Phase retrieval, or signal recovery from magnitude-only measurements, is a challenging signal processing problem. Recent progress has revealed that measurement- and computational-complexity challenges can be alleviated if the underlying signal belongs to certain low-dimensional model families, including sparsity, low-rank, or neural generative models. However, the remaining bottleneck in most of these approaches is the requirement of a carefully chosen initial signal estimate. In this paper, we assume that a portion of the signal is already known a priori as”side information” (this assumption is natural in applications such as holographic coherent diffraction imaging). When such side information is available, we show that a much simpler initialization can provably succeed with considerably reduced costs. We supplement our theory with a range of simulation results.