Closing the discount-factor gap in retained-bit complexity

Close the factor of order 1/(1−γ) between the upper and lower bounds for the retained-bit complexity of approximating all TD iterates, thereby establishing the sharp γ-dependent analogue of the context-accuracy law for arbitrary offline codes.

Background

The retained-bit theorem gives a Θ_γ expression with matching dependence on n, L, and log(1/ε) up to γ-dependent constants, but the proof’s explicit upper and lower bounds use different effective depths. The remark identifies a remaining multiplicative gap of order 1/(1−γ) when the accuracy scale is smaller than both n and L, and explicitly states that closing this gap is open.

References

Closing this factor, the analogue of Theorem~\ref{thm:gamma-law} for bits, is open.

— Provable Transformer Context Compression for In-Context Reinforcement Learning  (2610.20004 - Zhao et al., 14 Oct 2026) in Appendix C, subsection “Offline Retained-Bit Complexity,” Remark “Dependence on γ”