If an optimal (even intractable) protocol achieves utility α in k bits, a polynomial-time algorithm can find a protocol achieving α−ε using 2^{O(k)}/ε^2 bits, and this is tight up to a constant in the exponent.
Multical- ibration: Calibration for the (computationally-identifiable) masses
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Computationally Efficient Collaborative Communication Via Regularity-Based Coarsening
If an optimal (even intractable) protocol achieves utility α in k bits, a polynomial-time algorithm can find a protocol achieving α−ε using 2^{O(k)}/ε^2 bits, and this is tight up to a constant in the exponent.