Online Regularization towards Always-Valid High-Dimensional Dynamic Pricing

Devising a dynamic pricing policy with always valid online statistical learning procedures is an important and as yet unresolved problem. Most existing dynamic pricing policies, which focus on the faithfulness of adopted customer choice models, exhibit a limited capability for adapting to the online uncertainty of learned statistical models during the pricing process. In this paper, we propose a novel approach for designing a dynamic pricing policy based on regularized online statistical learning with theoretical guarantees. The new approach overcomes the challenge of continuous monitoring of the online Lasso procedure and possesses several appealing properties. In particular, we make the decisive observation that the always-validity of pricing decisions builds and thrives on the <i>online regularization</i> scheme. Our proposed online regularization scheme equips the proposed optimistic online regularized maximum likelihood pricing (OORMLP) pricing policy with three major advantages: encode market noise knowledge into pricing process optimism; empower online statistical learning with always-validity overall decision points; envelop prediction error process with time-uniform non-asymptotic oracle inequalities. This type of non-asymptotic inference results allows us to design more sample-efficient and robust dynamic pricing algorithms in practice. In theory, the proposed OORMLP algorithm exploits the sparsity structure of high-dimensional models and secures a logarithmic regret in a decision horizon. These theoretical advances are made possible by proposing an optimistic online Lasso procedure that resolves dynamic pricing problems at the <i>process</i> level, based on a novel use of non-asymptotic martingale concentration. In experiments, we evaluate OORMLP in different synthetic and real pricing problem settings and demonstrate that OORMLP advances the state-of-the-art methods.