Shapley-Based Progressive Quantization Estimation

• The paper introduces SPQE, a Shapley-value estimation method for post-training mixed-precision quantization that progressively reduces layer precision to capture inter-layer effects.
• SPQE quantifies layer sensitivity by measuring changes in per-token negative log-likelihood as layers are sequentially lowered from 4-bit to 2-bit.
• Empirical results on models like Llama-3.1-8B and Gemma-2 demonstrate that SPQE’s interaction-aware approach achieves up to 80% perplexity reductions compared to isolated layer evaluations.

Searching arXiv for the specified paper and closely related Shapley-quantization work to ground the article. Shapley-based Progressive Quantization Estimation (SPQE) is the Shapley-value estimation procedure introduced for post-training mixed-precision quantization of LLMs in “IMPQ: Interaction-Aware Layerwise Mixed Precision Quantization for LLMs” (Zhao et al., 18 Sep 2025). It is designed to measure both layer-wise sensitivity to precision and inter-layer interactions by treating Transformer layers as players in a cooperative game, defining coalitions as subsets of layers kept at high precision, and estimating marginal effects through progressive quantization rather than pruning. In the formulation used for LLM PTQ, SPQE starts from a uniformly 4-bit model, lowers individual layers to 2-bit along random permutations, and evaluates the resulting change in average per-token negative log-likelihood (NLL) on calibration data. The method is intended to keep the model inside a stable operating regime while making Shapley-based interaction modeling computationally and statistically feasible.

1. Motivation and problem setting

SPQE is introduced to address a specific failure mode of existing mixed-precision quantization heuristics at very low average precision. Earlier approaches such as LIM, Z-score, gradient-based sensitivity, and Hessian approximations evaluate layers largely in isolation through per-layer statistics including weight norms, activation norms, outlier counts, or Hessians. The paper argues that this neglect of inter-layer interactions becomes especially problematic when the average precision drops below four bits, because error propagation across layers begins to dominate behavior (Zhao et al., 18 Sep 2025).

A second motivation concerns the use of Shapley values themselves. Prior Shapley-based work on LLM layer importance had applied the cooperative-game framework to pruning, estimating contribution by removing layers altogether. In large LLMs, however, layer removal causes catastrophic loss spikes, high variance in payoff measurements, and unreliable Shapley estimates, which in turn restricts exploration of coalitions. SPQE replaces abrupt layer removal with progressive precision reduction. This change is not merely implementation detail: it redefines the estimation regime so that each sampled model remains functional and loss changes remain smooth enough to support Monte Carlo Shapley approximation.

A common misconception addressed by the SPQE formulation is that mixed-precision allocation can be solved adequately by ranking layers independently. The central claim of the method is that quantization damage is context-dependent: the effect of lowering one layer’s precision depends on which other layers remain at higher precision. SPQE is therefore constructed as an interaction-sensitive estimator rather than a purely layer-local sensitivity score.

2. Cooperative-game formulation

Let the model be an ordered set of layers

$T = \{1,2,\dots,L\}.$

SPQE casts mixed-precision quantization as a cooperative game in which the players are Transformer layers $i \in T$, and a coalition $S \subseteq T$ denotes the set of layers kept at high precision. For each layer $t \in T$, the bit assignment is

$$b_t = \begin{cases} b_{\text{high}}, & t \in S,\ b_{\text{low}}, & t \in T \setminus S. \end{cases}$$

In the estimation procedure described in the paper, $b_{\text{high}}=4$ and $b_{\text{low}}=2$.

The value function is defined through average per-token NLL on a validation or calibration corpus:

$$v_{\mathrm{NLL}}(S)=\mathbb{E}_{(x,t)\sim \mathcal{D}} \left[-\log p\!\left(x_{t+1}\mid x_{\le t}; S\right)\right].$$

The experiments use C4 as the calibration or validation corpus. Under this formulation, SPQE does not measure contribution through structural deletion; it measures the performance effect induced by a specific precision pattern over layers (Zhao et al., 18 Sep 2025).

The Shapley value for layer $i$ is given by the standard cooperative-game expression

$$\phi_i = \sum_{S \subseteq T \setminus \{i\}} \frac{|S|!(L-|S|-1)!}{L!}\Big(v(S\cup\{i\})-v(S)\Big).$$

In this setting, $i \in T$0 operationalizes the expected marginal effect of moving layer $i \in T$1 to low precision across many coalitional contexts. The interpretation given in the paper is straightforward: a high $i \in T$2 indicates that the layer is highly sensitive and should remain at higher precision if the memory budget permits.

3. Progressive estimation procedure

The exact evaluation of the Shapley definition is exponential in the number of layers and is therefore infeasible for LLMs. SPQE replaces exact evaluation with Monte Carlo permutation sampling combined with progressive quantization. For $i \in T$3 random permutations

$i \in T$4

each permutation defines an order in which layers are lowered from 4-bit to 2-bit.

For a given permutation, the procedure begins with all layers at 4-bit, corresponding to the coalition $i \in T$5. At step $i \in T$6, the layer $i \in T$7 is quantized from 4-bit to 2-bit, leaving the remaining high-precision set

$i \in T$8

The immediate marginal quantity recorded at that step is

$i \in T$9

where $S \subseteq T$0 is computed by running the quantized model on the calibration set and measuring NLL. Aggregating across permutations yields the empirical Shapley estimate

$S \subseteq T$1

The term “progressive” refers precisely to this one-layer-at-a-time reduction in precision. The paper contrasts it with pruning-based schemes in which entire layers are removed. For Llama 3.1-8B, Figure 1 shows that pruning-based Shapley estimation causes perplexity to diverge after removing approximately five layers, whereas SPQE maintains a smooth perplexity trajectory as more layers transition from 4-bit to 2-bit (Zhao et al., 18 Sep 2025).

SPQE also captures interactions implicitly through context-dependent marginals. Each $S \subseteq T$2 is measured under a different set of still-high-precision layers, so the estimator samples how the effect of quantizing a layer changes with the state of the rest of the network. The paper’s ablation over $S \subseteq T$3 on Llama 3.1-8B reports that even 10 samples already expose meaningful importance structure, including consistent sensitivity of first and last layers, while 50–100 samples show diminishing returns and yield relative improvement in average perplexity of about 2.8–2.9% and a geometric mean improvement of roughly 20% (Zhao et al., 18 Sep 2025).

4. Interaction modeling and integration into IMPQ

SPQE is the estimation engine for the broader IMPQ framework. Its output comprises a vector of empirical layer sensitivities,

$S \subseteq T$4

together with the collection of marginal-contribution samples accumulated across permutations. IMPQ converts these quantities into an interaction-aware surrogate loss model.

The paper defines a covariance-like matrix from deviations of sampled marginal contributions relative to mean Shapley estimates:

$S \subseteq T$5

This matrix is used as an empirical proxy for pairwise interaction structure. Because finite-sample off-diagonal estimates can be noisy, the method applies diagonal shrinkage:

$S \subseteq T$6

The paper reports that $S \subseteq T$7 gives the best perplexity in the ablation table, while $S \subseteq T$8 corresponds to ignoring interactions and $S \subseteq T$9 retains the full covariance estimate (Zhao et al., 18 Sep 2025).

To separate first-order and second-order effects, IMPQ computes

$t \in T$0

The resulting parameters define a binary quadratic objective over layerwise precision decisions. With $t \in T$1, where $t \in T$2 means the layer remains at low precision and $t \in T$3 means it is promoted to high precision, the surrogate loss increase is

$t \in T$4

Under memory budget $t \in T$5 and per-layer promotion costs $t \in T$6, the optimization problem is

$t \in T$7

Within the full pipeline, SPQE therefore serves two distinct roles. First, it supplies a Shapley-based ranking of layer sensitivity. Second, through the covariance of marginal effects, it enables explicit interaction modeling, which the paper treats as necessary for aggressive low-bit allocation.

5. Computational profile and empirical behavior

The naive complexity of exact Shapley evaluation is $t \in T$8 coalition-value computations, which is infeasible for LLMs. SPQE reduces this to approximately

$t \in T$9

where $$b_t = \begin{cases} b_{\text{high}}, & t \in S,\ b_{\text{low}}, & t \in T \setminus S. \end{cases}$$0 is the cost of evaluating NLL on the calibration corpus. For Llama-3.1-8B, the paper reports that SPQE with $$b_t = \begin{cases} b_{\text{high}}, & t \in S,\ b_{\text{low}}, & t \in T \setminus S. \end{cases}$$1 permutations takes about 18 hours on one NVIDIA A40. It also reports that 10–50 permutations can already provide useful estimates, reflecting an explicit trade-off between estimation quality and runtime (Zhao et al., 18 Sep 2025).

The empirical case for SPQE is presented mainly through the performance of IMPQ, which depends entirely on SPQE-derived sensitivities and interactions. Across Llama-3, Gemma-2, and Qwen-3, and across the three PTQ backends Quanto, HQQ, and GPTQ, the reported results show consistent superiority over baselines based only on isolated metrics. The abstract summarizes the overall outcome as perplexity reductions of 20 to 80 percent relative to the best baseline across average precisions ranging from 4 bit down to 2 bit, with larger margins as the bit-width tightens (Zhao et al., 18 Sep 2025).

The detailed examples emphasize the low-bit regime. On Gemma-2-2B under GPTQ at 2.01–2.5 bits, the reported perplexities are $$b_t = \begin{cases} b_{\text{high}}, & t \in S,\ b_{\text{low}}, & t \in T \setminus S. \end{cases}$$2 for Sensitivity, $$b_t = \begin{cases} b_{\text{high}}, & t \in S,\ b_{\text{low}}, & t \in T \setminus S. \end{cases}$$3 for LIM, and $$b_t = \begin{cases} b_{\text{high}}, & t \in S,\ b_{\text{low}}, & t \in T \setminus S. \end{cases}$$4 for IMPQ. On Gemma-2-9B under GPTQ at the same average precision, the corresponding values are 189.55, 214.03, and 48.52. On Qwen3-4B under GPTQ, the paper reports $$b_t = \begin{cases} b_{\text{high}}, & t \in S,\ b_{\text{low}}, & t \in T \setminus S. \end{cases}$$5, $$b_t = \begin{cases} b_{\text{high}}, & t \in S,\ b_{\text{low}}, & t \in T \setminus S. \end{cases}$$6, and 697.28. Similar trends are reported for Quanto and HQQ. These are not standalone SPQE numbers; rather, they are evidence that the SPQE estimates are sufficiently accurate and interaction-aware to support a more effective mixed-precision allocation.

The ablation on diagonal shrinkage supplies direct evidence for the benefit of interaction modeling. For Llama-3.2-3B with Quanto, averaged over 2–4 bits, the reported perplexities are $$b_t = \begin{cases} b_{\text{high}}, & t \in S,\ b_{\text{low}}, & t \in T \setminus S. \end{cases}$$7 for $$b_t = \begin{cases} b_{\text{high}}, & t \in S,\ b_{\text{low}}, & t \in T \setminus S. \end{cases}$$8, $$b_t = \begin{cases} b_{\text{high}}, & t \in S,\ b_{\text{low}}, & t \in T \setminus S. \end{cases}$$9 for $b_{\text{high}}=4$0, and $b_{\text{high}}=4$1 for $b_{\text{high}}=4$2. This supports the paper’s claim that interactions help, but that they must be regularized rather than taken from the raw covariance estimate without shrinkage (Zhao et al., 18 Sep 2025).

6. Relations, limitations, and extensions

SPQE belongs to a broader Shapley-based lineage, but it differs from adjacent formulations in the object of the game and the way coalitions are explored. Relative to generic sampling-based Shapley approximation, its Monte Carlo estimator follows the standard permutation-sampling view in which Shapley values are expectations of marginal contributions over random orderings. Non-asymptotic analyses of such estimators establish concentration guarantees under known variance or known range and show that stratified sampling by coalition size can improve the error bound in certain game classes (Maleki et al., 2013). This suggests a methodological avenue for SPQE beyond simple random permutation sampling, and the paper explicitly points to advanced sampling strategies such as stratified or importance sampling as potential extensions.

SPQE also differs from Shapley-based MPQ methods defined over supernet bit-width operations rather than Transformer layers. In “Where and How to Enhance: Discovering Bit-Width Contribution for Mixed Precision Quantization,” SMPQ models bit-width operations as players, uses validation accuracy or validation loss as the value function, and drives mixed-precision search through Shapley-based contribution weights inside a supernet framework (Kang et al., 5 Aug 2025). SPQE instead operates in post-training quantization for LLMs, treats layers as players, and uses progressive 4-bit to 2-bit reduction with NLL-based payoff. The contrast is significant: SMPQ addresses differentiable search and the “magnitude-based selection pitfall,” whereas SPQE is constructed to make layerwise interaction estimation stable in low-bit PTQ.

A further extension concerns the permutation sampler itself. Work on Shapley estimation over permutations shows that kernel herding, sequential Bayesian quadrature, orthogonal spherical codes, and Sobol permutations can improve convergence over standard Monte Carlo by constructing more informative or lower-discrepancy permutation sets (Mitchell et al., 2021). A plausible implication is that SPQE’s Monte Carlo permutation stage could be strengthened by importing such samplers, especially when the number of layers is moderate or when NLL evaluations are very costly.

The method also has clear limitations. It incurs substantial evaluation cost; its sensitivity estimates depend on the chosen calibration corpus, which is C4 in the reported experiments; it is formulated for binary 2-bit versus 4-bit decisions rather than multi-level bit assignment; and its implementation primarily concerns weight precision rather than explicitly modeling activation quantization (Zhao et al., 18 Sep 2025). The paper nevertheless identifies several extensions: applying the same cooperative-game framework to layer or head pruning and other structured compression strategies, generalizing to more precision levels such as 2/3/4/8 bits or microscaling formats like MXINT, and transferring the layer-as-player quantization game to other deep architectures including CNNs and ViTs.

In that sense, SPQE is best understood not as a generic Shapley estimator but as a domain-specific reformulation of Shapley sampling for low-bit LLM PTQ. Its defining feature is the replacement of catastrophic coalition perturbations with progressive precision reduction, which makes layer sensitivity and interaction estimation compatible with the operating constraints of large Transformer models.