Shuffle order can be a larger source of fine-tuning noise than a memoryless analysis predicts: fixed-clock optimizer memory makes local equal-multiset contrasts first order in the learning rate rather than second order, and the resulting order channel can be large enough for a single seed to flip a close A/B comparison. We isolate this mechanism and derive a fit-free way to size the noise it produces. For a memoryless optimizer, reordering an equal multiset has no first-order endpoint term; the leading local contrast is the $O(η^2)$ gradient bracket. Fixed-clock optimizers such as AdamW are different. Their moment buffers, preconditioner state, and de-biasing counters advance with the step index rather than with the learning-rate-scaled time $τ=ηk$, so the same gradient can receive a position-dependent endpoint weight. For any fixed finite measurement window, a lifted-state expansion gives an $O(η)$ equal-multiset contrast whenever the first-order replay coefficient is nonzero, while regular and clock-matched controls remain $O(η^2)$; a bare fixed-$β$ momentum buffer is already enough. A bitwise-deterministic replay from one warmed optimizer state isolates the mechanism, giving order-variance slopes 1.83 for AdamW, 2.00 for fixed-$β$ momentum, and 4.00 for SGD; matching the memory clock to $τ$ restores the regular exponent. For AdamW with a frozen preconditioner, the same impulse-weight kernel gives a closed-form asymptotic order-variance floor after the local potentials are measured, with no fitted coefficients. The result is local to the measurement window (independent reshuffling can average the channel across windows), but it yields order-noise error bars, positional attribution weights, and a seed-budget criterion for fine-tuning comparisons.

AdamW is the de facto optimizer for training large language models (LLMs), yet the theory behind it still lives mostly in finite-variance regimes. This is increasingly unsatisfying, as empirical evidence indicates that stochastic gradient noise in LLM pretraining is typically heavy-tailed. Recent work shows that sign-based optimizers such as Lion and Muon achieve sharp heavy-tailed rates, and that AdaGrad can also converge under heavy-tailed noise. However, no rigorous convergence theory for AdamW has yet been established in this regime. Can AdamW converge under the same heavy-tailed assumptions, or does its second-moment accumulator create a genuine obstruction? We formulate this as an open problem, prove a positive weighted-metric benchmark, and give a corridor lower-bound mechanism showing how denominator memory can hide large gradients.

While Large language models (LLMs) are driving the rapid advancement of artificial intelligence, effectively and reliably training these large models remains one of the field's most significant challenges. To address this challenge, we propose POET, a novel reParameterized training algorithm that uses Orthogonal Equivalence Transformation to optimize neurons. Specifically, POET reparameterizes each neuron with two learnable orthogonal matrices and a fixed random weight matrix. Because of its provable preservation of spectral properties of weight matrices, POET can stably optimize the objective function with improved generalization. We further develop efficient approximations that make POET flexible and scalable for training large-scale neural networks.

As the default optimizer for training large language models, AdamW has achieved remarkable success in deep learning. However, its convergence behavior is not theoretically well-understood.

Adam is a popular and widely used adaptive gradient method in deep learning, which has also received tremendous focus in theoretical research. However, most existing theoretical work primarily analyzes its full-batch version, which differs fundamentally from the stochastic variant used in practice. Unlike SGD, stochastic Adam does not converge to its full-batch counterpart even with infinitesimal learning rates. We present the first theoretical characterization of how batch size affects Adam's generalization, analyzing two-layer over-parameterized CNNs on image data. Our results reveal that while both Adam and AdamW with proper weight decay λ converge to poor test error solutions, their mini-batch variants can achieve near-zero test error. We further prove Adam has a strictly smaller effective weight decay bound than AdamW, theoretically explaining why Adam requires more sensitive λtuning.

Several recently introduced deep learning optimizers utilizing matrix-level preconditioning have shown promising speedups relative to the current dominant optimizer AdamW, particularly in relatively small-scale experiments. However, efforts to validate and replicate their successes have reported mixed results. To better understand the effectiveness of these optimizers at scale, in this work we investigate how to scale preconditioned optimizers via hyperparameter transfer, building on prior works such as $\mu$P. We study how the optimal learning rate and weight decay should scale with model width and depth for a wide range of optimizers, including Shampoo, SOAP, and Muon, accounting for the impact of commonly used techniques such as blocking and grafting. We find that scaling the learning rate according to $\mu$P improves transfer, but can still suffer from significant finite-width deviations that cause drifting optimal learning rates, which we show can be mitigated by blocking and explicit spectral normalization. For compute-optimal scaling, we find scaling independent weight decay as $1/\mathrm{width}$ is nearly optimal across optimizers. Applying these scaling rules, we show Muon, SOAP and Shampoo consistently achieve near $1.4\times$ speedup over AdamW for training Llama-architecture language models of sizes ranging from $190$M to $1.4$B, whereas the speedup vanishes rapidly with scale under incorrect scaling. Based on these results and further ablations, we argue that studying optimal hyperparameter transfer is essential for reliably comparing optimizers at scale given a realistic tuning budget.

Preconditioned optimizers are central to language model training, but their stochastic update rules are usually treated as direct approximations to population preconditioned descent. We show that this view misses two finite-sample biases. First, the gradient and preconditioner are typically estimated from the same minibatch, introducing gradient--preconditioner coupling bias. Second, even when the preconditioner estimate is unbiased, its inverse or inverse-root is generally biased because inversion is nonlinear. We propose a single-batch bias-correction framework that addresses both effects: cross-fitted preconditioning estimates the numerator and preconditioner from independent microbatch groups, while variance-corrected inversion uses microbatch variability to subtract the leading delta-method bias term. The framework applies to diagonal moment, diagonal curvature, and matrix preconditioning methods, instantiated in AdamW, Sophia, and Shampoo. Bias correction reduces held-out pretraining loss on Qwen2.5-0.5B by $0.15$, $0.07$, and $0.11$ nats, respectively; the effects on mixed-quality pretraining and downstream instruction tuning are consistently neutral-to-positive. Together, these results establish bias correction as a practical mechanism for reducing finite-sample update bias and improving the performance of preconditioned optimizers.

Suppose we have a non-zero solution θ which is a stationary point of f(θ,t) at t-th step and SGD finds θt = θ at t-th step. Theorem 2.2 of Shapiro and Wardi [9] told us that the learning rate should be small enough for convergence. Obviously, we have η < in practice. As ηt = ηt+1 does not hold, SGD cannot converging to any non-zero stationary point. The proof is now complete.

Spectral optimizers such as Muon have recently shown strong empirical performance in large-scale language model training, but the source and extent of their advantage remain poorly understood. We study this question through the linear associative memory problem, a tractable model for factual recall in transformer-based models. In particular, we go beyond orthogonal embeddings and consider Gaussian inputs and outputs, which allows the number of stored associations to greatly exceed the embedding dimension. Our main result sharply characterizes the recovery rates of one step of Muon and SGD on the logistic regression loss under a power law frequency distribution. We show that the storage capacity of Muon significantly exceeds that of SGD, and moreover Muon saturates at a larger critical batch size. We further analyze the multi-step dynamics under a thresholded gradient approximation and show that Muon achieves a substantially faster initial recovery rate than SGD, while both methods eventually converge to the information-theoretic limit at comparable speeds. Experiments on synthetic tasks validate the predicted scaling laws. Our analysis provides a quantitative understanding of the signal amplification of Muon and lays the groundwork for establishing scaling laws across more practical language modeling tasks and optimizers.