Abstract Recent years have seen advances in optimizing large scale statistical estimation problems. In statistical learning settings iterative optimization algorithms have been shown to enjoy geometric convergence. While powerful, such results only hold for the original dataset, and may face computational challenges when the sample size is large. In this paper, we study sketched iterative algorithms, in particular sketchedPGD (projected gradient descent) and sketchedSVRG (stochastic variance reduced gradient) for structured generalized linear model, and illustrate that these methods continue to have geometric convergence to the statistical error under suitable assumptions. Moreover, the sketching dimension is allowed to be even smaller than the ambient dimension, thus can lead to significant speed-ups. The sketched iterative algorithms introduced provide an additional dimension to study the trade-offs between statistical accuracy and time