Recently, there has been growing evidence that if the width and depth of a neural network are scaled toward the so-called rich feature learning limit ( µ P and its depth extension), then some hyperparameters -- such as the learning rate -- exhibit transfer from small to very large models. From an optimization perspective, this phenomenon is puzzling, as it implies that the loss landscape is consistently similar across very different model sizes. In this work, we study the landscape through the lens of the loss Hessian, with a focus on its largest eigenvalue (i.e. the sharpness), and find that certain spectral properties under µ P are largely independent of the size of the network, and remain consistent as training progresses. We name this property Super Consistency of the landscape. On the other hand, we show that in the Neural Tangent Kernel (NTK) and other scaling regimes, the sharpness exhibits very different dynamics at different scales.

Hyperparameter tuning can dramatically impact training stability and final performance of large-scale models. Recent works on neural network parameterisations, such as $μ$P, have enabled transfer of optimal global hyperparameters across model sizes. These works propose an empirical practice of search for optimal global base hyperparameters at a small model size, and transfer to a large size. We extend these works in two key ways. To handle scaling along most important scaling axes, we propose the Complete$^{(d)}$ Parameterisation that unifies scaling in width and depth -- using an adaptation of CompleteP -- as well as in batch-size and training duration. Secondly, with our parameterisation, we investigate per-module hyperparameter optimisation and transfer. We characterise the empirical challenges of navigating the high-dimensional hyperparameter landscape, and propose practical guidelines for tackling this optimisation problem. We demonstrate that, with the right parameterisation, hyperparameter transfer holds even in the per-module hyperparameter regime. Our study covers an extensive range of optimisation hyperparameters of modern models: learning rates, AdamW parameters, weight decay, initialisation scales, and residual block multipliers. Our experiments demonstrate significant training speed improvements in Large Language Models with the transferred per-module hyperparameters.

Bayesian optimization (BO) is a successful methodology to optimize black-box functions that are expensive to evaluate. While traditional methods optimize each black-box function in isolation, there has been recent interest in speeding up BO by transferring knowledge across multiple related black-box functions. In this work, we introduce a method to automatically design the BO search space by relying on evaluations of previous black-box functions. We depart from the common practice of defining a set of arbitrary search ranges a priori by considering search space geometries that are learnt from historical data. This simple, yet effective strategy can be used to endow many existing BO methods with transfer learning properties. Despite its simplicity, we show that our approach considerably boosts BO by reducing the size of the search space, thus accelerating the optimization of a variety of black-box optimization problems. In particular, the proposed approach combined with random search results in a parameter-free, easy-to-implement, robust hyperparameter optimization strategy. We hope it will constitute a natural baseline for further research attempting to warm-start BO.

This article reviews modern optimization methods for training neural networks with an emphasis on efficiency and scale. We present state-of-the-art optimization algorithms under a unified algorithmic template that highlights the importance of adapting to the structures in the problem. We then cover how to make these algorithms agnostic to the scale of the problem. Our exposition is intended as an introduction for both practitioners and researchers who wish to be involved in these exciting new developments.

Our findings reveal that the dynamics of standard SAM effectively reduce to applying SAM solely in the last layer in wide neural networks, even with optimal hyperparameters.

Recently, there has been growing evidence that if the width and depth of a neural network are scaled toward the so-called rich feature learning limit ( µ P and its depth extension), then some hyperparameters -- such as the learning rate -- exhibit transfer from small to very large models. From an optimization perspective, this phenomenon is puzzling, as it implies that the loss landscape is consistently similar across very different model sizes. In this work, we study the landscape through the lens of the loss Hessian, with a focus on its largest eigenvalue (i.e. the sharpness), and find that certain spectral properties under µ P are largely independent of the size of the network, and remain consistent as training progresses. We name this property Super Consistency of the landscape. On the other hand, we show that in the Neural Tangent Kernel (NTK) and other scaling regimes, the sharpness exhibits very different dynamics at different scales.

Recent years have seen a growing interest and adoption of LLMs, with Mixture-of-Experts (MoE) emerging as a leading architecture in extremely large models. Currently, the largest open-source models reach over $1$T parameters. At such scales, hyperparameter tuning becomes prohibitively expensive. Precisely for this reason, the $μ$Transfer is becoming a key technique. It allows for seamless transfer of optimal hyperparameters across model scales, resulting in a huge reduction in tuning costs. However, existing work has primarily focused on dense LLMs, leaving MoE architectures unexplored. In this work, we derive a $μ$-Parameterization for MoE, providing theoretical guarantees for feature learning across model widths. Our experiments demonstrate that the optimal learning rate reliably transfers across model sizes, establishing a foundation for efficient hyperparameter tuning in large-scale MoE models.

We present a comprehensive theoretical and empirical study of the Muon optimizer for training transformers only with a small to medium decoder (30M - 200M parameters), with an emphasis on its mathematical foundations, convergence properties and synergistic interactions with modern architectural optimizations. Building on recent work showing Muon's scalability, we provide rigorous theoretical analysis including: (i)showing the convergence rate under standard assumptions, (ii) spectral regularization properties that prevent gradient explosion, (iii) connection to natural gradient descent on the Stiefel manifold, and (iv) equivalence to steepest gradient descent under the spectral norm. Crucially, we demonstrate that Muon expands the Pareto frontier in the compute-time trade-off by maintaining superior data efficiency at large batch sizes, a key finding of~\cite{essentialai2025muon} that we validate across our model scales. Empirically, Muon reaches the target loss with 48-52\% of the training calculated by AdamW while maintaining or improving the final perplexity, consistent with larger-scale results. When combined with Multi-Head Latent Attention (MLA) and Mixture-of-Experts (MoE), we observe multiplicative efficiency gains: MLA+MoE+Muon achieves 68\% memory reduction and 3.2$\times$ inference speedup, while improving perplexity by 8-12\%. We provide detailed procedures on 15 architectural and optimizer components, stability analyzes across 100+ training runs, and practical implementation guidelines including Newton-Schulz coefficients $(3.4445, -4.7750, 2.0315)$ optimized by~\cite{su2024muonblog}. Our theoretical analysis and comprehensive experiments establish Muon as a principled, robust alternative to AdamW that particularly excels when combined with modern efficiency techniques and large-batch training regimes.

Optimizers play a decisive role in reducing pre-training times for LLMs and achieving better-performing models. In this study, we compare three major variants: the de-facto standard AdamW, the simpler Lion, developed through an evolutionary search, and the second-order optimizer Sophia. For better generalization, we train with two different base architectures and use a single- and a multiple-epoch approach while keeping the number of tokens constant. Using the Maximal Update Parametrization and smaller proxy models, we tune relevant hyperparameters separately for each combination of base architecture and optimizer. We found that while the results from all three optimizers were in approximately the same range, Sophia exhibited the lowest training and validation loss, Lion was fastest in terms of training GPU hours but AdamW led to the best downstream evaluation results.