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Faster Stochastic Quasi-Newton Methods.

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    This study introduces SpiderSQN, a novel stochastic quasi-Newton method that achieves optimal stochastic first-order oracle complexity for machine learning optimization. This new approach improves upon existing methods for nonconvex optimization problems.

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    Area of Science:

    • Machine Learning
    • Optimization Theory

    Background:

    • Stochastic optimization methods, particularly stochastic gradient descent (SGD), are widely used in machine learning due to their low computational cost per iteration.
    • While second-order methods like Newton or quasi-Newton (QN) offer better solutions, their stochastic variants (SQN) have not yet matched the optimal stochastic first-order oracle (SFO) complexity.
    • Existing SQN methods require further improvement to reach the best known SFO complexity for efficient optimization.

    Purpose of the Study:

    • To propose a novel, faster stochastic quasi-Newton (SQN) method that achieves the best known stochastic first-order oracle (SFO) complexity.
    • To enhance the practical performance of the proposed method by incorporating momentum schemes.
    • To generalize the algorithms to the online setting and establish their SFO complexity.

    Main Methods:

    • Development of a novel faster stochastic QN method (SpiderSQN) utilizing the variance reduction technique from SIPDER.
    • Theoretical analysis to prove the SFO complexity of SpiderSQN in the finite-sum setting.
    • Incorporation of momentum schemes and generalization to the online setting for improved performance and broader applicability.

    Main Results:

    • SpiderSQN achieves the best known SFO complexity of O(n+n^1/2ϵ^-2) for obtaining an ϵ-first-order stationary point in the finite-sum setting.
    • The generalized algorithms for the online setting achieve an SFO complexity of O(ϵ^-3), matching the existing best result.
    • Extensive experiments show that the proposed algorithms outperform state-of-the-art methods in nonconvex optimization tasks.

    Conclusions:

    • The proposed SpiderSQN method effectively bridges the gap in SFO complexity for stochastic quasi-Newton methods.
    • The integration of momentum and online generalization further enhances the practical utility and theoretical performance of the algorithms.
    • SpiderSQN represents a significant advancement in efficient and effective optimization for machine learning, particularly for nonconvex problems.