FLeX: Fourier-based Low-rank EXpansion for multilingual transfer

Large language models (LLMs) achieve high pass rates on code generation benchmarks, yet whether they can transfer this ability to languages absent from pretraining remains poorly understood. We introduce PyLang, a minimal imperative language absent from all pretraining corpora, and evaluate frontier models zero-shot and fine-tuned Qwen3 (4B, 8B, 32B) on 352 problems. We find that fine-tuning quickly teaches syntax but fails to transfer semantic competence: Python outperforms PyLang by up to 19% across all configurations, and no intervention (multi-task learning, preference tuning, code infilling, or latent-space objectives) closes the gap. An LLM judge reveals that frontier models select an identical algorithm to Python 80% of the time, yet cannot translate it into a working PyLang implementation., and CKA analysis confirms that fine-tuned models converge to nearly identical internal representations across languages (CKA > 0.97) while diverging at the output stage. We term this the implementation fidelity gap: models possess language-agnostic algorithmic understanding but cannot express it in an unfamiliar language. Our findings highlight the need for training methods that decouple reasoning from language-specific realization.

Transpilation, or code translation, aims to convert source code from one programming language (PL) to another. It is beneficial for many downstream applications, from modernizing large legacy codebases to augmenting data for low-resource PLs. Recent large language model (LLM)-based approaches have demonstrated immense potential for code translation. Among these approaches, training-based methods are particularly important because LLMs currently do not effectively adapt to domain-specific settings that suffer from a lack of knowledge without targeted training. This limitation is evident in transpilation tasks involving low-resource PLs. However, existing training-based approaches rely on a pairwise transpilation paradigm, making it impractical to support a diverse range of PLs. This limitation is particularly prominent for low-resource PLs due to a scarcity of training data. Furthermore, these methods suffer from suboptimal reinforcement learning (RL) reward formulations. To address these limitations, we propose CodePivot, a training framework that leverages Python as an intermediate representation (IR), augmented by a novel RL reward mechanism, Aggressive-Partial-Functional reward, to bootstrap the model's multilingual transpilation ability without requiring parallel corpora. Experiments involving 10 PLs show that the resulting 7B model, trained on Python-to-Others tasks, consistently improves performance across both general and low-resource PL-related transpilation tasks. It outperforms substantially larger mainstream models with hundreds of billions more parameters, such as Deepseek-R1 and Qwen3-235B-A22B-Instruct-2507, on Python-to-Others tasks and Others-to-All tasks, respectively. In addition, it outperforms its counterpart trained directly on Any-to-Any tasks on general transpilation tasks. The code and data are available at https://github.com/lishangyu-hkust/CodePivot.

Recent literature on fine-tuning Large Language Models highlights a fundamental debate. While Full Fine-Tuning (FFT) provides greater representational plasticity, Low-Rank Adaptation (LoRA) can match or surpass FFT performance while constraining updates to a low-rank space and potentially benefiting from additional regularization. Through empirical evaluation across diverse tasks (SQL, Medical QA, and Counterfactual Knowledge) and varying language models (Gemma-3-1B, Qwen2.5-1.5B, and Qwen2.5-3B), we observe both trends and find that the better static architecture depends on the task and model. Spectral and truncation analyses further show that endpoint compressibility alone does not explain these task differences, suggesting task-score sensitivity and constrained optimization trajectories as possible explanations. To address this challenge, we propose a Mixture of LoRA and Full (MoLF) Fine-Tuning, a unified framework that enables continuous navigation between both training regimes. MoLF dynamically routes updates between FFT and LoRA at the optimizer level to ensure that exact gradient signals are available to both experts throughout training, while only selected experts update their weights. For memory-constrained environments, we also introduce MoLF-Efficient, which freezes base weights and only routes updates among a pair of LoRA experts of potentially varying rank. Our evaluations show that MoLF either improves on or stays within $1.5$ percentage points of the better of FFT and LoRA across the nine tested settings, while MoLF-Efficient outperforms both AdaLoRA and AdaMix in eight of nine settings, with gains over the stronger baseline of up to $11.70$ percentage points on Fact, $3.13$ on Med, and $2.98$ on SQL.