PRISM: A Geometric Risk Bound that Decomposes Drift into Scale, Shape, and Head
Authors: Chieh-Yen Lin, Shao-Hua Sun
Organizations: Appier AI Research · National Taiwan University
Abstract
Comparing post-training LLM variants, such as quantized, LoRA-adapted, and distilled models, requires a diagnostic that identifies how a variant has drifted, not only whether it has degraded. Existing similarity scores such as CKA and SVCCA can flag degradation, but they do not directly link representation drift to risk or mechanism. We propose PRISM, Proxy Risk Inference via Structural Mapping, which exploits the linear output head of LLMs and the empirically near-isometric structure of their backbones to derive a closed-form upper bound on the cross-entropy risk gap between a target model and a post-training variant. The bound is calibrated for variant ranking and decomposes drift into three independently measurable axes: scale mismatch, shape mismatch, and head divergence. Each axis corresponds to a distinct failure mode, including shape distortion under low-bit quantization, scale separability under LoRA forgetting, and head divergence under GGUF k-quantization. As a result, the dominant axis suggests a remediation direction rather than merely raising a degradation flag. Because the shape term is differentiable, the same geometry can also serve as a training-time regularizer against catastrophic forgetting. Across two model families and five benchmarks, PRISM ranks variants with mean Spearman correlations of 0.820 for post-training quantization and 0.831 for LoRA forgetting, and its axis-guided shape regularizer outperforms experience replay in aggregate at mitigating downstream forgetting.
Deployment of Large Language Models (LLMs) on memory-constrained edge devices relies heavily on aggressive post-training quantization. However, evaluating these models is largely based on zero-shot task accuracy, which depends solely on argmax predictions and is insensitive to changes in the underlying predictive distribution. Consequently, accuracy can exhibit unstable, non-monotonic behavior under progressive quantization, masking substantial fidelity loss relative to the BFloat16 (BF16) uncompressed base model and providing misleading deployment signals. We introduce a distribution-sensitive evaluation framework quantifying information loss in quantized LLMs as the divergence between full-vocabulary predictive distributions at the token decision boundary. We compute statistical distances, including Jensen-Shannon Divergence and Total Variation Distance, between outputs of full-precision and quantized models, enabling a fine-grained analysis of distributional shift. Using this framework, we quantify probability mass displacement and distributional drift relative to the BF16 reference, capturing predictive distribution changes not reflected in top-1 accuracy. We conduct a 120-run experimental matrix across five foundation architectures and four reasoning benchmarks under progressive quantization regimes, from uncompressed BF16 to Q2_K, providing a systematic fidelity analysis. Our results show divergence metrics generally increase under stronger quantization, complementing task accuracy with a fidelity signal. Across tested llama-cpp schemes, mixed-precision Q4_K generally yields lower divergence than uniform Q4_0 at similar memory footprints. These findings motivate distribution-aware evaluation as a practical diagnostic complement to task accuracy; they do not directly establish correctness, calibration, safety, or user-perceived quality.
Shahzeb Qamar, Lorenz Sparrenberg, Christian Bauckhage +5
Post-training quantization is widely used to deploy large language models in resource-constrained settings, yet its evaluation relies almost exclusively on accuracy and perplexity. We show that these metrics fail to capture behavioral changes induced by quantization. We introduce correctness agreement, a decision-level metric that measures overlap in correct predictions between a base model and its quantized variants, independent of absolute accuracy. Across multiple models and quantization schemes from 8-bit to 2-bit, we find that behavioral divergence emerges under moderate quantization even when task performance appears preserved. To explain this effect, we analyze quantization as a structural operator on attention weights and quantify layer-wise distortions using statistical and distributional measures. Our results reveal non-linear breakpoints at low bit-widths and show that query and key projections are consistently more sensitive than value and output projections. These findings expose an illusion of equivalence between base and quantized models and motivate behavioral evaluation beyond conventional performance metrics.
Baha Rababah, Cuneyt Gurcan Akcora, Carson K. Leung
Post-training quantization compresses large language models (LLMs) by storing their weights at reduced precision, and each quantized weight introduces an error into the hidden states. Naively, these errors should accumulate with depth and corrupt next-token prediction; randomly initialized models accumulate these discrepancies rapidly, whereas quantized pretrained models accumulate much less hidden-state error and largely maintain downstream task performance, even though they were never trained with quantization noise. This raises the question we address: why does post-training quantization work? Comparing full-precision and quantized forward passes, we identify two mechanisms that characterize pretrained quantization robustness. First, the error a layer newly introduces tends to oppose the error it inherits from the layer's input. The two cancel partially such that the discrepancy between full-precision and quantized passes grows slowly. This counteracting residual interaction develops during pretraining. Our quantitative analysis identifies it as a major factor slowing hidden-error growth. Second, LM-head geometry preferentially preserves the scores and probabilities of high-ranked tokens, which typically represent the model's most confident predictions. Together, these mechanisms explain why quantization error that passes through numerous layers can still produce only small output changes, and we verify the findings across models and quantization settings.