Convergence of a stochastic gradient method with momentum for non-smooth non-convex optimization

• Stochastic gradient methods with momentum are widely used in applications and at the core of optimization subroutines in many popular machine learning libraries. However, their sample complexities have not been obtained for problems beyond those that are convex or smooth. This paper establishes the convergence rate of a stochastic subgradient method with a momentum term of Polyak type for a broad class of non-smooth, non-convex, and constrained optimization problems. Our key innovation is the construction of a special Lyapunov function for which the proven complexity can be achieved without any tuning of the momentum parameter. For smooth problems, we extend the known complexity bound to the constrained case and demonstrate how the unconstrained case can be analyzed under weaker assumptions than the state-of-The-Art. Numerical results confirm our theoretical developments.

Ämnesord

TEKNIK OCH TEKNOLOGIER  -- Elektroteknik och elektronik -- Reglerteknik (hsv//swe)

ENGINEERING AND TECHNOLOGY  -- Electrical Engineering, Electronic Engineering, Information Engineering -- Control Engineering (hsv//eng)

Nyckelord

Constrained optimization

Convex optimization

Gradient methods

Lyapunov functions

Machine learning

Momentum

Complexity bounds

Constrained optimi-zation problems

Convergence rates

Nonconvex optimization

Numerical results

Stochastic gradient methods

Sub-gradient methods

Theoretical development

Stochastic systems

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