Manifold constrained steepest descent
This work addresses optimization over manifolds for large-scale learning applications, offering a more efficient alternative to existing methods, though it is incremental as it builds on prior LMO-based approaches.
The paper tackles the challenge of extending norm-constrained linear minimization oracle (LMO)-based optimizers to manifold-constrained problems, which often require inefficient nested-loop schemes, by proposing Manifold Constrained Steepest Descent (MCSD), a single-loop framework that achieves improved stability and competitive performance in experiments on tasks like PCA and LLM adapter tuning.
Norm-constrained linear minimization oracle (LMO)-based optimizers such as spectral gradient descent and Muon are attractive in large-scale learning, but extending them to manifold-constrained problems is nontrivial and often leads to nested-loop schemes that solve tangent-space subproblems iteratively. We propose \emph{Manifold Constrained Steepest Descent} (MCSD), a single-loop framework for optimization over manifolds that selects a norm-induced steepest-descent direction via an LMO applied to the Riemannian gradient, and then returns to the manifold via projection. Under standard smoothness assumptions, we establish convergence guarantees for MCSD and a stochastic momentum variant. We further introduce \emph{SPEL}, the spectral-norm specialization of MCSD on the Stiefel manifold, which admits scalable implementations via fast matrix sign computations. Experiments on PCA, orthogonality-constrained CNNs, and manifold-constrained LLM adapter tuning demonstrate improved stability and competitive performance relative to standard Riemannian baselines and existing manifold-aware LMO methods.