Disciplined Geodesically Convex Programming
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911110583549952 |
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| author | Cheng, Andrew Dixit, Vaibhav Weber, Melanie |
| author_facet | Cheng, Andrew Dixit, Vaibhav Weber, Melanie |
| contents | Convex programming plays a fundamental role in machine learning, data science, and engineering. Testing convexity structure in nonlinear programs relies on verifying the convexity of objectives and constraints. Grant et al. (2006) introduced a framework, Disciplined Convex Programming (DCP), for automating this verification task for a wide range of convex functions that can be decomposed into basic convex functions (atoms) using convexity-preserving compositions and transformations (rules). Here, we extend this framework to functions defined on manifolds with non-positive curvature (Hadamard manifolds) by introducing Disciplined Geodesically Convex Programming (DGCP). In particular, this allows for verifying a broader range of convexity notions. For instance, many notable instances of statistical estimators and matrix-valued (sub)routines in machine learning applications are Euclidean non-convex, but exhibit geodesic convexity through a more general Riemannian lens. To define the DGCP framework, we determine convexity-preserving compositions and transformations for geodesically convex functions on general Hadamard manifolds, as well as for the special case of symmetric positive definite matrices, a common setting in matrix-valued optimization. For the latter, we also define a basic set of atoms. Our paper is accompanied by a Julia package SymbolicAnalysis.jl, which provides functionality for testing and certifying DGCP-compliant expressions. Our library interfaces with manifold optimization software, which allows for directly solving verified geodesically convex programs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05261 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Disciplined Geodesically Convex Programming Cheng, Andrew Dixit, Vaibhav Weber, Melanie Optimization and Control Machine Learning Mathematical Software Convex programming plays a fundamental role in machine learning, data science, and engineering. Testing convexity structure in nonlinear programs relies on verifying the convexity of objectives and constraints. Grant et al. (2006) introduced a framework, Disciplined Convex Programming (DCP), for automating this verification task for a wide range of convex functions that can be decomposed into basic convex functions (atoms) using convexity-preserving compositions and transformations (rules). Here, we extend this framework to functions defined on manifolds with non-positive curvature (Hadamard manifolds) by introducing Disciplined Geodesically Convex Programming (DGCP). In particular, this allows for verifying a broader range of convexity notions. For instance, many notable instances of statistical estimators and matrix-valued (sub)routines in machine learning applications are Euclidean non-convex, but exhibit geodesic convexity through a more general Riemannian lens. To define the DGCP framework, we determine convexity-preserving compositions and transformations for geodesically convex functions on general Hadamard manifolds, as well as for the special case of symmetric positive definite matrices, a common setting in matrix-valued optimization. For the latter, we also define a basic set of atoms. Our paper is accompanied by a Julia package SymbolicAnalysis.jl, which provides functionality for testing and certifying DGCP-compliant expressions. Our library interfaces with manifold optimization software, which allows for directly solving verified geodesically convex programs. |
| title | Disciplined Geodesically Convex Programming |
| topic | Optimization and Control Machine Learning Mathematical Software |
| url | https://arxiv.org/abs/2407.05261 |