arXiv2020
The ODE method has been a workhorse for algorithm design and analysis since the introduction of the stochastic approximation. It is now understood that convergence theory amounts to establishing robustness of Euler approximations for ODEs, while theory of rates of convergence requires finer analysis. This paper sets out to extend this theory to quasi-stochastic approximation, based on algorithms in which the "noise" is based on deterministic signals. The main results are obtained under minimal assumptions: the usual Lipschitz conditions for ODE vector fields, and it is assumed that there is a well defined linearization near the optimal parameter $θ^*$, with Hurwitz linearization matrix $A^*$. The main contributions are summarized as follows: (i) If the algorithm gain is $a_t=g/(1+t)^ρ$ with $g>0$ and $ρ\in(0,1)$, then the rate of convergence of the algorithm is $1/t^ρ$. There is also a well defined "finite-$t$" approximation: \[ a_t^{-1}\{Θ_t-θ^*\}=\bar{Y}+Ξ^{\mathrm{I}}_t+o(1) \] where $\bar{Y}\in\mathbb{R}^d$ is a vector identified in the paper, and $\{Ξ^{\mathrm{I}}_t\}$ is bounded with zero temporal mean. (ii) With gain $a_t = g/(1+t)$ the results are not as sharp: the rate of convergence $1/t$ holds only if $I + g A^*$ is Hurwitz. (iii) Based on the Ruppert-Polyak averaging of stochastic approximation, one would expect that a convergence rate of $1/t$ can be obtained by averaging: \[ Θ^{\text{RP}}_T=\frac{1}{T}\int_{0}^T Θ_t\,dt \] where the estimates $\{Θ_t\}$ are obtained using the gain in (i). The preceding sharp bounds imply that averaging results in $1/t$ convergence rate if and only if $\bar{Y}=\sf 0$. This condition holds if the noise is additive, but appears to fail in general. (iv) The theory is illustrated with applications to gradient-free optimization and policy gradient algorithms for reinforcement learning.