Time series foundation models (TSFMs) are increasingly pre-trained on synthetically generated time series trajectories, where the data generating process is known. Current pre-training recipes are based on loss objectives which compare TSFM outputs to realized future values of each trajectory. We instead propose loss objectives which compare TSFM outputs to the conditional forecast distribution of each trajectory, a procedure we call synthetic data distillation (SDD). SDD corresponds to a Rao-Blackwellization of the training objective, in that it leaves the expectation of stochastic gradients unchanged while provably reducing the covariance of the stochastic gradient under the Loewner partial ordering. We empirically validate SDD on a TSFM model family of sizes from $4$M to $2.5$B parameters, and observe faster convergence of validation loss at every model size: on Gaussian Process data, SDD attains or improves upon the Status Quo loss whilst requiring $10\%-40\%$ less training iterations.

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