Multi-Armed Bandits

This is an umbrella project for several related efforts at Microsoft Research Silicon Valley that address various Multi-Armed Bandit (MAB) formulations motivated by web search and ad placement. The MAB problem is a classical paradigm in Machine Learning in which an online algorithm chooses from a set of strategies in a sequence of trials so as to maximize the total payoff of the chosen strategies.

This page is inactive since the closure of MSR-SVC in September 2014.

The name “multi-armed bandits” comes from a whimsical scenario in which a gambler faces several slot machines, a.k.a. “one-armed bandits”, that look identical at first but produce different expected winnings. The crucial issue here is the trade-off between acquiring new information (exploration) and capitalizing on the information available so far (exploitation). While the MAB problems have been studied extensively in Machine Learning, Operations Research and Economics, many exciting questions are open. One aspect that we are particularly interested in concerns modeling and efficiently using various types of side information that may be available to the algorithm.

Contact: Alex Slivkins.

Research directions

  • MAB with similarity information
  • MAB in a changing environment
  • Explore-exploit tradeoff in mechanism design
  • Explore-exploit learning with limited resources
  • Risk vs. reward tradeoff in MAB

External visitors and collaborators

Prof. Sébastien Bubeck (Princeton)
Prof. Robert Kleinberg (Cornell)
Filip Radlinski (MSR Cambridge)
Prof. Eli Upfal (Brown)

Former interns
Yogi Sharma (Cornell —> Facebook; intern at MSR-SV in summer 2008)
Umar Syed (Princeton —> Google; intern at MSR-SV in summer 2008)
Shaddin Dughmi (Stanford —>USC; intern at MSR-SV in summer 2010)
Ashwinkumar Badanidiyuru (Cornell –> Google; intern at MSR-SV in summer 2012)

MAB problems with similarity information

  • Multi-armed bandits in metric spaces
    Robert Kleinberg, Alex Slivkins and Eli Upfal (STOC 2008)
    Abstract We introduce a version of the stochastic MAB problem, possibly with a very large set of arms, in which the expected payoffs obey a Lipschitz condition with respect to a given metric space. The goal is to minimize regret as a function of time, both in the worst case and for ‘nice’ problem instances.
  • Sharp dichotomies for regret minimization in metric spaces
    Robert Kleinberg and Alex Slivkins (SODA 2010)
    Abstract We focus on the connections between online learning and metric topology. The main result that the worst-case regret is either O(log t) or at least sqrt{t}, depending on whether the completion of the metric space is compact and countable. We prove a number of other dichotomy-style results, and extend them to the full-feedback (experts) version.
  • Learning optimally diverse rankings over large document collections
    Alex Slivkins, Filip Radlinski and Sreenivas Gollapudi (ICML 2010)
    Abstract We present a learning-to-rank framework for web search that incorporates similarity and correlation between documents and thus, unlike prior work, scales to large document collections.
  • Contextual bandits with similarity information
    Alex Slivkins (COLT 2011)
    Abstract In the ‘contextual bandits’ setting, in each round nature reveals a ‘context’ x, algorithm chooses an ‘arm’ y, and the expected payoff is µ(x,y). Similarity info is expressed by a metric space over the (x,y) pairs such that µ is a Lipschitz function. Our algorithms are based on adaptive (rather than uniform) partitions of the metric space which are adjusted to the popular and high-payoff regions.
  • Multi-armed bandits on implicit metric spaces
    Alex Slivkins (NIPS 2011)
    Abstract Suppose an MAB algorithm is given a tree-based classification of arms. This tree implicitly defines a “similarity distance” between arms, but the numeric distances are not revealed to the algorithm. Our algorithm (almost) matches the best known guarantees for the setting (Lipschitz MAB) in which the distances are revealed.

MAB problems in a changing environment

  • Adapting to a stochastically changing environment
    Alex Slivkins and Eli Upfal (COLT 2008)
    Abstract We study a version of the stochastic MAB problem in which the expected reward of each arm evolves stochastically and gradually in time, following an independent Brownian motion or a similar process. Our benchmark is a hypothetical policy that chooses the best arm in each round.
  • Adapting to the Shifting Intent of Search Queries
    Umar Syed, Alex Slivkins and Nina Mishra (NIPS 2009)
    Abstract Query intent may shift over time. A classifier can use the available signals to predict a shift in intent. Then a bandit algorithm can be used to find the new relevant results. We present a meta-algorithm that combines such classifier with a bandit algorithm in a feedback loop.
  • Contextual bandits with similarity information
    Alex Slivkins (COLT 2011)
    Abstract Interpreting the current time as a part of the contextual information, we obtain a very general bandit framework that (in addition to similarity between arms and contexts) can include slowly changing payoffs and variable sets of arms.
  • The best of both worlds: stochastic and adversarial bandits
    Sebastien Bubeck and Alex Slivkins (COLT 2012)
    Abstract We present a new bandit algorithm whose regret is optimal both for adversarial rewards and for stochastic rewards, achieving, resp., square-root regret and polylog regret.

Explore-exploit tradeoff in mechanism design

  • Characterizing truthful multi-armed bandit mechanisms
    Moshe Babaioff, Alex Slivkins and Yogi Sharma (EC 2009)
    Abstract We consider a multi-round auction setting motivated by pay-per-click auctions in the Internet advertising, which can be viewed as a strategic version of the MAB problem. We investigate how the design of MAB algorithms is affected by the restriction of truthfulness. We show striking differences in terms of both the structure of an algorithm and its regret. 
  • Truthful mechanisms with implicit payment computation
    Moshe Babaioff, Robert Kleinberg and Alex Slivkins (EC 2010 Best Paper Award)
    Abstract We show that any single-parameter, monotone allocation function can be truthfully implemented using a single call to this function. We apply this to MAB mechanisms.
  • Monotone multi-armed bandit allocations
    Alex Slivkins (Open Question at COLT 2011)
    Abstract The reduction in the EC’10 paper opens up a problem of designing monotone MAB allocation rules, a new twist on the familiar MAB problem.
  • Multi-parameter mechanisms with implicit payment computation
    Moshe Babaioff, Robert Kleinberg and Alex Slivkins (EC 2013)
    Abstract We generalize the main result of the EC’10 paper to the multi-parameter setting. We apply this to a natural multi-parameter extension of MAB mechanisms.

Explore-exploit learning with limited resources

  • Dynamic pricing with limited supply
    Moshe Babaioff, Shaddin Dughmi, Robert Kleinberg and Alex Slivkins (EC 2012)
    Abstract We consider dynamic pricing with limited supply and unknown demand distribution. We extend MAB techniques to the limited supply setting, and obtain optimal regret rates.
  • Bandits with Knapsacks
    Ashwinkumar Badanidiyuru, Robert Kleinberg and Alex Slivkins (FOCS 2013)
    Abstract We define a broad class of explore-exploit problems with knapsack-style resource constraints, which subsumes dynamic pricing, dynamic procurement, pay-per-click ad allocation, and many other problems. Our algorithms achieve optimal regret w.r.t. the optimal dynamic policy.

Risk vs. reward trade-off in MAB