A Reduction from Delayed to Immediate Feedback for Online Convex Optimization with Improved Guarantees

We develop a reduction-based framework for online learning with delayed feedback that recovers and improves upon existing results for both first-order and bandit convex optimization. Our approach introduces a continuous-time model under which regret decomposes into a delay-independent learning term and a delay-induced drift term, yielding a delay-adaptive reduction that converts any algorithm for online linear optimization into one that handles round-dependent delays. For bandit convex optimization, we significantly improve existing regret bounds, with delay-dependent terms matching state-of-the-art first-order rates. For first-order feedback, we recover state-of-the-art regret bounds via a simpler, unified analysis. Quantitatively, for bandit convex optimization we obtain $O(\sqrt{d_{\text{tot}}} + T^{\frac{3}{4}}\sqrt{k})$ regret, improving the delay-dependent term from $O(\min\{\sqrt{T d_{\text{max}}},(Td_{\text{tot}})^{\frac{1}{3}}\})$ in previous work to $O(\sqrt{d_{\text{tot}}})$. Here, $k$, $T$, $d_{\text{max}}$, and $d_{\text{tot}}$ denote the dimension, time horizon, maximum delay, and total delay, respectively. Under strong convexity, we achieve $O(\min\{σ_{\text{max}} \ln T, \sqrt{d_{\text{tot}}}\} + (T^2\ln T)^{\frac{1}{3}} {k}^{\frac{2}{3}})$, improving the delay-dependent term from $O(d_{\text{max}} \ln T)$ in previous work to $O(\min\{σ_{\text{max}} \ln T, \sqrt{d_{\text{tot}}}\})$, where $σ_{\text{max}}$ denotes the maximum number of outstanding observations and may be considerably smaller than $d_{\text{max}}$.

Reinforcement Learning with Action-Triggered Observations

We study reinforcement learning problems where state observations are stochastically triggered by actions, a constraint common in many real-world applications. This framework is formulated as Action-Triggered Sporadically Traceable Markov Decision Processes (ATST-MDPs), where each action has a specified probability of triggering a state observation. We derive tailored Bellman optimality equations for this framework and introduce the action-sequence learning paradigm in which agents commit to executing a sequence of actions until the next observation arrives. Under the linear MDP assumption, value-functions are shown to admit linear representations in an induced action-sequence feature map. Leveraging this structure, we propose off-policy estimators with statistical error guarantees for such feature maps and introduce ST-LSVI-UCB, a variant of LSVI-UCB adapted for action-triggered settings. ST-LSVI-UCB achieves regret $\widetilde O(\sqrt{Kd^3(1-\gamma)^{-3}})$, where $K$ is the number of episodes, $d$ the feature dimension, and $\gamma$ the discount factor (per-step episode non-termination probability). Crucially, this work establishes the theoretical foundation for learning with sporadic, action-triggered observations while demonstrating that efficient learning remains feasible under such observation constraints.

Capacity-Constrained Online Learning with Delays: Scheduling Frameworks and Regret Trade-offs

We study online learning with oblivious losses and delays under a novel ``capacity constraint'' that limits how many past rounds can be tracked simultaneously for delayed feedback. Under ``clairvoyance'' (i.e., delay durations are revealed upfront each round) and/or ``preemptibility'' (i.e., we have ability to stop tracking previously chosen round feedback), we establish matching upper and lower bounds (up to logarithmic terms) on achievable regret, characterizing the ``optimal capacity'' needed to match the minimax rates of classical delayed online learning, which implicitly assume unlimited capacity. Our algorithms achieve minimax-optimal regret across all capacity levels, with performance gracefully degrading under suboptimal capacity. For $K$ actions and total delay $D$ over $T$ rounds, under clairvoyance and assuming capacity $C = \Omega(\log(T))$, we achieve regret $\widetilde{\Theta}(\sqrt{TK + DK/C + D\log(K)})$ for bandits and $\widetilde{\Theta}(\sqrt{(D+T)\log(K)})$ for full-information feedback. When replacing clairvoyance with preemptibility, we require a known maximum delay bound $d_{\max}$, adding $\smash{\widetilde{O}(d_{\max})}$ to the regret. For fixed delays $d$ (i.e., $D=Td$), the minimax regret is $\Theta\bigl(\sqrt{TK(1+d/C)+Td\log(K)}\bigr)$ and the optimal capacity is $\Theta(\min\{K/\log(K),d\}\bigr)$ in the bandit setting, while in the full-information setting, the minimax regret is $\Theta\bigl(\sqrt{T(d+1)\log(K)}\bigr)$ and the optimal capacity is $\Theta(1)$. For round-dependent and fixed delays, our upper bounds are achieved using novel scheduling policies, based on Pareto-distributed proxy delays and batching techniques. Crucially, our work unifies delayed bandits, label-efficient learning, and online scheduling frameworks, demonstrating that robust online learning under delayed feedback is possible with surprisingly modest tracking capacity.