balanced fractional exchanges are introduced, which compress the policy mixture into a single fractional base while retaining the exchange information needed by the Poisson analysis, and lead to an polynomial time algorithm with the same regret guarantee.
Abstract
We study adversarial bandit maximization of monotone submodular functions under a matroid constraint. For a rank-$k$ matroid on $n$ elements, we give a randomized oracle-polynomial algorithm that makes one feasible value query per round and has expected $(1-1/e)$-regret $\widetilde O(n^{1/3}k^{2/3}T^{2/3})$. This is the first sublinear-regret algorithm for adversarial bandit submodular maximization under general matroid constraints. Technically, we view the problem as learning an exchange policy for the Poisson base walk. This connects the problem to contextual bandits and gives an information-theoretic sublinear-regret guarantee, but directly learning the exponentially many policies requires exponential time and space. We therefore introduce \emph{balanced fractional exchanges}, which compress the policy mixture into a single fractional base while retaining the exchange information needed by the Poisson analysis. This leads to an polynomial time algorithm with the same regret guarantee.
We study nonnegative submodular maximization on $n$ elements subject to a general matroid of rank $k$, when the offline algorithm is given an arbitrary controlled value oracle. Our main result is an adversarial resilience theorem for the Spiteful Greedy Swap Poisson Process (SGS-Poisson): without modifying its Poisson intensity, single-element exchange rule, or spiteful drop step, the algorithm retains limiting approximation factors $1/e$ for non-monotone objectives and $1-1/e$ for monotone objectives. More precisely, given an error bound $\xi\ge0$, under every controlled oracle $\widehat f$ satisfying $|\widehat f(S)-f(S)|\le \xi$ for every set $S$, our implementation returns a feasible set with expected value at least $(1/e-\varepsilon)OPT-O(k\xi)$ and $(1-1/e-\varepsilon)OPT-O(k\xi)$, respectively, where $OPT$ is the feasible optimum. The implementation uses a \emph{deterministically bounded} budget of $O(nk^{2}\varepsilon^{-2}\log n\log^{2}(1/\varepsilon))$ oracle calls. As a consequence, an offline-to-online reduction yields full-bandit combinatorial multi-armed bandit (CMAB) algorithms for general matroid-constrained submodular rewards with exact limiting approximation-regret factors $1/e$ and $1-1/e$ and $\widetilde O(n^{1/5}k^{4/5}T^{4/5})$ regret over $T$ rounds. These online guarantees allow exploration to play sets that become independent after deleting at most one element; exploitation and the benchmark remain matroid-feasible. For unit-capacity partition matroids we obtain $\widetilde O(n^{1/5}k^{3/5}T^{4/5})$ under the same exploration relaxation. We establish a deterministic query budget by truncating the Poisson process and identify the enlarged action set needed to answer its offline queries.
We study online maximization of nonnegative, non-monotone DR-submodular functions over compact convex down-closed subsets of the $d$-dimensional unit cube. The best known constructive offline approximation factor is $0.401$ under the corresponding meta-solvability assumptions, whereas comparable adversarial online guarantees had remained at $1/e$. We show that this factor is also achievable online. In the post-decision full-information value-oracle model, our algorithm attains factor $0.401$ with sublinear approximate regret when oracle feedback is conditionally unbiased and bounded. The online algorithm does not run the offline construction on a changing objective. Instead, it replaces the offline objective-dependent box step by a weighted online learner that controls the required residual terms cumulatively. An exact asymmetric balance theorem preserves the offline coefficients despite adversarial variation. The direct implementation has $O(T^{3/4})$ regret and uses $O(dT^{1/4})$ oracle calls per round. More generally, for every $\delta\in[0,1/4]$, batching gives $O(T^\delta)$ calls per round and $O(T^{4/5-\delta/5})$ regret, including a one-call $O(T^{4/5})$ endpoint. Under a positive-anchor condition, randomized blocking retains factor $0.401$ with $O(T^{5/6})$ one-point bandit regret.
We study switching regret in adversarial multi-armed bandits, where the learner competes with an arm sequence that changes at most $S$ times. When $S$ is known, an optimal expected regret of $\widetilde{\mathcal{O}}(\sqrt{(S+1)KT})$ is obtainable [Auer et al., 2002]. However, when $S$ is unknown, Marinov and Zimmert [2021] show that this guarantee is impossible under an adaptive adversary. In this paper, we show that a single algorithm achieves $\widetilde{\mathcal{O}}(\sqrt{(S+1)KT})$ expected regret for every $S$ against an oblivious adversary, resolving an open problem of Auer et al. [2019b]. Our algorithm combines a fixed-share learner initialized with a small learning rate and dyadic-interval subroutines that search for local improvements using randomized learning rates and implicit exploration. Importantly, a non-uniform prior favors following the main learner, keeping the cost of maintaining many subroutines small. When the subroutines accumulate sufficient improvement over the main learner, its learning rate doubles, allowing adaptation to the unknown number of comparator switches $S$.
$k$-submodularity generalizes submodularity by allowing each selected element to be assigned one of $k$ labels, rather than being merely selected or not selected. We study the problem of maximizing a nonnegative non-monotone $k$-submodular function, where $k\ge 2$, under classical support constraints, including a single matroid constraint and a single knapsack constraint. Previously, the best known approximation guarantees for non-monotone constrained $k$-submodular maximization had long remained at $1/3$ or $1/3-\varepsilon$, even in basic settings such as cardinality, matroid, and knapsack constraints. We show that this $1/3$ barrier is not inherent: for both the matroid and knapsack settings considered here, we give randomized polynomial-time algorithms achieving an approximation ratio of $\sqrt{2}-1\approx 0.4142$. The algorithms use a simple randomized greedy rule: once an element is selected, its label is chosen only from the two labels with the largest marginal gains, with probabilities proportional to the positive parts of these two gains. The value-oracle query complexity is $O(n^2k)$ in the matroid setting and $O(n^3k^2)$ in the knapsack setting. These results give the first approximation guarantees exceeding $1/3$ for non-monotone $k$-submodular maximization under matroid and knapsack constraints.
Si-Yuan Chen, Shengminjie Chen, Sui-Xiang Gao et al.· 0 citations
Setting $m=1$ proves that the $\log K$ for ordinary $K$-armed bandits against adaptive non-anticipating adversaries is unavoidable, closing the remaining $\sqrt{\log K}$ gap between confidence-tuned upper and lower bounds left by Gerchinovitz and Lattimore.
F. Bacchiocchi, Tommaso Cesari, Roberto Colomboni· 1 citation
The Lipschitz bandit problem extends the traditional multi-armed bandit framework to continuous action spaces by assuming that the reward functions satisfy a Lipschitz condition. This work investigates Lipschitz bandits under arbitrary feedback delays, where reward signals are not received immediately upon taking an action but after an arbitrarily chosen delay. We consider both stochastic and adversarial reward settings, proposing an elimination-based algorithm and an EXP3-based algorithm, respectively. For both settings, our algorithms achieve a regret bound of $\tilde{O}\left(T^{\frac{d_z+1}{d_z+2}}+\sqrt{D}\right)$ over a time horizon $T$ with total delay $D$, where the main difference between settings lies in the definition of the zooming dimension $d_z$. Our bounds match existing delay-free regret guarantees for Lipschitz bandits and characterize the additional $\tilde{O}(\sqrt{D})$ impact introduced by feedback delays.
Yuhao Liu, Yu Chen, Longbo Huang· 0 citations
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