MNL-Bandit with Knapsacks: a near-optimal algorithm

TL;DR

This paper introduces a near-optimal UCB-based algorithm for dynamic assortment selection with unknown demand modeled by MNL, achieving regret bounds that adapt to inventory size over time.

Contribution

The paper proposes MNLwK-UCB, a novel algorithm with regret bounds that are near-optimal and adapt to different inventory regimes in the MNL-bandit with knapsacks setting.

Findings

Achieves $ ilde{O}(N + ext{sqrt}(NT))$ regret for large inventories.

Attains regret bounds that scale with inventory growth rate for smaller inventories.

Guarantees a $ ilde{O}( ext{sqrt}(NT))$ regret rate over long horizons.

Abstract

We consider a dynamic assortment selection problem where a seller has a fixed inventory of $N$ substitutable products and faces an unknown demand that arrives sequentially over $T$ periods. In each period, the seller needs to decide on the assortment of products (satisfying certain constraints) to offer to the customers. The customer's response follows an unknown multinomial logit model (MNL) with parameter $\boldsymbol{v}$. If customer selects product $i \in [N]$, the seller receives revenue $r_i$. The goal of the seller is to maximize the total expected revenue from the $T$ customers given the fixed initial inventory of $N$ products. We present MNLwK-UCB, a UCB-based algorithm and characterize its regret under different regimes of inventory size. We show that when the inventory size grows quasi-linearly in time, MNLwK-UCB achieves a $\tilde{O}(N + \sqrt{NT})$ regret bound. We also show that for a smaller inventory (with growth $\sim T^{\alpha}$, $\alpha < 1$), MNLwK-UCB achieves a $\tilde{O}(N(1 + T^{\frac{1 - \alpha}{2}}) + \sqrt{NT})$. In particular, over a long time horizon $T$, the rate $\tilde{O}(\sqrt{NT})$ is always achieved regardless of the constraints and the size of the inventory.