  Ill-posed linear inverse problems appear in many scientific setups, and are
typically addressed by solving optimization problems, which are composed of
data fidelity and prior terms. Recently, several works have considered a
back-projection (BP) based fidelity term as an alternative to the common least
squares (LS), and demonstrated excellent results for popular inverse problems.
These works have also empirically shown that using the BP term, rather than the
LS term, requires fewer iterations of optimization algorithms. In this paper,
we examine the convergence rate of the projected gradient descent (PGD)
algorithm for the BP objective. Our analysis allows to identify an inherent
source for its faster convergence compared to using the LS objective, while
making only mild assumptions. We also analyze the more general proximal
gradient method under a relaxed contraction condition on the proximal mapping
of the prior. This analysis further highlights the advantage of BP when the
linear measurement operator is badly conditioned. Numerical experiments with
both $\ell_1$-norm and GAN-based priors corroborate our theoretical results.
