The Loss Does Not See the Basis, But Adam Does [R]

The recent paper "The Loss Does Not See the Basis, But Adam Does [R]" presents a fascinating and surprisingly impactful observation about the behavior of popular optimization algorithms, particularly Adam. It highlights a subtle but critical distinction: while Gradient Descent (GD) inherently respects the rotational invariance of loss functions in factored models (W = UV^T), Adam, due to its per-coordinate second-moment estimation, does not. This seemingly technical detail has significant implications for training low-rank models, and the paper’s systematic evaluation of several optimizers provides compelling evidence supporting this claim. It’s a reminder that even widely adopted techniques like Adam aren't universally optimal and that a deeper understanding of their underlying mechanics is crucial for achieving the best results, especially when dealing with underdetermined systems. This aligns with our ongoing exploration of efficient and insightful methods in machine learning, similar to the strategies detailed in [How to Place Vertiport Locations in Any City Using Geospatial Machine Learning], where understanding underlying data structures and constraints is key to successful outcomes. Furthermore, grasping the nuances of gradient descent, as explored in [Backpropagation Explained for Beginners (Part 3): How Backpropagation Really Works], is foundational to appreciating this new perspective on optimizer behavior.

The core finding—that Adam's per-coordinate nature disrupts the implicit low-rank bias of GD—is striking. The paper’s methodical approach, comparing optimizers at matched training loss to isolate the effect, strengthens the argument considerably. The observation regarding Muon, a relatively newer optimizer, is particularly intriguing. Its performance demonstrates a sharp transition between spectral simplicity bias and degradation as the spectral tail energy increases, highlighting the complexities of designing optimizers for specific model characteristics. The authors’ own experience with their earlier optimizer, and the subsequent improvement achieved through a global norm clip, further underscores the importance of carefully considering the structural properties that an optimizer is intended to preserve. This resonates with the emphasis on architectural efficiency discussed in [Presentation: Producing the World's Cheapest Tokens: A How-to Guide], where even seemingly small adjustments can lead to substantial performance gains.

The caveat regarding the train-only learning rate rule is well-acknowledged, and the authors appropriately temper their claims by noting the narrowing performance gap when each method is allowed to optimize its own learning rate. This demonstrates a commitment to rigorous scientific inquiry, prioritizing the underlying mechanism over absolute quantitative results. The theoretical limitations regarding momentum also provide a transparent view of the paper's scope. While the empirical evidence is strong, the lack of a complete theoretical underpinning for momentum’s effects leaves room for further investigation. The fact that the paper openly admits these limitations enhances its credibility and invites further research to build upon these findings. This is an important lesson for the field: recognizing the boundaries of current understanding is just as valuable as pushing those boundaries.

Ultimately, this research serves as a valuable reminder that optimization is not a one-size-fits-all problem. While Adam remains a popular and often effective choice, its inherent biases can be detrimental in specific scenarios, particularly when training low-rank models. The paper’s clear articulation of this issue, along with the comparative analysis of multiple optimizers, provides a useful framework for practitioners to consider when selecting an appropriate algorithm. A crucial question moving forward is whether these findings can be generalized to other model architectures and training paradigms. Could similar biases be present in other optimizers, or perhaps in the way we structure our loss functions? The answer to this question could significantly reshape our understanding of how machines learn and pave the way for more efficient and robust training techniques.