Towards Agnostic Feature-based Dynamic Pricing: Linear Policies vs Linear Valuation with Unknown Noise

Abstract

In feature-based dynamic pricing, a seller sets appropriate prices for a sequence of products (described by feature vectors) on the fly by learning from the binary outcomes of previous sales sessions ("Sold" if valuation ≥ price, and "Not Sold" otherwise). Existing works either assume noiseless linear valuation or preciselyknown noise distribution, which limits the applicability of those algorithms in practice when these assumptions are hard to verify. In this work, we study two more agnostic models: (a) a "linear policy" problem where we aim at competing with the best linear pricing policy while making no assumptions on the data, and (b) a "linear noisy valuation" problem where the random valuation is linear plus an unknown and assumption-free noise. For the former model, we show a Θ(d) minimax regret up to logarithmic factors. For the latter model, we present an algorithm that achieves an Õ(T 3 4 ) regret, and improve the best-known lower bound from Ω(T 3 5 ) to Ω(T 2 3 ). These results demonstrate that no-regret learning is possible for feature-based dynamic pricing under weak assumptions, but also reveal a disappointing fact that the seemingly richer pricing feedback is not significantly more useful than the bandit-feedback in regret reduction.Contextual pricing. For t = 1, 2, ..., T :revealed that describes a sales session (product, customer and context). 2. The customer valuates the product as yt using xt. 3. The seller proposes a price vt > 0 concurrently (according to xt and historical sales records).4. The transaction is successful if vt ≤ yt, i.e., the seller gets a reward (payment) of rt = vt • 1(vt ≤ yt).