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Abstract We consider the problem of estimating the factors of a rank-$$1$$ matrix with i.i.d. Gaussian, rank-$$1$$ measurements that are nonlinearly transformed and corrupted by noise. Considering two prototypical choices for the nonlinearity, we study the convergence properties of a natural alternating update rule for this non-convex optimization problem starting from a random initialization. We show sharp convergence guarantees for a sample-split version of the algorithm by deriving a deterministic one-step recursion that is accurate even in high-dimensional problems. Notably, while the infinite-sample population update is uninformative and suggests exact recovery in a single step, the algorithm—and our deterministic one-step prediction—converges geometrically fast from a random initialization. Our sharp, non-asymptotic analysis also exposes several other fine-grained properties of this problem, including how the nonlinearity and noise level affect convergence behaviour. On a technical level, our results are enabled by showing that the empirical error recursion can be predicted by our deterministic one-step updates within fluctuations of the order $$n^{-1/2}$$ when each iteration is run with $$n$$ observations. Our technique leverages leave-one-out tools originating in the literature on high-dimensional $$M$$-estimation and provides an avenue for sharply analyzing complex iterative algorithms from a random initialization in other high-dimensional optimization problems with random data.
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This content will become publicly available on July 14, 2026
Smart Starts: Accelerating Convergence Through Uncommon Region Exploration
Initialization profoundly affects evolutionary algorithm (EA) efficacy by dictating search trajectories and convergence. This study introduces a hybrid initialization strategy combining empty-space search algorithm (ESA) and opposition-based learning (OBL). OBL initially generates a diverse population, subsequently augmented by ESA, which identifies under-explored regions. This synergy enhances population diversity, accelerates convergence, and improves EA performance on complex, high-dimensional optimization problems. Benchmark results demonstrate the proposed method's superiority in solution quality and convergence speed compared to conventional initialization techniques.
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- Award ID(s):
- 2106434
- PAR ID:
- 10633346
- Publisher / Repository:
- ACM
- Date Published:
- ISBN:
- 9798400714641
- Page Range / eLocation ID:
- 547 to 550
- Subject(s) / Keyword(s):
- Evolutionary algorithms initialization opposition-based learning empty-space search
- Format(s):
- Medium: X
- Location:
- NH Malaga Hotel Malaga Spain
- Sponsoring Org:
- National Science Foundation
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