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The frequency parameter $$\theta_k$$ in Rotary Positional Embeddings (RoPE) is defined as an exponential function of its index $$k$$. The specific formula is: $$\theta_k = b^{-\frac{2(k-1)}{d}}$$ where $$b$$ is a predefined base, $$k$$ is the index for each pair of dimensions (ranging from $${}1$$ to $$d/2$$), and $$d$$ is the total dimensionality of the embedding. This formulation creates a geometric progression of frequencies.

Formula for RoPE Frequency Parameters (θ_k)

In Rotary Positional Embeddings (RoPE), the individual frequency parameters $\theta_k$ are calculated using a fixed base, commonly 10000. This approach is analogous to the frequency settings used in sinusoidal positional embeddings. The formula for the k-th frequency component is given by: $$\theta_k = 10000^{-\frac{2(k-1)}{d}}$$ where $d$ is the dimensionality of the embedding and $k$ is the index of the component, ranging from $1$ to $d/2$.

Calculation of RoPE Frequency Parameters

The period, $$T_k$$, of the sine and cosine functions used in Rotary Positional Embeddings (RoPE) is determined by the frequency parameter $$\theta_k$$. The formula for the period is given by: $$T_k = 2\pi \cdot b^{\frac{2(k-1)}{d}}$$ where $$b$$ is the base, $$k$$ is the component index, and $$d$$ is the embedding dimensionality. This formula is derived from the relationship $$T_k = 2\pi / \theta_k$$.

Formula for the Period of RoPE's Sine and Cosine Components

A set of frequency parameters is defined by the formula: $$\theta_k = b^{-\frac{2(k-1)}{d}}$$ If the dimensionality `d` is 4 and the frequency for the second component (`k=2`) is `θ_2 = 0.1`, what is the value of the base `b`?

Determining the Base from a Frequency Parameter

An AI engineer is fine-tuning a language model to summarize very long legal documents. The model is underperforming, and the engineer suspects it's failing to capture relationships between pieces of information that are far apart in the text. The model uses positional embeddings based on the frequency parameter formula: $$\theta_k = b^{-\frac{2(k-1)}{d}}$$ To improve the model's ability to handle these long-range dependencies, the frequencies ($\theta_k$) need to correspond to longer periods. Which parameter in the formula (`b` or `d`) should the engineer adjust, and in which direction (increase or decrease), to achieve this? Justify your reasoning.

Tuning Positional Embeddings for Long-Context Models

Consider the generalized formula for calculating a set of frequency parameters: $$ \theta_k = b^{-\frac{2(k-1)}{d}} $$ In this formula, `b` is a configurable base greater than 1, `d` is the dimensionality (a positive integer), and `k` is the component index, which is an integer greater than 1. How would increasing the value of the base `b` affect the calculated frequency `θ_k` for any given `k` and `d`?

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