We study the spectral perturbation of the empirical Fisher Information Matrix (FIM) of a parametric statistical model under two structured perturbations: departure of the input from a reference (in-distribution) ensemble, and finite-precision (quantized) perturbation of the model's parameters. For the first, under an explicit local curvature-monotonicity hypothesis on the dominant eigenvalue lambda_max of the FIM, we show departure from a reference manifold provably elevates lambda_max relative to a calibration baseline (Proposition 3.2), and discuss why this hypothesis is required, since curvature need not increase monotonically under every perturbation. Our principal result is a directional eigenvalue perturbation bound, via Weyl's inequality, showing lambda_max under a quantization noise perturbation is lower bounded by its unperturbed value up to a third-order remainder, and, under a mild genericity condition, strictly exceeds it at leading order (Theorem 4.3). We give two tractable approximations to lambda_max -- one heuristic, one with a rigorous two-sided bound -- and a completeness result for a threshold-based partition of an augmented state space. These results motivate using sigma_t = lambda_max(F_t)/lambda_base as a runtime monitoring statistic for deployed language models: the quantization result offers a mechanism for an empirical observation of our own, where a calibration threshold for this statistic was approximately 244 times larger than a preliminary full-precision estimate on a 4-bit quantized model, a single measurement rather than a value derived in closed form. We report supporting measurements (twelve models, n=1,080 trajectories) broadly consistent with our predictions, discuss the scope and limitations of every result, and state as an open problem the closed-form prediction of the quantization inflation magnitude our bound does not supply.
Recent years have witnessed remarkable achievements of Large Language Models (LLMs) in multiple domains, while the excessive resource requirements of LLMs hinder the deployment on resource-constrained devices. Although model quantization stands out as an effective approach, conventional quantization approaches typically incur severe performance degradation due to uniform bit-width or simple heuristic sensitivity evaluation. In this paper, we propose a novel Fisher information-based Adaptive Mixed Precision Weight Quantization approach, i.e., FAMPWQ, which performs layer-adaptive weight quantization for effective LLM inference on commodity GPUs. First, we propose a system model with a novel Fisher information metric to measure the layer-wise sensitivity to quantization. Second, we propose a reinforcement learning-based bit-width allocator in FAMPWQ, which generates an adaptive bit-width allocation strategy based on the Fisher information sensitivity metric. Extensive experiments on 7 models and 5 benchmarks demonstrate that FAMPWQ significantly outperforms 7 baseline approaches in terms of PPL (up to 3.39 smaller), accuracy (up to 6.87% higher), and LLM-as-a-judge comparison (up to 76% win rate).
Post-training quantization reduces the deployment cost of large language models, yet how severely a quantized model degrades is not determined by bit-width alone. We systematically study weight-only post-training quantization across bit-widths, quantization methods, model scales and downstream tasks on multiple model families. We observe that such degradation varies substantially across these factors: 4-bit quantization usually preserves performance, 2-bit often causes broad degradation, and at 3-bit, degradation becomes apparent but varies markedly with task type, quantization method and model scale. To explain this variability, we use the signal-to-noise ratio (SNR) to measure how strongly quantization perturbs full-precision representations. We trace degradation back to two linked processes: how quantization errors arise within individual modules, and how they accumulate across layers. First, a source SNR decomposition shows that newly introduced errors depend on three factors: the magnitude of the weight error, the strength of the task-specific signal, and how strongly the quantization error aligns with task-specific activations. Different factors affect these components in distinct ways. Second, a cross-layer propagation analysis shows that these errors can be attenuated, preserved, or amplified as they pass across layers, and that larger models benefit from weaker error amplification. Together, these results establish that quantization degradation is governed by how errors are introduced at the source and how they accumulate across the network.
Low-precision pretraining (FP8, MXFP4, NVFP4) is now standard for frontier language models, yet the literature is almost entirely achievability -- algorithms and empirical scaling laws -- with no matching characterization of what is information-theoretically possible. We study a B-bit quantized stochastic first-order oracle: an optimizer interacts for T rounds and receives, each round, a B-bit adaptive public-coin description of its stochastic gradient. Our main contribution is an exact reduction from optimizing a strongly convex quadratic family to interactively compressed Gaussian mean estimation -- under the B-bit oracle the query carries no information, so optimization collapses exactly onto a sequential distributed-estimation problem. This yields two unconditional lower bounds, a communication bound TB = Omega(d) and a statistical bound T = Omega(sigma^2 d / eps^2), and the sharp product-form bound T = Omega((sigma^2 d / eps^2) max{1, d/B}). The product form is also unconditional: a B-bit transcript carries at most O(TB / sigma^2) of Fisher trace about the mean, so bits rather than dimension limit the recoverable information, and combined with the multivariate van Trees inequality this gives the bound directly, without bounded-likelihood-ratio truncation. We give a near-matching achievability result with exact per-round bit accounting under a bounded-dynamic-range oracle, tight up to a logarithmic factor; the lower bound is for truly Gaussian (unbounded) gradients, and closing this oracle gap is left open. A sequential rate-distortion perspective extends the reduction to correlated and drifting oracles and corrects an earlier conjecture: positive noise correlation raises the bound by (1+rho)/(1-rho) rather than relaxing it. The bounds give an information-theoretic baseline for any low-bit gradient path, not an optimality claim about deployed FP4 systems.