Online Combinatorial Allocations and Auctions with Few Samples

In online combinatorial allocations/auctions, n bidders sequentially arrive, each with a combinatorial valuation (such as submodular/XOS) over subsets of m indivisible items. The aim is to immediately allocate a subset of the remaining items to maximize the total welfare, defined as the sum of bidder valuations. A long line of work has studied this problem when the bidder valuations come from known independent distributions. In particular, for submodular/XOS valuations, we know 2-competitive algorithms/mechanisms that set a fixed price for each item and the arriving bidders take their favorite subset of the remaining items given these prices. However, these algorithms traditionally presume the availability of the underlying distributions as part of the input to the algorithm. Contrary to this assumption, practical scenarios often require the learning of distributions, a task complicated by limited sample availability. This paper investigates the feasibility of achieving O(1)-competitive algorithms under the realistic constraint of having access to only a limited number of samples from the underlying bidder distributions. Our first main contribution shows that a mere single sample from each bidder distribution is sufficient to yield an O(1)-competitive algorithm for submodular/XOS valuations. This result leverages a novel extension of the secretary-style analysis, employing the sample to have the algorithm compete against itself. Although online, this first approach does not provide an online truthful mechanism. Our second main contribution shows that a polynomial number of samples suffices to yield a $(2+\epsilon)$-competitive online truthful mechanism for submodular/XOS valuations and any constant $\epsilon>0$. This result is based on a generalization of the median-based algorithm for the single-item prophet inequality problem to combinatorial settings with multiple items.

Bandit Algorithms for Prophet Inequality and Pandora's Box

The Prophet Inequality and Pandora's Box problems are fundamental stochastic problem with applications in Mechanism Design, Online Algorithms, Stochastic Optimization, Optimal Stopping, and Operations Research. A usual assumption in these works is that the probability distributions of the $n$ underlying random variables are given as input to the algorithm. Since in practice these distributions need to be learned, we initiate the study of such stochastic problems in the Multi-Armed Bandits model. In the Multi-Armed Bandits model we interact with $n$ unknown distributions over $T$ rounds: in round $t$ we play a policy $x^{(t)}$ and receive a partial (bandit) feedback on the performance of $x^{(t)}$. The goal is to minimize the regret, which is the difference over $T$ rounds in the total value of the optimal algorithm that knows the distributions vs. the total value of our algorithm that learns the distributions from the partial feedback. Our main results give near-optimal $\tilde{O}(\mathsf{poly}(n)\sqrt{T})$ total regret algorithms for both Prophet Inequality and Pandora's Box. Our proofs proceed by maintaining confidence intervals on the unknown indices of the optimal policy. The exploration-exploitation tradeoff prevents us from directly refining these confidence intervals, so the main technique is to design a regret upper bound that is learnable while playing low-regret Bandit policies.

Online Learning with Vector Costs and Bandits with Knapsacks

We introduce online learning with vector costs (\OLVCp) where in each time step $t \in \{1,\ldots, T\}$, we need to play an action $i \in \{1,\ldots,n\}$ that incurs an unknown vector cost in $[0,1]^{d}$. The goal of the online algorithm is to minimize the $\ell_p$ norm of the sum of its cost vectors. This captures the classical online learning setting for $d=1$, and is interesting for general $d$ because of applications like online scheduling where we want to balance the load between different machines (dimensions). We study \OLVCp in both stochastic and adversarial arrival settings, and give a general procedure to reduce the problem from $d$ dimensions to a single dimension. This allows us to use classical online learning algorithms in both full and bandit feedback models to obtain (near) optimal results. In particular, we obtain a single algorithm (up to the choice of learning rate) that gives sublinear regret for stochastic arrivals and a tight $O(\min\{p, \log d\})$ competitive ratio for adversarial arrivals. The \OLVCp problem also occurs as a natural subproblem when trying to solve the popular Bandits with Knapsacks (\BwK) problem. This connection allows us to use our \OLVCp techniques to obtain (near) optimal results for \BwK in both stochastic and adversarial settings. In particular, we obtain a tight $O(\log d \cdot \log T)$ competitive ratio algorithm for adversarial \BwK, which improves over the $O(d \cdot \log T)$ competitive ratio algorithm of Immorlica et al. [FOCS'19].