Jiachun Li
Publications
An Isotropy-Preserving Spectral Cap for Muon: Theory and Three Case Studies
Muon and related matrix-sign optimizers are increasingly used to pre-train large language models, but their effect on the internal geometry of individual weight matrices is not well understood. This preliminary report proposes a unified framework built on a single idealizing assumption -- exact scale invariance of the loss under weight rescaling, which holds approximately in normalization-heavy networks. Under this assumption, plain SGD carries a built-in 1/||W|| brake on its update size, whereas Muon's matrix-sign step removes that brake, so both the Frobenius and spectral norms drift outward faster (t^{1/2} versus t^{1/4}). We further observe that the spectral-norm perturbation has a non-negative second-order term. This implies that a lightweight "spectral cap" -- which projects out only the first-order growth of the single top singular direction from each update -- can control the output covariance W K_X W^T without freezing training: the weight keeps learning through non-top directions, top-direction rotation, and top switching. We relate this cap to the min-entropy (H-infinity) of the singular-value spectrum. We then study three systems trained with Muon: a nanoGPT feed-forward projection, a 64-expert mixture-of-experts router, and the query/key projections of a bf16 FlashAttention block. In each case the cap increases isotropy and, at the margins -- a router collapsing to a single expert, and the near-divergence of one attention head -- prevents a concrete failure, while leaving validation loss essentially unchanged. We emphasize that the scale-invariance assumption is strong and that these small-scale results are preliminary; comments are welcome.
Transformers as Bayesian In-Context Experimenters: Smoothness-Adaptive Efficient ATE Estimation
Adaptive experiments for average treatment effects (ATE) require randomized allocations balancing valid inference with statistical efficiency. The oracle design is a covariate-dependent Neyman rule governed by unknown arm-conditional outcome variances. We investigate whether this sequential variance-estimation and allocation process can be amortized via in-context learning. We introduce Bayesian in-context experimenters: transformer policies trained to imitate a Bayesian posterior Neyman teacher. The teacher updates nonparametric beliefs over potential outcomes using experimental history to assign posterior Neyman treatment probabilities. This design converges to the oracle rule, supporting efficient ATE inference. Transformers constructively implement this mapping through attention-based sufficient statistics and projected gradient descent, imitating Bayesian updating for Gaussian-series priors. To address unknown outcome smoothness, we combine smoothness-indexed experimenters using a mixture-of-experts transformer. The gate acts as a hierarchical posterior over smoothness classes, concentrating on near-oracle experts. By bounding the complexity of the transformer class, we prove this amortized policy can be learned via empirical risk minimization using supervised pretraining. Experiments confirm accurate teacher imitation, adaptive allocation, and improved ATE precision over baselines.
LLM Evaluation as Tensor Completion: Low Rank Structure and Semiparametric Efficiency
Large language model (LLM) evaluation platforms increasingly rely on pairwise human judgments. These data are noisy, sparse, and non-uniform, yet leaderboards are reported with limited uncertainty quantification. We study this as semiparametric inference for a low-rank latent score tensor observed through pairwise comparisons under Bradley-Terry-Luce-type models. This places LLM evaluation in a new tensor completion setting with structured observations, non-uniform sampling, and pairwise contrasts. Our target is a smooth functional $ψ(T^\star)$, including linear estimands such as ability gaps and nonlinear ones such as win probabilities. We derive the information operator on the low-rank tangent space, the efficient influence function, and the semiparametric efficiency bound, then construct a one-step debiased estimator with asymptotic normality. A central challenge is that the information operator is anisotropic and does not commute with the tangent-space projection, creating a bottleneck absent from isotropic models. We introduce a score-whitening method that equalizes local Fisher information and restores stable inference at the optimal sample-complexity scale. Our results provide a principled framework for uncertainty quantification in LLM evaluation and more broadly for inference on low-rank structures from pairwise data.