Source
Saligrama (2026), LIT-1880, §§2–4 (Lemma 1, Definitions 3–6, Theorem 1, Corollaries 2–4), pp. 4–18 of arXiv 2607.16895v1; read in NOTE-1466.
What was actually shown
Setting. A causal policy must keep the probability of any constraint violation at most η_T under every model θ in the initial class (uniform chance safety). Regret is measured against the safe oracle that knows θ.
Commitment. For a target θ and an alternative θ′, a commitment is the first action after which, under θ′, no safe continuation remains. It is decided from the history and the action, before that action’s observation, so τ is a stopping time for σ(H_{t−1}, U_t).
Three quantities. α_T is the largest probability, over uniformly safe policies, of committing under θ′; if commitment forces a violation there, α_T ≤ η_T. G_T(q) is the least target regret of a safe policy that fails to commit with probability at least q. I_pre is the largest KL divergence between the records under θ and θ′ stopped at commitment.
The bound. Data processing on the event “no commitment” gives d_bin(P_θ(no commit) ∥ P_θ′(no commit)) ≤ I_pre. Bretagnolle–Huber then gives P_θ(no commit) ≥ ½e^{−I_pre} − α_T. Hence R_T(π; θ) ≥ G_T(½e^{−I_pre} − α_T). With a linear profile G_T(q) ≥ G_T q − ζ_T, if β = sup_T I_pre < ∞, α_T → 0 and ζ_T = o(G_T), the worst-case regret of every uniformly safe policy is at least ½e^{−β}G_T asymptotically. The explicit bound is positive until I_pre ≥ log(1/(2α_T)), which is 2 log T − log 2 at α_T = T⁻².
Instances. In a device whose commands spend headroom θu² and are observed through θu + noise, I_pre ≤ (θ_d − θ_b)²d₀/(2σ²θ_d) for every T, and against a healthy target the gap is Θ(T), so regret is linear. Two constrained linear-regulation variants give linear regret too (Corollary 4).
Checked. The Bretagnolle–Huber relaxation of ψ_α over twenty (α, D) pairs; the finite-headroom bound and threshold by hand; the boundary-target optimization (44)–(45) numerically, converging to the stated T^{2/3} correction.
What could have failed. The evidence produced by the committing action could have been usable, as it is in an analysis over the full trajectory. The stopping-time construction is what excludes it.
What this does not say
- That unbounded information gives recovery. The certificate is sufficient for a gap, not necessary. One safe policy must still gather the information and then enter the selected regime safely (Remark 6, §9).
- That the gap is total. With many alternatives, finite information can still certify a smaller candidate list and recover part of the gap (Proposition 1).
- Anything about a different safety criterion. A budget on cumulative violations, rather than a per-model bound on violation probability, falls outside the premise (Remark 4).
- That a better estimator helps. When pre-commitment information is bounded, the experiment, the sensor or the degree of specialization has to change (§9).