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版本更新 · v3arXiv:2508.05844

Online Budget Allocation with Censored Semi-Bandit Feedback

François Bachoc, Nicolò Cesa-Bianchi, Tommaso Cesari, Roberto Colomboni

原文摘要

We study a stochastic budget-allocation problem over $K$ tasks. At each round $t$, the learner chooses an allocation $X_t \in Δ_K$. Task $k$ succeeds with probability $F_k(X_{t,k})$, where $F_1,\dots,F_K$ are nondecreasing budget-to-success curves, and upon success yields a random reward with unknown mean $μ_k$. The learner observes which tasks succeed, and observes a task's reward only upon success (censored semi-bandit feedback). This model captures, for instance, splitting payments across crowdsourcing workers or distributing bids across simultaneous auctions, and subsumes stochastic multi-armed bandits and semi-bandits. We design an optimism-based algorithm that operates under censored semi-bandit feedback and assumes access to an exact maximization oracle. Our main result is an instance-dependent bound showing that in diminishing-returns regimes, the regret of this algorithm scales polylogarithmically with the horizon $T$ without any ad hoc tuning. For general nondecreasing curves, we prove that the same algorithm (with the same tuning) achieves a worst-case regret upper bound of $\tilde O(K\sqrt{T})$. Finally, we establish a matching worst-case regret lower bound of $Ω(K\sqrt{T})$ that holds even for full-feedback algorithms, highlighting the intrinsic hardness of our problem outside diminishing returns.

cs.GT · cs.LG · stat.ML

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新论文 · v1arXiv:2611.06159

Regret Bounds for Market Making with Persistent Orders

Nicolò Cesa-Bianchi, Tommaso Cesari, Roberto Colomboni, Luigi Foscari, Vinayak Pathak

原文摘要

We study adversarial market making with orders that remain active until execution. The maker operates on two markets: a primary market where orders are made and a secondary market where the resulting positions are hedged. In each round, the maker posts a quote on the primary market before observing both primary and secondary market prices. An order executes when the primary price reaches its quote, and the resulting position is immediately offset at the secondary price. When both price sequences are chosen by an adversary over $T$ rounds, we give a simple deterministic policy with regret of order $\log T$ against the best fixed quote (static comparator) and $\sqrt T$ against the best quote for each submission date (dynamic comparator), plus two terms measuring exposure to future primary and secondary prices. These rates remain optimal even when both exposure terms vanish and lower bounds show that the exposure terms are necessary. Our policy naturally extends to simultaneous bids and asks, where we show that static and dynamic regret are both characterized (up to constant factors) by the exposures terms (where the primary price exposure $V_T^{\mathrm{two}}$ now accounts for both bids and asks) and a term of order $\sqrt{V_T^{\mathrm{two}}\log T}$. Our policy is simple, efficient and fully parameterless. We extend this setting by demonstrating how natural assumptions on the price processes lead to improved guarantees without changing the policy.

cs.GT · cs.LG

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版本更新 · v2arXiv:2606.03851

The Price of Partial Information with Switching Costs: A Complete Characterization for Feedback Graphs

Nicolò Cesa-Bianchi, Tommaso Cesari, Roberto Colomboni, Emmanuel Esposito, Federico Fusco, Tomer Koren

原文摘要

We study how switching costs affect minimax regret under partial feedback. Which features of the feedback preserve the $\sqrt T$ rate, and which force the $T^{2/3}$ rate? We address this question in the framework of feedback graphs, which interpolate between full information and bandits: playing an action reveals the losses of its out-neighbors in a known directed graph. Prior work on feedback graphs with switching costs gave algorithms with regret of order $T^{2/3}$, but matching lower bounds only for restricted classes of graphs, leaving the question open. We establish a complete characterization of the minimax regret with switching costs for all feedback graphs: $\widetildeΘ(\sqrt T)$ if the graph is strongly revealing, meaning that every pair of distinct actions has an endpoint that observes the losses of both; $\widetildeΘ(γ^{1/3}T^{2/3})$ for every other observable graph, where $γ$ is the domination number; and linear otherwise. In particular, tournaments, in which every action observes its own loss and each pair of actions is joined by one directed edge, sit at the phase transition: they behave like full information, but deleting the edge of any pair forces the rate $T^{2/3}$. Hence an action that observes all losses is neither necessary nor sufficient for the rate $\sqrt T$. Our algorithm TournamentSteering for the strongly revealing case rests on a new principle: each observed comparison of two actions serves both to estimate losses and to steer play, and the loss reduction from steering pays for the variance of the estimates. For every strongly revealing graph, distributing the comparisons among observing actions keeps this variance, and hence the regret, polylogarithmic in the number of actions, by an argument similar to the greedy construction of logarithmic dominating sets in tournaments.

cs.LG

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版本更新 · v3arXiv:2608.12231

An Efficient Minimax-Optimal Algorithm for Adversarial $m$-Set Bandits

Francesco Bacchiocchi, Tommaso Cesari, Roberto Colomboni

原文摘要

We study adversarial combinatorial bandits with $m$-set actions, where at each round the learner selects $m$ out of $d$ items and observes only the aggregate loss of the selected items. The resulting action set contains $K=\binom{d}{m}$ elements and can therefore be exponentially large. We propose a computationally efficient algorithm that exploits this structure without explicitly enumerating the action set. Against adaptive non-anticipating adversaries, it guarantees, with probability at least $1-δ$, regret against the best action in hindsight of $R_T =O(\sqrt{dT\log(K/δ)}). $ This matches the high-probability regret bound of the finite-action EXP3 with Kiefer-Wolfowitz exploration algorithm of Zimmert and Lattimore, whose direct implementation may require exponential space. Our algorithm instead represents each sampling distribution with $d$ parameters and runs in polynomial time without enumerating the action set. This resolves an open problem posed by Maiti et al. We complement this upper bound with a matching high-probability lower bound. For all sufficiently small $δ$ and all sufficiently large horizons, every randomized algorithm admits an adaptive non-anticipating adversary for which, with probability at least $δ$, $R_T=Ω( \sqrt{dT\log(K/δ)}). $ Thus, the rate is minimax optimal up to universal constants in this regime. In particular, setting $m=1$ shows that the $\log K$ factor in the high-probability regret bound for ordinary $K$-armed bandits against adaptive non-anticipating adversaries is unavoidable. This strengthens the lower bound of Gerchinovitz and Lattimore, closing the remaining $\sqrt{\log K}$ gap between high-probability upper and lower regret bounds.

cs.LG

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arXiv ↗PDF ↗