Orthogonal Directions Constrained Gradient Method: from non-linear equality constraints to Stiefel manifold

• URL: http://arxiv.org/abs/2303.09261v1
• Date: Thu, 16 Mar 2023 12:25:53 GMT
• Title: Orthogonal Directions Constrained Gradient Method: from non-linear equality constraints to Stiefel manifold
• Authors: Sholom Schechtman, Daniil Tiapkin, Michael Muehlebach, Eric Moulines
• Abstract summary: We propose a novel algorithm, the Orthogonal Directions Constrained Method (ODCGM) ODCGM only requires computing a projection onto a vector space. We show that ODCGM exhibits the near-optimal oracle complexities.
• Score: 16.099883128428054
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the problem of minimizing a non-convex function over a smooth manifold $\mathcal{M}$. We propose a novel algorithm, the Orthogonal Directions Constrained Gradient Method (ODCGM) which only requires computing a projection onto a vector space. ODCGM is infeasible but the iterates are constantly pulled towards the manifold, ensuring the convergence of ODCGM towards $\mathcal{M}$. ODCGM is much simpler to implement than the classical methods which require the computation of a retraction. Moreover, we show that ODCGM exhibits the near-optimal oracle complexities $\mathcal{O}(1/\varepsilon^2)$ and $\mathcal{O}(1/\varepsilon^4)$ in the deterministic and stochastic cases, respectively. Furthermore, we establish that, under an appropriate choice of the projection metric, our method recovers the landing algorithm of Ablin and Peyr\'e (2022), a recently introduced algorithm for optimization over the Stiefel manifold. As a result, we significantly extend the analysis of Ablin and Peyr\'e (2022), establishing near-optimal rates both in deterministic and stochastic frameworks. Finally, we perform numerical experiments which shows the efficiency of ODCGM in a high-dimensional setting.

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