CubicQuant: Parametric Non-Uniform Codebooks for High-Throughput LLM Inference with 1-8-Bit Weights

Weight quantization for large-language-model inference must balance adaptive reconstruction levels with representations regular enough for efficient GPU execution. Uniform integers constrain each group to a linear grid. Low-bit floating-point formats use a fixed exponent-mantissa structure, while learned codebooks gain flexibility at the cost of irregular decoding and additional metadata. We introduce CubicQuant, a parametric non-uniform scalar format that preserves a dense integer code stream while adapting reconstruction levels within each weight group. A monotonic cubic curve, specified by two shape parameters and one scale, maps uniformly spaced magnitude codes to non-uniform levels. The family spans 1-8-bit weight payloads, contains symmetric uniform integer quantization as an exact special case, and has effective width B + 64/G bits per weight for payload width B and group size G. We derive population distortion under Uniform, Gaussian, and Laplace distributions, formulate continuous and Dynamic-A8-carrier-aware fitting objectives, and describe direct packed-weight GPU execution. For finite groups of G=128 with 15,360 samples per distribution, W4 CubicQuant reduced reconstruction RMSE relative to optimally clipped four-bit uniform integer quantization by 3.90% on Uniform, 13.49% on Gaussian, and 28.14% on Laplace samples. Relative to the best enumerated four-bit finite floating-point format, the reductions were 3.90%, 9.44%, and 6.27%. Preliminary H200 kernel measurements show a workload-dependent crossover: model-dtype execution is faster for narrow GEMV, while Dynamic A8 becomes favorable as row count grows. The results establish the format's representational promise and direct executability; downstream model quality and cross-device end-to-end performance remain open evaluation questions.

Neural Network Quantization by Learning Low-Loss Subspaces

Neural network quantization aims to find a discrete representation of parameters that preserves the performance of a full-precision (FP) model as faithfully as possible. Enforcing discrete constraints perturbs parameters away from a well-optimized minimum, generally resulting in performance degradation. Recent studies indicate that low-loss FP solutions are not isolated, but instead belong to connected low-loss subspaces of the loss landscape, where the loss maintains nearly the same minimum value. Models sampled from these subspaces are diverse and retain high accuracy. This raises the question: can a quantized model be constructed to lie within a low-loss subspace of the FP model, thereby automatically preserving performance? We address this question by learning quantization-aware linear paths in weight space optimized to minimize loss. We demonstrate that the midpoint of the resulting subspace is, by design, quantization-friendly and that its direct quantization yields performance comparable to that of quantization-aware training. The proposed procedure offers a novel perspective on weight quantization and, in contrast to conventional methods, neither relies on the straight-through estimator nor involves explicit discretization during training.

Optimal Post-Training Quantization Scales and Where to Find Them

Post-training quantization (PTQ) compresses large language models by mapping weights to low-bit representations. The scaling factor that defines the quantization grid is typically chosen using simple, data-free heuristics. In this work, we present PiSO (Piecewise Scale Optimization), an algorithm that leverages calibration data to compute the optimal channel-wise weight scales exactly and efficiently under round-to-nearest quantization. PiSO partitions the scale search space into finitely many intervals on which the objective admits a closed-form minimizer. We extend PiSO to group-wise quantization via principled heuristics and propose effective strategies for interleaving scale optimization with error correction. Experiments on Llama and Qwen models across multiple model sizes and target weight bit-widths demonstrate consistent improvements in perplexity and downstream zero-shot accuracy, both standalone and combined with error correction. In particular, we observe increased benefits as the target bit-width narrows and quantization becomes more challenging.