arxiv: 2604.06253 · v1 · submitted 2026-04-06 · 💻 cs.LG · cs.AI· cs.PL

Recognition: 2 theorem links

· Lean Theorem

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

Gaurav Narasimhan

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:45 UTC · model grok-4.3

classification 💻 cs.LG cs.AIcs.PL

keywords cross-lingual transferLoRAFourier regularizationcode generationparameter-efficient fine-tuningmultilingual LLMsCode Llama

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The pith

Fourier-based regularization during low-rank fine-tuning raises Java code generation accuracy from 34.2 percent to 42.1 percent.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Large language models for code typically require separate fine-tuning for each programming language, which becomes expensive when multiple languages must be supported. The paper tests whether a regularization method that operates in the frequency domain, added to low-rank adaptation, can produce stronger transfer from Python training data to Java tasks. Experiments on Code Llama 7B show that the combination reaches higher pass rates on Java problems than either a broader Python fine-tune or standard low-rank training without the frequency penalty. The gains appear even when using a compact, high-quality dataset rather than large-scale multilingual corpora. If the result holds, organizations could maintain one base model and adapt it efficiently across languages instead of retraining full models for each new language.

Core claim

The central claim is that applying regularization in the Fourier domain to the updates of low-rank adapter matrices during fine-tuning enables a model trained primarily on Python to generate correct Java code at a higher rate than either the baseline low-rank method or a more extensively fine-tuned Python model, with the Fourier term producing the clearest lift on the target language.

What carries the argument

Fourier-based regularization, which adds a penalty on selected frequency components of the low-rank weight updates to encourage adaptations that transfer better across languages.

If this is right

LoRA fine-tuning on the compact MBPP dataset alone exceeds the cross-lingual performance of the released Code Llama-Python-7B model.
The Sophia optimizer reaches competitive final accuracy faster than Adam, although the end scores remain close.
The largest measured gain in Java transfer comes from adding the Fourier regularization during the low-rank updates.
Parameter-efficient adaptation with frequency-domain constraints can substitute for full multilingual fine-tuning in at least the Python-to-Java direction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same frequency penalty might reduce language-specific overfitting and therefore help transfer to additional programming languages beyond the tested pair.
Combining the Fourier term with other efficient adaptation methods could lower the total compute needed to support many languages at once.
Repeating the protocol on larger base models or different source-target language pairs would test whether the regularization effect scales.

Load-bearing premise

The reported improvement on Java tasks is produced by the Fourier regularization itself rather than by choices of dataset, optimizer settings, or other training details that were not varied in the experiments.

What would settle it

Re-run the identical LoRA fine-tuning schedule on the same MBPP data but remove the Fourier regularization term, then measure whether the Java pass@1 score falls back to the 34.2 percent baseline.

Figures

Figures reproduced from arXiv: 2604.06253 by Gaurav Narasimhan.

**Figure 4.** Figure 4: Fourier Transform regularization substan [PITH_FULL_IMAGE:figures/full_fig_p002_4.png] view at source ↗

**Figure 3.** Figure 3: Java: Improvement of earlier code In this paper, I systematically investigate the efficacy of Low-Rank Adaptation (LoRA) and optimization strategies in enhancing cross-lingual code generation capabilities using the Code Llama 7B model. I demonstrate that parameter-efficient LoRA fine [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗

**Figure 6.** Figure 6: Python: Corrections of prior failures 3 Approach 3.1 Parameter-Efficient Fine-Tuning My approach utilizes the Code Llama 7B model [1], a decoder-only transformer-based large language model designed explicitly for generating programming code. Given the computational constraints inherent in enterprise environments, I employ parameter-efficient fine-tuning via Low-Rank Adaptation (LoRA) [2]. LoRA introduces … view at source ↗

**Figure 7.** Figure 7: Training loss comparison between AdamW and Sophia optimizers. Sophia demonstrates more stable convergence and ultimately reaches a lower final loss. 3.3 Fourier-Based Regularization Drawing inspiration from signal-processing principles, I integrate a lightweight Fourier-based regularization technique into the LoRA fine-tuning process. The key insight is that different frequency components of model para… view at source ↗

**Figure 8.** Figure 8: Parameter exploration for Fourier Trans [PITH_FULL_IMAGE:figures/full_fig_p005_8.png] view at source ↗

**Figure 10.** Figure 10: Beam size evaluation showing performance improvements with larger beam sizes up to 10, after which results plateaued. parameter-efficient methods can outperform models specifically pre-trained for a language while modifying only 0.2% of the parameters. The hyperparameter analysis revealed that the alphato-rank ratio significantly impacted performance. The optimal 2:1 ratio (alpha=16, rank=8) provided s… view at source ↗

**Figure 9.** Figure 9: Temperature evaluation results showing that higher temperatures (0.8-1.0) and very low temperatures (0.0-0.2) produced better results than midrange values. The LoRA adaptation significantly reduced trainable parameters to approximately 11.9 million parameters, representing less than 0.2% of the model’s original 7 billion parameters. A crucial finding was that unmerged LoRA weights consistently outperfo… view at source ↗

**Figure 13.** Figure 13: Perplexity comparison showing Sophia achieved lower perplexity more consistently than AdamW. Sophia consistently achieved faster convergence, requiring approximately 30% fewer gradient update steps to reach equivalent validation loss levels, and exhibited more stable gradient norms throughout training. However, final pass@1 performance on the APPS dataset was comparable between the two optimizers, sugge… view at source ↗

**Figure 14.** Figure 14: Gradient norm comparison showing Sophia maintained smaller, more stable gradient norms compared to AdamW’s larger fluctuations. Notably, merged LoRA models performed worse compared to unmerged LoRA weights, reinforcing the finding from Round 1 that LoRA adapters retain bet12 [PITH_FULL_IMAGE:figures/full_fig_p012_14.png] view at source ↗

**Figure 16.** Figure 16: Effect of beam size on cross-lingual trans [PITH_FULL_IMAGE:figures/full_fig_p013_16.png] view at source ↗

**Figure 17.** Figure 17: 3D visualization of model performance across beam size, temperature, and regularization strength parameters, highlighting the multidimensional nature of hyperparameter optimization [PITH_FULL_IMAGE:figures/full_fig_p014_17.png] view at source ↗

**Figure 18.** Figure 18: Performance distribution across different [PITH_FULL_IMAGE:figures/full_fig_p014_18.png] view at source ↗

**Figure 21.** Figure 21: Three-dimensional prediction surface showing the relationship between key hyperparameters and model performance. Examples 15 [PITH_FULL_IMAGE:figures/full_fig_p015_21.png] view at source ↗

read the original abstract

Cross-lingual code generation is critical in enterprise environments where multiple programming languages coexist. However, fine-tuning large language models (LLMs) individually for each language is computationally prohibitive. This paper investigates whether parameter-efficient fine-tuning methods and optimizer enhancements can improve cross-lingual transfer from Python to languages like Java. We fine-tune the Code Llama 7B model using low-rank adaptation (LoRA) to optimize a small subset of parameters and compare Adam and Sophia optimizers, while exploring a novel Fourier-based regularization technique. Our contributions include: (1)demonstrating that LoRA fine-tuning on a small, high-quality dataset (MBPP) can exceed the pass@1 performance of the more broadly fine-tuned Code Llama-Python-7B model (40.1% vs. 38.4%); (2) showing that while Sophia achieves faster convergence than Adam, final pass@1 scores show marginal differences; and (3) presenting evidence that Fourier-based regularization during fine-tuning significantly improves cross-lingual transfer, achieving 42.1% pass@1 on Java tasks compared to the 34.2% baseline. These findings suggest that combining LoRA, optimized training methods, and frequency-domain regularization can efficiently adapt single-language LLMs to perform well across multiple programming languages.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The reported jump from 34.2% to 42.1% Java pass@1 is tied to the Fourier term in the abstract, but the comparison does not isolate that term from other training choices.

read the letter

The paper tests whether LoRA plus a Fourier regularization term can improve cross-lingual transfer when fine-tuning Code Llama 7B on Python data for Java code generation. The clearest practical finding is that LoRA on the small MBPP set already beats the official Code Llama-Python-7B checkpoint on Java tasks (40.1% vs 38.4%). That result is useful on its own for anyone trying to adapt a single-language model cheaply. Sophia converges faster than Adam but ends at similar final scores, which matches what we have seen with other optimizers on small fine-tunes. The new piece is the Fourier-based regularization, presented as the driver of the further lift to 42.1% pass@1. The abstract gives concrete numbers, which is better than many transfer papers that stay at qualitative claims. The soft spot is exactly the one the stress-test flags: the 34.2% baseline is not described in enough detail to confirm it matches the LoRA rank, optimizer, step count, and data order used in the Fourier run. Without those controls or ablations, the gain could come from any of several unstated differences rather than the frequency term itself. No error bars or statistical tests are mentioned either. The work is aimed at practitioners who need quick multilingual code models rather than theorists working on fundamental transfer limits. It is honest empirical reporting, but the central causal claim would need tighter experimental design to hold up under review. I would send it to referees with a request for the missing ablations and baseline details; the practical angle is worth checking.

Referee Report

1 major / 2 minor

Summary. The paper proposes FLeX, which augments LoRA-based fine-tuning of Code Llama 7B on the MBPP dataset with a Fourier-based regularization term and optimizer comparisons (Adam vs. Sophia). It claims three contributions: (1) LoRA on MBPP alone yields 40.1% pass@1 on Java, exceeding the 38.4% of the broader Code Llama-Python-7B model; (2) Sophia converges faster than Adam with comparable final performance; and (3) the Fourier regularization further improves cross-lingual transfer to 42.1% pass@1 on Java tasks versus a 34.2% baseline.

Significance. If the reported gains from the Fourier regularization can be isolated through controlled ablations, the approach would offer a computationally efficient route to multilingual code generation without per-language full fine-tuning. The combination of parameter-efficient adaptation and frequency-domain regularization is a plausible direction for low-resource language transfer in LLMs.

major comments (1)

[Abstract / Experimental Results] Abstract, contribution (3): the 42.1% vs. 34.2% Java pass@1 lift is presented as evidence for the Fourier regularization, yet the manuscript does not state whether the 34.2% baseline uses the identical LoRA rank, optimizer, training steps, and MBPP data as the proposed run. Because contribution (1) already demonstrates that LoRA on MBPP alone improves over broader baselines, any additional gain cannot be attributed to the frequency-domain term without an ablation that holds all other factors fixed.

minor comments (2)

[Abstract] No error bars, number of random seeds, or statistical tests accompany the pass@1 figures, limiting assessment of whether the reported differences are reliable.
[Methods] The precise definition of the Fourier regularization term (e.g., which frequencies are penalized and how the strength hyper-parameter is chosen) should be stated explicitly in the methods section rather than left to the abstract.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for identifying an ambiguity in how the contributions are presented. We address the major comment below and will revise the manuscript to improve clarity and experimental rigor.

read point-by-point responses

Referee: [Abstract / Experimental Results] Abstract, contribution (3): the 42.1% vs. 34.2% Java pass@1 lift is presented as evidence for the Fourier regularization, yet the manuscript does not state whether the 34.2% baseline uses the identical LoRA rank, optimizer, training steps, and MBPP data as the proposed run. Because contribution (1) already demonstrates that LoRA on MBPP alone improves over broader baselines, any additional gain cannot be attributed to the frequency-domain term without an ablation that holds all other factors fixed.

Authors: We appreciate the referee highlighting this important point regarding the attribution of improvements to the Fourier regularization. The 34.2% figure represents the pass@1 performance of the base Code Llama 7B model on Java tasks from the MBPP benchmark, prior to any fine-tuning. Contribution (1) shows that applying LoRA fine-tuning on the MBPP dataset alone raises this to 40.1%, surpassing even the Code Llama-Python-7B model. The 42.1% is achieved by incorporating the Fourier-based regularization into this LoRA fine-tuning process. Nevertheless, to ensure the gain from the regularization is isolated, we agree that a controlled ablation is necessary. In the revised version, we will add such an ablation experiment, maintaining identical settings for LoRA rank, optimizer, number of training steps, and the MBPP training data. We will update the abstract and the experimental section to clearly present these comparisons. This revision will allow readers to directly assess the impact of the Fourier term. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical results from controlled fine-tuning runs

full rationale

The paper reports direct experimental measurements of pass@1 scores on Java code generation tasks after fine-tuning Code Llama 7B with LoRA adapters, Adam/Sophia optimizers, and a Fourier-based regularization term. Contributions (1)–(3) consist of observed performance deltas (e.g., 42.1 % vs. 34.2 % baseline, 40.1 % vs. 38.4 % for LoRA on MBPP) obtained from training runs. No equations, parameter-fitting procedures, or self-citations are presented that would reduce any claimed improvement to a quantity defined by the result itself. The derivation chain is therefore self-contained and consists solely of empirical observation rather than any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim rests on the unproven assumption that frequency-domain regularization captures transferable features across programming languages and that the MBPP dataset plus the chosen evaluation protocol are sufficient to demonstrate this.

free parameters (1)

Fourier regularization strength
A coefficient controlling the weight of the frequency-domain penalty must be chosen or tuned to obtain the reported 42.1% score.

axioms (2)

domain assumption LoRA updates are sufficient to achieve meaningful cross-lingual transfer in code models
The paper assumes low-rank adaptation can capture the necessary language-specific knowledge without full fine-tuning.
domain assumption MBPP is a high-quality and representative dataset for both fine-tuning and cross-lingual evaluation
All reported numbers depend on this dataset choice.

invented entities (1)

Fourier-based regularization term no independent evidence
purpose: To improve cross-lingual transfer by penalizing updates in the frequency domain
A new regularization mechanism introduced in the paper with no independent evidence of its general utility provided in the abstract.

pith-pipeline@v0.9.0 · 5530 in / 1538 out tokens · 68949 ms · 2026-05-10T19:45:57.112718+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Fourier-based regularization ... LFourier(w) = sum ρ(k,n,T) |ŵk|^2 with ρ(k) = 1 - ϕ_low + (ϕ_high - ϕ_low) min(1, k/(n T))
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

LoRA fine-tuning on MBPP achieving 40.1% pass@1 vs Code Llama-Python-7B 38.4%

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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Figure 9: Temperature evaluation results showing that higher temperatures (0.8-1.0) and very low tem- peratures (0.0-0.2) produced better results than mid- range values

consisting of 974 Python programming problems. Figure 9: Temperature evaluation results showing that higher temperatures (0.8-1.0) and very low tem- peratures (0.0-0.2) produced better results than mid- range values. The LoRA adaptation significantly reduced trainable parameters to approximately 11.9 million parame- ters, representing less than 0.2% of th...
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Isolation of updates more effectively constrained regularization to preserve cross-lingual knowl- edge without disrupting base model capabilities Figure 19: Performance comparison across different Fourier regularization parameter combinations, re- vealing clear patterns in effectiveness. 8.4.2 Optimal Configuration and Results The optimal configuration us...