Constrained Online Learning with Noisy Constraint Values

• URL: http://arxiv.org/abs/2609.06921v2
• Date: Sun, 13 Sep 2026 20:13:35 GMT
• Title: Constrained Online Learning with Noisy Constraint Values
• Abstract summary: Under common feasibility, our LEDGER algorithm attains $O(sqrt T)$ expected regret and $O(sqrtTlog(eT))$ expected budget violation.<n>Neither a Slater condition, independence between feedback channels, nor an absolute constraint-value bound is needed.
• Score: 55.29259818039367
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study constrained online convex optimization with adversarial constraints and conditionally unbiased, finite-variance observations of constraint values and gradients. Under common feasibility, our \LEDGER\ algorithm attains $O(\sqrt T)$ expected regret and $O(\sqrt{T\log(eT)})$ expected budget violation, the largest cumulative overspend over any window. It uses a reflected exponential potential, clipped signed observations, and predictable adaptive regularization, with one feedback triple and one projection per round. Neither a Slater condition, independence between feedback channels, nor an absolute constraint-value bound is needed. A Gaussian testing lower bound proves that the budget rate has optimal horizon dependence under square-root regret at fixed positive noise, including the logarithm. The same obstruction holds for terminal violation, so the logarithm is not a cost of maximizing over windows; an $O(\sqrt T)$ budget bound instead forces linear regret. In contrast, fixed positive Gaussian value noise yields a joint regret--hard-violation lower bound of $Ω(\min\{σ,1\}T/\log^2 T)$, even with exact gradients in one dimension. The hard-violation construction matches arbitrarily many moments while preserving a feasible-endpoint gap and constant endpoint probabilities. Together, the bounds separate uncertainty about hard feasibility from learnable signed budgets. Deterministic restarts remove the horizon input without changing either upper rate.

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