Artin Tajdini

Artin Tajdini, Lalit Jain, Kevin Jamieson

We consider maximizing an unknown monotonic, submodular set function $f: 2^{[n]} \rightarrow [0,1]$ with cardinality constraint under stochastic bandit feedback. At each time $t=1,\dots,T$ the learner chooses a set $S_t \subset [n]$ with $|S_t| \leq k$ and receives reward $f(S_t) + η_t$ where $η_t$ is mean-zero sub-Gaussian noise. The objective is to minimize the learner's regret with respect to an approximation of the maximum $f(S_*)$ with $|S_*| = k$, obtained through robust greedy maximization of $f$. To date, the best regret bound in the literature scales as $k n^{1/3} T^{2/3}$. And by trivially treating every set as a unique arm one deduces that $\sqrt{ {n \choose k} T }$ is also achievable using standard multi-armed bandit algorithms. In this work, we establish the first minimax lower bound for this setting that scales like $\tildeΩ(\min_{L \le k}(L^{1/3}n^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$. For a slightly restricted algorithm class, we prove a stronger regret lower bound of $\tildeΩ(\min_{L \le k}(Ln^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$. Moreover, we propose an algorithm Sub-UCB that achieves regret $\tilde{\mathcal{O}}(\min_{L \le k}(Ln^{1/3}T^{2/3} + \sqrt{{n \choose k - L}T}))$ capable of matching the lower bound on regret for the restricted class up to logarithmic factors.

Artin Tajdini, Jonathan Scarlett, Kevin Jamieson

We study stochastic linear bandits with heavy-tailed rewards, where the rewards have a finite $(1+ε)$-absolute central moment bounded by $\upsilon$ for some $ε\in (0,1]$. We improve both upper and lower bounds on the minimax regret compared to prior work. When $\upsilon = \mathcal{O}(1)$, the best prior known regret upper bound is $\tilde{\mathcal{O}}(d T^{\frac{1}{1+ε}})$. While a lower with the same scaling has been given, it relies on a construction using $\upsilon = \mathcal{O}(d)$, and adapting the construction to the bounded-moment regime with $\upsilon = \mathcal{O}(1)$ yields only a $Ω(d^{\fracε{1+ε}} T^{\frac{1}{1+ε}})$ lower bound. This matches the known rate for multi-armed bandits and is generally loose for linear bandits, in particular being $\sqrt{d}$ below the optimal rate in the finite-variance case ($ε= 1$). We propose a new elimination-based algorithm guided by experimental design, which achieves regret $\tilde{\mathcal{O}}(d^{\frac{1+3ε}{2(1+ε)}} T^{\frac{1}{1+ε}})$, thus improving the dependence on $d$ for all $ε\in (0,1)$ and recovering a known optimal result for $ε= 1$. We also establish a lower bound of $Ω(d^{\frac{2ε}{1+ε}} T^{\frac{1}{1+ε}})$, which strictly improves upon the multi-armed bandit rate and highlights the hardness of heavy-tailed linear bandit problems. For finite action sets, we derive similarly improved upper and lower bounds for regret. Finally, we provide action set dependent regret upper bounds showing that for some geometries, such as $l_p$-norm balls for $p \le 1 + ε$, we can further reduce the dependence on $d$, and we can handle infinite-dimensional settings via the kernel trick, in particular establishing new regret bounds for the Matérn kernel that are the first to be sublinear for all $ε\in (0, 1]$.

Junyan Liu, Arnab Maiti, Artin Tajdini et al.

We initiate the study of a repeated principal-agent problem over a finite horizon $T$, where a principal sequentially interacts with $K\geq 2$ types of agents arriving in an adversarial order. At each round, the principal strategically chooses one of the $N$ arms to incentivize for an arriving agent of unknown type. The agent then chooses an arm based on its own utility and the provided incentive, and the principal receives a corresponding reward. The objective is to minimize regret against the best incentive in hindsight. Without prior knowledge of agent behavior, we show that the problem becomes intractable, leading to linear regret. We analyze two key settings where sublinear regret is achievable. In the first setting, the principal knows the arm each agent type would select greedily for any given incentive. Under this setting, we propose an algorithm that achieves a regret bound of $O(\min\{\sqrt{KT\log N},K\sqrt{T}\})$ and provide a matching lower bound up to a $\log K$ factor. In the second setting, an agent's response varies smoothly with the incentive and is governed by a Lipschitz constant $L\geq 1$. Under this setting, we show that there is an algorithm with a regret bound of $\tilde{O}((LN)^{1/3}T^{2/3})$ and establish a matching lower bound up to logarithmic factors. Finally, we extend our algorithmic results for both settings by allowing the principal to incentivize multiple arms simultaneously in each round.