arXiv subjects

Michael Helcig

Publications and source records attributed to Michael Helcig.

6 recordsLinked to original sources

arXiv2026

SHq: Learnable Grids for LLM Weight Quantization

Block-scaled lookup-table (LUT) formats, which represent weights as indices into a small value set (the grid), have become standard for low-bit LLM quantization, ranging from NF4 and BOF4 to the 3-bit LUT instruction in NVIDIA Rubin. While these formats leave the grid definition flexible, existing implementations rely on fixed grids derived either from numerical data types (INT, FP) or assumed weight distributions (NF, BOF). We demonstrate that no single fixed grid is universally optimal; the best fixed grid varies with bit-width: BOFS at 2 bits, NF at 4 and 5 bits, and INT at 6 bits. We introduce SHq, a closed-form family of grids parameterized by two critical degrees of freedom: a companding strength α that interpolates level spacing from uniform to floating-point-like, and an offset c that introduces asymmetry. This offset allows the sign bit of the FP8 block scale, otherwise unused, to select between the grid and its mirror image, up to doubling the number of distinct values available per block. SHq modifies only the lookup table contents and serves as a drop-in replacement for the grid in any such format. Across five models ranging from 3B to 35B parameters, including a mixture-of-experts architecture, evaluated at 2-6 bits using RTN and GPTQ, SHq achieves the lowest KL divergence in every setting at 3 and 4 bits and in most settings at 2, 5 and 6 bits. It outperforms the best fixed grid by up to 34% and remains superior under mixed-precision configurations. At 4.5 bits per weight, the SHq format reduces the KL divergence of NVFP4 by 41-67% while improving downstream accuracy. Grid families incorporating both spacing and offset parameters perform comparably, whereas those lacking an offset exhibit clearly higher error, confirming that these two degrees of freedom drive the observed gains. We therefore propose SHq as the standard grid for block-scaled lookup-table formats.

cs.LG↗

arXiv2026

Non-Acceleration of the Heavy-Ball Method in 1D

Despite having an asymptotically optimal rate of convergence on convex quadratics, the acceleration of Polyak's heavy-ball method and a characterization of its dynamics within more general settings have remained long standing open problems. Recently, the work of Goujaud et al. [Math. Program. (2025)] made progress regarding acceleration in the context of smooth, strongly convex minimization: for a small enough inverse condition number, either the method does not accelerate on quadratics, or there exists a smooth strongly convex function such that heavy-ball cycles over the roots of unity. The constructed functions have two variables, thus leaving the question of acceleration over univariate functions open. In this work, we use a Fourier-based analysis to i) provide a proof for the non-acceleration of the heavy-ball method over the class of univariate smooth strongly convex functions. We also ii) provide a strict enlargement of the previous root-of-unity cycling region, with insights regarding the "fractional" sets of Goujaud et al. [arXiv:2502.19916], and iii) give a higher dimensional cycling instance which is not covered by root-of-unity cycling.

math.OC↗

arXiv2026

Self-Repulsive Sampling for Diffusion Language Models

Sampling several responses and voting over their answers can improve a language model's accuracy, but repeated answers limit the benefit of additional samples. Raising temperature increases diversity at a potential cost to per-sample accuracy. We introduce Self-Repulsion (SR), a sampler for masked diffusion language models that uses peer commitments to diversify the pool. At each penalized denoising step, each path lowers a token's logit according to how many peers have committed that token at the same position. Paths share a batched forward pass and then commit in sequence, so later paths observe choices made earlier in the same step. This coupling requires no training or additional forward or backward pass and can produce distinct paths even at temperature zero. When all paths commit a position together from identical logits, the update exactly maximizes total logit minus a convex duplication cost. On LLaDA-8B-Instruct with ten paths and 128 denoising steps, deterministic SR reaches 80.38% plurality accuracy on GSM8K, compared with 70.17% for the unpenalized greedy decoder. At temperature 0.6 and matched model-evaluation budgets, the count penalty improves over self-consistency by 2.06 percentage points in blocks of 32 and 14.50 under pure diffusion. Experiments on GSM8K, MATH and TruthfulQA show that voting gains arise mainly from higher coverage of correct answers, with gains that vary by benchmark and decoding regime.

cs.LG↗

arXiv2026

Model Compression with Exact Budget Constraints via Riemannian Manifolds

Assigning one of K options to each of N groups under a total cost budget is a recurring problem in efficient AI, arising in mixed-precision quantization, non-uniform pruning, and expert selection. The objective (model loss) depends on all assignments jointly and does not decompose across groups, so combinatorial solvers can only optimize proxy objectives. Evolutionary search evaluates the actual loss but lacks gradients, while penalty-based methods enforce the budget only approximately and often require heavy hyperparameter tuning. We show that under softmax relaxation, the budget constraint defines a smooth Riemannian manifold in logit space with unusually clean geometry: the normal vector is available in closed form, shifting logits along the cost vector changes expected cost monotonically, and vector transport reduces to a single inner product. Building on this, we propose Riemannian Constrained Optimization (RCO), which wraps tangent projection, binary-search retraction, and momentum transport around a standard Adam step. Combined with Gumbel straight-through estimation and budget-constrained dynamic programming for discrete feasibility, RCO optimizes the actual loss with first-order methods, enforces the expected budget exactly at every iterate, and introduces no constraint-related hyperparameters. The same construction handles multiple simultaneous budgets with no additional coefficients. RCO matches or exceeds state-of-the-art methods on synthetic problems and realistic LLM compression settings, often at considerably lower wall-clock cost. Source code is available at https://github.com/IST-DASLab/RCO.

cs.LG↗

arXiv2026

Statistically-Lossless Quantization of Large Language Models

Model quantization has become essential for efficient large language model deployment, yet existing approaches present clear trade-offs: methods such as GPTQ and AWQ achieve practical compression but are lossy, while lossless techniques preserve fidelity but lack inference acceleration. This paper explores the middle ground of statistically-lossless compression, examining three complementary aspects of what losslessness means for quantized LLMs. First, task-lossless compression preserves zero-shot benchmark accuracy within natural sampling variance and is achievable at aggressive bitwidths. Second, we formalize the stricter notion of distribution-lossless compression, requiring the quantized model's next-token distribution to be practically indistinguishable from the original, and propose the Expected Acceptance Rate (EAR), the maximum token-agreement probability under optimal coupling, as a directly interpretable fidelity metric. For example, EAR >= 0.99 means 99% agreement. Third, we prove a gamma-squared variance law showing that symmetric quantization inflates noise variance by gamma^2 relative to asymmetric quantization, making asymmetric quantization a prerequisite for distribution-lossless fidelity but not for task-level preservation. Through SLQ, a layer-wise non-uniform method with asymmetric quantization and wide bitwidth search, we obtain task-lossless compression at well below 4 bits per parameter, as low as 3.3 bits depending on the model, distribution-lossless compression at 5-6 bits per parameter on average, and inference speedups of 1.7-3.7x compared to FP16 using optimized kernels. Source code is available at [https://github.com/IST-DASLab/SLQ](https://github.com/IST-DASLab/SLQ).

cs.LG↗

arXiv2026

GSQ: Highly-Accurate Low-Precision Scalar Quantization for LLMs via Gumbel-Softmax Sampling

Quantization has become a standard tool for efficient LLM deployment, especially for local inference, where models are now routinely served at 2-3 bits per parameter. The state of the art is currently split into simple scalar quantization techniques, such as GPTQ or AWQ, which are widely deployed but plateau in accuracy at 3-4 bits per parameter (bpp), and "second-generation" vector- or trellis-quantized methods, such as QTIP, GPTVQ and AQLM, which push the accuracy frontier but are notoriously hard to implement and to scale. In this paper, we ask whether this gap is fundamental, or whether a carefully optimized $\textit{scalar}$ quantizer can recover most of it. We answer in the affirmative, by introducing GSQ (Gumbel-Softmax Quantization), a post-training scalar quantization method which jointly learns the per-coordinate grid assignments and the per-group scales using a Gumbel-Softmax relaxation of the discrete grid. GSQ matches the cardinality of the relaxation to the small number of levels available in the target bit-width regime (e.g., 3-8 levels for ternary and 3 bpp, respectively), making optimization tractable. Practically, on the standard Llama-3.1-8B/70B-Instruct models, GSQ closes most of the gap between scalar quantization and the QTIP frontier at 2 and 3 bits, while using a symmetric scalar grid with group-wise quantization, and thus remains compatible with existing scalar inference kernels. We further show that the same discrete-assignment optimization can be applied to practical GGUF K-Quant checkpoints: starting from publicly released GGUF models, GSQ improves accuracy while projecting the result back into the same deployment format. Finally, GSQ scales to trillion-scale Mixture-of-Experts models such as Kimi-K2.5, where vector-quantized methods are difficult to apply. The source code is publicly available at https://github.com/IST-DASLab/GSQ.

cs.CL↗