Positional LSH: Binary Block Matrix Approximation for Attention with Linear Biases
Authors: Daniel Wolfson, Tal Wagner
Organizations: Blavatnik School of Computer Science and AI, Tel Aviv University
Abstract
Positional encoding in transformers is commonly implemented through positional embeddings, attention masks, or bias terms, but formal connections between these mechanisms remain limited. We study attention with positional bias through the lens of locality-sensitive hashing (LSH), focusing on Attention with Linear Biases (ALiBi). We show that the ALiBi bias matrix is the expectation of contiguous block-diagonal binary masks induced by a ``positional LSH'' scheme. The empirical mean of masks sampled from this scheme yields spectral norm and max-norm approximation guarantees with bounded block sizes with high probability. This structural theorem implies a uniform approximation theorem for ALiBi-biased attention: with high probability over the sampled masks, the approximate attention output is accurate simultaneously for all query-key-value inputs and can be computed in near-linear time in the context length, reducing long-context ALiBi to a collection of randomized short-context regular (positionally unbiased) attention operations. Conceptually, this connects positional bias, masks, and positional embeddings in a single formal framework and suggests an approach to efficient ALiBi-biased attention. Experiments on large language models validate our theoretical findings.
We identify a previously overlooked failure mode of ALiBi positional encoding: its linear bias scaling underflows floating-point precision, which zeroes out a large fraction of attention weights and renders the affected attention heads partially blind. We analyze this failure mode, characterize its impact, and examine four mitigation strategies. We further demonstrate its occurrence in state-of-the-art pretrained models based on ALiBi. Comprehensive pretraining experiments with 148M-parameter decoder models help us to disentangle its effects from out-of-context degradation. We find that ALiBi's failure mode can substantially impair token retrieval while having only a minor effect on standard decoder benchmarks. We propose four training-time mitigation strategies and evaluate them individually and in combinations, finding that log-scaled distances yield the most consistent improvements in passkey retrieval. Despite this problem, default ALiBi slopes remain a surprisingly strong baseline, particularly for needle-in-a-haystack retrieval. Based on these findings we provide concrete recommendations on how to train models with ALiBi.
Christopher Schröder, Lukas Gienapp, Ferdinand Schlatt +2
Large Language Models (LLMs) still struggle with the ``lost-in-the-middle'' problem, where critical information located in the middle of long-context inputs is often underrepresented or lost. While existing methods attempt to address this by combining multi-scale rotary position embeddings (RoPE), they typically suffer from high latency or rely on suboptimal hand-crafted scaling strategies. To overcome these limitations, we introduce a layer-specific positional embedding scaling~(LPES) method that assigns distinct scaling factors to each layer. LPES achieves a more balanced attention distribution without fine-tuning model parameters or increasing inference delay. A specially designed genetic algorithm is employed to efficiently select the optimal scaling factors for each layer by incorporating Bézier curves to significantly reduce the search space. Extensive experiments demonstrate that LPES effectively mitigates positional attention bias and delivers consistent improvements across multiple long-context benchmarks, yielding up to an 11.2% accuracy gain on the key-value retrieval dataset.
Long-context language models suffer from position bias, where information in middle positions is underutilized. Attention Sorting addresses this by iteratively reordering documents based on attention patterns, but its multiple sort-and-generate cycles increase deployment cost. We hypothesize that position bias is the primary bottleneck and propose Debiased One-Pass Attention Sorting, which estimates a per-prompt position-bias curve from the low-attention majority of documents and uses it to correct raw attention scores (via subtraction or division) to enable single-pass sorting. Our experiments on two models refute this hypothesis in the tested setting: on LLaMA-2-7B-32K-Instruct, debiasing produces identical results to uncalibrated single-pass sorting (94.83% containment accuracy), while on YaRN-Llama-2-7b-64k, debiasing improves accuracy by 8.67 percentage points but remains 14.84pp behind iterative sorting, closing only 37% of the gap. These results suggest that position-bias correction is insufficient to match iterative sorting, and that repeated reordering provides additional benefits beyond bias correction.