  This paper provides a unified treatment to the recovery of structured signals
living in a star-shaped set from general quantized measurements
$\mathcal{Q}(\mathbf{A}\mathbf{x}-\mathbf{\tau})$, where $\mathbf{A}$ is a
sensing matrix, $\mathbf{\tau}$ is a vector of (possibly random) quantization
thresholds, and $\mathcal{Q}$ denotes an $L$-level quantizer. The ideal
estimator with consistent quantized measurements is optimal in some important
instances but typically infeasible to compute. To this end, we study the
projected gradient descent (PGD) algorithm with respect to the one-sided
$\ell_1$-loss and identify the conditions under which PGD achieves the same
error rate, up to logarithmic factors. For multi-bit case, these conditions
only ensure local convergence, and we further develop a complementary approach
based on product embedding. When applied to popular models such as 1-bit
compressed sensing with Gaussian $\mathbf{A}$ and zero $\mathbf{\tau}$ and the
dithered 1-bit/multi-bit models with sub-Gaussian $\mathbf{A}$ and uniform
dither $\mathbf{\tau}$, our unified treatment yields error rates that improve
on or match the sharpest results in all instances. Particularly, PGD achieves
the information-theoretic optimal rate $\tilde{O}(\frac{k}{mL})$ for recovering
$k$-sparse signals, and the rate $\tilde{O}((\frac{k}{mL})^{1/3})$ for
effectively sparse signals. For 1-bit compressed sensing of sparse signals, our
result recovers the optimality of normalized binary iterative hard thresholding
(NBIHT) that was proved very recently.
