Source
Saligrama (2026), LIT-1880, §5 and Appendix B, pp. 20–25 and 38–40 of arXiv 2607.16895v1; read in NOTE-1466.
What was actually shown
Assumption 1. A scalar θ in [θ_lo, θ_hi]. An experiment κ_ϑ with ϑ ≤ θ is safe under θ and returns the system to a common state (P2); its value loss is at most ω(θ − ϑ) (P3); each episode yields y = θa + ξ, ξ ~ N(0, σ²), and V_t = Σa²/σ² accumulates.
Lemma 5. Mixing exp(λM_t − λ²V_t/2) over λ ~ N(0, 1/ν) gives a nonnegative martingale √(ν/(V_t + ν))·exp(M_t²/(2(V_t + ν))). Ville’s inequality makes |θ̂_t − θ| ≤ b(V_t, δ; ν) = √(2(V_t + ν)log(√(1 + V_t/ν)/δ))/V_t hold at every t at once with probability 1 − δ.
Theorem 2. Run κ at ϑ_t = θ̂_{t−1} − b(V_{t−1}), clipped. On the confidence event every ϑ_t ≤ θ, so every episode is safe, and θ − ϑ_t ≤ 2b, so the loss is ω(2b). Regret ≤ r_max + Σ E ω(2b(V_{t−1}))
- r_max Tδ. With V_t ≍ t and ω linear this is O(√(T log T)); in the braking example O(σ√(D/θ)·√(T log T)).
Certainty equivalence. Using the sample mean, the next entry speed overshoots the safe one exactly when θ̂ > θ, which for Gaussian noise has probability ½ for every n. A one-episode risk δ < ½ needs a margin of at least σΦ⁻¹(1 − δ)/√n, and a guarantee over the whole trajectory needs a time-uniform one.
What could have failed. Persistent information could have been unusable: the controller might have had to leave the common safe regime to exploit it. The common-reset condition rules that out here.
What this does not say
- That recovery is generic. The author: “These are sufficient conditions, not necessary ones. Vector parameters, multiple control-equivalence classes, non-resetting trajectories, and dynamic latent environments require additional transition and exploitation arguments” (Remark 9).
- That growth of information suffices. Information must also be convertible into a safe transition to the selected regime (§6, §9).
- Anything when information is bounded. That is THEORY-1254’s regime, where no rule recovers the gap.