dataset, and this procedure is repeated in each subsequent epoch to iteratively update the training dataset. This mechanism helps alleviate the inaccuracy of lo- In particular, we compute the matrix J using the empirical mean of gradient outer products in line 10, with the loss function E restricted to the negative log-likelihood, following widely adopted approach known the empirical Fisher [16 [8]
Algorithm OTQMS: Training
1: Input: Target data Dy = {(=7, 14) mm source data {Ds, {(Es, v's, 1 model
type fg and its parameters d for target task and {ex for source tasks, parameter dimension d.
Il represents the feature and y represents the label
2: Parameter: Learning rate 17.
3: Initialize: randomly initialize fo initialize {0 |= with parameters of pretrained source models.
4: Output: well-trained Bo for target task model fe,
5: Dtrain + Dr
// Initialize the trainning dataset by target task samples I/ Use dynamic strategy to train the target task
6: repeat
7: Lirain + Derecsl
Σ
E (o,
8: Oo +00 00 Vo, Ctrain
9: O+ + [0] l.....,
-212
10:
J(Oo)
Σ
Vege(v) -u,,, Se. (z1)) (Ve, E(V Se, (x'))
Drl (z) JEDT
(s", (²) + arg min migmin
2² (No+s)=
J J( 10g
Nots
d
(s.a)
rand
12:
Deource + U D3s
D3.
S Ds..DDS s
13:
Dtrain + Dsource
UDT
Update the trainning dataset
14: until On converges;

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Experiments
5.1
Experiments Settings
Benchmark Datasets. DomainNet contains 586,575 samples of 345 classes from 6 domains (i.e., C: Clipart, I: Infograph, P: Painting, Q: Quickdraw, R: Real and S: Sketch). Office-Home benchmark contains 15588 samples of classes, with adaptation scenarios constructed from domains: Art, Clipart, Product and Real World (abbr. Ar, Cl. Pr and Rw). Digits contains four-digit sub-datasets: MNIST(mt). Synthetic(sy), SVHN(sv) and USPS(up), with each sub-dataset containing samples of numbers ranging from 0 to 9.
Implementation Details. We employ the ViT-Small model pre-trained on ImageNet-2Ik as the backbone for all datasets. The Adam optimizer is employed with learning rate of 1e-b. We allocate 20% of the dataset the set, and report the highest accuracies within 5 epoch early stops in all experiments. Following the standard few-shot learning protocol, the training data for k-shot consists randomly selected samples per class from the target task. All experiments are conducted on Nvidia A800 GPUs.
Baselines. For general performance evaluation, we take SOTA works under similar settings as baselines. The scope of compared methods includes: 1) Unsupervised Methods: MSFDA [21].
DATE 19, M3SDA [9]. 2) Few-Shot Methods Based Model(Parameter)) aarrametrrr--eehhiinng H-ensemble
31. MOW [1]. 3) Few-Shot Methods Based on Sample: MADA [32), WADN 12 4) Source Sew-SSouue-vvginnglllee
Ablating Methods: Target-Only,
Best (average/best performance
of single-source transfer), AllSources V Target (all source & target data in multi-source transfer).
Note that MADA [32 leverages all unlabeled target data in conjunction with limited amount of labeled target data, which a hybrid approach combining unsupervised and supervised learning. Due the page limit, provide detailed information the experimental design and the results of an experiment adapted to the WADN settings on the Digits dataset in Appendix D.2
Table Multi-Source Transfer Performance on DomainNet and Office-Home. The ar- rows indicate transfering from the rest tasks. The highest/second-highhee highest accuracy is marked in Bold/Underscore form respectively.
DomainNet
Office-Home
Method
Backbone
+QQ
Avg
-+
Avg
Unsupervise--ll--hhee
MSFDA
ResNe150
66.5
21.6
56.7
20.4
70.5
54.4
48.4
75.6
62.8
84.8
85.3
77.1
(
ReaNet50
75.2
85.2
84.0
76.3
M3SDA 19
ResNet101
$7.2
24.2
51.6
5.2
61.6
49.6
41.5
Supervised-10--hhe 10-shots
Sew-Shot Methods:
H-ensemble 31
VIT-S
53.4
21.3
54.4
19.0
70.4
44.0
43.8
71.8
47.5
77.6
79.1
69.0
MADA
VIT-S
22.7
78.4
82.3
MADA 32
ResNet50
66.1
23.9
60.4
31.9
75.4
52.5
$1.7
72.2
64.4
82.9
81.9
75.4
MCW
VIT-S
21.0
53.6
20.4
70.8
42.4
43.9
68.9
77.4
70.1
WADN 22
VIT-S
68.0
29.7
59.1
16.8
74.2
55.1
50.5
60.3
39.7
66.2
68.7
58.7
Source-Ablation
Methods:
Target-Only
VIT-S
14.2
3.3
23.2
7.2
41.4
10.6
16.7
40.0
33,3
54.9
52.6
45.2
Single-Souree-Ar
VIT-S
50.4
22.1
44.9
24.7
58.8
42.5
40.6
65.2
53.3
74.4
72.7
66.4
Single-Source-Bes
VIT-S
55.4
28.4
66.0
49,7
48.0
72.9
80.7
74.8
72.3
AllSources U Target
VIT-S
21.7
32.4
60.0
31.4
71.7
58.5
$4.3
77.0
62.3
84.9
84.5
77.2
OTQMS (Ours)
59.8
78.1
5.2 Main Result
We evaluated algorithm, OTQMS, alongside baseline methods the few-shot multi-source transfer learning tasks using the Domainñett and -emmme datasets. The quantitative results summarized in Table Since the unsupervised baselines are not designed for the supervised few-shot setting, we report their original results from the respective papers for reference. We make the following observations:
Overall Performance. In general, compared to baseline methods, OTQMS achieves the best performance almost all the transfer scenarios the two datasets. Specifically, OTQMS outper-

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forms the sfate-of-the- -hrt CaaeeeartttlllSSSie (AllSources U Target) by an average of 1.5% on DomainNet and 1.0% on Office-Hool Home.
Data Speaks Louder Than Model Weighting, is worth noting that on both datasets, sample-based methods utilizing both target and source samples to jointly train the model, such WADN, MADA approaches which construct the larget model by weighting source models, such H-ensemble and MCW. This observation suggests that sample-based approaches offer greater advantages over model-based methods, because they can fully leverage the relevant information from the source data for the target task.
and OTQMS, generally outperform
Take, But Only as Much as You Need. Comparing results in Table 2 among Target-only, AllSources U Target, and OTQMS, we observe that OTQMS achieves the best performance in both datasets by leveraging only a subset of data selected from all available sources based on model preference. This result validates the source tasks, we could train the target model more accurately, and we give an analysis on the domain preference of transfer quantity in Appendix F Furthermore, Figure 4 shows that OTQMS also significantly reduces the training time and sample usage, which validates its superiority in terms of data efficiency.
theory. By choosing the right quantities of samples from
Few-Shot Labels, Big Gains, We make a comparison of the results of unsupervised and supervised methods. While other conditions remain the same, Table demoonsrates that even if unsupervised methods like MSFDA and M3SDA take all the target data into account (up to 1.3x10 samples on Real domain of DomainNet)., their performance still falls short compared to the supervised methods, which rely on only a limited number of samples (3450 samples). This illustrates the importance of having supervised information in multi-source transfer learning.
5.3 Static vs. Dynamic Transfer Quantity
In our proposed Algorithm we employ "Dy- namic" strategy that dynamically determines the op- timal transfer quantities and updates the resampled dataset during the joint training of target task. To val- idate the effectiveness of this strategy, we conducted comparative experiments using the "Static-*" meth- ods. "Static-*" "" methods first simulate the distribution of target on target dataset only, and different types of Static such as "Under, Exact and Over" stands for different fitting levels. In "Static-*" methods, we only compute the optimal transfer quantity once make the resampled dataset, and evaluated on demonstrate OTQMS using dynamic transfer
Table 3: Static vs. Dynamic Transfer Quantity in OTQMS on Home.
Office Ollee-Homee
Method
-Ar
Prr
-Rw
Avg
Sepervised- /0
Stoii--Under
VTES
77.0
60.3
84.9
84.5
73.2
Suntc Exact
VITS
46.0
59.5
85.1
83.7
68.7
Staric Over
WITS
76.3
61.9
75.6
68.6
71.5
Dynamic
WTS
78.1
64.5
85.2
$6.9
78.2
until target model converges. The results on Table 3 quantity achieved the best performance.
5.4 Generality across Different Shot Settings
As discussed in the theoretical analysis of Theorem[4 our theoretical framework is applicable to any quantity of target samples. Therefore, OTQMS exhibits shot generality, enabling it to avoid negative transfer across different shot settings. To validate this, we increase the number of shots from 5 to 100 across methods including AllSources U Target, Target-Only, and OTQMS. As shown in Figure 3) experimental results demonstrate that OTQMS consistently outperforms other approaches across all shot settings. This highlights the generality and scalability of OTQMS in terms of data utilization.
0.65
0.35
0.60
0.50
00
50.25
0.30
50
15
60.35
0.45
00. 00
0.65
0.20
0.
20.25
20.45
20.50
2.330
20.20
0.10
0.25
0.35
20 40
organngeed
60 dis 160
hayer
0.20
20 40
orggnnttent
E0 als TDU)
0.03
20 30
organggent
dis TG0
noye
0.30
0.25
20 40
El BD TOO
Turget Sumplex Per Class
Turqet Sumples Per Class
Turqet Sumples Per Clasn
Turqet Sumples Pee Class
Infograph
(b) Painting
(c) Quickdraw
Real
Figure 3: Performance comparison with increasing target shots up to 100 per class on DomainNet
dataset (I, P. Q and R domains). OTQMS (blue) outperforms other methods.
8

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5.5 Data Efficiency Test
In this section, we demonstrate the advantage of OTQMS in terms data efficiency. Specifically, we conduct experiments with MADA, AllSources U Target, and OTQMS across different shot settings, and for each shot setting. we accumulate the total sample used and time consumed until the highest accuracy is reached. To better visualize the results, we present the average sample usage and training time across all shot settings in Figure 4 To be specific, OTQMS reduces the average training time by 35. 19% and the average sample usage by 47.85% on DomainNet, compared to the AllSources Target method.
Data Used (natural log scaled)
18
Second loturallog scaled)
rmggssetass
MAD
12
16
11
10
14-
12
Clipart
Info
Paint
Quick
Real
Sketch
Avg
Figure 4: Data efficiency comparison of average sample usage and training time DomainNet dataset, the left vertical axis represents the amount of sample usage, with green bars indicating AllSources U Target data counts, blue bars about OTQMS, red bars about MADA(ViT-S) and azury bars about MADA(Res50), while the right orange vertical axis and lines represent training time.
5.6 Compatibility to Parameter-Efficien Training
To evaluate the applicability with parameter-efficien training, used ViT-Base model (6 inte- grated with the LoRA framework [1]] as the backbone. This approach significantly reduces the number of trainable parameters for downstream tasks while ensuring model quality and improving training efficiency. In the experiments, we consider the trainable low-rank matrices and the classi- fication head the parametric model in our theoretical framework, while treating the remaining parameters as constants. Other experimental settings are the same as default. Experiments were conducted on the Office- Home dataset. The results of Table 6 in Appendix demonstrate that our method remains effective.
5.7 Compatibility to Multi Task Learning
To demonstrate the applicability of our method to multi-task learning scenarios, we conduct experi- ments the Office-Home dataset using the ViT- Small model. In multi-task learning, each task simul- taneously serves both source and target task. In the experiments, each task is treated as the target task in OTQMS in during training, while the transfer quantities from all source tasks are computed in turn. This setup enables us to evaluate how effectively OTQMS leverages information across tasks. We compare the performance of our method against single-task training to evaluate its effectiveness in Table
Table Multi-task performance on four tasks of Office-Home.
Office-Home
Method
Buckbone
Ar
CI
Pr
Rw
Avg
Single-task OTOMS
VIT-S
66.7
62.3
87.8
68.6
71.4
VITS
81.7
76.0
88.6
87.5
83.5
Conclusion
In this work, we propose theoretical framework determine the optimal transfer quantities in multi-source transfer learning. Our framework reveals that by optimizing the transfer quantity of each source task, we can improve target task training while reducing total transfer quantity. Based on this theoretical framework, we develop an architcture-agnoose OTQMS for jointly training the target model. We evaluated the proposed algorithm through extensive experiments and demonstrated its superior accuracy and enhanced data efficiency.
and data-efficient practical algorithm

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Acknowledge
The research is supported in part by National Key R&D Program of China under Grant 202IYFA075500,, the National Natural Science Foundation of China under Grants 62571296, the Shenzhen Science and Technology Program under KJZDD0240003102700001, and the Natural Science Foundation of China (Grant 62371270).
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A Notations
Symbol
Description
T
target task source tasks quantity of target samples
(S
Sx}
No
NI NK
maximum sample quantity of each transfer quantity of each souree model parameter
n1 TK
vectorized model parameter
Co
vectorized model parameter target task vectorized model parameter of i-th source task,
0 i E [1, K]
J(0)
Fisher information (scalar) of 0
[(Oaad axd
Fisher information (matrix) of d-dimensional 0
Px:0
distribution of X parameterized by 0
|
1-2 norm of vector x
a2
transfer proportion in multi-source case from i-th source task transfer proportion vector in multi-source case whose i-th entry is dz total transfer quantity in multi-source case estimator
Egg
expectation of
Table Notations
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B Extended Related Work
B.1 Transfer Learning Theory
Existing theoretical works can be categorized into two groups. The first group focuses on proposing measures to quantify the similarity between the larget and source tasks. Within this group, some measures have been introduced, including ly-distance [1]. optimal transport cost [3, LEEP [17],
Wasserstein distance [22]. and maximal correlations [2]. This work belongs to the second group focusing on developing new generalization enror measures. Within this group, the measures having been introduced include f-divergence [10], mutual information [2], X -ivveeeeee [25]. H-score [I
31. However, the potential of K-L divergence a generalization error measure has not been sufficiently explored.
B.2 Multi-source Transfer Learning
Classified by the object of transfer, existing multi-source transfer learning methods mainly focus on two types: model transfer vs sample transfer [36]. Model transfer assumes there one or more pre- trained models on the source tasks and transfers their parameters to the target task via fine-tuning (27). This work focuses on the latter, which is based on joint training of the source task samples with those of the target task (32 22 14] Classified by the strategy of transfer, existing methods mainly focus on two types: alignment strategy and matching- strategy [55]. Alignment strategy aims to reduce
the domain shift among source and target domains [K 34 14). This work is more similar to the latter,
focusing on determining which source domains or samples should be selected or assigned higher weights for transfer (8 22 25 31]. However, most existing works either utilize all samples from all sources or perform task-level selection, whereas this work explores framework that optimizes the transfer quantity of each source task. Moreover, many works are restricted to specific target tasks, such as classification, which limits their task generality. In addition, many studies are mainly applicable to fho-shot scenarios, and may suffer from negative transfer in son-few-shot settings, which limits their shot generality, as illustrated in Table
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C Proofs
C.1 Proof of Lemma
Lemma 8. In the asymptotic case, the proposed measure (6) and the mean squared error have the relation as follows.
E D Px:00 Px.00
1 M(O)))))()) + of No
(16)
Proof. In this section, for the sake of clarity, we will write in its parameterized form ((xN)) N when necessary, and these two forms are mathematically equivalent. By taking Taylor expansion of at lo. we can get
D Px:00
Px Px18(x00))
D (x:00 Cx Px)((xN)))
(x)
Σ (x) log P Px;0(XNo)(0) (2x)
xeX
P (x)((
Σ (x) log zEX
px (xr)
Px Px;0(XM))(1) (x) Px (x)
Σ (x) log
1+
Px(9(
(Pxidixmo (x))) (xpo0(0))
Px-0(XNo)(1)) Px.eo (2) p ))
Σ (x) zEX
+ o(10(xx - 0013)
Px.m((t)
Px.00 (x)
(Pxi(xMo)(sT) (x) (x)
2
1
+ ol (() - 2)
(17)
2 REX
We denote small constant, and can rewrite (6 as
E
D Px...
| Px16(x00))]
Σ
Px (X DD (xx:00 PX(8))N)))
No
P Xx (x) (x)
Σ Px
Σ
+ =((((xx0) -bol
(x)
X No
FEX
aax (2) (0(X (0(x)))-00))
Σ
Px No)
Σ DEX
890
+0(1((x -ll1))
(x)
x No
(Pxx d8
(0(X No) - 00))
Σ
+ o((((x 001
Px)
xEX
aPx (x) (0(xNo) -80) Bb))
Σ
Px No)
1 Σ
00,
011((x
0o13)
(x)
(18)
22

---

To facilitate the subsequent proof, we introduce the concept of "Dot Equal".
Definition 9. (Dot Equal (=)) Specifically, given two functions f(n) and g(n), the notation
f(n)=g(n) is defined as
log f(n) g(n)
f(n)=g(n) lim
0,
(19)
which shows that f(n and g(n) have the same exponential decaying rate.
We denote Px as the empirical distribution of XNo. Applying Sanov's Theorem to (18 we can know that
Px (X- e Px.0))
(20)
Then, aim to establish connection between (20 and [((X 0013. From
can know
that the D (x Px:00) in (20) can be transformed to
(PxNo(0) - Px:00 (z))
D [xxoo| Px:00) - Σ
1
+ 0013)
(x)
(PxNo (x) (x)
2 Σ
+o - -bul
(21)
(x)
ZEX
From the characteristics of MIFE, we can know that
log P (xx6(xNo)(x))
EpyNo No
00
=0
=EpxNo
Olo Px (x)
+ PxxN
O log (x)
(0(X 00)
+ -00
+0(10(xx)) - 0ol3),
00g
008
(22)
which can be transform to
(o(XNo) 00)
0(10(x -
Epym
1og Px.e, (x)
86o
Ep Exxro
8 log Px.dn (2r) 866
O.Px,e, JMgg
E Epym No
Px: (2r)
Ep No
8a log Px (2)
800
0PX (x)
Σ (Pxmo(x)- -x, (2))
D0g
(x)
ZEX
(23)
J(Hu)
Using the Cauchy-Schwarz inequality, obtain
(Pxno (z) - Px)
o Px (ar) 2 00g
0P (x)
2
Σ
(Pxmo(s) Px (1))
20p
(x)
Px:0n (I:)
Px:00 (x)
x
ZEX
(24)
23

---

where
Px (m)
)2
xx (x)
@blo Px (x))
Σ Σ
80%
0r (x)
Σ
pxi (2)
= ((0))
Px:00 (x)
Px;00 (2)
000
300
(25)
Combining with 21
23 and 25
the inequality (4 can be transformed to
D Px Px;00)
(x))o(x= (x) - (x))(=))²
1 Σ
+ +0(10(x2))- -60l3)
2 REX
(x)
2 ((h) (0(xNNo) 0 +((((xxN)) - -Bo13) + +0(10(x00) - -
1
2
1 ((0o) 1(00)10(XX)) 8o12 o((((xx))) 0013)
(26)
2
Combining (20 and (26). we can know that
px (X c (N | Px.00) Se
X6 Noy ep
(27)
For the first term in 18). the []²(x - 0o || is small enough for us to omit the term of of 0(0(XN)))
0) As for the second term in in (8)). even though the magnitude of (0 0oll is
longer negligible, the probability of such sequences O -No/1000111xx -o)--xl by (27) which
exponentially decaying with No such that the second term is ol N. By transfering 18 we can get
E [ (Px;80 / P
8Px (0(X:N)) - -Bol) Ul
Σ
Px (X.NN
Σ
+ o(10(x -611)
(x(
(x00
#EX
0Px (x) (0(XNo) - 00) 2
Σ
(X No)
880
+ o 1
+
2
(x
No
{
[x - 121200
=ex
0Px (1) (8(X 800
1
80a
E
+ ol
(28)
2
Px:0, (x)
No
ZEX
We then transform
28 with 25
(0( 80))
E
2
Px(90
ZEX
aPx )2
2
E
Σ
800
2
Px.0,
1
2
E
0n)
x((0)
(29)
2
Combining 28
29
we can get
E
D
(Px.ee
1 -
E
(o ( j(Bb) + +ol
1
(30)
2
No
24

---

From 16,, we can establish the relationship between the proposed proposed measure O and the mean squared error. Then, by using the ( the K-L measure
o
(31)
2No
No
C.2 Proof of Theorem
In this theorem, we are using samples from both target task and source task for our maximum likelihood estimation, optimization problem becomes
i arg max L,, (0),
(32)
where, when using No samples from target task and n samples from source task
(n
log (x; + No T1
Σ log Px;o(x))
(33)
No
T1 ZEXNO
#EX"1
And, we also define the expectation of our estimator, which is somewhere between Og and 01 to be Eg
Fgg = arg max L(0).
(34)
where L(0) denotes the expectation of (0)
L(
No
Exx,, [log Px,o(x)]+ +
M1
Exx,, [log (x.o(x)).
(35)
No + nr
No + n
We could equivalently transform 34
into
No 1
n1 D (Px.0, PP
Eg arg min
(36)
8
No n1
No n1
Lemma 10. By taking argmin of 36 we can get
Px-E, No + n. Px.0,
(37)
No 121
By doing Taylor Expansion of 37 around 0', which is in the neighbourhood of 0,, O2 and Eg. can get
Nol. + n1
No + m {
E
+0
(38)
No + n2
Proof.
From 36),
we know that
No (Px;.o Px:00
D px Px.0)
Fgg arg min
No + n
No + m
No
Px:00 (x)
Σ Px.o, (x) log Px:0, (x) Px. (x) (39)
arg min
Σ Px,0, (o) log ZEX
No nn
Px(0(x)
No + n Nex
To minimize the weighted K-L divergence. We treat (x.,(r), Vo constraint
X' the variable with the
Σ (x) = 1, then we form the Lagrangian:
Lagrangian(P.. (P,A) = Σ
(
No
Px.0, (x) +
M1 No + ny
Px:o, (x) log Px:o + Σ
(x; -1
No + nı
x
(40)
Taking the derivative with respect to (x; and setting it zero gives:
Ol aggaanngan P,A)
No No
Px (x) + Notai Px,,, (x)
+ = 0.
(41)
(z)
Px;
25

---

So
Px.a(x) = Notm: Px.0, (x)
No
Notn, Pxio, (2)
(42)
Normalizing (x;o(z) gives = 1, hence the optimal solution is:
Px;E, (z)
No (x)+
ni
Pxie, (2), X,
(43)
No + n1
No + m
which corresponds to 37
Then we begin to prove (38
By doing Taylor Expansion of 43 around 0g, we can get
Px (x) + 0Px10. (x)
+ +O - -bol²) (Eg- -8-60) 60)
00
No + (x) +
No
n
(x) +
0 Pxx:0 (x)
(01 0o) + 0(16 0,19)
(44)
No n
00
From we can get
No6o + n111
Eg
+ 0(10g 0113),
(45)
No n1
So get
No0o + n101
+0 1111 1
No n2 (
Eg No + n2
(46)
Lemma 11. We assume that the following regularity conditions hold:
The log-likelihood function twice continuously differentiable the neighborhood of 0o.
2. The Fisher information J(fo) is positive and finite.
Then, the estimator 0 is asymptotically normal,i.e..
NNo + n -Eg) AN N (o. )
(47)
Since is a maximizer of (0). (0) = 0 = (Egg + (x,)) Eg)) + + (
Proof.
Therefore,
NNo + ni (Fg)))
No + ( FEg)
+
(48)
(Eg)
VNo
26

---

Since Fig maximizes I((0), T' (Eg) = No+n1 Therefore,
No
Edo
algg PxiNg()) (x)
+
n1
Bo,
a log Px: (16)
ci
No
No + Toi (Eg)
No V No +
Oog Px,Eg (x)
R2
Σ
Ölog Px:E, (4)
+
No
00
No
mi
00
DEXNO
sEX"1
No
Σ
a log Px,E, (z)
n
Σ
olog Ex:Eg (2)
V No +
No
00
No +
n1
00
NEXNO
#EX"1
No
Olog Px.Fg (x)
Flog Px.Fg (x)
Beg
Es,
VNo + n
Or
VNu n1
Ox
No
i Olog Px,Ea (z)
i Ölog Px.Eg (x)
NNoBo
V No +
No
00
00
0 log Px,E, (1)
Ölog Px,E, (x)
71
+
Me,
(49)
No +1
00
00
REX"1
Applying the Central Limit Theorem to 49
get
No + ni (Fig) a.e, v(1
No
Olog Px; (x)
27
[Olo Px:E, (x) (x.28((0)))
)1
Euoo
Eoo
+
No + n
00
00
@Olog Px,E (x)
(ålog Ex (x)
71
Eg,
Eg,
No + 71
00
00
(50)
By taking Taylor expansion of Eg at dg. we can get
log Px:E, (x)
Eeg
00
Ölog Px;0, (x)
θ θ log Px;0o (2)
27
Eeg
(Eg (Fg-0h) 0o) 00
00
00
00
No + n1
Olog Px.F, F (x) 82 log Px.F, (I)
1
=((b) + (Eg
Ego
0
00
092
No + ni
((0)) + O
(51)
VNo +
and
log Px,E, (x)
2
Eoo
00
(Olog Px (x)
① Ölog Px,0, (I)
1
Egg
(Eg 0o) +0
00
00
No + n1
② ① log Px.0, (x)
Eeg
(Eg 0o)
+0
00
00
No + m
( 00
θ log Px (x)
2
(Eg - 2
+ ol No +
00
0( No + n1
(52)
27

---

By combining (50 551 and
(2). we can get
var No + 1 No + n1
No
J(00) +
R1 J(01) + 0
(53)
No +1
VNo +
Additionally, we know 1'n (Fi) 2. (( Combining with 48 (33) know that
Noo + n1 Eg) 4 N (0, Nota;
[(Oo) + Notor J(B1) No
(54)
(2
Under the assumption that 0o, 01; Fig are sufficiently close to each other, we can easily deduce that difference among J and (EE) O O (Nottt))
can easily get
v No + nz(0 Eg) SN (o. _((6))
(55)
(0 Fg)) 27
Therefore, the limit of E
is
(56)
(No + ((0)
Combining 6 and 29
we know that
E
Px:00
P X
[O
(0 - 0o)
27
+ o(
No
= 2 (0o)
(0- Eg)
27
+ E
2 (0 - Eg) (Eg -bo) +E
(E, - -66)
+ + =
No
- (00)
LE (0- Eg)
+ E (Eg - 0o) + ol No
(57)
Combining (77.
38
and (56).
we know that the proposed measure is
1
n²
+ 2
(0b)
(01 - -0)) + 0
(58)
2 No ni
(No n1
No
C.3
Proof of Proposition S
Similar to (1). we can get
D Px.e, Pxxi0]
E
tr (S(B)) (0-00) (B- 8o) 7)) + ol 1
No
(59)
Combining with Lemma
I we will know that the K-L measure is
マ tr (((b)) (S(eN)
No + of bo))))))
2No + o -
d
(60)
No
28

---

C.4 Proof of Proposition 6
Similar to 57 , we can get
D
Px Px
Px Px10)
1
(Q- (o) (2 20) )"]) + ol No
[(
2
1 tr (QQ))) (2 Eig) (e 1i8) '1) tr (
(me 20) (Eg 2b) )))
t No
2
(61)
Similar to 47 we can get
No + m (e Fig S N (0. -(B))=
(62)
(o E3)
So we can know that E
has the limit
J(Bo) -1
(63)
(No + n1)
The same to (38) we can get
Nolo + nill
Eg
+ +0
(64)
No +
No + mi
Combining 61
63
and (64).
we can know that the K-L measure is
d
1
nf
+
(No + )1 +o
(65)
2
No +
No mi
where we denote
(0 e,)" (1 - -l)
t
(66)
d
C,5
Proof of Theorem
Proof. Similar
44
get
No +
m
S N (0. J(BN)) Fig
(67)
So know that E
0 E3)
has the limit
1
J(OO)
(68)
(No + s)
Similar to
38 we can get
we
& Nob + Σ n,O,
No
n,O;
Eg
O
1
1=1
0
1
(69)
No +
No +
m No + Σ im1
No
Σ
nz
Combining 61
68
and
69
for the d-demension parameter, can know that the K-L measure is
k
No +
(No + s)ad" ir
J(O.)
Σ aill -=:(e-2))
20)
Σ
aa - 2n)
i=1
d 1
s²
aT J(L,)(a)
+
+o
2
No
+
(No s)2
d
No +
(70)
29

---

where a = (e......kk] T is a K-dimensional vector, and
Odxk = = -00- ... 0x 8g)-
(71)
30

---

D Experiment Details
D.1
Details information of LoRA framework experiments.
Table 6: Multi-Source Transfer with LoRA on Office-Home. We apply LoRA on ViT-B backbone for PEFT.
Office
Backhone
-AA
-CI
-Pr
-ww
Avg
Supervisnd- fo-shotss Soutoe-dblatio::
Only
Sinple Sourceavg
72.3
Single Source-bett best
VTT-B
74.4
61.8
81.9
81.9
75.8
AllSonmces U Target
VIT-B
81.1
66.0
85.0
89.2
X1.1
OTQMS (Ouns)
VI-B
81.5
68.0
89.2
90.3
82.3
D.2 Experimental Design and Model Adaptation
To ensure consistency in the experimental setup, we first evaluate the performance of different methods on the DomainNet and Office-Home datasets by adapting their settings align with ours, such as the backbone, dataset, and early stopping criteria. Specifically, for the MADA method, we adjusted the preset of keeping 5% of labeled target samples 10-shots per class target samples while maintaining other conditions. And in turn, ours is adapted to the WADN settings, equipped with 3-layer ConvNet and evaluated on Digits dataset.
Table 7: Performance on Digits Dataset. The arrows indicate transferring from the rest tasks. "3Conv" denotes the backbone with 3 convolution layers.
Digits
Method
Backhone
-Mmt
N2.7 -WVV
-20y
Avg
Following vellings WADN:
WADN 22]
3Conv
88.3
70.6
81.5
90.5
Target
OTQMS (Ours)
3Coev
89.1
31

---

D.3
Single-Source transfer performance details.
Table 8: Single-Source Transfer Performance DomainNet. The details accuracy information of "Single-Source- lines of Table
Source Domain
Target Domain
Backbone
Clipart
Infograph
Painting
Quickdraw
Real
Sketch
Ave
Best
Clipart
VIT-S
46.5
55.4
30.3
60.2
59.7
50.4
60.2
Infograph
VIT-S
25.6
25.3
7.3
28.0
24.4
22.1
28.0
Painting
VIT-S
Quickdraw
VIT-S
26.9
18.1
23.9
25.9
28.4
24.7
28.4
Real
VIT-S
64.6
62.2
38.5
62.7
58.8
Sketch
VIT-S
49.7
40.9
48.1
26.1
47.8
42.5
49.7
Table 9: Single-Source Transfer Performance on Office-Home. The details accuracy information of the "Single-Source-=" lines Table 2
Source Domain
Target Domain
Backbone
Art
Clipurt
Product
Roal World
Avg.
Best
Ari
VIES
61.7
61.1
72.9
65.2
72.9
Clipart
WEE
4908
49.4
60.9
53.3
601-9
Real World
VITSS
70.9
72.
74.8
72.7
74.8
Table 10: Single-Source Transfer Performance on Office-Home of LoRA. The details accuracy information the "Single-Source-** lines of Table ViT-B backbone is already frozen and equipped with small trainable LoRA layers.
Source Domaln
Target Domain
Backbone
Clipart
Product
Real World
Avg
Best
Art
VIT-B
68.4
74.4
73.8
72.2
74,4
Clipast Product Real World
VIT-B
58.3
59.6
61.8
59.9
61.8
VIT-B
82.0
80.8
84.9
82.6
84.9
80.3
81.9
D.4 Baselines experiments settings in Table 2
Since the significant difference from unsupervised methods and supervised methods, the results of MSFDA 21), D and M3SDA are directly supported by their own article.
On all the experiments of the baselines, we take all the source samples of different domains from trainset into account. And types of the backbone all pretrained on ImageNet21k33
As for based on poodel-parameterweeee since they have not taken experiments DomainNet and Office-Home datasets with ViT-Small backbone and 10-shot per class training samples, we take it by changing the backbone and samples condition while maintaining other configurations supported by their own work. And for fairness and efficiency, the well-trained source models from different domains these methods are directly equipped with the same models trained by the first stage of our OTQMS method respectively.
few-shot methods: MCW[[2] and H-ensemble 31]
As for based on samples few-shot methods: We exam the WADN (22) method under the condition of limited label target domain as this setting is similar to ours. And we also change the backbone and samples condition while maintaining other configurations of its report to realize the experiments on DomainNet and Office-Home datasets. It is apparent that the settings of MADA 32) are quite different of ours. leveraging all the labeled source data and all the unlabeled target data with the few-shot labeled ones that are difficult to classify, which means all the target samples have been learned to some extent. So we not only decrease the labeled target samples to 10-shot per class but also the unlabeled target samples. And since MADA is the most SOTA and comparable method, realize it with our fair 10-shot setting under both VÍT-S and ResNet50 backbone on DomainNet and Office-Home datasets while maintaining other configurations of its report.
32

---

D.5 Data efficient test details in 5.5
Table 11: Data Efficient Test Results on DomainNet. domains.
The capital letters represent the target
Data Counts (x10")
Training Consumed Time (x10) Second)
Method
Backhone
Avg.
MADA MADA
8.70
19.76
21.32
12.58
12.52
Re4Ner50
7.83
8.21
4.57
3.69
4.35
8.78
6.24
7.56
9.58
4.36
4,05
4.21
9.58
6.59
AlISourva OTQMM
Target
VIES
2,17
1.42
1.42
4.20
1.42
1.90
2.09
0.56
0.37
0.39
1,09
0.36
0.49
0.54
Table 12: Data Efficient Log Scaled Test Results on DomainNet. The capital letters represent the target domains.
Counts
Trsining Consumed Time srale)
Method
Backbone
Avg
MADA MADA
ResNet50
15.87
15.92
15.33
15.12
15.29
15.99
15.85
11.23
11.47
10.73
10.61
10.65
11.47
11.30
AllSounces U Tirgel
VIT-S
14.59
14.17
14.18
15.25
14.17
14.46
14.55
8.63
8.21
8.21
9.30
8.19
8.50
8.59
OTQMS (Ouns)
VITS
13.98
13.95
13.70
13.78
13.99
13.96
13.90
8.17
8.10
7.93
8.12
8.46
8.18
8.17
D.6 Domain choosing analysis details.
To compute the heatmap matrix visualizing domain preference in Figure b), for each source domain, we count the samples selected from it until the target model converges under the 10-shot condition. Since the quantities of available samples varies significantly across domains, we normalize counts by these quantities. The final domain preference is then determined by computing the importance of these normalized values.
Similarly, to compute the heatmap matrix for domain selection in Figure [a), we calculate the importance of the counts selected samples from different source domains throughout the training epochs.
33

---

E
Method to get s" and a which minimize (14) in Theorem 7
The minimization problem of proposed measure (14
T T J(
(s",'' Qt + arg min
(72)
No +
(No
+ s)
(s.ge)
We decompose this problem and explicitly formulate the constraints as follows.
s² (No + s) aEAls)
(s*,a*) +- arg min (0, Nil
arg min o(",""""
(73)
2 Nots
where
A(s)
Q
Σ a = 1 Na,; 0, -....,,K
(74)
Due to the complex constraints between and a, obtaining analytical solution to this problem is challenging. Therefore, we propose a numerical approach to get the optimal solution.
This problem requires optimizing the objective function over two variables: scalar variable representing the total transfer quantity, and a vector variable a representing the proportion of samples drawn from each source domain. For which is restricted to integer values, we perform exhaustive
search over its feasible domain [0, 8max), where maax
Σ N,. For each candidate s' in the search, ==1
we compute the optimal a' under the constraint A(s'), which is a K x K quadratic programming problem with respect a
a'= arg min oc ((x)
J(O,)=)
The quadratic coefficient matrix this optimization problem is given by θ ((2,) Since the Fisher information matrix ((,) is positive semi-definite, the quadratic coefficient matrix is also positive semi-definite. This guarantees the existence global optimal solution. After getting s' and a'. the function is then evaluated at (s',a').
In brief, for each s', we solve for the corresponding optimal a'. yielding finite collection of candidate solutions (s',a') a') and their associated objective function values. After completing the search, the optimal solution (s" _=" chosen as the pair that achieves the lowest objective function values among all candidate (s', a'). Since the feasible set of is finite and enumerable, and for each fixed s' the optimization over a' has solution, the overall optimization problem is guaranteed to have at least one global solution. Hence, the optimal pair (s",a") exists.
It is worth noting that, for the sake of computational efficiency, do exhaustively enumerate all possible values of s in our experiments. Instead, we perform a grid search using 1000 uniformly spaced steps over the feasible range of s, where the number of steps denoted as stepnumber 1000. Experimental results in Section demonstrate that this strategy does not compromise the effectiveness or stability of the method.
34

---

Analysis on Domain-specific Transfer Quantity
To understand the domain preference of OTQMS, visualize the proportion of each source in the selected samples for each target domain. As shown in Fjg) when the target domain is Clipart, OTQMS primarily leverages samples from Real, Painting, and Sketch. addition, Quickdraw contributes minimally to any target domain. These observations align with the findings in [9]]. To further clarify the selection process of OTQMS during training, we visualize observe that in the Office Home dataset, Clipart initially selects all source domains but later focuses on Art and Real World. the DomainNes dataset, Sketch predominantly selects Clipart, Painting, and Real throughout the training process.
Figure a)) We
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(a) Domain Selection
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Figure Visualization of domain-speciff transfer quantity under 10-shot setting. (a) Domain selection during training epochs (from left to right), where the blue upper part represents the selection and the orange lower part represents the selection of
of target domain Clipart on target domain Sketch DomainNet. Darker colors indicate stronger tendencies throughout the training process. (b) Source domain preferences of different target domains on DomainNet. Each row corresponds to target domain while each column represents source domain.
We additionally report the raw average transfer quantities between domains on the Office- Home and DomainNet datasets. Each row denotes target domain and each column denotes source domain.
Table 13: Average transfer quantities on the Office-Home dataset. Each row denotes a target domain and each column denotes source domain.
Target \\ Source
A
C
P
R
A
0
2546
2956
3341
C
1325
0
753
2192
P
736
1320
0
2378
R
1879
3371
3300
Table 14: Average transfer quantities on the DomainNet dataset. Each row denotes a target domain and each column denotes source domain.
Target Source
C
I
P
Q
R
S
C
0
18391
57901
47641
138497
43100
I
28962
28979
17581
138497
41558
P
30907
41380
0
41945
83092
44356
Q
12380
11305
12883
30498
18050
R
24168
31066
36179
51749
0
27716
S
23591
11302
36897
50182
88133
0
35

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G A Detailed Discussion on the Limitations
Sampling Method. Since our theoretical analysis focuses on the transfer quantity from each source task, we adopt straightforward random sampling strategy in the algorithm imple- mentation. Given that our theoretical results are derived under average-cass assumptions, random sampling is sufficient to demonstrate the robustness of both our theoretical analysis and the proposed algorithm. Nevertheless, we anticipate that more sophisticated sampling strategies, such as active sampling, may further improve the algorithm's 'ss performance.
Weight or Quantity. Many existing work of multi-source transfer learning assign weights to source tasks and utilize all available samples. In contrast, this work focuses on optimizing the transfer quantity from each source task. We anticipate that future work could further improve algorithmic performance by jointly optimizing both the sample weights and the transfer quantity from each source task.
Possible Extensions to Other Loss Functions. Our theoretical analysis developed under the assumption of negative log-likelihood objective, which aligns with the cross-entropy loss commonly used in classification tasks with softmax outputs and one-hot labels. However, for other learning objectives such the mean squared error regression problems, our framework does not yet provide direct theoretical guarantee. We believe, nevertheless, that the core idea can be generalized broader loss functions under suitable regularity conditions. We leave more rigorous theoretical extension and empirical validation with alternative loss functions as future work demonstrate the robustness and wider applicability of our method.
H Broader Impacts
We first develop a theoretical framework for optimizing transfer quantities transfer learning, and subsequently propose architecture-agnost and data-efficient algorithm based on this theoretical framework. The proposed theoretical framework and algorithm have broad applicability in various transfer learning scenarios. including domains such as medical image analysis, recommendation systems, and anomaly detection. There are no negative social impacts foreseen.
36


Reviews:
[{"review_id": "56CzkuavrY", "reviewer": "Reviewer_rZbt", "summary": "This paper tackles a fundamental problem in multi-source transfer learning (MSTL): how to eliminate negative transfer from a sample-level perspective. The authors aim to determine the optimal number of samples to transfer from each source task to best benefit the target task. Unlike existing approaches that transfer all available data or rely on heuristic selection strategies, the paper proposes a theoretically grounded framework for transfer quantity optimization.\n\nSpecifically, the authors introduce a generalization error measure based on the expected Kullback-Leibler (KL) divergence between the true target distribution and the learned model. They leverage asymptotic high-dimensional statistical analysis to derive closed-form expressions for the optimal transfer quantities in both single-source and multi-source settings. This theoretical foundation leads to the development of OTQMS, a data-efficient and architecture-agnostic algorithm. Furthermore, the paper proposes a dynamic training strategy that iteratively updates both the estimation of the target model parameters and the corresponding optimal transfer quantities during training.", "questions": "1. Fisher Information Estimation:\nGiven that the Fisher information matrix is estimated using only a small number of target samples, how do the authors ensure the stability and reliability of this estimate? Have they evaluated the sensitivity of the algorithm to noise or variance in the empirical Fisher?\n\n2. Negative Transfer Evidence:\nWhile the theoretical analysis suggests that transferring too many source samples may lead to negative transfer, the experimental results do not explicitly show this. Can the authors provide empirical evidence demonstrating performance degradation when the transfer quantity exceeds the theoretical optimum?\n\n3. Optimal Transfer Quantity in Practice:\nThe framework is designed to compute the optimal number of samples to transfer from each source, but the paper does not state what these optimal quantities actually are in practical settings. Could the authors report or visualize the learned transfer quantities to enhance interpretability?\n\n4. Existence and Solvability of the Optimum:\nHas the existence and uniqueness of the solution to the optimization problem in Theorem 7 been formally analyzed or empirically verified? Are there conditions under which the solution may not exist?", "limitations": "One key limitation of the proposed framework is its reliance on asymptotic statistical assumptions, which may not hold in extreme few-shot regimes. In such cases, the empirical estimation of the Fisher information matrix based on a small number of target samples can suffer from high variance and numerical instability, potentially affecting the reliability of the computed transfer quantities", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strength\n\nThe paper presents a theoretically grounded formulation for optimizing transfer quantities in multi-source transfer learning. By leveraging KL-divergence-based generalization error and asymptotic high-dimensional statistical analysis, it offers a novel and mathematically rigorous approach. A key strength of the work lies in its focus on sample-level optimization, which allows for fine-grained control over transfer and directly addresses the problem of negative transfer, unlike prior approaches that operate at the domain or model level. The proposed algorithm, OTQMS, is both architecture-agnostic and data-efficient, enabling effective knowledge transfer while minimizing unnecessary sample usage. \nEmpirically, OTQMS demonstrates strong performance on challenging benchmarks such as DomainNet and Office-Home with notable improvements in accuracy. The method exhibits robustness across a wide range of shot settings from 5-shot to 100-shot.\n\nWeakness\n\nWhile the proposed framework is theoretically well-grounded, it relies on asymptotic analysis, and the paper does not offer non-asymptotic guarantees or finite-sample bounds. This limits our understanding of how well the method performs when the number of target samples is extremely limited. A central step in the algorithm is estimating the Fisher information matrix using a small set of target samples. While this is a practical design choice, it raises concerns about the variance and numerical stability of the estimate—especially in few-shot settings where data scarcity may result in unreliable curvature information. Furthermore, although the authors argue that transferring too many samples can lead to negative transfer, the experiments do not clearly show this behavior. The paper does not quantify what the optimal number of transferred samples actually is, nor does it provide empirical evidence of performance degradation when that number is exceeded. This weakens the practical insight into when and how negative transfer occurs. In addition, the optimization problem in Theorem 7, involving the total transfer quantity \\$s^*$  and the proportion vector \\$\\alpha^∗$, is said to be solved numerically, but Appendix F provides insufficient detail about how this is actually done. It is unclear whether a unique or stable solution always exists, and the lack of discussion on the solvability or initialization sensitivity may hinder reproducibility and theoretical completeness.\n\nOne minor notation issue: the paper uses underlines to denote high-dimensional vectors. It is more standard and clearer to use boldface notation to indicate vector quantities.", "was_revised": true, "final_justification": "I appreciate the authors' efforts in addressing my questions. For Q1, the stability of the Fisher information estimate in the initial stage appears to be demonstrated only through empirical results, but I think its robustness remains unclear. Nevertheless, the proposed method demonstrates clear novelty and technical soundness, and the authors have sufficiently addressed our other concerns (Q2–Q4).\n\nGiven the overall quality of the work and the thoughtful responses, I have decided to raise our score.", "rebuttal": "Thank you for your insightful feedback on our paper. Below, we address each of your concerns point by point.\n\n>Q1:Fisher Information Estimation: Given that the Fisher information matrix is estimated using only a small number of target samples, how do the authors ensure the stability and reliability of this estimate? \n\n\nThe dynamic strategy proposed in lines 210–212 can address this issue. Specifically, in the first epoch, we train a $\\underline{\\theta}_0$ using only the target data, and get the $J(\\underline{\\theta}_0)$ using the gradient of loss. \nThis $\\underline{\\theta}_0$ and $J(\\underline{\\theta}_0)$ is then used to determine the optimal transfer quantity from each source task, and we use random sampling to form a new resampled training dataset. Finally, we continue training $\\underline{\\theta}_0$ and get new $J(\\underline{\\theta}_0)$ on this new dataset, and this procedure is repeated in each subsequent epoch to iteratively update the training dataset. In brief, only in the algorithm’s initial epoch do we use a small number of target samples to train the Fisher information matrix; in subsequent epochs, we incorporate additional source samples to assist in training the Fisher information matrix, and the inclusion of more samples helps ensure its stability. Introducing source samples to assist in training the Fisher information matrix can ensure its stability without incurring significant error, because, as stated in line~533, the difference between ${J}(\\theta_0)$ and ${J}(\\theta_1)$ is of the order $O\\left(\\frac{1}{\\sqrt{N_0}}\\right)$. We will include the proof of this result in the revision version of the paper.\n\n>Q2:Negative Transfer Evidence: While the theoretical analysis suggests that transferring too many source samples may lead to negative transfer, the experimental results do not explicitly show this. Can the authors provide empirical evidence demonstrating performance degradation when the transfer quantity exceeds the theoretical optimum?\n\nWe have addressed this issue at two points in the paper. First, in Figure 1, we observe that when the quantity of target samples is large, training with both target task samples and all source samples may perform worse than using target task samples only. Second, in Table 2, the method \"AllSources $\\cup$ Target\", which uses all available source and target data for training, gain worse performance than our proposed optimal transfer quantity method, OTQMS.\n\nFurthermore, to better address your concern, we conducted additional experiments. Specifically, over the feasible domain of the total transfer quantity $s \\in [0, s_{\\max}]$, where $s_{\\max}= \\sum\\limits_{i=1}^{K} N_i$, we evaluated the training performance corresponding to the transfer quantity at each decile point of $s_{\\max}$.\n The results demonstrate that none of these transfer quantities achieve better performance than our proposed optimal transfer quantity $s^{*}$. This experiment is conducted on the Office-Home dataset, and the table reports the average accuracy across all domains when each is used as the target domain.\n\n\n| $s = 0$ | $s = 0.1s_{\\max}$ | $s = 0.2s_{\\max}$ | $s = 0.3s_{\\max}$ | $s = 0.4s_{\\max}$ | $s = 0.5s_{\\max}$ | $s = 0.6s_{\\max}$ | $s = 0.7s_{\\max}$ | $s = 0.8s_{\\max}$ | $s = 0.9s_{\\max}$ | $s = s_{\\max}$ | $s^{*} \\approx 0.67s_{\\max}$ |\n|:-------:|:-----------------:|:-----------------:|:-----------------:|:-----------------:|:-----------------:|:-----------------:|:-----------------:|:-----------------:|:-----------------:|:--------------:|:---------------------------:|\n| 45.2%   | 57.2%              | 67.1%              | 68.6%              | 70.1%              | 73.1%              | 75.2%              | 77.5%              | 77.3%              | 77.1%              | 77.2%          | 78.2%                        |\n\n\n\n>Q3:Optimal Transfer Quantity in Practice: The framework is designed to compute the optimal number of samples to transfer from each source, but the paper does not state what these optimal quantities actually are in practical settings. Could the authors report or visualize the learned transfer quantities to enhance interpretability?\n\nYour suggestion is very helpful. In Figure 5(b) of Appendix G, we provide a domain preference heatmap for transfer on the DomainNet dataset, which reflects the normalized transfer proportion of each source domain. Your comment made us realize the importance of also reporting the raw transfer quantities. We hereby supplement the actual average transfer quantities on both the Office-Home and DomainNet datasets, and will include them in the revised version of the paper. In the table below, each row corresponds to a target domain, while each column represents a source domain.\n\n| Office-Home | A    | C    | P    | R    |\n|-----|------|------|------|------|\n| A   | 0    | 2546 | 2956 | 3341 |\n| C   | 1325 | 0    | 753  | 2192 |\n| P   | 736  | 1320 | 0    | 2378 |\n| R   | 1879 | 3371 | 3300 | 0    |\n\n\n\n\n|  DomainNet   | C     | I     | P     | Q     | R      | S     |\n|-----|-------|-------|-------|-------|--------|-------|\n| C   | 0     | 18391 | 57901 | 47641 | 138497 | 43100 |\n| I   | 28962 | 0     | 28979 | 17581 | 138497 | 41558 |\n| P   | 30907 | 41380 | 0     | 41945 | 83092  | 44356 |\n| Q   | 12380 | 11305 | 12883 | 0     | 30498  | 18050 |\n| R   | 24168 | 31066 | 36179 | 51749 | 0      | 27716 |\n| S   | 23591 | 11302 | 36897 | 50182 | 88133  | 0     |\n\n\n>Q4:Existence and Solvability of the Optimum: Has the existence and uniqueness of the solution to the optimization problem in Theorem 7 been formally analyzed or empirically verified? Are there conditions under which the solution may not exist?\n\n\nHere, we provide a more detailed explanation of the optimization method of Theorem 7 described in Appendix F, and prove the existence of a globally optimal solution. The minimization problem of the objective function in Theorem 7 is Eq. (74), i.e.,\n\n$$\n(s^\\*, \\underline{\\alpha}^\\*) \\gets \\arg \\min_{(s, \\underline{\\alpha})} \\frac{d}{2}\\left(\\frac{1}{N_0+s}+\\frac{s^2}{(N_0+s)^2}\\frac{\\underline{\\alpha}^T\\Theta^T{J}({{\\underline{\\theta}}_0})\\Theta\\underline{\\alpha}}{d}\\right).\n$$\n\nWe decompose this problem and explicitly formulate the constraints as Eq. (75), i.e., \n\n$$\n(s^\\*, \\underline{\\alpha}^\\*) \\gets \\arg\\min_{s\\in[0,\\sum\\limits_{i=1}^{K}N_i]} \\frac{d}{2}\\left(\\frac{1}{N_0+s}+\\frac{s^2}{(N_0+s)^{2}d}\\arg\\min\\limits_{\\underline{\\alpha}\\in\\mathcal{A}(s)}\\underline{\\alpha}^T\\Theta^T{J}({{\\underline{\\theta}}_0})\\Theta\\underline{\\alpha}\\right),\n$$\n\nwhere\n\n$$\n\\mathcal{A}(s)=\\bigg\\\\{\\underline{\\alpha}\\Bigg|\\sum_{i=1}^{K}\\alpha_i=1, s*\\alpha_i \\le N_i, \\alpha_i \\ge0,i=1,\\dots,K \\bigg\\\\}.\n$$\n\nThis problem requires optimizing the objective function over two variables: a scalar variable $s$ representing the total transfer quantity, and a vector variable $\\underline{\\alpha}$ representing the proportion of samples drawn from each source domain. For $s$ which is restricted to integer values, we perform a exhaustive search over its feasible domain $[0, s_{\\max}]$，where $s_{\\max}= \\sum\\limits_{i=1}^{K} N_i$. For each candidate $s'$ in the search, we compute the optimal $\\underline{\\alpha}'$ under the constraint $\\mathcal{A}(s')$, which is a $K\\times K$ quadratic programming problem with respect to $\\underline{\\alpha}$\n\n$$\n\\underline{\\alpha}'=\\arg\\min\\limits_{\\underline{\\alpha}\\in\\mathcal{A}(s')}\\underline{\\alpha}^T\\Theta^T{J}({{\\underline{\\theta}}_0})\\Theta\\underline{\\alpha}. \n$$\n\n\nThe quadratic coefficient matrix in this optimization problem is given by $\\Theta^\\top J(\\underline{\\theta}_0) \\Theta$. Since the Fisher information matrix $J(\\underline{\\theta}_0)$ is positive semi-definite, the quadratic coefficient matrix is also positive semi-definite. This guarantees the existence of a global optimal solution. After getting $s'$ and $\\underline{\\alpha}'$, the function is then evaluated at $(s', \\underline{\\alpha}')$. \n\nIn brief, for each $s'$, we solve for the corresponding optimal $\\underline{\\alpha}'$, yielding a finite collection of candidate solutions $(s', \\underline{\\alpha}')$ and their associated objective function values.\nAfter completing the search, the optimal solution $(s^\\*, \\underline{\\alpha}^\\*)$ is chosen as the pair that achieves the lowest objective function values among all candidate $(s', \\underline{\\alpha}')$. Since the feasible set of $s$ is finite and enumerable, and for each fixed $s'$ the optimization over $\\underline{\\alpha}'$ has a solution, the overall optimization problem is guaranteed to have at least one global solution. Hence, the optimal pair $(s^\\*, \\underline{\\alpha}^\\*)$ exists.\n\nIt is worth noting that, for the sake of computational efficiency, we do not exhaustively enumerate all possible values of $s$ in our experiments. Instead, we perform a grid search using 1000 uniformly spaced steps over the feasible range of $s$, where the number of steps is denoted as $stepnumber = 1000$. Experimental results in Section 5 demonstrate that this strategy does not compromise the effectiveness or stability of the method.\n\nThank you again for your valuable insights. We kindly request that the reviewer considers these points when assigning the final scores."}, {"review_id": "dEYlGrr4A2", "reviewer": "Reviewer_43HZ", "summary": "This paper presents a novel theoretical framework for determining the optimal number of samples to transfer from each source task in a multi-source transfer learning setting. The authors formulate the problem as a parameter estimation task and leverage high-dimensional statistical analysis to derive the optimal transfer quantities by minimising a generalisation error measure based on Kullback-Leibler (K-L) divergence. This framework is implemented in a practical, architecture-agnostic algorithm named OTQMS. The method is empirically evaluated on standard benchmarks, demonstrating superior performance in both accuracy and data efficiency compared to existing approaches.", "questions": "My questions are related to the weaknesses mentioned in the critical review above. I will summarise my main concerns in the form of questions for the authors:\n- The theoretical framework relies on the assumption that task parameters are asymptotically close ($|\\theta_{0}-\\theta_{1}|=O(\\frac{1}{\\sqrt{N_0}})$). While the empirical results appear robust, what are the theoretical implications when this assumption is strongly violated, for instance, in cases of far-domain transfer? How does the framework behave, and could it be extended to formally account for larger parameter distances?\n- The framework models task divergence as a unified parametric distance ($\\underline{\\theta}_i - \\underline{\\theta}_0$), which aggregates different sources of domain shift. Can the framework be decomposed to provide more granular insights into how different types of shift (e.g., covariate shift vs. concept shift) individually impact the optimal transfer quantity?\n- The dynamic OTQMS algorithm initializes the target parameter $\\underline{\\theta}_0$ using only the limited target data before iteratively refining it. How sensitive is the final model performance to the quality of this initial, and potentially noisy, estimation? Have you investigated whether a poor initialization could lead the optimization to a suboptimal solution?\n- The paper highlights significant reductions in training time. However, the OTQMS algorithm introduces a selection step in each epoch that involves solving a constrained quadratic program. Could you provide an analysis of the computational overhead of this selection process itself, particularly as the number of source domains ($K$) increases?\n- In the high-dimensional case, the task discrepancy term $t$ is an average over the parameter dimension $d$. Does this averaging potentially mask the impact of large discrepancies in a small subset of critical parameters? Would a dimension-weighted approach offer further improvements?\n- On page 4, you state the assumption $||\\theta_i-\\theta_0||=O(1/\\sqrt{N_0})$ is \"supported by related studies [20]\". Could you elaborate on how the cited work [20] specifically justifies this scaling law for parameter distance in the context of transfer learning?", "limitations": "My main comments on the limitations of this work are detailed in the \"Weaknesses\" section of the review above. To summarise, the primary limitations are: (1) the restrictive theoretical assumption that source and target task parameters are asymptotically close, which may not hold in cases of significant domain shift; (2) the framework's inability to distinguish between different types of transfer (e.g., covariate vs. concept shift), limiting its explanatory power; and (3) unanalyzed sensitivities in the practical OTQMS algorithm, namely its dependence on a potentially noisy initial estimate of the target model and the unexamined computational overhead of the sample selection step.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "# Strengths\n\n- The paper addresses a critical and often overlooked question in transfer learning: not just which sources to use, but how much data to use from each. \n- At its core, the paper proposes a solid theoretical analysis. It introduces the expected K-L divergence as a measure of generalisation error, which is well-justified by its connection to cross-entropy loss. The use of high-dimensional statistical analysis to derive the optimal transfer quantities is rigorous, and the proofs are clear.\n- The theoretical findings are successfully translated into a practical algorithm, OTQMS. A key feature is the dynamic strategy that iteratively updates the estimate of the target parameter $\\theta_0$​ and re-computes the optimal quantities, mitigating the issue of having limited target data. The algorithm is architecture-agnostic nature and its demonstrated compatibility with both full model training and parameter-efficient methods like LoRA.\n- The authors conduct extensive experiments on multiple real-world datasets (DomainNet, Office-Home, Digits) and across various settings. OTQMS consistently outperforms a range of state-of-the-art baselines in accuracy.\n\n# Weaknesses\n\n- The theoretical derivation for the optimal transfer quantity in Theorem 4 and its extensions relies on the assumption that the parameter distance between source and target tasks is small, i.e., $||\\theta_i-\\theta_0||=O(1/\\sqrt{N_0})$. The paper argues this is a reasonable assumption supported by related work and that its conclusions hold even when the distance is large. However, this is a strong condition that may not hold in cases of significant domain shift. The theoretical guarantees are strongest for closely related tasks, and the robustness in practice does not fully alleviate this theoretical limitation.\n- The framework models the difference between tasks solely as a parametric distance, $\\theta_i-\\theta_0$, within a given model class $P(X|\\theta)$​​. While this general formulation is a strength, it also prevents a deeper analysis of why tasks differ. The framework does not distinguish between different transfer learning scenarios, such as covariate shift (change in the input distribution $P(Z)$) versus concept shift (change in the conditional distribution $P(Y|Z)$). Consequently, it cannot provide insights into how these distinct types of domain shifts affect the optimal transfer quantity. The parameter vector $\\theta$​ conflates these potentially disparate effects.\n- The OTQMS algorithm requires computing the optimal transfer quantities $(s,\\alpha)$ in each training epoch. This is achieved by solving a quadratic programming problem for $\\alpha$ within a numerical search for $s$. While the paper reports a net reduction in training time due to using fewer samples, it does not analyse the computational overhead of this selection step itself. In scenarios with a very large number of source tasks (large $K$), this step could introduce a non-trivial computational burden that might offset the gains from training on a smaller dataset.\n- The dynamic strategy for estimating the target parameter $\\theta_0$ begins with a model trained only on the (often scarce) target samples. The accuracy of this initial estimate can be poor, potentially influencing the subsequent iterative optimisation process. While the dynamic approach is shown to be superior to a static one, the sensitivity of the final performance to this initial, potentially noisy, estimate of $\\theta_0$ is not explored.\n\n# Conclusion\n\nThe formulation of a method to optimise transfer quantity is a significant contribution, and the empirical results are compelling. However, the theoretical claims are narrower than the empirical results might suggest due to restrictive assumptions. I believe that the paper deserves acceptance for its novelty and practical impact, but its limitations should be clearly acknowledged. The theoretical framework is more of a good motivation for an optimisation strategy than a general theory of transferability.", "was_revised": true, "final_justification": "I maintain my recommendation to accept this paper.\n\n​The work addresses a novel and practical question in multi-source transfer learning: how much data to transfer from each source. The proposed theoretical framework, based on minimizing KL divergence, provides a principled motivation for the OTQMS algorithm. The algorithm itself is architecture-agnostic and demonstrates compelling empirical performance, outperforming several state-of-the-art methods on standard benchmarks.\n\n​I have read the authors' rebuttal and find their responses to my questions satisfactory. They have adequately clarified the limitations of the theoretical assumption regarding parameter distance, provided a reasonable analysis of the computational overhead of their method, and addressed concerns about the algorithm's sensitivity to initialization. ​While the theoretical guarantees are confined to a specific regime, the paper's primary contribution lies in formulating this problem and providing an effective practical solution.", "rebuttal": "We sincerely appreciate the reviewer’s thoughtful observation. Below, we address each of your concerns point by point.\n\n\n>Question1：How does the framework behave, and could it be extended to formally account for larger parameter distances?\n\nThank you for the valuable feedback. We agree that the assumption of a small parameter distance (i.e., $|\\theta_0 - \\theta_1| = O(\\frac{1}{\\sqrt{N_0}})$) is a strong condition underlying the asymptotic derivation in Theorem 4 and its extensions. This assumption is necessary for controlling higher-order terms in the theoretical analysis. Nevertheless, the optimal transfer quantity derived from the framework remains consistent with intuition even when the assumption does not strictly hold. In particular, as shown in lines 160–162, when the parameter distance becomes large, the optimal transfer quantity naturally reduces to zero, reflecting the expected behavior under significant domain shift. This suggests that the theoretical formulation still captures the correct transfer tendency even for less related tasks. Moreover, our empirical results in Section 5 demonstrate that the proposed algorithm remains robust across a wide range of transfer scenarios.\n\n>Question2：Can the framework be decomposed to provide more granular insights into how different types of shift (e.g., covariate shift vs. concept shift) individually impact the optimal transfer quantity?\n\nThank you for the insightful comment. Our framework models task divergence as a unified parametric distance between $\\underline{\\theta}_0$ and $\\underline{\\theta}_i$, which implicitly captures different types of domain shift (e.g., covariate shift and concept shift). This design reflects our goal of constructing a task-agnostic framework that determines the optimal transfer quantity without relying on prior knowledge of task-specific structure or decomposition.\n\nOn the other hand, in scenarios where such prior knowledge is available and the model architecture explicitly separates covariate and concept shifts, for example by assigning them to different layers for processing, the parametric difference vector $\\underline{\\theta}_0 - \\underline{\\theta}_i$ naturally reflects these distinctions  across different subsets of parameters. In such cases, our method can effectively incorporate both types of shift during optimization without requiring explicit separation.\n\n>Question3：How sensitive is the final model performance to the quality of this initial, and potentially noisy, estimation?\n\nAs discussed in Section 5.4, we investigate the performance of OTQMS under different shot settings. The results show that increasing the number of target samples indeed improves the quality of the initial estimation, which in turn leads to better model performance. Nevertheless, across the vast majority of settings, our method consistently outperforms the baselines, demonstrating its robustness to the quality of the initial estimation.\n\n>Question4：The paper highlights significant reductions in training time. However, the OTQMS algorithm introduces a selection step in each epoch that involves solving a constrained quadratic program. Could you provide an analysis of the computational overhead of this selection process itself, particularly as the number of source domains $K$ increases?\n\nThank you for the question. The detailed procedure of the selection step is provided in Appendix F. Here, we provide an analysis of its computational complexity. Specifically, the procedure involves optimizing over two variables: a scalar variable $s$ representing the total transfer quantity, and a vector variable $\\underline{\\alpha}$ representing the proportion of samples drawn from each source domain. We perform a uniform grid search over the feasible range of $s$, i.e., $[0, s_{\\max}]$, using 1000 uniform steps (denoted as $stepnumber = 1000$), where $s_{\\max}= \\sum\\limits_{i=1}^{K} N_i$. For each candidate value $s'$, we solve a constrained optimization problem over $\\underline{\\alpha}$ under the constraint set $A(s')$, which corresponds to a $K \\times K$ quadratic program (QP) with complexity $\\mathcal{O}(K^3)$. Thus, the overall time complexity of the selection process is $\\mathcal{O}(stepnumber \\cdot K^3)$. In our experiments on the Office-Home dataset, where $stepnumber = 1000$ and $K = 3$, the total computational cost of the selection step is approximately $2.7 \\times 10^4$ basic operations per epoch. Empirically, the selection step takes around 3.19 seconds per epoch on average, accounting for only 1.1% of the total training time. We acknowledge the reviewer’s concern regarding the growth of computational overhead with respect to $K$. In practice, the number of source domains in standard transfer learning benchmarks is typically small (e.g., $K \\leq 10$), making the $\\mathcal{O}(stepnumber \\cdot K^3)$ complexity acceptable for most practical applications. Moreover, for scenarios with a large number of source domains, one possible extension is to cluster or merge similar domains before applying our algorithm.\n\n\n\n>Question5：Does this averaging potentially mask the impact of large discrepancies in a small subset of critical parameters? Would a dimension-weighted approach offer further improvements?\n\nUnder the high-dimensional setting in Eq. (13) and Eq. (15), the Fisher information matrix $J$ within the $t$ term naturally plays a role similar to dimension-weighted averaging. Specifically, the Fisher matrix captures the model’s sensitivity to perturbations in different dimensions of the parameter vector, thereby implicitly assigning greater importance to more influential parameters.\n\n>Question6：Could you elaborate on how the cited work [20] specifically justifies this scaling law for parameter distance in the context of transfer learning?\n\nThank you for the question. [20] provides empirical evidence that supports this assumption. The paper shows that during fine-tuning, model parameters—especially in the lower layers—change very little from their pretrained values. This suggests that for related domains or similar data distributions, the corresponding optimal parameters lie close in parameter space. Since transfer learning typically involves source and target tasks with some degree of similarity, we interpret this as indirect support for our assumption that the parameter distance between source and target models is small in transfer learning settings.\n\nThank you for your insightful comment!"}, {"review_id": "yLKKVv9H4k", "reviewer": "Reviewer_ZUcy", "summary": "This paper discusses the optimal quantity of source samples from each source task. A generalization error measure based on KL divergence is introduced. The quantity of source samples can be solved based on high-dimensional statistical analysis. Experiment results demonstrate the effectiveness of the proposed method.", "questions": "The questions are consistent with the \"Weaknesses\".\n\n1.\tAs the learning objective of many problems are not limited to negative log likelihood, I am interested in the optimal quantities with general learning objectives, such as cross-entropy loss in classification problems and mean square error in regression problems, or at least some error analysis when the proposed method is applied in general settings.\n\n2.\tHow can the proposed method combine with other parts of the transfer learning? For examples, the sample selection and domain alignment. I understand these may be a bit  out of the scope of this paper, so it's only a suggestion and the answer will not decrease the score.", "limitations": "The authors have mentioned the sampling method in the discussion of limitations, but the limitations in the general training objective have not been discussed.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: \n\n1. The task that estimates the optimal quantity of source samples holds broad application scenarios. And the theoretical results are intuitive yet not trivial.\n\n2. The derivation of the method is reasonable, and the training process is clear and is easy to generalize to other tasks.\n\n3. The authors have conducted sufficient evaluation and analysis experiments, demonstrating the effectiveness and efficiency of the proposed method.\n\nWeaknesses:\n\n1. The main theorems are derived based on the MLE estimation of the model parameters. In learning problems with other objectives, the effectiveness of the results is unclear.\n\n2. Optimizing the quantities of source samples should be considered as part of the whole transfer learning paradigm. Therefore, further discussions about the relation with sample selection and domain alignment will be helpful.", "was_revised": true, "final_justification": "Despite existing limitations, I still believe that the proposed method to optimize transfer quantity is a meaningful and novel contribution to the field of transfer learning.", "rebuttal": "We are grateful to you for taking the time to review our paper and provide thoughtful insights. Below, we address each of your concerns point by point.\n\n\n>Question1：As the learning objective of many problems are not limited to negative log likelihood, I am interested in the optimal quantities with general learning objectives, such as cross-entropy loss in classification problems and mean square error in regression problems, or at least some error analysis when the proposed method is applied in general settings.\n\nThank you for your valuable suggestion. For the case of cross-entropy loss in classification problems, it is essentially equivalent to the negative log-likelihood when softmax outputs and one-hot encoded labels are used. This corresponds precisely to the setup adopted in the main experiments of our paper.\n\nFor other learning objectives, such as mean squared error in regression tasks, we acknowledge that there is a gap compared to the negative log-likelihood formulation assumed in our theoretical analysis. While our framework does not directly cover these cases, we believe the core idea can be extended to broader loss functions under certain regularity conditions. We will clarify this limitation in the revised version of paper, and include additional experiments with other loss functions to empirically demonstrate the robustness and broader applicability of our method.\n\n>Question2：How can the proposed method combine with other parts of the transfer learning? For examples, the sample selection and domain alignment. \n\nThank you for the insightful suggestion, which points to a valuable direction for future work. In this paper, after obtaining the optimal transfer quantity of each source domain, we adopt a straightforward random sampling strategy to construct the joint dataset in our algorithm implementation. As our theoretical analysis is based on average-case assumptions, random sampling is sufficient to validate the robustness of both the theoretical framework and the proposed algorithm. Nonetheless, we anticipate that more advanced sample selection strategies, such as active sampling, could further enhance the algorithm’s performance. We plan to explore this possibility in future work.\n\nOnce again, we thank you for the constructive feedback."}, {"review_id": "NRCJPj6968", "reviewer": "Reviewer_2Vcv", "summary": "This paper aims to answer the question of “what is the optimal sample size for each sources when conducting transfer learning.” The authors proposes a parametric model-agnostic framework to find out the optimal size for each source by minimizing the expected KL divergence between true target distribution and the pooled-version estimated distribution w.r.t. source sample sizes. Practical algorithm is presented and the effectiveness of the proposed algorithm is evaluated through experiments.", "questions": "The authors provide a numerical approach in Appendix F to get the solution to objective (14). The authors proposed to do the grid search by first fixing $s$ in some feasible domain of $s$. However, what is the feasible domain of $s$ in general? What is it order? If no prior knowledge kicks in, the range of the feasible domain of $s$ can go quite large and ruin the computation.", "limitations": "Yes, although authors only put limitations in the appendix.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "## Strength\n\n1. To my knowledge, while most existing transfer learning work focus on finding the optimal sources, this paper goes with another direction by focusing on finding the optimal sample size the algorithm should use from each source.\n2. The methodology is theory-driven.\n\n## Weakness\n\n1. **Terminology**: In theoretical statistics or machine learning community, the term “high-dimensional” usually refer to the cases where the dimension of the data fast excess the sample size, i.e., $d \\gg n$, or the dimension grows as sample size increase. In such regime, many theoretical results in fixed dimension no longer hold and require different techniques or mathematical tools. For example, the asymptotic normality, i.e., eq (2), could fail under high-dimensional regime. I believe what authors consider is a fixed dimension $d$, which aligns with the classical multivariate analysis. Using “high-dimensional” is rather misleading. \n2. **The set-up**: Although I don’t appreciate the idea that the authors consider the data are generated from a specific network parameterized by $\\theta$ cuz it can produce some specification problem, it generally does not affect the core idea of the paper. However, the assumption that all source tasks and the target task share the same parametric model is worrying, which is practically impossible. This assumption is essential for the derivation in Section 4, as if it does not hold, the proposed methodology does not stand.\n3. **The estimator**: The transfer learning estimator considered in this manuscript is rather simple, which is the minimizer of the pooled log-likelihood. It has been too simple to provide helpful insight into the current transfer learning paradigm, i.e., pre-training and fine-tuning. Pooling both source and target data to obtain an estimator is never effective and efficient, as has been long demonstrated in the community. \n    \n    Generally, I can understand why authors need to consider such estimator. This pooled version allows it corresponding distribution be the weighted combination of all source and target distribution (where the weights are determined by the sample size ratio). This allows the authors to derive the expression of expected KL divergence easier under the (1) same parametric model assumption (point 2) and (2) sufficient close models assumption (line 139). However, with so many restriction, again, it is too hard to have the theory provide insights.\n    \n4. **Assumption**:  The bounds in this paper are asymptotic and the authors require a “sufficient small models” assumption (line 139 or line 145) when derive these asymptotic results. However, in asymptotic regime, the “sufficient small models” actually indicates the source and target tasks/models are identical. Therefore, it is not surprised why the assumptions on same parametric model and pooled estimator are needed.\n5. **Practical algorithm**: The practical algorithm (QTAMS) replaced the true source parameters by their empirical version. While this is a compromise with reality where the true parameters are unknown in practice, we have to remember that, since the bounds are asymptotic, there could be error terms induced by the approximation that are not lower order of $1/(N_0 + n_1)$ in eq (8), (12), (14). This will make the optimal solution for $n_{k}$ no longer follows the solutions derived from the objective in eq (8), (12), (14).\n6. **Experiments**: I do not closely work on applications, so I typically have no idea what SOTA results would be, therefore, please view my comments on this point dialectically. For me, QTQMS only improved very limited compared to existing approaches/baselines, i.e., the improvement over the runner-up is rather small. What is more interesting to me is that the efficiency improved by QTQMS.", "was_revised": true, "final_justification": "After reading authors' response and other reviewers' comment, I will raise my score. Please make sure to revised the camera-ready version based on the discussion period accordingly.", "rebuttal": "We appreciate your detailed comments. Below, we address each of your concerns point by point.\n\n>Question1:what is the feasible domain of s?  \n\nThank you for the question. The feasible domain of the total transfer quantity $s$ is defined as the interval from 0 to the total number of available source samples, i.e., $s\\in[0, s_{\\max}]$ where $s_{\\max} = \\sum\\limits_{i=1}^{K} N_i$. In our experiments, we perform a uniform grid search over this interval using 1000 equally spaced steps (denoted as $stepnumber = 1000$). The computational complexity of this procedure depends solely on the value of $stepnumber$, and is independent of the size of the feasible domain. Therefore, a large feasible range does not pose a challenge to the efficiency of the algorithm. A detailed analysis of the computational complexity of this process can be found in our response to Reviewer 43HZ, Question 4.\n\n\n>Weakness1: Using “high-dimensional” is rather misleading.\n\nThank you for your suggestion. We agree that the term “high-dimensional” is misleading in our context, as our analysis is conducted in a fixed-dimensional regime. In the revised version of the paper, we will remove the term \"high-dimensional statistics\" and replace it with the more precise term \"asymptotic analysis\".\n\n>Weakness2: The assumption that all source tasks and the target task share the same parametric model is worrying, which is practically impossible. \n\nThank you for raising this important concern. We would like to clarify that our framework only assumes that the source and target tasks share the same parametric model class, but each task is associated with its own set of model parameters. As described in Section 3, the underlying model for the target task is $P\\_{X;\\underline{\\theta}\\_0}$, while the source tasks $\\mathcal{S}\\_1, \\dots, \\mathcal{S}\\_K$ correspond to models $P\\_{X;\\underline{\\theta}\\_1}, \\dots, P\\_{X;\\underline{\\theta}\\_K}$, respectively.\n\nIn transfer learning, it is common to select source and target tasks with the same input/output structure and similar feature spaces. This makes the use of a shared parameterized model a reasonable assumption, which is also widely adopted in the multi-source transfer learning literature [1][2].\n\nMoreover, we do not impose any specific structural assumptions on the form of the underlying model $P_{X;\\underline{\\theta}}$. In fact, by the Universal Approximation Theorem [5], a feedforward neural network with sufficient width can approximate any continuous function on a compact domain. This ensures that our framework is applicable to a broad class of tasks and models, without being limited to a particular architecture.\n\nIn summary, while our theoretical derivation requires a shared model class, it allows full flexibility in parameterization across tasks. This assumption enables a general and tractable formulation, and our experiments further demonstrate the robustness and effectiveness of the proposed algorithm in realistic settings.\n\n>Weakness3:It has been too simple to provide helpful insight into the current transfer learning paradigm, i.e., pre-training and fine-tuning. Pooling both source and target data to obtain an estimator is never effective and efficient, as has been long demonstrated in the community.\n\nThank you for the comment. We respectfully clarify that the model-based paradigm you mentioned (e.g., fine-tuning) and the sample-based paradigm adopted in this paper represent two established approaches to transfer learning. While model-based methods have been more extensively studied, recent works—as well as our own results—demonstrate that sample-based strategies can also achieve competitive performance [3][4], both effectively and efficiently when properly designed.\n\nWe would also like to draw your attention to Table 2, where the runner-up method “AllSources $\\cup$ Target” is a simple pooled estimator that uses all available source and target data for training. While naive, this baseline performs surprisingly well on certain datasets. The fact that more complex baselines do not consistently outperform such a simple pooling strategy suggests that they may not be universally suitable across all datasets.\n\n>Weakness4:The bounds in this paper are asymptotic and the authors require a “sufficient small models” assumption (line 139 or line 145) when derive these asymptotic results. However, in asymptotic regime, the “sufficient small models” actually indicates the source and target tasks/models are identical.\n\nThank you for the comment. We would like to clarify that the assumption of a “sufficiently small model distance” (i.e., $|\\theta_0 - \\theta_1| = O(1/\\sqrt{N_0})$) is a technical condition introduced to control higher-order terms in the asymptotic analysis. Given that the source and target tasks used in transfer learning often share similar input-output structures and semantic spaces, this assumption is also reasonable in practice.\n\nNevertheless, it does not imply that the source and target tasks are identical. Importantly, even under this assumption, our framework allows for meaningful differences between tasks. If the tasks were truly identical, all source samples would always benefit the target task. In contrast, both our theoretical analysis (lines 155–164) and experimental results (Section 5) demonstrate that using all available source samples is not always optimal. Furthermore, in the multi-source setting, the optimal transfer quantity from each source depends explicitly on its distance to the target task, as formally characterized in Theorem 7 and empirically validated in Appendix G.\n\n>Weakness5:Practical algorithm: The practical algorithm (QTAMS) replaced the true source parameters by their empirical version. While this is a compromise with reality where the true parameters are unknown in practice, we have to remember that, since the bounds are asymptotic, there could be error terms induced by the approximation.\n\nThank you for pointing out this important issue. We fully agree that the substitution of true source parameters $\\theta_i$ with their empirical estimates $\\hat{\\theta}_i$ introduces approximation error. We acknowledge that this is a limitation of our current formulation, as the true source parameters are inaccessible in real-world scenarios. Despite this, we argue that our framework still provides a principled approximation scheme, and our empirical results (Section 5.2) show that it yields consistently strong performance across diverse settings. We will explicitly state this limitation in the revised version of paper. A more refined theoretical treatment that accounts for estimation error in source parameters is indeed an interesting direction for future work.\n\nMoreover, it is important to note that our definition of $N_i$ refers only to the number of source samples available during the target training stage, rather than the full dataset used to obtain the empirical estimates $\\hat{\\theta}_i$. In many real-world scenarios—such as the use of large pretrained models including GPT-3 [6] and PaLM [7]—the parameters $\\hat{\\theta}_i$ are obtained by training on datasets that are significantly larger than the accessible source sample sizes $N_i$, often incorporating non-public or proprietary data sources. In such cases, the estimation error of $\\hat{\\theta}_i$ is smaller than the order of $1 / (N_0 + n_1)$, making it a highly reliable approximation to $\\theta_i$ and thereby justifying our substitution.\n\n>Weakness6:The improvement over the runner-up is rather small. What is more interesting to the reviewer is that the efficiency improved by QTQMS.\n\nThank you for your thoughtful and balanced comment. We acknowledge that the performance gains of QTQMS over the strongest baselines are relatively modest in some cases. However, our goal is not only to pursue absolute accuracy, but also to develop a theoretically principled and practically efficient framework for multi-source transfer learning.\n\nIn fact, several components of our current implementation remain intentionally simple, in order to better isolate and verify the robustness of the theoretical framework. For instance, after obtaining the optimal transfer quantity for each source domain, we adopt a straightforward random sampling strategy to construct the joint dataset. This design choice allows us to focus on validating the effectiveness of the core theory. Nevertheless, we believe that incorporating more advanced sampling strategies—such as active sampling—could further improve the performance of the algorithm.\n\nIn addition, as shown in Section 5.4, QTQMS exhibits stable and superior performance across different shot settings, further demonstrating the robustness of the proposed method. As the reviewer kindly noted and as demonstrated in Section 5.5, QTQMS consistently achieves comparable or better accuracy while significantly reducing computational overhead, which we believe is one of the key advantages of our approach.\n\nThank you again for your valuable insights. We kindly request that the reviewer considers these points when assigning the final scores.\n\n## References：\n[1]Wu, Y., Wang, J., Wang, W., and Li, Y. H-ensemble: An information theoretic approach to reliable few-shot multisource-free transfer. AAAI 2024\n\n[2]Li Y, Yuan L, Chen Y, et al. Dynamic transfer for multi-source domain adaptation[C] CVPR2021\n\n[3]Shui C, Li Z, Li J, et al. Aggregating from multiple target-shifted sources[C]//International conference on machine learning. PMLR, 2021: 9638-9648.\n\n[4]Zhang W, Lv Z, Zhou H, et al. Revisiting the domain shift and sample uncertainty in multi-source active domain transfer[C] CVPR 2024\n\n[5] Hornik K. Approximation capabilities of multilayer feedforward networks[J]. Neural networks, 1991\n\n[6] Brown T, Mann B, Ryder N, et al. Language models are few-shot learners[J]. NeurIPS 2020\n\n[7] Chowdhery A, Narang S, Devlin J, et al. Palm: Scaling language modeling with pathways[J]. Journal of Machine Learning Research, 2023, 24(240): 1-113."}]

Author rebuttal:

Meta-review: This paper received generally positive ratings (5, 5, 4, 4). The reviewers appreciated the novel problem formulation of optimizing transfer quantities from each source task in multi-source transfer learning, with a theoretically-grounded KL-divergence framework. The main concerns centered on restrictive theoretical assumptions (small parameter distances between tasks, asymptotic analysis), misleading "high-dimensional" terminology, and the simplicity of the pooled MLE approach compared to modern transfer learning paradigms. The authors provided comprehensive rebuttals addressing most concerns, including computational complexity analysis, clarification of assumptions, and commitment to terminology corrections. All reviewers maintained or raised their scores post-rebuttal, acknowledging that while theoretical limitations exist, the practical contributions and consistent empirical improvements justify acceptance. Given the consensus from four reviewers supporting acceptance, the paper's novel contribution to an underexplored problem, and good empirical validation across multiple settings, the AC recommends acceptance. The authors should address remaining concerns in the final version, particularly clarifying the scope of theoretical assumptions and correcting the "high-dimensional" terminology.
Decision: Accept (poster)