The essential best and average rate of convergence of the exact line search gradient descent method Article Swipe
It is very well known that when the exact line search gradient descent method is applied to a convex quadratic objective, the worst-case rate of convergence (ROC), among all seed vectors, deteriorates as the condition number of the Hessian of the objective grows. By an elegant analysis due to H. Akaike, it is generally believed – but not proved – that in the ill-conditioned regime the ROC for almost all initial vectors, and hence also the average ROC, is close to the worst case ROC. We complete Akaike’s analysis by determining the essential best case ROC (defined in a measure-theoretic way) by using a dynamical system approach, facilitated by the theorem of center and stable manifolds. Our analysis also makes apparent the effect of an intermediate eigenvalue in the Hessian by establishing the following amusing result: In the absence of an intermediate eigenvalue, the average ROC gets arbitrarily fast – not slow – as the Hessian gets increasingly ill-conditioned. We discuss in passing some contemporary applications of exact line search GD to well-conditioned polynomial optimization problems arising from imaging and data sciences. In particular, we observe that a tailored exact line search GD algorithm for a POP arising from the phase retrieval problem is only 50% more expensive per iteration than its constant step size counterpart, while promising a ROC only matched by the optimally tuned (constant) step size which can rarely be achieved in practice.
Related Topics
Concepts
Hessian matrix
Mathematics
Gradient descent
Line search
Rate of convergence
Quadratic equation
Applied mathematics
Convergence (economics)
Eigenvalues and eigenvectors
Line (geometry)
Polynomial
Constant (computer programming)
Descent direction
Descent (aeronautics)
Theory of computation
Convex optimization
Mathematical optimization
Exact solutions in general relativity
Stability (learning theory)
Regular polygon
Algorithm
Gradient method
Quasi-Newton method
Numerical analysis
Quadratic function
Combinatorics
Time complexity
Broyden–Fletcher–Goldfarb–Shanno algorithm
Stochastic gradient descent
Matching (statistics)
Convex function
Frank–Wolfe algorithm
Phase retrieval
Optimization problem
Search algorithm
Phase (matter)
Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1007/s11075-025-02244-0
- https://link.springer.com/content/pdf/10.1007/s11075-025-02244-0.pdf
- OA Status
- hybrid
- References
- 22
- OpenAlex ID
- https://openalex.org/W7105897086
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
-
https://openalex.org/W7105897086Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.1007/s11075-025-02244-0Digital Object Identifier
- Title
-
The essential best and average rate of convergence of the exact line search gradient descent methodWork title
- Type
-
articleOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2025Year of publication
- Publication date
-
2025-11-18Full publication date if available
- Authors
-
Thomas YuList of authors in order
- Landing page
-
https://doi.org/10.1007/s11075-025-02244-0Publisher landing page
- PDF URL
-
https://link.springer.com/content/pdf/10.1007/s11075-025-02244-0.pdfDirect link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
hybridOpen access status per OpenAlex
- OA URL
-
https://link.springer.com/content/pdf/10.1007/s11075-025-02244-0.pdfDirect OA link when available
- Concepts
-
Hessian matrix, Mathematics, Gradient descent, Line search, Rate of convergence, Quadratic equation, Applied mathematics, Convergence (economics), Eigenvalues and eigenvectors, Line (geometry), Polynomial, Constant (computer programming), Descent direction, Descent (aeronautics), Theory of computation, Convex optimization, Mathematical optimization, Exact solutions in general relativity, Stability (learning theory), Regular polygon, Algorithm, Gradient method, Quasi-Newton method, Numerical analysis, Quadratic function, Combinatorics, Time complexity, Broyden–Fletcher–Goldfarb–Shanno algorithm, Stochastic gradient descent, Matching (statistics), Convex function, Frank–Wolfe algorithm, Phase retrieval, Optimization problem, Search algorithm, Phase (matter)Top concepts (fields/topics) attached by OpenAlex
- Cited by
-
0Total citation count in OpenAlex
- References (count)
-
22Number of works referenced by this work
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| primary_location.landing_page_url | https://doi.org/10.1007/s11075-025-02244-0 |
| publication_date | 2025-11-18 |
| publication_year | 2025 |
| referenced_works | https://openalex.org/W1833752784, https://openalex.org/W2041187281, https://openalex.org/W1996595848, https://openalex.org/W4247591450, https://openalex.org/W2182304189, https://openalex.org/W653291882, https://openalex.org/W2143075842, https://openalex.org/W2138505091, https://openalex.org/W2142949395, https://openalex.org/W2611328865, https://openalex.org/W2078397124, https://openalex.org/W2963841451, https://openalex.org/W2542482481, https://openalex.org/W2473376220, https://openalex.org/W3015834454, https://openalex.org/W2329370823, https://openalex.org/W1967344706, https://openalex.org/W404764422, https://openalex.org/W2768927505, https://openalex.org/W4283276019, https://openalex.org/W2070503827, https://openalex.org/W2283023501 |
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