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<p>This paper introduces a novel double-momentum projected gradient method for solving monotone variational inequalities in real Hilbert spaces. The proposed algorithm is computationally efficient, requiring only one projection onto the feasible set and one evaluation of the Lipschitz-continuous operator per iteration. It incorporates a two-term inertial extrapolation step, based on both velocity and acceleration from past iterates, to accelerate convergence. A key advantage over existing methods is the use of a simple, self-adaptive step-size rule that avoids the computational overhead of evaluating multiple norms. We prove weak convergence of the generated sequence under standard assumptions and establish a non-asymptotic convergence rate of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal{O}(1/j)\)</EquationSource> </InlineEquation> for the ergodic sequence. Extensive numerical experiments on large-scale machine learning problems, including standard benchmarks such as MNIST (60,000 samples) and CIFAR-10 (50,000 samples), as well as on signal processing tasks (non-negative compressed sensing), demonstrate that our method consistently outperforms state-of-the-art algorithms in both iteration count and CPU time.</p>

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Double momentum projected gradient method for large-scale machine learning and signal processing

  • 期刊论文

  • 发表日期：

    2026年4月7日

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  • Jian-Wen Peng,
  • Abubakar Adamu,
  • Yekini Shehu,
  • Jen-Chih Yao

摘要

This paper introduces a novel double-momentum projected gradient method for solving monotone variational inequalities in real Hilbert spaces. The proposed algorithm is computationally efficient, requiring only one projection onto the feasible set and one evaluation of the Lipschitz-continuous operator per iteration. It incorporates a two-term inertial extrapolation step, based on both velocity and acceleration from past iterates, to accelerate convergence. A key advantage over existing methods is the use of a simple, self-adaptive step-size rule that avoids the computational overhead of evaluating multiple norms. We prove weak convergence of the generated sequence under standard assumptions and establish a non-asymptotic convergence rate of \(\mathcal{O}(1/j)\) for the ergodic sequence. Extensive numerical experiments on large-scale machine learning problems, including standard benchmarks such as MNIST (60,000 samples) and CIFAR-10 (50,000 samples), as well as on signal processing tasks (non-negative compressed sensing), demonstrate that our method consistently outperforms state-of-the-art algorithms in both iteration count and CPU time.