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Abstract We present a new feasible proximal gradient method for constrained optimization where both the objective and constraint functions are given by summation of a smooth, possibly nonconvex function and a convex simple function. The algorithm converts the original problem into a sequence of convex subproblems. Formulating those subproblems requires the evaluation of at most one gradient-value of the original objective and constraint functions. Either exact or approximate subproblems solutions can be computed efficiently in many cases. An important feature of the algorithm is the constraint level parameter. By carefully increasing this level for each subproblem, we provide a simple solution to overcome the challenge of bounding the Lagrangian multipliers and show that the algorithm follows a strictly feasible solution path till convergence to the stationary point. We develop a simple, proximal gradient descent type analysis, showing that the complexity bound of this new algorithm is comparable to gradient descent for the unconstrained setting which is new in the literature. Exploiting this new design and analysis technique, we extend our algorithms to some more challenging constrained optimization problems where (1) the objective is a stochastic or finite-sum function, and (2) structured nonsmooth functions replace smooth components of both objective and constraint functions. Complexity results for these problems also seem to be new in the literature. Finally, our method can also be applied to convex function constrained problems where we show complexities similar to the proximal gradient method.
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Optimizing NOTEARS Objectives via Topological Swaps
Recently, an intriguing class of non-convex optimization problems has emerged in the context of learning directed acyclic graphs (DAGs). These problems involve minimizing a given loss or score function, subject to a non-convex continuous constraint that penalizes the presence of cycles in a graph. In this work, we delve into the optimality challenges associated with this class of non-convex programs. To address these challenges, we propose a bi-level algorithm that leverages the non-convex constraint in a novel way. The outer level of the algorithm optimizes over topological orders by iteratively swapping pairs of nodes within the topological order of a DAG. A key innovation of our approach is the development of an effective method for generating a set of candidate swapping pairs for each iteration. At the inner level, given a topological order, we utilize off-the-shelf solvers that can handle linear constraints. The key advantage of our proposed algorithm is that it is guaranteed to find a local minimum or a KKT point under weaker conditions compared to previous work and finds solutions with lower scores. Extensive experiments demonstrate that our method outperforms state-of-the-art approaches in terms of achieving a better score. Additionally, our method can also be used as a post-processing algorithm to significantly improve the score of other algorithms. Code implementing the proposed method is available at https://github.com/duntrain/topo.
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- Award ID(s):
- 1956330
- PAR ID:
- 10542251
- Publisher / Repository:
- Proceedings of the 40th International Conference on Machine Learning
- Date Published:
- Volume:
- 202
- Page Range / eLocation ID:
- 7563-7595
- Subject(s) / Keyword(s):
- nonconvex optimization nonlinear constraints KKT conditions directed acyclic graphs Bayesian networks
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
