ShapLoRA: Shapley-Driven LoRA Rank Allocation

• The paper introduces a Shapley sensitivity metric that systematically assesses each rank’s contribution for efficient pruning and reallocation in LLMs.
• It details a Monte Carlo-based approach that reliably estimates rank importance by averaging over diverse coalition configurations.
• Experimental evaluations across multiple LLM backbones demonstrate superior parameter efficiency and accuracy improvements with ShapLoRA.

ShapLoRA is a framework for allocating low-rank adaptation (LoRA) ranks in LLMs using a Shapley value-inspired importance estimation. It addresses the limitations of prior rank allocation methods by introducing a game-theoretically-motivated metric, termed Shapley sensitivity, to assess the contribution of each rank within low-rank adapters. The method systematically prunes and reallocates ranks to maximize effective parameter usage, yielding superior parameter efficiency and accuracy on a wide range of benchmarks, while incurring minimal additional training overhead (Zhao et al., 25 Jan 2026).

1. Limitations of Prior Rank Allocation Methods

Traditional LoRA techniques add a low-rank update $\Delta W = P\,\Lambda\,Q$ to each linear module, typically using a fixed rank $r$ across the entire Transformer backbone. This uniform allocation does not account for variance in layer or module importance for specific downstream tasks. Adaptive schemes such as AdaLoRA and SoRA/SaLoRA, which prune ranks by local sensitivity $\mathrm{ipt}(w) = \|w\nabla_w \mathcal{L}\|$, fail to incorporate interaction effects between ranks, leading to unreliable importance estimates. AutoLoRA and allied NAS-style approaches learn architectural weights through bi-level optimization but these indicators can be unstable and lack interpretability.

ShapLoRA is motivated by the need to (a) employ a principled importance metric accounting for all possible coalitions of ranks and (b) decouple rank allocation from retraining, guarding against biased comparisons driven by initialization or optimization artifacts.

2. Formulation of Shapley Sensitivity

The Shapley sensitivity measure generalizes gradient-based sensitivity to a coalitional setting. For a given LoRA parameterization,

$$x' = x\,W^{\ell,m} + x\,P^{\ell,m}\,\Lambda^{\ell,m}\,Q^{\ell,m},$$

with singular values $\Lambda^{\ell,m} = \operatorname{diag}(\lambda_i)$, each rank $i$ constitutes a 'player' in the cooperative game formalism.

The classical Shapley value for rank $k$ with respect to coalitions $\mathcal S_k$ is

$$\Phi_k = \frac{1}{|\mathcal S_k|}\sum_{A\in\mathcal S_k}\bigl[V(A\cup\{k\})-V(A)\bigr].$$

ShapLoRA adapts this by (i) masking ranks outside a coalition $S$ (zeroing $\lambda_{i'}$ for $i'\notin S$), and (ii) computing coalition-conditional sensitivity,

$$\mathrm{SAN}(\mathcal G_i^{\ell,m} \mid S) = \mathrm{ipt}(\lambda_i^{\ell,m}) + \frac1{d_1} \sum_{j=1}^{d_1}\mathrm{ipt}(P_{j,i}^{\ell,m}) + \frac1{d_2} \sum_{j=1}^{d_2} \mathrm{ipt}(Q_{i,j}^{\ell,m}),$$

averaged over all sampled coalitions containing $i$. The full Shapley sensitivity is

$$\mathrm{SAN}(\mathcal G_i^{\ell,m}) = \frac{1}{|\mathcal S_i^{\ell,m}|} \sum_{S \in \mathcal S_i^{\ell,m}} \mathrm{SAN}(\mathcal G_i^{\ell,m} \mid S).$$

Because enumerating all $2^R$ coalitions is infeasible, a Monte Carlo estimator averages over $N_3$ random coalitions (practically $N_3 \approx 90$), ensuring complementary masking so each rank is masked/unmasked equally often.

3. ShapLoRA Workflow and Algorithmic Procedure

The ShapLoRA process comprises two main stages:

• Stage 1 (Rank Allocation): Start from full-rank LoRA ($r_{\mathrm{init}}=16$), fine-tune on the training set, and compute Shapley sensitivities for all ranks using a held-out validation set. Prune the lowest $$R^{\mathrm{prune}} = R^{\mathrm{init}} - R^{\mathrm{target}}$$ ranks globally, based on the computed sensitivities.
• Stage 2 (Retraining): Remove pruned ranks, reinitialize the remaining ranks, and retrain LoRA from scratch on the training data at the reduced rank.

Pseudocode for sensitivity estimation:

Input: fine-tuned LoRA params {P, Λ, Q}, validation set D_v, init ranks R_init, target ranks R_target, sample size N3, mask-probs {p_k} Output: rank importance scores SAN[i] for all i Initialize SAN[i] = 0 for all ranks i for t = 1…N3: Sample a mask probability p ∼ Uniform{0.1,0.9} Generate coalition S_t by masking each rank independently with prob p For each rank i: Zero Λ entries for ranks ∉ S_t Compute gradients ∇ℒ_v on D_v under current masking For each rank i: SAN_cond[i] = ipt(λ_i) + avg_j ipt(P_{j,i}) + avg_j ipt(Q_{i,j}) SAN[i] += SAN_cond[i] end For each i: SAN[i] /= N3 Sort ranks by SAN[i], prune lowest R_init–R_target

Averaging over varied coalitions ensures that the resulting importance scores robustly reflect each rank’s marginal impact across a representative set of contexts.

4. Experimental Evaluation

ShapLoRA’s empirical evaluation covers multiple LLM backbones including LLaMA-3 8B, distilled LLaMA-3 8B, and Qwen 3B, over a broad suite of tasks:

• Commonsense QA: BoolQ, OBQA, ARC-e/c, PIQA, AQuA, GSM8k
• NLP/NLG: SST-2, RTE, QNLI, E2E, WikiSQL
• LLM instruction and evaluation: UltraChat, Alpaca $\rightarrow$ MT-Bench (GPT-4 score), MMLU, BBH

Comparison baselines encompass LoRA, AdaLoRA, AutoLoRA, MOELoRA, DoRA, and other PEFT strategies such as Adapter, P-tuning v2, IAPT, BitFit, (IA)$^3$, and SSP.

Key quantitative outcomes (LLaMA-3 8B, $\approx$22.8M tunable parameters, median over 5 seeds):

Task/Setting	Best LoRA Variant	ShapLoRA	Absolute Gain
Commonsense & Math QA	70.6 (DoRA)	72.1	+1.5
Instr. Tuning & Eval	7.39/57.1/47.8	7.56/58.7/48.7	+0.17/1.6/0.9
NLP & NLG	94–95%/72–74/86–87	96.1/74.8/88.5	up to +2.1

ShapLoRA demonstrates consistent improvements over state-of-the-art rank allocation and PEFT baselines. On distilled LLaMA-3 8B, improvements are observed on BoolQ (84.1 vs 83.2), PIQA (86.4 vs 85.8), and MMLU (60.2 vs 59.4).

5. Performance Analysis and Computational Trade-offs

The adoption of Shapley sensitivity effectively captures the marginal utility of each rank under a variety of coalition configurations. This comprehensive approach yields more reliable selection and allocation of ranks, resulting in superior parameter efficiency: higher accuracy at fixed parameter budgets.

The primary trade-off is an estimated 20–30% increase in training duration (4.8 h for ShapLoRA vs 2.1 h for MOELoRA, against a typical 8–10 h LoRA fine-tune). This overhead is caused by the requisite forward+backward passes across $N_3$ masked configurations on the validation set. Inference costs remain negligible.

Task-specific analysis reveals that rank necessity is not uniform—e.g., Q/V modules may require greater rank allocation—indicating the adaptability of ShapLoRA’s data-driven allocation.

6. Implementation Guidelines and Best Practices

Validation data selection: Use a held-out validation set $\mathcal D_v$ reflecting deployment data distribution (minimally 1,000 examples to ensure stable sensitivity estimates); avoid using training data for sensitivity calculations, as this can induce bias.

Parameter budgeting: Begin with $r_{\mathrm{init}}$ in the range 16–32; target pruning to 50–75% of initial ranks based on desired parameter efficiency. For extreme compression (≤1% of model size), decrease $N_3$ while maintaining a minimum of 20 samplings for valid estimates.

Hyperparameters: $N_3 ≈ 90$ coalition samples, with masking probability $p$ sampled from $\{0.1,\ldots,0.9\}$ and five repetitions per level. One pass of pruning is typically sufficient; repeated prune-retrain cycles yield minimal further gains. Always retrain from scratch after pruning to avoid overfitting or “lucky” initializations.

Recommended practices and pitfalls:

• Sensitivities should never be computed on the training set (overfitting, per ShapLoRA-4 ablation).
• Insufficient ($<$18) or excessive ($>$900) coalition samples negatively impact reliability and computational efficiency.
• Masking distribution must be balanced, ensuring each rank is equally represented in masked/unmasked conditions.

7. Broader Context and Significance

ShapLoRA extends the parameter-efficient fine-tuning paradigm in LLMs, providing a theoretically grounded and empirically validated methodology for rank allocation. Unlike magnitude- or gradient-based heuristics, ShapLoRA employs Shapley value principles to measure each rank’s marginal contribution in co-adaptation scenarios, mitigating issues of over-pruning or misallocation.

By focusing model capacity on ranks and subspaces with empirically validated utility for the downstream objective, ShapLoRA strengthens accuracy under constrained parameter budgets. An observed implication is task-dependent variability in “critical” modules, underscoring the need for data-driven rather than static design. The approach is compatible with major foundation models and facilitates democratization of LLM adaptation through increased tuning efficiency and robust generalization (Zhao et al., 25 Jan 2026).