Multi-armed Bandit Problems with Dependent Arms Sandeep Pandey (spandey@cs.cmu.edu) Deepayan Chakrabarti (deepay@yahoo-inc.com) Deepak Agarwal (dagarwal@yahoo-inc.com)

Background: Bandits Bandit “arms” μ1 μ2 μ3 (unknown reward probabilities) Pull arms sequentially so as to maximize the total expected reward Show ads on a webpage to maximize clicks Product recommendation to maximize sales

“Skiing, snowboarding” Dependent Arms Reward probabilities μi are generally assumed to be independent of each other What if they are dependent? E.g., ads on similar topics, using similar text/phrases, should have similar rewards “Skiing, snowboarding” “Skiing, snowshoes” “Snowshoe rental” “Get Vonage!” μ1=0.3 μ2=0.28 μ2=0.31 μ3=10-6

Dependent Arms Reward probabilities μi are generally assumed to be independent of each other What if they are dependent? E.g., ads on similar topics, using similar text/phrases, should have similar rewards A click on one ad  other “similar” ads may generate clicks as well Can we increase total reward using this dependency?

Cluster Model of Dependence Arm 1 Arm 2 Arm 3 Arm 4 Cluster 1 Cluster 2 # pulls of arm i Successes si ~ Bin(ni, μi) μi ~ f(π[i]) No dependence across clusters Some distribution (known) Cluster-specific parameter (unknown)

Cluster Model of Dependence Arm 1 Arm 2 Arm 3 Arm 4 μi ~ f(π1) μi ~ f(π2) Total reward: Discounted: ∑ αt.E[R(t)], α = discounting factor Undiscounted: ∑ E[R(t)] t=0 ∞ T

Discounted Reward Arm 2 x’1 x’2 The optimal policy can be computed using per-cluster MDPs only. MDP for cluster 1 Pull Arm 1 x1 x2 x”1 x”2 Optimal Policy: Compute an (“index”, arm) pair for each cluster Pick the cluster with the largest index, and pull the corresponding arm Arm 4 x’3 x’4 MDP for cluster 2 Pull Arm 3 x3 x4 x”3 x”4

Discounted Reward Arm 2 x’1 x’2 The optimal policy can be computed using per-cluster MDPs only. MDP for cluster 1 Pull Arm 1 Reduces the problem to smaller state spaces Reduces to Gittins’ Theorem [1979] for independent bandits Approximation bounds on the index for k-step lookahead x1 x2 x”1 x”2 Optimal Policy: Compute an (“index”, arm) pair for each cluster Pick the cluster with the largest index, and pull the corresponding arm Arm 4 x’3 x’4 MDP for cluster 2 Pull Arm 3 x3 x4 x”3 x”4

Cluster Model of Dependence Arm 1 Arm 2 Arm 3 Arm 4 μi ~ f(π1) μi ~ f(π2) Total reward: Discounted: ∑ αt.E[R(t)], α = discounting factor Undiscounted: ∑ E[R(t)] ∞ t=0 T t=0

Undiscounted Reward “Cluster arm” 1 “Cluster arm” 2 Arm 1 Arm 2 Arm 3 Arm 4 All arms in a cluster are similar  They can be grouped into one hypothetical “cluster arm”

Undiscounted Reward Two-Level Policy In each iteration: Pick “cluster arm” using a traditional bandit policy Pick an arm within that cluster using a traditional bandit policy “Cluster arm” 1 “Cluster arm” 2 Each “cluster arm” must have some estimated reward probability Arm 1 Arm 2 Arm 3 Arm 4

Issues What is the reward probability of a “cluster arm”? How do cluster characteristics affect performance?

Reward probability of a “cluster arm” What is the reward probability r of a “cluster arm”? MEAN: r = ∑si / ∑ni, i.e., average success rate, summing over all arms in the cluster [Kocsis+/2006, Pandey+/2007] Initially, r = μavg = average μ of arms in cluster Finally, r = μmax = max μ among arms in cluster “Drift” in the reward probability of the “cluster arm”

Reward probability drift causes problems Best (optimal) arm, with reward probability μopt Arm 1 Arm 2 Arm 3 Arm 4 Cluster 1 Cluster 2 (opt cluster) Drift  Non-optimal clusters might temporarily look better  optimal arm is explored only O(log T) times

Reward probability of a “cluster arm” What is the reward probability r of a “cluster arm”? MEAN: r = ∑si / ∑ni MAX: r = max( E[μi] ) PMAX: r = E[ max(μi) ] Both MAX and PMAX aim to estimate μmax and thus reduce drift for all arms i in cluster

Reward probability of a “cluster arm” Bias in estimation of μmax MEAN: r = ∑si / ∑ni MAX: r = max( E[μi] ) PMAX: r = E[ max(μi) ] Both MAX and PMAX aim to estimate μmax and thus reduce drift Variance of estimator High Unbiased Low High

Comparison of schemes 10 clusters, 11.3 arms/cluster MAX performs best

Issues What is the reward probability of a “cluster arm”? How do cluster characteristics affect performance?

Effects of cluster characteristics We analytically study the effects of cluster characteristics on the “crossover-time” Crossover-time: Time when the expected reward probability of the optimal cluster becomes highest among all “cluster arms”

Effects of cluster characteristics Crossover-time Tc for MEAN depends on: Cluster separation Δ = μopt – μmax outside opt cluster Δ increases  Tc decreases Cluster size Aopt Aopt increases  Tc increases Cohesiveness in opt cluster 1-avg(μopt – μi) Cohesiveness increases  Tc decreases

Experiments (effect of separation) Circle mean, triangle max, square indep. Δ increases  Tc decreases  higher reward

Experiments (effect of size) Aopt increases  Tc increases  lower reward

Experiments (effect of cohesiveness) Cohesiveness increases  Tc decreases  higher reward

Related Work Typical multi-armed bandit problems Do not consider dependencies Very few arms Bandits with side information Cannot handle dependencies among arms Active learning Emphasis on #examples required to achieve a given prediction accuracy Citations!

Conclusions We analyze bandits where dependencies are encapsulated within clusters Discounted Reward  the optimal policy is an index scheme on the clusters Undiscounted Reward Two-level Policy with MEAN, MAX, and PMAX Analysis of the effect of cluster characteristics on performance, for MEAN

Discounted Reward 1 2 3 4 x”1 x”2 x’1 x’2 x3 x4 Pull Arm 1 success failure Change of belief for both arms 1 and 2 x1 x2 Estimated reward probabilities x3 x4 Create a belief-state MDP Each state contains the estimated reward probabilities for all arms Solve for optimal

Background: Bandits Regret = optimal payoff – actual payoff Bandit “arms” p1 p2 p3 (unknown payoff probabilities) Regret = optimal payoff – actual payoff

Reward probability of a “cluster arm” What is the reward probability of a “cluster arm”? Eventually, every “cluster arm” must converge to the most rewarding arm μmax within that cluster since a bandit policy is used within each cluster However, “drift” causes problems

Experiments Simulation based on one week’s worth of data from a large-scale ad-matching application 10 clusters, with 11.3 arms/cluster on average

Comparison of schemes MAX performs best 10 clusters, 11.3 arms/cluster Cluster separation Δ = 0.08 Cluster size Aopt = 31 Cohesiveness = 0.75 MAX performs best

Reward probability drift causes problems Best (optimal) arm, with reward probability μopt Arm 1 Arm 2 Arm 3 Arm 4 Cluster 1 Cluster 2 (opt cluster) Intuitively, to reduce regret, we must: Quickly converge to the optimal “cluster arm” and then to the best arm within that cluster