Deep Gaussian processes (DGPs) are flexible surrogate models for computer experiments with nonstationary response surfaces, but inference over their latent representations remains computationally challenging. Existing methods typically rely on variational inference or Markov chain Monte Carlo (MCMC), which impose variational-family assumptions or require costly posterior sampling. We propose the latent maximum-a-posteriori deep Gaussian process (LaMP-DGP), which uses gradient-based optimization to obtain MAP estimates of the latent representations and enables efficient probabilistic prediction through linked-GP calculations. To scale LaMP-DGP to large datasets, we develop a mini-batch optimization algorithm and, under a polynomial mini-batch bias-decay condition, establish a stationarity bound for latent-state optimization that motivates a growing batch-size schedule. Across numerical experiments and real-data analyses, LaMP-DGP provides consistently competitive predictive performance relative to doubly stochastic variational inference, stochastic gradient Hamiltonian Monte Carlo, and fully Bayesian Vecchia DGP inference. Computationally, LaMP-DGP is substantially faster than the Vecchia MCMC approach while remaining broadly competitive with the stochastic-gradient alternatives. These results position LaMP-DGP as a competitive and computationally efficient alternative for DGP inference. An R package, "LaMPDGP", is available for implementing the proposed method.

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