We consider experimentation in the presence of non-stationarity, inter-unit (spatial) interference, and carry-over effects (temporal interference), where we wish to estimate the global average treatment effect (GATE), the difference between average outcomes having exposed all units at all times to treatment or to control. We suppose spatial interference is described by a graph, where a unit's outcome depends on its neighborhood's treatments, and that temporal interference is described by an MDP, where the transition kernel under either treatment (action) satisfies a rapid mixing condition. We propose a clustered switchback design, where units are grouped into clusters and time steps are grouped into blocks, and each whole cluster-block combination is assigned a single random treatment. Under this design, we show that for graphs that admit good clustering, a truncated Horvitz-Thompson estimator achieves a $\tilde O(1/NT)$ mean squared error (MSE), matching the lower bound up to logarithmic terms for sparse graphs. Our results simultaneously generalize the results from \citet{hu2022switchback,ugander2013graph} and \citet{leung2022rate}. Simulation studies validate the favorable performance of our approach.

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