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Table of Contents Introduction The Setting The Optimization Problem Subgradient of the Loss Function The Family of Geometric Optimizers A Convergence Theorem Convergence Bounds for Specific Scenarios Normalized Gradient Descent Muon Adam/SignSGD A Note on Bound Tightness and Rate Variation Analysis of Rate Variation Analysis of Bound Tightness Using This In Practice: A Comparison of Optimizers The Problem with Vanilla Gradient Descent Geometric Optimizers as the Solution Comparing the structure for Normalized Gradient Descent, Muon, and Adam/SignSGD The Nature of the Updates Building Intuition from the Convergence Bounds The Meaning of \(C_{\text{lower}}\) and \(C_{\text{upper}}\) Rate Stability and Comparing Optimizers Further reading Introduction We want to answer the question: how do optimizers fundamentally differ in their approach to finding a minimum? To explore this question in a controlled environment, we focus on a simple problem: minimizing the matrix error term \(\min \lVert AX - B \rVert\). ...

Figure 1: Attention score similarity with our 3D RoPE. These plots show how positional similarity changes across an axis. Introduction RoPE has become the de facto positional embedding for transformer models. Its popularity mainly stems from its performance, but the “derivation” in the paper is also quite elegant (but flawed). Implementing high dimensional RoPE also pushes us to think about generalizing the underlying ideas as much as possible (alongside using signal processing intuition) - there’s code at the end of the post that implements things based on the ideas we develop here. ...

Motivation Let’s start by doing some arithmetic about large language models (LLMs). These are neural networks with huge parameter counts, with state-of-the-art open-weights models (i.e., ones you can download) having parameter counts of the order of 100B (\(10^{11}\)) or so (and usable ones around one order of magnitude smaller). Take the latest SOTA release Qwen 3 235B-A22B, for instance, which has roughly 235B parameters. If all these parameters were to be stored in a naive array of 32-bit (4 byte) floating point numbers, this model would require around 940 GB of storage as well as memory for a usable speed. Running this model purely on CPU with dual channel DDR4 RAM (which is likely the kind of RAM you have on your computer) would take you multiple seconds to output a single token/word (and even this is quite fast for the total size of the model because the architecture is what is called a Mixture of Experts, more on that later, so don’t worry yet). ...