Steady-State Convergence of Stochastic Approximation

For constant-stepsize stochastic approximation (SA), the iterates converge in distribution to a stationary law that depends on the stepsize $α.$ Steady-state convergence (SSC) concerns the limit of the scaled stationary distribution as $α\downarrow 0.$ Existing SSC theory requires i.i.d. or additive noise and global differentiability of the mean operator, and yields suboptimal rates. We develop a unified SSC theory for constant-stepsize contractive SA driven by Markovian, multiplicative noise, covering both locally differentiable and locally nondifferentiable mean operators. A key methodological contribution is a multi-step universality framework that progressively reduces the original st...

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