Rotary Positional Encodings (RoPE) are currently the most popular positional encodings used in modern language models. RoPE rotates two-dimensional chunks of query and key vectors, operating as a function of their relative positional offset. The position-wise rates of rotation in RoPE typically follow a geometric sequence specified by a fixed base-frequency hyperparameter. Prior work has improved performance by either increasing this parameter to slow rotation or by applying RoPE to only a subset of QK dimensions. In this work we modify RoPE by learning a scalar per frequency, treating frequencies as learnable parameters rather than hyperparameters. We validate Learned RoPE by training a ladder of language models from scratch, ranging from 52M to 2.5B parameters. We observe and analyze the emergence of a high-norm, positional LeRoPE band. LeRoPE consistently outperforms RoPE and partial RoPE across all scales, with RoPE requiring 3.4% more compute (FLOPs) to match LeRoPE at the largest scale.
Rotary Position Embeddings (RoPE) provide transformers with a fixed grid of positional frequencies, yet trained models use these frequencies highly non-uniformly. We study what determines this frequency usage and propose a data-centered explanation: RoPE frequencies are selected to match the relative-distance structure of the training data. Viewing each frequency as a positional lens, we formalize a field-resolution tradeoff and show that, for a data-induced dependency profile of width W, the optimal frequency scales as 1/W. This frequency-matching principle explains controlled observations on synthetic and text-based data, and suggests that the mid-low frequency bands observed in language models arise from the multi-scale dependency structure of natural language. We further connect frequency selection to position-interpolation-based length generalization: scaling frequencies down expands the effective field while reducing resolution. This helps when longer-context dependencies are approximate dilations of those seen during training, but can fail when relevant dependencies do not scale with context length. Empirically, we show that natural language exhibits approximate self-similarity across positional scales, explaining why test-time frequency scaling can support long-context generalization. Overall, our results identify a data-driven mechanism behind emergent RoPE frequency usage and show that long-context generalization depends on two forms of scale matching: between learned frequencies and training-time dependencies, and between frequency scaling and how those dependencies extend to longer contexts.
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.
Position information is essential for language modeling. In softmax transformers, Rotary Position Embeddings (\textit{RoPE}) encode positions through \textit{fixed-angle} rotations, while in linear transformers, order is handled via input-dependent (selective) gating that decays past key-value associations. Selectivity has generally been shown to improve language-related tasks. Inspired by this, we introduce \textit{Selective RoPE}, an \textit{input-dependent} rotary embedding mechanism, that generalizes \textit{RoPE}, and enables rotation in \textit{arbitrary angles} for both linear and softmax transformers. We show that softmax attention already performs a hidden form of these rotations on query-key pairs, uncovering an implicit positional structure. We further show that in state-space models and gated linear transformers, the real part manages forgetting while the imaginary part encodes positions through rotations. We validate our method by equipping gated transformers with \textit{Selective RoPE}, demonstrating that its input-dependent rotations improve performance in language modeling and on difficult sequence tasks like copying, state tracking, and retrieval.