Two Years of Bayesian Bandits for E-Commerce

NYC College of Technology • April, 18 2019 • @AustinRochford

About Me

Chief Data Scientist, Monetate

@AustinRochford • austinrochford.com • github.com/AustinRochford

arochford@monetate.com • austin.rochford@gmail.com

About Monetate

• Founded 2008, web optimization and personalization SaaS
• Observed 5B impressions and $4.1B in revenue during Cyber Week 2017

Non-technical marketer-focused

About this talk

Outline

• Web optimization
  • A/B testing
  • Multi-armed bandits
• Bayesian bandits
  • Thompson sampling
• Bandit bias
  • Inverse propensity weighting

Web Optimization

A/B testing

A/B testing machinery

Ronald Fisher

Whom the Gods Would Destroy, They First Give Real-time Analytics

Sequential testing

Abraham Wald

Sequential optimization

Multi-armed bandits

Multi-armed bandit systems

Bayesian Bandits

Beta-binomial model

$$ \begin{align*} x_A, x_B & = \textrm{number of rewards from users shown variant } A, B \\ x_A & \sim \textrm{Binomial}(n_A, r_A) \\ x_B & \sim \textrm{Binomial}(n_B, r_B) \\ r_A, r_B & \sim \textrm{Beta}(1, 1) \end{align*} $$

$$ \begin{align*} r_A\ |\ n_A, x_A & \sim \textrm{Beta}(x_A + 1, n_A - x_A + 1) \\ r_B\ |\ n_B, x_B & \sim \textrm{Beta}(x_B + 1, n_B - x_B + 1) \end{align*} $$

Thompson sampling

Thompson sampling randomizes user/variant assignment according to the probabilty that each variant maximizes the posterior expected reward.

The probability that a user is assigned variant A is

$$ \begin{align*} P(r_A > r_B\ |\ \mathcal{D}) & = \int_0^1 P(r_A > r\ |\ \mathcal{D})\ \pi_B(r\ |\ \mathcal{D})\ dr \\ & = \int_0^1 \left(\int_r^1 \pi_A(s\ |\ \mathcal{D})\ ds\right)\ \pi_B(r\ |\ \mathcal{D})\ dr \\ & \propto \int_0^1 \left(\int_r^1 s^{\alpha_A - 1} (1 - s)^{\beta_A - 1}\ ds\right) r^{\alpha_B - 1} (1 - r)^{\beta_B - 1}\ dr \end{align*} $$

Monte Carlo Methods

N = 5000

x, y = np.random.uniform(0, 1, size=(2, N))

fig