Low precision training and inference affect both the quality and cost of language models, but current scaling laws do not account for this. In this work, we devise "precision-aware" scaling laws for both training and inference. We propose that training in lower precision reduces the model's "effective parameter count," allowing us to predict the additional loss incurred from training in low precision and post-train quantization. For inference, we find that the degradation introduced by post-training quantization increases as models are trained on more data, eventually making additional pretraining data actively harmful. For training, our scaling laws allow us to predict the loss of a model with different parts in different precisions, and suggest that training larger models in lower precision can be compute optimal. We unify the scaling laws for post and pretraining quantization to arrive at a single functional form that predicts degradation from training and inference in varied precisions. We fit on over 465 pretraining runs and validate our predictions on model sizes up to 1.7B parameters trained on up to 26B tokens.

Recent works show we can linearize large language models (LLMs)—swapping the quadratic attentions of popular Transformer-based LLMs with subquadratic analogs, such as linear attention—avoiding the expensive pretraining costs. However, linearizing LLMs often significantly degrades model quality, still requires training over billions of tokens, and remains limited to smaller 1.3B to 7B LLMs. We thus propose Low-rank Linear Conversion via Attention Transfer (LoLCATs), a simple two-step method that improves LLM linearizing quality with orders of magnitudes less memory and compute. We base these steps on two findings. First, we can replace an LLM's softmax attentions with closely-approximating linear attentions, simply by *training* the linear attentions to match their softmax counterparts with an output MSE loss (“attention transfer”). Then, this enables adjusting for approximation errors and recovering LLM quality simply with *low-rank* adaptation (LoRA). LoLCATs significantly improves linearizing quality, training efficiency, and scalability. We significantly reduce the linearizing quality gap and produce state-of-the-art subquadratic LLMs from Llama 3 8B and Mistral 7B v0.1, leading to 20+ points of improvement on 5-shot MMLU. Furthermore, LoLCATs does so with only 0.2% of past methods' model parameters and 0.04-0.2% of their training tokens. Finally, we apply LoLCATs to create the first linearized 70B and 405B LLMs (50$\times$ that of prior work). When compared with prior approaches under the same compute budgets, LoLCATs significantly improves linearizing quality, closing the gap between linearized and original Llama 3.1 70B and 405B LLMs by 77.8\% and 78.1\% on 5-shot MMLU.

Machine learning models are increasingly being scaled in both sequence length and model dimension to reach longer contexts and better performance. However, existing architectures such as Transformers scale quadratically along both these axes. We ask: are there performant architectures that can scale sub-quadratically along sequence length and model dimension? We introduce Monarch Mixer (M2), a new architecture that uses the same sub-quadratic primitive along both sequence length and model dimension: Monarch matrices, a simple class of expressive structured matrices that captures many linear transforms, achieves high hardware efficiency on GPUs, and scales sub-quadratically. As a proof of concept, we explore the performance of M2 in three domains: non-causal BERT-style language modeling, ViT-style image classification, and causal GPT-style language modeling. For non-causal BERT-style modeling, M2 matches BERT-base and BERT-large in downstream GLUE quality with up to 27% fewer parameters, and achieves up to 9.1$\times$ higher throughput at sequence length 4K. On ImageNet, M2 outperforms ViT-b by 1% in accuracy, with only half the parameters. Causal GPT-style models introduce a technical challenge: enforcing causality via masking introduces a quadratic bottleneck. To alleviate this bottleneck, we develop a novel theoretical view of Monarch matrices based on multivariate polynomial evaluation and interpolation, which lets us parameterize M2 to be causal while remaining sub-quadratic. Using this parameterization, M2 matches GPT-style Transformers at 360M parameters in pretraining perplexity on The PILE—showing for the first time that it may be possible to match Transformer quality without attention or MLPs.

Several definitions of logical consequence have been proposed in many-valued logic, which coincide in the two-valued case, but come apart as soon as three truth values come into play. Those definitions include so-called pure consequence, order-theoretic consequence and mixed consequence. In this article, we examine whether those definitions together carve out a natural class of consequence relations. We respond positively by identifying a small set of properties that we see instantiated in those various consequence relations, namely truth-relationality, value-monotonicity, validity-coherence and a constraint of bivalence-compliance, provably replaceable by a structural requisite of nontriviality. Our main result is that the class of consequence relations satisfying those properties coincides exactly with the class of mixed consequence relations and their intersections, including pure consequence relations and order-theoretic consequence. We provide an enumeration of the set of those relations in finite many-valued logics of two extreme kinds: those in which truth values are well-ordered and those in which values between 0 and 1 are incomparable.