On $f$-Divergence Principled Domain Adaptation: An Improved Framework

Summary

This study addresses the gap in the theory and algorithms of unsupervised domain adaptation based on f-divergence proposed by Acuna et al. 2021. Specifically, while the theory uses absolute values, the algorithms do not, and this issue is resolved by introducing a single scaling factor. The newly proposed f-DD generalization bound is derived based on Rademacher complexity, and tighter bounds are obtained using the localization technique. As a specific domain adaptation algorithm, an adversarial type algorithm is proposed, yielding favorable results in benchmarks.

Strengths

This study bridges the gap between theory and practice in existing UDA methods based on f-divergence, advancing the foundational research in domain adaptation. Furthermore, the derivation of sharper bounds using the recently introduced localization technique for DA is highly commendable as a contribution to the theoretical framework of DA. The authors validate their theoretical contributions with empirical results, showing superior performance on popular benchmarks.

Weaknesses

While the empirical validation is strong, it is limited to specific benchmarks. Broader validation across diverse datasets and tasks would strengthen the findings. It is nice to present some insight into what kind of dataset the proposed f-DD works well (and why), and also into what kind of dataset it does not work well (and why).

Questions

Please provide the source of the technique that resolves the absolute value with a scaling parameter. Additionally, clarify whether the claim in this paper—that there is no need to adjust the scaling parameter—holds generally.

Rating

6

Confidence

3

Soundness

3

Presentation

3

Contribution

3

Limitations

Not applicable.

Summary

This paper studies the learning theory aspect of the domain adaptation problem, where the key is to bound the estimation errors between expectations over shifting distributions. Specifically, this work improves the recently developed $f$-divergence-based generalization analysis, where the main results ensure a tighter generalization upper bound and the consistency between theory and method. For finite sample setting, a sharp bound is provided to accelerate the asymptotic rate. Numerical simulation is conducted to demonstrate the superiority of the theory-guided method over the existing discrepancy-based framework.

Strengths

+ The motivation is clear, i.e., improving the $f$-divergence-based bound and bridging the gap between method and theory, and the presentation is easy to follow. + The technical part is generally sound and the justifications are sufficient. + The experiment results are superior compared with recently developed generalization bounds.

Weaknesses

+ Some notations are inconsistent in theoretical analysis. + The proposed algorithm needs further justifications. + The experiment comparison could be improved.

Questions

There seem no major faults in this submission, and I only have the following minor concerns. Q1. Theory and methodology. The major result for the target error bound is provided in Eq. (4) in Thm. 4.1 and the specific bound w.r.t. KL-divergence is presented in line 162, where the induced learning objective consists of source risk and the square root of cross-domain KL-divergence. However, it seems that the optimization objective Eq. (5) considers the divergence without the square root directly. I understand the optimal solutions are the same for these two objectives (if the optimal solutions ensure 0 cross-domain discrepancies). But considering Eq. (4) is closely related to the major merit of this work, i.e., the tight bound, the consistency between Eq. (4) and Eq. (5) seems to be important. Some justifications are highly expected. Q2. Method application. As far as I understand this work, the derived $f$-DD measure can be applied to existing works whose primary goal is discrepancy minimization. Thus, it could serve as a plug-and-play module for existing SOTA DA methods. Thus, some detailed discussions on the capability of $f$-DD w.r.t. existing methods are highly expected. Q3. Following Q2, apart from the experiments in the current version, some comparisons between SOTA DA methods and their combination with $f$-DD objective are highly expected. Q4. The clarity w.r.t. definitions could be improved, e.g., $K_{h',\mu}(t)$ depends on the hypothesis $h$ while the justification (i.e., line 178) is provided after the definition (i.e., line 176). A thorough check for these issues could improve the readability. Q5. Some notations seem to be inconsistent. For example, the notation $I_{\nu}^{\phi}(h,h')$ in line 132 is inconsistent with $I$ in line 129; the notations $\mathbb{E}_{\nu}$ in line 132 seems to be incorrect (probably should be expectation over $\mu$?).

Rating

7

Confidence

2

Soundness

3

Presentation

3

Contribution

3

Limitations

The limitations are discussed in the checklist, and there seems no potential negative societal impact.

Summary

This paper aims to develop an improved version of f-divergence-based unsupervised domain adaptation (UDA) learning theory. In particular, the authors introduce a novel f-divergence-based domain discrepancy measure (f-DD) by combining the two existing concepts, which are f-divergence and domain discrepancy. Based on that f-DD measure, the paper next provides a generalization bound on the target domain, which is shown to be sharper than the existing related bound. The experimental results consistently demonstrate that f-DD outperforms the original f-DAL in three popular UDA benchmarks, with the best performance achieved by Jeffereys-DD.

Strengths

The paper is well-written and easy to follow. The idea of introducing f-divergence-based UDA, targeting a better risk-bound on the target domain is novel and interesting. All the main statements of the paper are theoretically supported, though I did not have enough time to verify all of those propositions/theorems carefully. The experimental results consistently demonstrate that f-DD outperforms the original f-DAL in three popular UDA benchmarks, with the best performance achieved by Jeffereys-DD.

Weaknesses

The novelty of the paper is quite limited since the f-divergence-based domain discrepancy measure (f-DD) is proposed by combining the two existing concepts, which are f-divergence and domain discrepancy. In Theorem 5.2, the authors claim that the application of the localization technique gives a fast-rate generalization, they do not provide a concrete evidence. Could the author give some explanations/clarifications for that. Moreover, the experimental part of the paper seems not be very convincing since it only provides experiments with quite small datasets (Office31, Office-Home, MNIST & USPS) and simple model (e.g., Lenet). It raises the concern about capability of f-DD in more complicated settings with large datasets and backbone network.

Questions

Please refer to my comments about the weaknesses of the paper.

Rating

6

Confidence

3

Soundness

3

Presentation

3

Contribution

2

Limitations

N/A

Summary

This paper improves the theoretical foundations of UDA proposed by previous work, named f-DD. By removing the absolute value function and incorporating a scaling parameter, f-DD yields novel target error and sample complexity bounds, allowing us to recover previous KL-based results and bridging the gap between algorithms and theory presented in Acuna et al. Leveraging a localization technique, this paper also develops a fast-rate generalization bound. Empirical results demonstrate the superior performance of f-DD-based domain learning algorithms over previous works in popular UDA benchmarks.

Strengths

1) This paper holds significant theoretical significance in the field of UDA (Unsupervised Domain Adaptation); 2) The proof of the theorem is very solid; 3) The experiments are also sufficient.

Weaknesses

1) The readability of the paper is poor. It is almost entirely composed of definitions, remarks, lemmas and theorems, lacking a figure to introduce the motivation of this paper and explain why the improved framework is effective. 2) It is difficult to reproduce the results, as the training objective (5) is very abstract and unclear how to implement it experimentally. 3) This paper requires a substantial foundation of reading other papers in order to be understood.

Questions

How to implement the training objective (5)?

Rating

4

Confidence

3

Soundness

3

Presentation

2

Contribution

4

Limitations

Please refer to the weakness.

Summary

In this paper, new expected risk analysis based on f-divergence is provided for the unsupervised domain adaptation problem. Although there are prior researches on expected risk analysis based on f-divergence, several issues have been pointed out, such as the fact that the variational representation of f-divergence used in these studies does not recover the Donsker-Varadhan representation of KL-divergence, and the use of the absolute value of the variational representation as a measure of domain discrepancy. In this paper, to address these issues, the authors adopt an alternative variational representation of f-divergence and, based on this, provide an upper bound evaluation of the expected risk in the target domain, namely ``target risk $\le$ source risk + marginal domain discrepancy + joint error''. Additionally, a sample approximation version of the derived upper bound is also provided, allowing it to be estimated from the data (excluding the joint error part, as in conventional bounds).

Strengths

- The paper clearly discusses what are difficulties with the conventional DA theory using f-divergence and explains how it is solved by the proposed approach. Especially, this paper provides a solid theoretical foundation, with detailed assumptions and rigorous proofs that are well-documented in the appendix. - Previous expected risk bounds in UDA have often been given by relatively simple inequality evaluations, following the formulation given by Ben-David et al. In contrast, a similar upper bound evaluation using the f-DD proposed in this paper requires an inequality evaluation for ``change of measure" (as given in Lemma 4.1), and it can be seen that this is not an incremental extension of the conventional DA theory.

Weaknesses

- I don't think there is enough information needed when trying to calculate the derived upper boundary from the sample. For example, $t_0$ in Lemma 4.2 and the construction of the Rashomon set in Sec 5 should be discussed in more detail.

Questions

- Are no assumptions specifically placed on the hypothetical set $\mathcal{H}$? - In Lemma 4.2, is there a way to estimate the value of t_0 when t_0 cannot be written in closed form (e.g. for KL or Jeffereys employed in the Experiments?)? - The Rashomon set used in Section 5 appears to be in fact the set to be estimated (as the true expected risk of the source domain is unknown). How exactly is the Rashomon set constructed in this paper? - In the experiments, three types of discrepancies are evaluated for the proposed method, namely KL-DD, $\chi^2$-DD and Jeffereys-DD. Then, do you have any insights on which discrepancy measure should be used for which type of problem? I think the question of which measure to use to evaluate domain discrepancy is critical not only for theorists but also for practitioners. - Is the f-DD proposed in Definition 4.1 always 'better' than the existing f-divergence-based discrepancy (Definition 3.1)? I am wondering whether there are cases where using absolute values to define the discrepancy is an advantage.

Rating

7

Confidence

3

Soundness

3

Presentation

3

Contribution

3

Limitations

The authors have adequately addressed the limitations and potential negative societal impact of their work.

Upgrading my score to 6

Thanks to the authors for the detailed responses, especially for clarifying the scope of the paper. Since most of my concerns have been addressed, I upgraded the paper's score to 6.