ACL (Annual Meeting of the Association for Computational Linguistics) is the top international academic conference in the field of computational linguistics and natural language processing (NLP). It is a CCF-A conference and the highest-level international academic conference in the field of natural language processing recommended by the China Computer Federation. The ACL 2026 Outstanding Paper Award (Outstanding Paper) was announced in San Diego. A total of 18 papers from around the world were selected. “GeoRA: Geometry-Aware Low-Rank Adaptation for RLVR” by Meituan’s contract fulfillment technical team is one of them.
RLVR has become a key paradigm for improving the reasoning capabilities of large models, but its training overhead is high, so a natural technology is to use efficient parameter fine-tuning methods such as LoRA to do RLVR. However, LoRA and its variants are almost always established and verified in SFT scenarios. A series of recent mechanism studies have shown that there are essential differences in the optimization geometries of RLVR and SFT: RLVR is more like a constrained optimization, with effective updates scattered in sparse subspaces and tending to avoid the main direction of pre-trained weights. Therefore, directly moving the prior designed for SFT to RLVR constitutes a geometric misalignment, with the consequences of suboptimal results, forgotten abilities, and even training collapse.
In response to this problem, researchers from Meituan and Peking University proposed GeoRA (Geometry-Aware Low-Rank Adaptation), which explicitly aligns low-rank adaptation to the update geometry of RLVR: first use geometric priors to locate the sparse update area preferred by RLVR, and then use SVD (Singular Value Decomposition) to compress it into a low-rank dense adapter. On the Qwen and Llama models from 1.5B to 32B, GeoRA stably outperforms the popular low-rank baselines on three types of RLVR tasks: mathematics, medicine, and coding. This work has been accepted as an Outstanding Paper by ACL 2026.
This article introduces a low-rank training method specially designed for RLVR, and its implementation experience in business Agentic RL.
Large-scale reasoning models represented by OpenAI o1 and DeepSeek-R1 have established RLVR as a key paradigm for unlocking complex reasoning capabilities. To understand why existing PEFT methods are not good enough in this paradigm, we must first see what the update of RLVR looks like.
A recent batch of mechanism analysis work has outlined the distinction between SFT and RLVR, indicating that there are essential differences in the optimization geometries of the two: SFT explicitly injects new information by rewriting the main direction of weights, while RLVR is more like a constrained optimization. It relies on reward-induced sampling bias to amplify model reasoning behavior, effective updates are scattered in sparse subspaces, and tends to avoid the main direction of pre-trained weights. There is another side to this picture – RLVR is sensitive to update stability. It requires policy updates within the boundaries of the class trust region. Updates that are too aggressive or fall in the main direction can easily cause spectral drift, exploration diversity collapse and even training collapse.
Geometric misalignment. LoRA and its variants are the current de facto standard. A key difference between them is what spectral prior is used: LoRA is initialized with Gaussians and all zeros, regardless of pre-training weights; PiSSA does SVD on the original weights and allocates the trainable budget to the main singular direction; MiLoRA takes the tail singular direction. These priors are mainly designed and verified in the SFT scenario, without considering the RLVR update law described in the previous section. Taking PiSSA as an example, it forces updates to the strong inductive bias on the principal components, which happens to conflict with the subspace preferred by RLVR, which also indicates a phenomenon that is more prone to collapse in experiments.
Engineering misalignment. An intuitive approach is: since RLVR only updates a small part of the parameters, just do sparse fine-tuning on this subspace. This method is indeed more in line with RLVR’s preference in update mode, but modern GPUs are not efficient for unstructured sparse calculations, and reducing the number of parameters does not mean improved efficiency. In the efficiency experiment of this article, sparse fine-tuning reduced the training parameters by 68.0%, but increased the training time by 10.8%.
One side is computationally friendly but not geometrically correct, and the other side is geometrically correct but not computationally fast. GeoRA wants to solve these two misalignments at the same time. The goal is to locate the sparse subspace that is suitable for RLVR and train it in a low-rank dense way.
GeoRA can be broken down into three steps:
- Construct geometric subspace: Based on spectral prior and Euclidean prior, more stable and plastic parameter regions are screened out from pre-trained weights.
- Low-rank approximate construction adaptationDevice: Use SVD to make a low-rank approximation to the geometric subspace to initialize the low-rank adapter.
- Create residual anchor points: Freeze the residual weights so that the function remains unchanged during model initialization, smooth cold start and protect the pre-training structure.
The entire process is only executed once before the start of training, which is a one-time preprocessing; the calculation graph of the training phase is completely consistent with standard LoRA.
We use two complementary masks to filter parameter regions from the pre-trained weight matrix W, both of which share the same sparsity rate ρ.
- The spectral prior M_Spec emphasizes stability. First, perform a low-rank approximation of rank r on W to obtain Ŵ_r, and then select the ρ proportional position with the smallest amplitude in Ŵ_r. Intuitively, high-amplitude components in the principal components are also high-curvature areas, and changing them can easily destroy the pre-training structure; limiting updates to low-amplitude areas can improve spectral stability under RLVR.
- The Euclidean prior M_Euc emphasizes plasticity. Select the ρ proportional position with the smallest amplitude directly on the original weight W. These parameters close to zero are “less used” in pre-training, have a large space for adjustment, and can retain enough degrees of freedom for training.
The final geometric constraint matrix is the union of the two: W_Geo = W ⊙ (M_Spec ∪ M_Euc).
The two masks appear to have similar rules, but the actual selected parameter sets rarely overlap. For example, taking ρ = 0.2 on Qwen3-8B, 20.0% of the parameters are selected for each, but the intersection is only 4.55%, and the Jaccard similarity is only 0.128. This shows that they capture two different types of parameter areas, which are complementary rather than redundant relationships. The ablation experiments in the paper also confirm this: removing any prior will lead to performance degradation.
After getting W_Geo, do SVD on it:
W_Geo = U_Geo Σ_Geo V_Geo
Take the first r singular components to initialize two low-rank matrices:
A_Geo = Σ_Geo[:r,:r]^(1/2) · V_Geo[:,:r]ᵀ
B_Geo = U_Geo[:,:r] · S_Geo[:r,:r]^(1/2)
In this way, B_Geo A_Geo is the optimal rank-r approximation of W_Geo (the Eckart–Young theorem guarantees optimality under the Frobenius norm).
There is an easily overlooked point here: GeoRA takes the top-r direction of W_Geo, not the top-r of the original weight W. The change of adaptation objects is the core difference between GeoRA and PiSSA/MiLoRA.
From this perspective, the difference between the four methods can be summarized in one sentence: LoRA does not look at the weight, PiSSA looks at the main direction, MiLoRA looks at the tail direction, and GeoRA first changes to a more suitable adaptation object, and then takes the main direction.
The final step is to freeze the remaining weights. Calculate the residual matrix according to LoRA’s scaling convention:
W_res = W − (α/r) · B_Geo A_Geo
During forward pass, W_res freezes and only the adapter is trainable:
h = W_res x + (α/r) · B_Geo A_Geo x
This design brings two properties. One isThe function remains unchanged during initializationbecause the sum of the two items is exactly equal to W x, the model output will not mutate when training starts, avoiding the policy jitter in the cold start stage – this is very important for RLVR, as the initial output disturbance will contaminate the rollout sampling distribution. The second isStructural constraints imposed: The optimizer can only be updated on the A_Geo and B_Geo parameterized manifolds, and W_res serves as a stable anchor point to prevent the pre-training representation from being destroyed.
The main experiment uses the GRPO algorithm on DeepMath-103K to perform mathematical RLVR on Qwen3-8B and Llama-3.1-8B. The comparison objects include LoRA, PiSSA, MiLoRA, sparse fine-tuning (SparseFT) and full-parameter fine-tuning (FullFT). At the same time, GSM8K was used for scale verification on the 4B and 1.5B models.
- On the in-distribution (ID) mathematical benchmark, GeoRA achieved the strongest overall performance on both backbone models, and its lead over the low-rank baseline was particularly consistent on the competitive difficulty benchmark – Qwen3-8B’s AIME24 reached 23.75%, even slightly higher than full-parameter fine-tuning.
- The out-of-distribution (OOD) results are more worthy of attention: full-parameter fine-tuning showed significant regression in IFEval and TruthfulQA, while GeoRA was basically lossless, and HumanEval increased from 76.83 to 82.93.
This shows that geometrically aligned subspace updates can significantly reduce the forgetting of existing abilities while improving the reasoning ability of the target domain.
To confirm that the benefits are not limited to mathematics, we conducted extended experiments in medical and coding scenarios.
The conclusions in both fields are consistent: GeoRA is stably better than low-rank baselines and comparable to fully parametric fine-tuning.
- Convergence efficiency is higher. In terms of training dynamics, GeoRA remains ahead throughout the entire process, and reaches high performance significantly earlier than other low-rank baselines.
- Superparameters are more robust. Taking the learning rate as an example, GeoRA maintains high rewards within a wide range, while other methods decline significantly under large learning rates. This means that in practice GeoRA does not require careful hyperparameter tuning.
Compared with full-parameter fine-tuning, GeoRA reduces trainable parameters by 99.5%, single-step time consumption by 19.9%, and video memory usage by 28.5%. Comparing the data of SparseFT, we can see it more clearly: although it reduced the parameters by 68.0%, the single-step time consumption increased by 10.8%. The key engineering value of GeoRA lies inImplementing geometric priors into dense low-rank computationsso parameter efficiency can truly translate into speed and memory gains.
We further conducted structural analysis: W_Geo is a sparse matrix, compress it into a form of rank r, why is the loss controllable? We conducted three sets of analyzes on the singular value spectrum to answer.
- Sparseness itself does not bring about low rank. As a comparison, we first constructed two random noise matrices with different densities. The singular value spectra of the two almost overlap, and both show relatively gentle attenuation, indicating that the sparseness itself is still approximately isotropic, with no main direction available.
- W_Geo has low rank properties. The spectral shape of GeoRA Weight (W_Geo) in the figure is similar to the pre-trained weight, with most of the spectral energy concentrated on a few leading components. This shows that the selected region inherits a structured and compressible low-rank morphology.
- The actual update of fully parameterized RLVR is also low-rank. We directly examine ΔW = W_FullFT − W_Pretrain, which also exhibits a highly compressible heavy-tailed spectrum.
The three sets of analyzes combined give the structural basis of GeoRA: low-rank characteristics are possessed by RLVRintrinsic properties. This also explains why GeoRA can achieve the same effect as full parameter fine-tuning by training only 0.5% of the parameters – it approximates the shape of the RLVR update itself.
The starting point of GeoRA’s work is not to publish papers, but to meet the needs for efficient reinforcement learning in actual business. The scenario we are in is AI rider recruitment: the Agent actively reaches and follows up with candidates, identifies their wishes, concerns and decision-making points in multiple rounds of communication, formulates targeted next-step strategies, and then gradually advances candidates to interviews and onboarding. In the process, this requires emotional intelligence, strategy, speaking skills and on-the-spot responses similar to sales; the goals and results are clear and verifiable (appointments/onboarding/ROI), and are very suitable for optimization with RLVR; and this vertical domain scenario also has a special combination of training requirements:
- Large schema and long context. This Agent has high requirements for basic intelligence and long-range capabilities: it needs to be built on a large-size base model, and the context of a single trajectory is very long, and the resource overhead of reinforcement learning is considerable.
- Incremental capacity is not large. The scale of knowledge and capabilities that need to be additionally constructed in the vertical domain is usually not large, the amount of training data is much smaller than the base model, and the capacity of low-rank adaptation is basically sufficient.
On the one hand, the training overhead is high, and on the other hand, the incremental capacity is small. This is where low-rank and efficient training for RLVR comes in. In fact, what first made us suspect that “the low-rank prior designed for SFT may not be suitable for RLVR” was also the somatosensory experience in business training: there are obvious differences in the effects between LoRA, PiSSA, and MiLoRA. When the learning rate is increased, the training becomes unstable or even collapses. This shows that the initialized geometric prior is a large variable under RLVR, and there is room for optimization. GeoRA is currently being implemented in AI recruitment scenarios, and the experimental results of Agentic RL are:
Effect: Equivalent to full participation training, about 12% higher than LoRA.
efficiency: Comparable to LoRA, the video memory is reduced by 54% compared with full-parameter training. It can be further reduced if used with quantification methods such as QLoRA.
It can be seen that GeoRA has achieved the optimization effect of all parameters with the training cost of LoRA, which is its application value.
There are two other points of experience when implementing it, which are also for reference by students who also want to apply them in RLVR:
- Random SVD initialization. SVD initialization is a one-time preprocessing, and since only the first r singular components are required, it can be accelerated by Randomized SVD, which uses random projection to compress the matrix into a much smaller space for decomposition. Even the 72B model processing only takes less than 1 minute.
- Omit reference model storage. RLVR needs a reference strategy to calculate KL, and the common practice is to save another complete weight. The invariant nature of the function in Section 2.3 provides a free alternative: original weight = residual weight + initial adapter. Therefore, as long as an extra frozen initial adapter (low rank, small overhead) is left in the actor, an accurate reference strategy can be calculated immediately.
GeoRA is the first low-rank training method for optimized geometric design for RLVR. Its core contributions can be summarized into three points:
- Revealed structure of RLVR updates: The update subspace of RLVR is sparse but anisotropically compressible, which is the prerequisite for low-rank adaptation.
- Propose an adaptation method for positioning and compression: Use spectral prior and Euclidean prior to locate the update area preferred by RLVR, then use truncated SVD to compress the low-rank adapter and freeze the residual as the anchor point, which avoids the efficiency bottleneck of geometric misalignment and sparse calculation.
- Extensive experiments verify superior performance: From 1.5B to 32B models, the verification method on RLVR in mathematics, medicine, and coding scenarios is more stable than the baseline, has less out-of-distribution forgetting, and has the advantages of both speed and video memory.
In addition to the paper experiments, GeoRA has also been verified in actual business scenarios Agentic RL, achieving better training results at lower costs. We will continue to promote it to more similar scenarios, and hope that this method can provide a new reference for efficient training of RLVR.
