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Abstract A parameter identification inverse problem in the form of nonlinear least squares is considered.In the lack of stability, the frozen iteratively regularized Gauss–Newton (FIRGN) algorithm is proposed and its convergence is justified under what we call a generalized normal solvability condition.The penalty term is constructed based on a semi-norm generated by a linear operator yielding a greater flexibility in the use of qualitative and quantitative a priori information available for each particular model.Unlike previously known theoretical results on the FIRGN method, our convergence analysis does not rely on any nonlinearity conditions and it is applicable to a large class of nonlinear operators.In our study, we leverage the nature of ill-posedness in order to establish convergence in the noise-free case.For noise contaminated data, we show that, at least theoretically, the process does not require a stopping rule and is no longer semi-convergent.Numerical simulations for a parameter estimation problem in epidemiology illustrate the efficiency of the algorithm.
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On the Convergence of Continuous Constrained Optimization for Structure Learning
Recently, structure learning of directed acyclic graphs (DAGs) has been formulated as a continuous optimization problem by leveraging an algebraic characterization of acyclicity. The constrained problem is solved using the augmented Lagrangian method (ALM) which is often preferred to the quadratic penalty method (QPM) by virtue of its standard convergence result that does not require the penalty coefficient to go to infinity, hence avoiding ill-conditioning. However, the convergence properties of these methods for structure learning, including whether they are guaranteed to return a DAG solution, remain unclear, which might limit their practical applications. In this work, we examine the convergence of ALM and QPM for structure learning in the linear, nonlinear, and confounded cases. We show that the standard convergence result of ALM does not hold in these settings, and demonstrate empirically that its behavior is akin to that of the QPM which is prone to ill-conditioning. We further establish the convergence guarantee of QPM to a DAG solution, under mild conditions. Lastly, we connect our theoretical results with existing approaches to help resolve the convergence issue, and verify our findings in light of an empirical comparison of them.
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- Award ID(s):
- 2134901
- PAR ID:
- 10380982
- Editor(s):
- Camps-Valls, Gustau; Ruiz, Francisco J.; Valera, Isabel
- Date Published:
- Journal Name:
- Proceedings of Machine Learning Research
- Volume:
- 2022
- ISSN:
- 2640-3498
- Page Range / eLocation ID:
- 8176 - 8198
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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