Prompting and context-based fine-tuning methods, which we call Prefix Learning, have been proposed to enhance the performance of language models on various downstream tasks.They are empirically efficient and effective, matching the performance of full parameter fine-tuning, but the theoretical understandings are limited.In this paper, we aim to address this limitation by studying their ability from the perspective of prefix length.In particular, we provide a convergence guarantee for training an ultra-long prefix in a stylized setting using the Neural Tangent Kernel (NTK) framework.Based on this strong theoretical guarantee, we design and implement an algorithm that only needs to introduce and fine-tune a few extra trainable parameters instead of an infinite-long prefix in each layer of a transformer, and can approximate the prefix attention to a guaranteed polynomial-small error.Preliminary experimental results on vision, natural language, and math data show that our method achieves superior or competitive performance compared to existing methods like full parameters finetuning, P-Tuning V2, and LoRA.This demonstrates our method is promising for parameterefficient fine-tuning.

Extrapolating ultra-long contexts (text length $>$128K) remains a major challenge for large language models (LLMs), as most training-free extrapolation methods are not only severely limited by memory bottlenecks, but also suffer from the attention sink, which restricts their scalability and effectiveness in practice. In this work, we propose ParallelComp, a parallel long-context compression method that effectively overcomes the memory bottleneck, enabling 8B-parameter LLMs to extrapolate from 8K to 128K tokens on a single A100 80GB GPU in a training-free setting. ParallelComp splits the input into chunks, dynamically evicting redundant chunks and irrelevant tokens, supported by a parallel KV cache eviction mechanism. Importantly, we present a systematic theoretical and empirical analysis of attention biases in parallel attention—including the attention sink, recency bias, and middle bias—and reveal that these biases exhibit distinctive patterns under ultra-long context settings. We further design a KV cache eviction technique to mitigate this phenomenon. Experimental results show that ParallelComp enables an 8B model (trained on 8K context) to achieve 91.17% of GPT-4’s performance under ultra-long contexts, outperforming closed-source models such as Claude-2 and Kimi-Chat. We achieve a 1.76x improvement in chunk throughput, thereby achieving a 23.50x acceleration in the prefill stage with negligible performance loss and pave the way for scalable and robust ultra-long contexts extrapolation in LLMs. We release the code at https://github.com/menik1126/ParallelComp.

We consider the task of interpolating a $k$-sparse band–limited signal from a small collection of noisy time-domain samples. Exploiting a new analytic framework for hierarchical frequency decomposition that performs systematic noise cancellation, we give the first polynomial-time algorithm with a provable $(3+\sqrt{2}+\epsilon)$-approximation guarantee for continuous interpolation. Our method breaks the long-standing $C > 100$ barrier set by the best previous algorithms, sharply reducing the gap to optimal recovery and establishing a new state of the art for high-accuracy band–limited interpolation. We also give a refined ``shrinking-range'' variant that achieves a $(\sqrt{2}+\varepsilon+c)$-approximation on any sub-interval $(1-c)T$ for some $c \in (0,1)$, which gives even higher interpolation accuracy.

Large language models (LLMs) and generative AI have played a transformative role in computer research and applications. Controversy has arisen as to whether these models output copyrighted data, which can occur if the data the models are trained on is copyrighted. LLMs are built on the transformer neural network architecture, which in turn relies on a mathematical computation called Attention that uses the softmax function. In this paper, we observe that large language model training and optimization can be seen as a softmax regression problem. We then establish a method of efficiently performing softmax regression, in a way that prevents the regression function from generating copyright data. This establishes a theoretical method of training large language models in a way that avoids generating copyright data.