arXiv subjects

Mads Toftrup

Publications and source records attributed to Mads Toftrup.

5 recordsLinked to original sources

arXiv2026

A Unified Hessian-Guided Framework for Post-Training Quantization

Low-bit quantization enables efficient deployment of large language models by reducing memory footprint and computational cost. Achieving high accuracy at low precision requires selecting quantization ranges, linear transformations, and weight rounding decisions. Existing approaches typically optimize these components independently using local reconstruction objectives, which fail to account for the varying impact of quantization errors on downstream predictions. Direct optimization of task-level metrics would provide a more faithful objective, but is challenging due to the discontinuous nature of quantization operations and the high computational cost of evaluating their downstream effects. In this work, we derive a Hessian-guided approximation of the output change induced by joint weight and activation quantization. Based on a tractable Fisher approximation, we formulate a unified post-training quantization framework that enables the joint optimization of quantization ranges, linear transformations, and weight rounding under a common output-sensitivity objective. The resulting method remains practical for modern large language models and consistently outperforms state-of-the-art transformation and rounding methods across both channel-wise and block-wise quantization settings.

cs.LG↗

arXiv2026

SS-ESOAP: Self-Scaled Adaptive Preconditioning for Physics-Informed Learning

Physics-informed neural networks (PINNs) often face ill-conditioned objectives that limit high-accuracy training. Dense quasi-Newton methods improve local conditioning but require expensive optimizer state, while Kronecker-factored methods such as SOAP scale to larger networks but rely on periodic basis updates. We introduce \method, which augments SOAP-style preconditioning with a scalar secant-energy correction adapted to Kronecker geometry and an adaptive basis update followed by variance-state downscaling. We characterize the directional secant matching induced by the scalar correction and give a bound on variance-state mismatch across basis changes. Across eight PDE benchmarks, \method attains the lowest final residual on six, including Burgers and Boussinesq, while SOAP-family baselines perform better on Gray-Scott and Ginzburg-Landau. On Boussinesq, \method reaches a residual of $10^{-5}$ in 4.1 hours with 9.2 GB peak VRAM, while Adam does not reach this target within 14 hours. Three-seed $L^2$ and $H^1$ errors on four representative PDEs support the link between lower residuals and improved solution accuracy. These results position \method as a scalable option for stiff, high-accuracy physics-informed training, rather than a uniform replacement for existing optimizers.

cs.LG↗

arXiv2026

M+Adam: Low-Precision Training via Additive-Multiplicative Optimization

Training with quantized weights can reduce costs but often results in degraded accuracy, especially when optimization is carried out in low precision, without storing high-precision copies. We identify a key failure mode: under low precision, standard optimizers can get stuck and not make progress, especially at large weight magnitudes due to coarse mantissa resolution. To overcome this, multiplicative updates have been previously proposed, in place of additive updates in standard optimizers. While successful under extremely low precision, such as under the logarithmic number system, they suffer from failures near zero and across sign changes. The failure modes of additive and multiplicative updates are therefore complementary. To exploit this, we propose M+Adam, which combines both update types: additive steps handle sign changes and small magnitudes, while multiplicative steps ensure progress at large magnitudes when additive updates are zeroed out under rounding. We prove monotone descent for M+Adam under standard smoothness assumptions. Across LLaMA-style pretraining with 60M-1B models, 1x-8x Chinchilla budgets, and using only BF16, FP8, and FP4 master weights, M+Adam consistently improves low-precision training.

cs.LG↗

arXiv2024

LoQT: Low-Rank Adapters for Quantized Pretraining

Despite advances using low-rank adapters and quantization, pretraining of large models on consumer hardware has not been possible without model sharding, offloading during training, or per-layer gradient updates. To address these limitations, we propose Low-Rank Adapters for Quantized Training (LoQT), a method for efficiently training quantized models. LoQT uses gradient-based tensor factorization to initialize low-rank trainable weight matrices that are periodically merged into quantized full-rank weight matrices. Our approach is suitable for both pretraining and fine-tuning models. We demonstrate this for language modeling and downstream task adaptation, finding that LoQT enables efficient training of models up to 7B parameters on a 24GB GPU. We also demonstrate the feasibility of training a 13B model using per-layer gradient updates on the same hardware.

cs.LG↗

arXiv2021

A reproduction of Apple's bi-directional LSTM models for language identification in short strings

Language Identification is the task of identifying a document's language. For applications like automatic spell checker selection, language identification must use very short strings such as text message fragments. In this work, we reproduce a language identification architecture that Apple briefly sketched in a blog post. We confirm the bi-LSTM model's performance and find that it outperforms current open-source language identifiers. We further find that its language identification mistakes are due to confusion between related languages.

cs.CL↗