  Driven by several successful applications such as in stochastic gradient
descent or in Bayesian computation, control variates have become a major tool
for Monte Carlo integration. However, standard methods do not allow the
distribution of the particles to evolve during the algorithm, as is the case in
sequential simulation methods. Within the standard adaptive importance sampling
framework, a simple weighted least squares approach is proposed to improve the
procedure with control variates. The procedure takes the form of a quadrature
rule with adapted quadrature weights to reflect the information brought in by
the control variates. The quadrature points and weights do not depend on the
integrand, a computational advantage in case of multiple integrands. Moreover,
the target density needs to be known only up to a multiplicative constant. Our
main result is a non-asymptotic bound on the probabilistic error of the
procedure. The bound proves that for improving the estimate's accuracy, the
benefits from adaptive importance sampling and control variates can be
combined. The good behavior of the method is illustrated empirically on
synthetic examples and real-world data for Bayesian linear regression.
