Many statistical $M$-estimators are based on convex optimization problems\nformed by the combination of a data-dependent loss function with a norm-based\nregularizer. We analyze the convergence rates of projected gradient and\ncomposite gradient methods for solving such problems, working within a\nhigh-dimensional framework that allows the data dimension $\\pdim$ to grow with\n(and possibly exceed) the sample size $\\numobs$. This high-dimensional\nstructure precludes the usual global assumptions---namely, strong convexity and\nsmoothness conditions---that underlie much of classical optimization analysis.\nWe define appropriately restricted versions of these conditions, and show that\nthey are satisfied with high probability for various statistical models. Under\nthese conditions, our theory guarantees that projected gradient descent has a\nglobally geometric rate of convergence up to the \\emph{statistical precision}\nof the model, meaning the typical distance between the true unknown parameter\n$\\theta^*$ and an optimal solution $\\hat{\\theta}$. This result is substantially\nsharper than previous convergence results, which yielded sublinear convergence,\nor linear convergence only up to the noise level. Our analysis applies to a\nwide range of $M$-estimators and statistical models, including sparse linear\nregression using Lasso ($\\ell_1$-regularized regression); group Lasso for block\nsparsity; log-linear models with regularization; low-rank matrix recovery using\nnuclear norm regularization; and matrix decomposition. Overall, our analysis\nreveals interesting connections between statistical precision and computational\nefficiency in high-dimensional estimation.\n