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Approximate message passing (AMP) emerges as an effective iterative paradigm for solving high-dimensional statistical problems. However, prior AMP theory --- which focused mostly on high-dimensional asymptotics --- fell short of predicting the AMP dynamics when the number of iterations surpasses o(log n / log log n) (with n the problem dimension). To address this inadequacy, this paper develops a non-asymptotic framework for understanding AMP in spiked matrix estimation. Built upon new decomposition of AMP updates and controllable residual terms, we lay out an analysis recipe to characterize the finite-sample behavior of AMP in the presence of an independent initialization, which is further generalized to allow for spectral initialization. As two concrete consequences of the proposed analysis recipe: (i) when solving Z_2 synchronization, we predict the behavior of spectrally initialized AMP for up to O(n / polylog n) iterations, showing that the algorithm succeeds without the need of a subsequent refinement stage (as conjectured recently by Celentano et al.); (ii) we characterize the non-asymptotic behavior of AMP in sparse PCA (in the spiked Wigner model) for a broad range of signal-to-noise ratio.