Post on 29-Mar-2015
Multi Armed Bandits
chalpert@meetup.com
Survey
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Click-through Rate (Clicks / Impressions)20%
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Click-through Rate20% ?
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Click-through Rate20% ?
AB Test• Randomized Controlled Experiment• Show each button to 50% of users
AB Test Timeline
AB Test
AB TestAfter Test (show winner)Before Test
Time
Exploration Phase(Testing)
Exploitation Phase(Show Winner)
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Click-through Rate20% ?
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Click-through Rate20% 30%
• 10,000 impressions/month
• Need 4,000 clicks by EOM• 30% CTR won’t be enough
Need to keep testing (Exploration)
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ABCDEFG... TestEach variant would be assigned with probability 1/N
N = # of variants
Not everyone is a winner
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ABCDEFG... TestEach variant would be assigned with probability 1/N
N = # of variants
Need to keep testing (Exploration)
Need to minimize regret(Exploitation)
Multi Armed Bandit
Balance of Exploitation & Exploration
Bandit Algorithm Balances Exploitation & Exploration
Multi Armed BanditBefore Test
Time
AB Test
AB TestAfter TestBefore Test
Discrete Exploitation & Exploration Phases
Continuous Exploitation & Exploration
Bandit Favors Winning Arm
Bandit Algorithm Reduces Risk of Testing
AB TestBest arm exploited with probability 1/N– More Arms: Less exploitation
BanditBest arm exploited with determined probability– Reduced exposure to suboptimal arms
Demo
Borrowed fromProbabilistic Programming & Bayesian Methods for Hackers
Split Test
Bandit
Winner Breaks Away!
Still sending losers
AB test would have cost 4.3 percentage points
How it works
Epsilon Greedy Algorithmε = Probability of Exploration
ε
1 - ε
Exploration
Exploitation(show best arm)
Start of round
1 / N
1 / N
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ε / N
1-ε
ε / N
Epsilon Greedy with ε = 1 = AB Test
Epsilon Greedy Issues
• Constant Epsilon:– Initially under exploring– Later over exploring– Better if probability of exploration decreases with
sample size (annealing)• No prior knowledge
Some Alternatives
• Epsilon-First• Epsilon-Decreasing• Softmax• UCB (UCB1, UCB2)• Bayesian-UCB• Thompson Sampling (Bayesian Bandits)
Bandit Algorithm Comparison
Regret:
Thompson Sampling
Setup: Assign each arm a Beta distribution with parameters (α,β) (# Success, # Failures)
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Beta(α,β) Beta(α,β) Beta(α,β)
Thompson Sampling
Setup: Initialize priors with ignorant state of Beta(1,1) (Uniform distribution)- Or initialize with an informed prior to aid convergence
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Beta(1,1) Beta(1,1) Beta(1,1)
For each round:
Thompson Sampling
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Beta(1,1) Beta(1,1) Beta(1,1)
1: Sample random variable X from each arm’s Beta Distribution
2: Select the arm with largest X3: Observe the result of selected arm
4: Update prior Beta distribution for selected arm
X
Success!
0.7 0.2 0.4
For each round:
Thompson Sampling
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Beta(2,1) Beta(1,1) Beta(1,1)
1: Sample random variable X from each arm’s Beta Distribution
2: Select the arm with largest X3: Observe the result of selected arm
4: Update prior Beta distribution for selected arm
X
Success!
0.7 0.2 0.4
For each round:
Thompson Sampling
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Beta(2,1) Beta(1,1) Beta(1,1)
1: Sample random variable X from each arm’s Beta Distribution
2: Select the arm with largest X3: Observe the result of selected arm
4: Update prior Beta distribution for selected arm
X
Failure!
0.4 0.8 0.2
For each round:
Thompson Sampling
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Beta(2,1) Beta(1,2) Beta(1,1)
1: Sample random variable X from each arm’s Beta Distribution
2: Select the arm with largest X3: Observe the result of selected arm
4: Update prior Beta distribution for selected arm
X
Failure!
0.4 0.8 0.2
Posterior after 100k pulls (30 arms)
Bandits at Meetup
Meetup’s First Bandit
Control: Welcome To Meetup! - 60% Open RateWinner: What?Winner: Hi - 75% Open Rate (+25%)
76 Arms
Control: Welcome To Meetup! - 60% Open RateWinner: What?Winner: Hi - 75% Open Rate (+25%)
76 Arms
Control: Welcome To Meetup! - 60% Open RateWinner: What?Winner: Hi - 75% Open Rate (+25%)
76 Arms
Avoid Linkbaity Subject Lines
Control: Save 50%, start your Meetup Group – 42% Open RateWinner: Here is a coupon – 53% Open Rate (+26%)
16 Arms
Coupon Email
398 Arms
210% Click-through Difference:
Best:Looking to start the perfect Meetup for you?We’ll help you find just the right people
Start the perfect Meetup for you!We’ll help you find just the right people
Worst:Launch your own Meetup in January and save 50%Start the perfect Meetup for you50% off promotion ends February 1st.
Choose the Right Metric of Success
• Success tied to click in last experiment• Sale end & discount messaging had bad
results• Perhaps people don’t know that hosting a
Meetup costs $$$?– Better to tie success to group creation
More Issues
• Email open & click delay• New subject line effect– Problem when testing notifications
• Monitor success trends to detect weirdness
Seasonality
• Thompson Sampling should naturally adapt to seasonal changes– Learning rate can be added for faster adaptation
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Winner all other times
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Bandit or Split Test?
AB Test good for:- Biased Tests- Complicated Tests
Bandit good for:- Unbiased Tests- Many Variants- Time Restraints- Set It And Forget It
Thanks!
chalpert@meetup.com