Probe-then-Commit Multi-Objective Bandits: Theoretical Benefits of Limited Multi-Arm Feedback
Ming Shi
TL;DR
This work introduces Probe-then-Commit (PtC) for multi-objective online learning under limited multi-arm feedback, formalizing a PtC protocol where up to q probes observe vector rewards before a single arm is executed. It proposes PtC-P-UCB, a frontier-aware algorithm that maximizes a hypervolume-based coverage potential for probe selection and uses marginal hypervolume gain or a scalarizer for commit decisions, with frontier pruning to focus on Pareto-relevant arms. Theoretical results quantify a clean 1/√q acceleration: the dominated-hypervolume frontier gap decays as \tilde{O}(K_P d / \sqrt{qT}) and scalarized regret as \tilde{O}(d L_\phi \sqrt{(K/q)T}), interpolating between bandit and full-information regimes. The paper also extends to multi-modal feedback (MM-PtC-P-UCB) with variance-adaptive improvements, and validates the theory with numerical experiments showing faster frontier discovery and improved scalarized performance under modest probing budgets and multi-modal fusion, highlighting practical benefits for wireless/edge decision problems.
Abstract
We study an online resource-selection problem motivated by multi-radio access selection and mobile edge computing offloading. In each round, an agent chooses among $K$ candidate links/servers (arms) whose performance is a stochastic $d$-dimensional vector (e.g., throughput, latency, energy, reliability). The key interaction is \emph{probe-then-commit (PtC)}: the agent may probe up to $q>1$ candidates via control-plane measurements to observe their vector outcomes, but must execute exactly one candidate in the data plane. This limited multi-arm feedback regime strictly interpolates between classical bandits ($q=1$) and full-information experts ($q=K$), yet existing multi-objective learning theory largely focuses on these extremes. We develop \textsc{PtC-P-UCB}, an optimistic probe-then-commit algorithm whose technical core is frontier-aware probing under uncertainty in a Pareto mode, e.g., it selects the $q$ probes by approximately maximizing a hypervolume-inspired frontier-coverage potential and commits by marginal hypervolume gain to directly expand the attained Pareto region. We prove a dominated-hypervolume frontier error of $\tilde{O} (K_P d/\sqrt{qT})$, where $K_P$ is the Pareto-frontier size and $T$ is the horizon, and scalarized regret $\tilde{O} (L_φd\sqrt{(K/q)T})$, where $φ$ is the scalarizer. These quantify a transparent $1/\sqrt{q}$ acceleration from limited probing. We further extend to \emph{multi-modal probing}: each probe returns $M$ modalities (e.g., CSI, queue, compute telemetry), and uncertainty fusion yields variance-adaptive versions of the above bounds via an effective noise scale.