PaTH Attention showed that replacing RoPE's position-indexed rotations with accumulated data-dependent Householder reflections yields strong length extrapolation, though performance degrades at extreme context lengths. We ask whether this depends on Householder-specific structure or reflects a general property of accumulated transformations along source-to-query paths. We study a simpler variant keeping RoPE's block-diagonal SO(2) rotations but replacing position-indexed angles with accumulated token-dependent ones. It shows the same pattern: improved extrapolation then degradation at long contexts. We prove the result extends to accumulated orthogonal transformations satisfying certain regularity conditions: their products become incoherent after finitely many steps, suppressing attention to distant tokens. Accumulated rotations of queries and keys create a finite mixing window independent of context length; per-token suppression learned in training transfers unchanged to any evaluation length, and high-dimensional concentration produces a score gap suppressing far tokens while near-route transport preserves the target signal. Conversely, a lower bound shows accumulated rotations must eventually degrade: as the far set grows, no rotations preserve the near signal without explicit far-mass control. For SO(2) rotations, rotating values too makes residual far contributions combine incoherently, extending the range. Controlled experiments support these predictions: random accumulated rotations substantially improve extrapolation over RoPE, learned token-dependent rotations maintain near-training-length perplexity far beyond the training context, and rotating values helps over queries and keys alone. Rotation-only models still degrade at extreme lengths, while ALiBi stays length-stable, consistent with the need for far-mass control.
We identify intrinsic limitations of Rotary Positional Embeddings (RoPE) in Transformer-based long-context language models. Our theoretical analysis abstracts away from the specific content of the context and depends only on its length. We prove that as context length increases, RoPE-based attention becomes unpredictable and loses two properties that are central to its effectiveness. First, it loses its locality bias: RoPE is no more likely to favor nearer positions than substantially farther ones. Second, it loses consistency in token relevance: a key vector that receives a higher attention score than an alternative at one position may receive a lower score at another. In both cases, the probability of failure approaches 0.5, no better than random guessing. We further prove that the attention score can remain unchanged when a key token is moved to a different position, or even replaced by a different token, indicating a failure to distinguish positions or tokens. Adjusting the RoPE base trades off distinguishing positions against distinguishing tokens but cannot preserve both at the same time. Increasing the RoPE base hyperparameter, a common practice in today's long-context models, helps distinguish different tokens, but inevitably sacrifices the ability to distinguish positions. Our empirical analysis shows that multi-head, multi-layer architectures are insufficient to overcome these limitations. Our findings suggest that fundamentally new mechanisms for encoding position and token order may be needed in future Transformer long-context language models.
Transformers with relative positional encodings often extrapolate to sequences longer than those seen during training, whereas transformers with learned absolute encodings typically do not. This is a robust empirical regularity, and the explanations offered for it so far are chiefly about expressivity, that is, about whether a length-generalizing solution exists. We give an optimization explanation. On a minimal fixed-offset retrieval task that isolates positional selection, the gap is governed by the implicit bias of the trained attention head: among the many solutions that fit short sequences, which one gradient descent actually selects. We prove that rotary encodings make the attention logit a function of relative offset alone, an exact equivariance, so whatever selection rule is learned at training lengths is reproduced verbatim at every longer length. Learned absolute encodings instead leave out-of-range positions unconstrained, and the trained head pins to a fixed absolute position inside the training range. We characterize the learned rotary rule as a low-rank ``carrier'' kernel aligned with the target offset, and we derive the resulting graceful accuracy decay as an attention-dilution law; both predictions are confirmed across seeds and offsets. A linear-attention control shows the mechanism is specific to softmax: without normalization, training selects a min-norm interpolant that does not extrapolate. The phenomenon, the equivariance, and the carrier all transfer to a multi-layer, multi-head transformer trained on a full-sequence length-generalization task. The account connects the implicit bias of attention, implicit bias for extrapolation in recurrent models, and the learning side of the RASP-L conjecture.
Rotary Position Embedding (RoPE) is widely adopted in Transformers to encode positional information, yet standard implementations enforce a uniform frequency schedule and scaling across all attention heads. Using simplified retrieval tasks and length generalization scenarios, we show -- both empirically and theoretically -- that heads with different functional roles require distinct frequency ranges and attention scaling factors to operate effectively. Ignoring this structure leads to suboptimal utilization of embedding dimensions and degraded performance, particularly under long-context settings. To address these limitations, we propose AdaRoPE, which equips each attention head with learnable rotation frequencies and attention scaling factors. Pretrained LLMs with AdaRoPE consistently outperform existing RoPE variants, including partial RoPE and NoPE baselines. For context extension, we further show that uniform frequency and attention scaling, used in methods such as YaRN, are suboptimal. By applying head-specific scaling, AdaRoPE enables better context extension while better preserving short-context performance in both the extrapolation setting and the long-context continued pretraining setting. These results highlight the importance of optimizing rotary position embedding at the level of individual attention heads.