This paper considers a multi-environment factor model in which high-dimensional covariates are collected from heterogeneous environments, with auxiliary labels available in a subset of these environments. The joint distribution of the covariates may vary across environments, whereas the latent structure is decomposed into invariant factors with shared loadings and heterogeneous factors with environment-specific loadings. Such a model is motivated by transfer learning and latent factor regression, where one seeks stable low-dimensional representations for both interpretation and robust out-of-sample prediction of the response $Y$. Leveraging the invariance principle, we show that the invariant and heterogeneous factors are disentangled under a minimal structural condition. Based on this, we propose ATLAS, an Auxiliary-label and invariance-guided Transfer via Latent Alignment across heterogeneous environmentS. ATLAS is a unified procedure that leverages the invariance principle to separate aligned invariant and unaligned heterogeneous factors, and further exploits supervision from auxiliary labels to extract prediction-invariant and transferable factors from those unaligned heterogeneous factors. ATLAS yields near-oracle performance for downstream latent factor regression, enables transferable prediction in new environments through the full latent signal when auxiliary labels are available, and reduces to robust invariant-factor-only prediction otherwise. We establish sharp non-asymptotic error bounds for recovering invariant and heterogeneous factors, identifying all the response-invariant factors, and estimating the invariant signal in $Y$.
Large Language Models (LLMs) have significantly propelled the advancement of edge intelligence and have been widely deployed across various scenarios, including autonomous driving, industrial inspection, and personalized IoT services. However, the collaborative adaptation of LLMs on edge devices continues to face formidable challenges due to strict data privacy constraints, highly heterogeneous computing and communication resources, and the non-independent and identically distributed (non-IID) nature of local data. Federated Fine-Tuning (FFT) enables the collaborative optimization of distributed models without exposing raw data. Yet, traditional synchronous aggregation suffers from a severe straggler effect, resulting in high system latency and low resource utilization. Existing asynchronous federated learning methods are predominantly designed for small-to-medium-scale models and struggle to address the specific challenges inherent in LLM fine-tuning namely, model drift caused by stale updates, aggravated client drift stemming from data heterogeneity, and aggregation fairness imbalance resulting from the dominance of fast clients. To address these issues, this paper proposes AlignFed, an asynchronous federated fine-tuning framework for LLMs tailored to heterogeneous edge environments. AlignFed employs a lightweight multi-stage semantic alignment mechanism comprising three core modules: version-aware update grouping, cross-version semantic alignment based on a mini-batch calibration set, and fairness-aware aggregation that integrates both update freshness and client participation frequency. This framework effectively mitigates cross-version model drift and client drift while enhancing aggregation fairness, thereby achieving stable and efficient asynchronous federated optimization in scenarios characterized by high heterogeneity and significant update staleness.