Search papers, labs, and topics across Lattice.
This paper introduces a modular language for auditing reachable gradient methods, termed geometric--nongeometric optimizer calculus, which integrates various mechanisms such as metrics, memory, and stochasticity to optimize descent directions. The authors establish a direction-expressivity theorem demonstrating that full positive-definite geometry accurately captures strict descent directions away from critical points. Additionally, they propose a Pareto optimization framework that emphasizes module budgets over a singular optimizer hierarchy, providing a new perspective on optimizer design and evaluation.
Full positive-definite geometry can precisely express descent directions, reshaping our understanding of optimizer effectiveness in gradient-based methods.
Adaptive optimizers mix several mechanisms: a metric or preconditioner maps gradients to descent directions, while estimation, memory, step-size control, constraints, stochasticity, target modification, and discretization determine which directions are available and how they are used. We introduce geometric--nongeometric optimizer calculus, a modular language for auditing reachable gradient methods under explicit oracle, budget, state, and rule constraints. The geometric module is a positive cometric family that maps covectors to parameter-space directions; the nongeometric modules are information, memory, control, operator, noise, target, and discretization mechanisms. The main formal result is a direction-expressivity theorem: away from critical points, full positive-definite geometry expresses exactly the strict descent directions. We then define restricted direction residuals for admissible metric families, prove exact expressivity conditions for diagonal and block geometries, and separate this direction-level diagnostic from condition-number geometric complexity. The resulting design problem is a Pareto optimization over module budgets, not a single universal optimizer ordering. We also lift pointwise residuals to a trajectory-level residual complexity that couples direction mismatch with the variation of the explaining geometry. We include diagnostic prototypes only as evidence for the language: a high-information full-metric probe solves deterministic quadratic benchmarks to numerical precision, while a practical Muon-style PyTorch candidate gives small-scale evidence that matrix-operator updates can be audited by the calculus. The paper is a theory and benchmark-language manuscript; it does not claim large-scale optimizer state-of-the-art performance.
