Bandits & Online Learning

Instance-Dependent Robustness Guarantees Beyond Worst-Case Regret Bounds for Constrained Online Learning

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Possible candidate · 2/4 runs6 papers report this67% from 2025+

Generated automatically from the limitations stated in 6 papers (ICLR, AISTATS, ICML), listed under Evidence. It is not a paper, and it does not come from papers submitted to CSPaper.

The problem

Today, online learning guarantees are only available under restrictive structural conditions—honest agents dominating malicious ones, known adversary classes, Slater's condition, smoothness assumptions—and when these fail, the bounds become vacuous or degrade catastrophically. Practitioners cannot distinguish between "the bound is loose" and "the problem is genuinely hard," because no instance-dependent analysis exists that quantifies what is actually achievable on a given problem instance. This blocks deployment in settings with few honest agents, near-zero constraint slacks, or moderately hard function classes, precisely where robust methods matter most.

Why it matters

Tight instance-dependent characterizations of when robust online learning is possible versus information-theoretically impossible, replacing vacuous worst-case bounds with actionable per-instance guarantees.

Ways to approach it

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1
Derive instance-dependent regret lower bounds for the constrained online learning setting that explicitly depend on the Slater parameter ρ, the number of constraints m, and the margin M_H_t − M_A_t, then compare against achievable upper bounds on synthetic instances where these parameters are varied. Measure: gap between lower and upper bounds as a function of ρ, m, and the honest/malicious margin.
2
Characterize the phase transition: for the known failure regimes (ρ near zero, Γ near Δ, honest/malicious ratio near 1), empirically map where current methods degrade to linear regret versus where they remain competitive despite vacuous theory. Measure: cumulative regret and constraint violation on synthetic bandit instances with swept parameters.
3
Design an adaptive algorithm whose regret bound depends on instance-level quantities (effective Slater parameter, effective honest-agent mass) rather than their worst-case values, validated on the same synthetic suite. Measure: regret improvement over worst-case-tuned baselines at fixed violation levels.

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Why it might fail

The instance-dependent parameters (ρ_eff, m_eff, the honest/malicious margin) may be fundamentally unestimable from bandit feedback faster than the regret horizon itself, making any "adaptive" bound unimplementable; or the lower bounds may already match the existing upper bounds, meaning the current bounds are tight and no improvement exists.

Evidence

Each paper's own statement of the limitation, verbatim.

Oracle-efficient Hybrid Learning with Constrained AdversariesICLR 2026
Requires the adversary to be constrained to a known, fixed label class R; regret degrades by T·rad_T(ℓ∘H×R), so expressiveness of R directly inflates the statistical bound
Open Multi-agent Multi-armed Bandit with Applications in Permissionless BlockchainAISTATS 2026
Regret analysis holds only on a high-probability event requiring M_H_t > 2·M_A_t (honest agents must outnumber malicious agents throughout), and requires √λ_H_A − √(2λ_A_A) ≥ 1 constraints on arrival rates; failure of these conditions voids the bounds
No-Regret is not enough! Bandits with General Constraints through Adaptive Regret MinimizationICML 2025
All guarantees degrade polynomially in 1/ρ and m (e.g., Õ(m²/ρ² √KT) regret, m³/ρ³ in the contextual case), so instances with small Slater's parameter or many constraints get weak bounds
Unified Projection-Free Algorithms for Adversarial DR-Submodular OptimizationICLR 2024
Regret bounds hold for oblivious adversaries and require smooth, Lipschitz, bounded DR-submodular functions; constants depend on M0, M1, M2 and oracle bounds
Multi-armed bandits with guaranteed revenue per armAISTATS 2024
SPOC's constant violation bound can be vacuous when some λ_k μ_k is very small (large D_{λ,μ} constant), and its regret bound is vacuous for μ_k ≤ T^{-1/2}
On the Interplay Between Misspecification and Sub-optimality Gap in Linear Contextual BanditsICML 2023
When ζ ≥ Ω̃(Δ/√d), no gap-dependent bound is possible (lower bound gives linear regret), so the method provides no benefit; a too-large Γ (e.g., Γ≈Δ) causes failure to learn and linear regret (1789.5±1918.8 in synthetic experiments)