Paul Dütting

Gagan Aggarwal, Ashwinkumar Badanidiyuru, Paul Dütting et al.

Motivated by online retail, we consider the problem of selling one item (e.g., an ad slot) to two non-excludable buyers (say, a merchant and a brand). This problem captures, for example, situations where a merchant and a brand cooperatively bid in an auction to advertise a product, and both benefit from the ad being shown. A mechanism collects bids from the two and decides whether to allocate and which payments the two parties should make. This gives rise to intricate incentive compatibility constraints, e.g., on how to split payments between the two parties. We approach the problem of finding a revenue-maximizing incentive-compatible mechanism from an online learning perspective; this poses significant technical challenges. First, the action space (the class of all possible mechanisms) is huge; second, the function that maps mechanisms to revenue is highly irregular, ruling out standard discretization-based approaches. In the stochastic setting, we design an efficient learning algorithm achieving a regret bound of $O(T^{3/4})$. Our approach is based on an adaptive discretization scheme of the space of mechanisms, as any non-adaptive discretization fails to achieve sublinear regret. In the adversarial setting, we exploit the non-Lipschitzness of the problem to prove a strong negative result, namely that no learning algorithm can achieve more than half of the revenue of the best fixed mechanism in hindsight. We then consider the $σ$-smooth adversary; we construct an efficient learning algorithm that achieves a regret bound of $O(T^{2/3})$ and builds on a succinct encoding of exponentially many experts. Finally, we prove that no learning algorithm can achieve less than $Ω(\sqrt T)$ regret in both the stochastic and the smooth setting, thus narrowing the range where the minimax regret rates for these two problems lie.

Paul Dütting, Federico Fusco, Silvio Lattanzi et al.

Consistency is an important property in dynamic submodular maximization and entails maintaining a near-optimal solution at all times, making only a small number of adjustments to the solution in each step. Prior work has explored this question for the insertion-only case, where the algorithm faces a stream of $n$ insertions, and has established lower and upper bounds for the cardinality-constrained version of the problem. We consider this question in the fully dynamic setting, where the stream of operations may contain both insertions and deletions. We develop a general framework for designing algorithms for this setting, and instantiate it to obtain the first constant-factor approximations with sublinear consistency. For cardinality constraints, we propose a $\frac 12 - O(\varepsilon)$ approximation that is $O\left(\frac{1}{\varepsilon^2}\right)$ consistent. For rank-$k$ matroid constraints, we construct a $\frac 14 - O(\varepsilon)$ approximation to the dynamic optimum that is $O\left(\frac{\log k}{\varepsilon^2}\right)$ consistent.

Simone Di Gregorio, Paul Dütting, Federico Fusco et al.

Bilateral trade models the task of intermediating between two strategic agents, a seller and a buyer, who wish to trade a good. We study this problem from the perspective of a profit-maximizing broker within an online learning framework, where the agents' valuations are generated by a smooth adversary. We devise a learning algorithm that guarantees a $\tilde{O}(\sqrt{T})$ regret bound, which is tight in the time horizon $T$ up to poly-logarithmic factors. This matches the minimax rate for the stochastic i.i.d. case, and is also well separated from the adversarial setting, where sublinear-regret is unattainable. By extending the strong regret guarantees from the i.i.d. case to the smooth adversary, we significantly broaden the scope of settings where such fast rate is achievable, while closing an important gap in the regret landscape of this fundamental economic problem. To overcome the challenges posed by this adversary, we leverage a continuity property of smooth instances and combines this with a hierarchical net-construction of the broker's action space, which is analyzed via algorithmic chaining. We showcase the applicability of these techniques by deriving a similarly tight $\tilde{O}(\sqrt{T})$ regret bound for a related mechanism design model: the joint ads problem.

Paul Dütting, Federico Fusco, Silvio Lattanzi et al.

In this work, we study online submodular maximization, and how the requirement of maintaining a stable solution impacts the approximation. In particular, we seek bounds on the best-possible approximation ratio that is attainable when the algorithm is allowed to make at most a constant number of updates per step. We show a tight information-theoretic bound of $\tfrac{2}{3}$ for general monotone submodular functions, and an improved (also tight) bound of $\tfrac{3}{4}$ for coverage functions. Since both these bounds are attained by non poly-time algorithms, we also give a poly-time randomized algorithm that achieves a $0.51$-approximation. Combined with an information-theoretic hardness of $\tfrac{1}{2}$ for deterministic algorithms from prior work, our work thus shows a separation between deterministic and randomized algorithms, both information theoretically and for poly-time algorithms.