Stochastic Bandit Models for Delayed Conversions

TL;DR

This paper introduces a stochastic multi-armed bandit model for delayed conversions in online advertising, addressing the challenge of learning from delayed and censored reward signals with new algorithms and theoretical bounds.

Contribution

It proposes a novel bandit model for delayed and censored rewards, along with two efficient algorithms based on UCB and KLUCB frameworks, and provides performance lower bounds.

Findings

The KLUCB-based algorithm performs well with low conversion rates.

Performance lower bounds for delayed conversion bandit models are established.

The Poissonization technique aids in aggregating Bernoulli observations with different success probabilities.

Abstract

Online advertising and product recommendation are important domains of applications for multi-armed bandit methods. In these fields, the reward that is immediately available is most often only a proxy for the actual outcome of interest, which we refer to as a conversion. For instance, in web advertising, clicks can be observed within a few seconds after an ad display but the corresponding sale --if any-- will take hours, if not days to happen. This paper proposes and investigates a new stochas-tic multi-armed bandit model in the framework proposed by Chapelle (2014) --based on empirical studies in the field of web advertising-- in which each action may trigger a future reward that will then happen with a stochas-tic delay. We assume that the probability of conversion associated with each action is unknown while the distribution of the conversion delay is known, distinguishing between the (idealized) case where the conversion events may be observed whatever their delay and the more realistic setting in which late conversions are censored. We provide performance lower bounds as well as two simple but efficient algorithms based on the UCB and KLUCB frameworks. The latter algorithm, which is preferable when conversion rates are low, is based on a Poissonization argument, of independent interest in other settings where aggregation of Bernoulli observations with different success probabilities is required.