Finite-Horizon Input-Output Dynamics of Minibatch Perturbations in AdamW

Adaptive optimizers retain gradient history in moment variables, allowing a local change in loss weighting to alter later updates. We examine whether this delayed transport is large enough to change prospective short-horizon decisions. On committed future-minibatch sequences, we differentiate eight-step AdamW trajectories through the complete model--optimizer state and select exposure-matched Math--Code loss schedules before independent evaluation. Across 12 unused 0.3M Transformer histories, full transport lowers token-disjoint loss relative to an optimizer-aware immediate derivative in 10/12 histories (mean benefit $4.71\times10^{-4}$; exact one-sided sign test, $p=0.0193$). The two controllers act equally often but select different schedules in 60/96 windows. Crossed checkpoint--future-path tests attribute this reordering to the interaction between optimizer state and near-future data, while an independent Ising--CNN experiment shows that deleting moment-state transport destroys accurate response prediction. Full-transport scores also concentrate exact-rollout winners in larger candidate libraries, focusing finite-amplitude evaluation on a shortlist. On these committed short paths, optimizer memory and near-future data order are therefore actionable components of the training state, providing a mechanism-based criterion for when finite-horizon rather than one-step intervention is required.

AdamW is a default optimizer for deep learning, but its moment states add two parameter-sized buffers to training memory, increasing the cost of large-scale pretraining. We propose Gefen, a memory-efficient optimizer that automatically shares second-moment estimates across parameter blocks and quantizes the first moment using a learned codebook. Gefen reduces AdamW's optimizer memory footprint by up to 8x while maintaining performance, saving 6.5 GiB per billion parameters. Prior work shares second moments across parameters grouped along the Hessian's block-diagonal structure, but relies on hand-specified architectural rules and leaves unexplained why such grouping works. We prove that large mixed Hessian entries constrain the ratio of squared gradients toward one, explaining why shared second moments are accurate when the squared gradients they pool are similar. The Hessian need not be computed: its block structure is inherited by squared gradients, allowing blocks to be found directly. Gefen therefore infers block structure from initial squared gradients, requiring no architecture-specific metadata or user-tuned hyperparameters beyond AdamW defaults. Gefen learns an exact histogram-based dynamic-programming quantization codebook and reuses the blocks for first-moment scaling. Across diverse pretraining experiments, Gefen achieves the lowest peak optimizer memory among compared methods that maintain AdamW-level performance. In single-machine and distributed training, the reduced footprint enables larger microbatches and substantially improves throughput over AdamW, making Gefen a drop-in replacement that can train larger models or use larger global batch sizes. We provide the complete Python implementation, including fused CUDA kernels at https://github.com/ndvbd/Gefen

The edge-of-stability (EoS) phenomenon of full-batch Adam has been widely observed, yet its underlying dynamical mechanism remains poorly understood. In this paper, we identify Adam's second-moment adaptation as a negative-feedback mechanism that drives the dynamics toward the stability boundary. We characterize this mechanism through the active curvature, namely, the preconditioned curvature along the preconditioned gradient direction, and establish rigorous characterizations in progressively richer settings: rank-one quadratics with momentum, diagonal quadratics, on which the active curvature separates from the sharpness, and general objectives. Importantly, the mechanism predicts gradient reversal of full-batch Adam near the edge: consecutive gradients repeatedly point in nearly opposite directions, as we observe across fully connected networks, ResNets, ViTs, LSTMs, GPT-2 medium, and Adam-family optimizers. Consistent with this picture, averaging iterates suppresses these fast oscillations and produces smoother and lower loss curves. Together, these results provide an important first step towards fully understanding the dynamical behavior of Adam's EoS through active curvature and gradient reversal.