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.
Augustina C. Amakor, Konstantin Sonntag, Sebastian Peitzcs.LG math.OC
In multi-task learning, handling an increasing number of objectives can quickly become challenging, both in terms of the computational resources and the decision maker's capacity to choose appropriate trade-offs. A widely used approach is thus to aggregate the individual losses in a single loss function by a weighted sum. This often fails to capture either the decision maker's preferences as a result of the shape of the Pareto front, or requires multiple adjustments and computations which becomes prohibitively expensive in deep learning applications. To address these issues, we introduce a novel framework, Preference Pareto Exploration (PPE), which enforces the decision maker's preferences while accounting for the geometry of the Pareto set in an interactive exploration process. PPE is based on a predictor-corrector method that performs predictor steps tangential to the manifold of Pareto-optimal solutions, following the decision maker's preference. The subsequent corrector step results in a new trade-off reflecting this preference. To avoid explicit Hessian computations when characterizing the tangent space of the manifold, we employ a Krylov subspace method that relies solely on matrix-vector products. These products can be efficiently obtained via automatic differentiation, ensuring both efficiency and robustness throughout the optimization process. The method's functionality and performance are demonstrated using both toy problems and examples from deep learning.