Transfer learning is a powerful paradigm for leveraging knowledge from source domains to enhance learning in a target domain. However, traditional transfer learning approaches often focus on scalar or multivariate data within Euclidean spaces, limiting their applicability to complex data structures such as probability distributions. To address this limitation, we introduce a novel transfer learning framework for regression models whose outputs are probability distributions residing in the Wasserstein space. When the informative subset of transferable source domains is known, we propose an estimator with provable asymptotic convergence rates, quantifying the impact of domain similarity on transfer efficiency. For cases where the informative subset is unknown, we develop a data-driven transfer learning procedure designed to mitigate negative transfer. The proposed methods are supported by rigorous theoretical analysis and are validated through extensive simulations and real-world applications. The code is available at https://github.com/h7nian/WaTL.

Theoretical works on supervised transfer learning (STL)--where the learner has access to labeled samples from both source and target distributions--have for the most part focused on statistical aspects of the problem, while efficient optimization has received less attention. We consider the problem of designing an SGD procedure for STL that alternates sampling between source and target data, while maintaining statistical transfer guarantees without prior knowledge of the quality of the source data. A main algorithmic difficulty is in understanding how to design such an adaptive sub-sampling mechanism at each SGD step, to automatically gain from the source when it is informative, or bias towards the target and avoid negative transfer when the source is less informative. We show that, such a mixed-sample SGD procedure is feasible for general prediction tasks with convex losses, rooted in tracking an abstract sequence of constrained convex programs that serve to maintain the desired transfer guarantees. We instantiate these results in the concrete setting of linear regression with square loss, and show that the procedure converges, with 1/ T rate, to a solution whose statistical performance on the target is adaptive to the a priori unknown quality of the source. Experiments with synthetic and real datasets support the theory.

Most existing transfer learning algorithms for high-dimensional models employ a two-step regularization framework, whose success heavily hinges on the assumption that the pre-trained model closely resembles the target. To relax this assumption, we propose a co-regularization process to directly exploit beneficial knowledge from the source domain for high-dimensional generalized linear models. The proposed method learns the target parameter by constraining the source parameters to be close to the target one, thereby preventing fine-tuning failures caused by significantly deviated pre-trained parameters. Our theoretical analysis demonstrates that the proposed method accommodates a broader range of sources than existing two-step frameworks, thus being more robust to less similar sources. Its effectiveness is validated through extensive empirical studies.

While data-driven methods such as neural operator have achieved great success in solving differential equations (DEs), they suffer from domain shift problems caused by different learning environments (with data bias or equation changes), which can be alleviated by transfer learning (TL). However, existing TL methods adopted in DEs problems lack either generalizability in general DEs problems or physics preservation during training. In this work, we focus on a general transfer learning method that adaptively correct the domain shift and preserve physical relation within the equation. Mathematically, we characterize the data domain as product distribution and the essential problems as distribution bias and operator bias. APhysics-preserved Optimal Tensor Transport (POTT) method that simultaneously admits generalizability to common DEs and physics preservation of specific problem is proposed to adapt the data-driven model to target domain, utilizing the pushforward distribution induced by the POTT map. Extensive experiments in simulation and real-world datasets demonstrate the superior performance, generalizability and physics preservation of the proposed POTT method.

Transfer learning is an essential technique for many machine learning/AI models of complex structures such as large language models and generative AI. The essence of transfer learning is to leverage knowledge from resolved source tasks for a new target task, especially when the sample size $m$ of the training data for the latter is low. In this work, we rigorously analyze the potential benefit of transfer learning in terms of sample efficiency. Specifically, taking an optimal transport viewpoint of transfer learning, we find that when the data dimension $d$ is higher than $3$, the sample complexity for transfer learning is $O(m^{-(α+1)/d})$, with $α$ indicating the smoothness of the data distribution, as opposed to the $O(m^{-p/d})$ sample complexity for direct learning with $p$ indicating the smoothness of the optimal target model. Our finding theoretically supports a better sample efficiency for transfer learning, when the target task is optimizing over a family of not-so-smooth models (i.e., highly complex networks with the possible use of non-smooth activation functions). Using image classification as an example, we numerically demonstrate the sample efficiency for transfer learning, that is, in the data hungry regime, the model performance can be significantly improved by transfer learning.

Large-scale population-level datasets, such as the UK Biobank and the All of Us Research Program, often lack covariates needed for a specific analysis, such as genetic or lifestyle measures, while related studies measure them. This creates a cross-population missing data problem in which covariates are completely unobserved in the target population, rather than partially missing within one dataset. We propose an augmented transfer regression learning method for this setting. The key identifying condition is a sub-population shift assumption: the joint distribution of the outcome and observed covariates may differ across source and target populations, but the conditional distribution of the missing covariates given observed variables is invariant. We combine importance-weighted estimating equations with imputation terms for first- and second-order moments of the missing covariates. The resulting estimator is doubly robust, remaining consistent if either the density ratio model or both imputation models are correctly specified. It is $n^{1/2}$-consistent and asymptotically normal, and attains the semiparametric efficiency bound when both nuisance models are correctly specified.

We investigate a practical domain adaptation task, called source-free unsupervised domain adaptation (SFUDA), where the source pretrained model is adapted to the target domain without access to the source data. Existing techniques mainly leverage self-supervised pseudo-labeling to achieve class-wise global alignment [1] or rely on local structure extraction that encourages the feature consistency among neighborhoods [2]. While impressive progress has been made, both lines of methods have their own drawbacks - the "global" approach is sensitive to noisy labels while the "local" counterpart suffers from the source bias. In this paper, we present Divide and Contrast (DaC), a new paradigm for SFUDA that strives to connect the good ends of both worlds while bypassing their limitations. Based on the prediction confidence of the source model, DaC divides the target data into source-like and target-specific samples, where either group of samples is treated with tailored goals under an adaptive contrastive learning framework. Specifically, the source-like samples are utilized for learning global class clustering thanks to their relatively clean labels. The more noisy target-specific data are harnessed at the instance level for learning the intrinsic local structures. We further align the sourcelike domain with the target-specific samples using a memory-based maximum mean discrepancy (MMD) loss to reduce the distribution mismatch. Extensive experiments on VisDA, Office-Home, and the more challenging DomainNet have verified the superior performance of DaC over current state-of-the-art approaches.

Fine-tuning is a widely used strategy for adapting pre-trained models to new tasks, yet its methodology and theoretical properties in high-dimensional nonparametric settings with variable selection have not yet been developed. This paper introduces the fine-tuning factor augmented neural Lasso (FAN-Lasso), a transfer learning framework for high-dimensional nonparametric regression with variable selection that simultaneously handles covariate and posterior shifts. We use a low-rank factor structure to manage high-dimensional dependent covariates and propose a novel residual fine-tuning decomposition in which the target function is expressed as a transformation of a frozen source function and other variables to achieve transfer learning and nonparametric variable selection. This augmented feature from the source predictor allows for the transfer of knowledge to the target domain and reduces model complexity there. We derive minimax-optimal excess risk bounds for the fine-tuning FAN-Lasso, characterizing the precise conditions, in terms of relative sample sizes and function complexities, under which fine-tuning yields statistical acceleration over single-task learning. The proposed framework also provides a theoretical perspective on parameter-efficient fine-tuning methods. Extensive numerical experiments across diverse covariate- and posterior-shift scenarios demonstrate that the fine-tuning FAN-Lasso consistently outperforms standard baselines and achieves near-oracle performance even under severe target sample size constraints, empirically validating the derived rates.