Rethinking the Flow-Based Gradual Domain Adaption: A Semi-Dual Optimal Transport Perspective

TL;DR

The paper addresses the brittleness of traditional flow-based Gradual Domain Adaptation under absent or ineffective intermediate domains by introducing E-SUOT, an entropy-regularized, semi-dual unbalanced OT framework that generates intermediate domains directly from samples without explicit target PDFs. By reformulating flow evolution as a semi-dual optimization and adding entropy regularization, the method stabilizes the notorious min–max training and guarantees uniqueness of the optimization landscape, enabling a robust, offline workflow that yields a sequence of transport maps toward the target domain. The authors provide theoretical results on stability and generalization, and demonstrate substantial empirical gains over state-of-the-art GDA and UDA baselines across multiple datasets, with strong ablations highlighting the importance of entropy regularization and early transport steps. Overall, E-SUOT unifies flow-based and OT-based ideas under a PDF-free, theoretically principled framework that improves both the reliability and performance of gradual domain adaptation in practice.

Abstract

Gradual domain adaptation (GDA) aims to mitigate domain shift by progressively adapting models from the source domain to the target domain via intermediate domains. However, real intermediate domains are often unavailable or ineffective, necessitating the synthesis of intermediate samples. Flow-based models have recently been used for this purpose by interpolating between source and target distributions; however, their training typically relies on sample-based log-likelihood estimation, which can discard useful information and thus degrade GDA performance. The key to addressing this limitation is constructing the intermediate domains via samples directly. To this end, we propose an Entropy-regularized Semi-dual Unbalanced Optimal Transport (E-SUOT) framework to construct intermediate domains. Specifically, we reformulate flow-based GDA as a Lagrangian dual problem and derive an equivalent semi-dual objective that circumvents the need for likelihood estimation. However, the dual problem leads to an unstable min-max training procedure. To alleviate this issue, we further introduce entropy regularization to convert it into a more stable alternative optimization procedure. Based on this, we propose a novel GDA training framework and provide theoretical analysis in terms of stability and generalization. Finally, extensive experiments are conducted to demonstrate the efficacy of the E-SUOT framework.

Figures (5)

• Figure 1: The illustration of the proposed E-SUOT: (a) the unbalanced OT formulation used to solve the transport map $\boldsymbol{T}_\theta(\cdot)$ at time $t$, where thicker arrows and larger points indicate higher mass flows, and (b) the evolution process from the source to the target domain. This figure is conceptually inspired by previous works on GDA and OT tasks zhuang2024gradualhe2024gradualwangunbiased.
• Figure 2: Ablation study of E-SUOT performance vary the UMAP embedding dimension on the Office-Home dataset. For dimension 256, the vanilla backbone features are used without UMAP. The shaded area indicates the $\pm$ 5.0 times standarad deviation error.
• Figure 3: Sensitivity analysis results on Portrait Dataset.
• Figure 4: Computational time (s).
• Figure :

Theorems & Definitions (20)

• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4
• Theorem 3.5
• Theorem 3.6
• Proposition : \ref{['thm:firstThmResultsDualProblem']}
• proof
• Proposition : \ref{['thm:notHaveUniqueSolution']}
• proof