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<p>We propose Spectral Regularization Dynamics (SRD), a continuous-time system for unconstrained non-convex optimization. Unlike discrete iterations that solve regularized subproblems, SRD employs an autonomous feedback control mechanism coupled to the Hessian’s minimum eigenvalue. This mechanism guarantees a descent direction by driving the regularization parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha (t)\)</EquationSource> </InlineEquation> strictly positive in regions of negative curvature. We prove global convergence to the set of critical points via LaSalle’s invariance principle. Furthermore, using the stable manifold theorem, we establish that the system dynamics almost surely avoid strict saddle points. In local neighborhoods of strict minima, the regularization vanishes asymptotically, recovering the convergence rate of the continuous Newton flow. Numerical benchmarks illustrate the saddle-avoidance mechanism and local convergence properties.</p>

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Spectral regularization dynamics: a continuous-time framework for non-convex optimization

  • 期刊论文

  • 发表日期：

    2026年5月28日

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  • Hasan Dalman,
  • Sara Badur Dalman

摘要

We propose Spectral Regularization Dynamics (SRD), a continuous-time system for unconstrained non-convex optimization. Unlike discrete iterations that solve regularized subproblems, SRD employs an autonomous feedback control mechanism coupled to the Hessian’s minimum eigenvalue. This mechanism guarantees a descent direction by driving the regularization parameter \(\alpha (t)\) strictly positive in regions of negative curvature. We prove global convergence to the set of critical points via LaSalle’s invariance principle. Furthermore, using the stable manifold theorem, we establish that the system dynamics almost surely avoid strict saddle points. In local neighborhoods of strict minima, the regularization vanishes asymptotically, recovering the convergence rate of the continuous Newton flow. Numerical benchmarks illustrate the saddle-avoidance mechanism and local convergence properties.