36 research outputs found
Adversarial Bandits with Knapsacks
We consider Bandits with Knapsacks (henceforth, BwK), a general model for
multi-armed bandits under supply/budget constraints. In particular, a bandit
algorithm needs to solve a well-known knapsack problem: find an optimal packing
of items into a limited-size knapsack. The BwK problem is a common
generalization of numerous motivating examples, which range from dynamic
pricing to repeated auctions to dynamic ad allocation to network routing and
scheduling. While the prior work on BwK focused on the stochastic version, we
pioneer the other extreme in which the outcomes can be chosen adversarially.
This is a considerably harder problem, compared to both the stochastic version
and the "classic" adversarial bandits, in that regret minimization is no longer
feasible. Instead, the objective is to minimize the competitive ratio: the
ratio of the benchmark reward to the algorithm's reward.
We design an algorithm with competitive ratio O(log T) relative to the best
fixed distribution over actions, where T is the time horizon; we also prove a
matching lower bound. The key conceptual contribution is a new perspective on
the stochastic version of the problem. We suggest a new algorithm for the
stochastic version, which builds on the framework of regret minimization in
repeated games and admits a substantially simpler analysis compared to prior
work. We then analyze this algorithm for the adversarial version and use it as
a subroutine to solve the latter.Comment: Extended abstract appeared in FOCS 201
Approximately Stationary Bandits with Knapsacks
Bandits with Knapsacks (BwK), the generalization of the Bandits problem under
global budget constraints, has received a lot of attention in recent years.
Previous work has focused on one of the two extremes: Stochastic BwK where the
rewards and consumptions of the resources of each round are sampled from an
i.i.d. distribution, and Adversarial BwK where these parameters are picked by
an adversary. Achievable guarantees in the two cases exhibit a massive gap:
No-regret learning is achievable in the stochastic case, but in the adversarial
case only competitive ratio style guarantees are achievable, where the
competitive ratio depends either on the budget or on both the time and the
number of resources. What makes this gap so vast is that in Adversarial BwK the
guarantees get worse in the typical case when the budget is more binding. While
``best-of-both-worlds'' type algorithms are known (single algorithms that
provide the best achievable guarantee in each extreme case), their bounds
degrade to the adversarial case as soon as the environment is not fully
stochastic.
Our work aims to bridge this gap, offering guarantees for a workload that is
not exactly stochastic but is also not worst-case. We define a condition,
Approximately Stationary BwK, that parameterizes how close to stochastic or
adversarial an instance is. Based on these parameters, we explore what is the
best competitive ratio attainable in BwK. We explore two algorithms that are
oblivious to the values of the parameters but guarantee competitive ratios that
smoothly transition between the best possible guarantees in the two extreme
cases, depending on the values of the parameters. Our guarantees offer great
improvement over the adversarial guarantee, especially when the available
budget is small. We also prove bounds on the achievable guarantee, showing that
our results are approximately tight when the budget is small
Adversarial blocking bandits
We consider a general adversarial multi-armed blocking bandit setting where each played arm can be blocked (unavailable) for some time periods and the reward per arm is given at each time period adversarially without obeying any distribution. The setting models scenarios of allocating scarce limited supplies (e.g., arms) where the supplies replenish and can be reused only after certain time periods. We first show that, in the optimization setting, when the blocking durations and rewards are known in advance, finding an optimal policy (e.g., determining which arm per round) that maximises the cumulative reward is strongly NP-hard, eliminating the possibility of a fully polynomial-time approximation scheme (FPTAS) for the problem unless P = NP. To complement our result, we show that a greedy algorithm that plays the best available arm at each round provides an approximation guarantee that depends on the blocking durations and the path variance of the rewards. In the bandit setting, when the blocking durations and rewards are not known, we design two algorithms, RGA and RGA-META, for the case of bounded duration an path variation. In particular, when the variation budget B_T is known in advance, RGA can achieve O(\sqrt{T(2\tilde{D}+K)B_{T}}) dynamic approximate regret. On the other hand, when B_T is not known, we show that the dynamic approximate regret of RGA-META is at most O((K+\tilde{D})^{1/4}\tilde{B}^{1/2}T^{3/4}) where \tilde{B} is the maximal path variation budget within each batch of RGA-META (which is provably in order of o(\sqrt{T}). We also prove that if either the variation budget or the maximal blocking duration is unbounded, the approximate regret will be at least Theta(T). We also show that the regret upper bound of RGA is tight if the blocking durations are bounded above by an order of O(1)
Clustered Linear Contextual Bandits with Knapsacks
In this work, we study clustered contextual bandits where rewards and
resource consumption are the outcomes of cluster-specific linear models. The
arms are divided in clusters, with the cluster memberships being unknown to an
algorithm. Pulling an arm in a time period results in a reward and in
consumption for each one of multiple resources, and with the total consumption
of any resource exceeding a constraint implying the termination of the
algorithm. Thus, maximizing the total reward requires learning not only models
about the reward and the resource consumption, but also cluster memberships. We
provide an algorithm that achieves regret sublinear in the number of time
periods, without requiring access to all of the arms. In particular, we show
that it suffices to perform clustering only once to a randomly selected subset
of the arms. To achieve this result, we provide a sophisticated combination of
techniques from the literature of econometrics and of bandits with constraints
Towards Thompson Sampling for Complex Bayesian Reasoning
Paper III, IV, and VI are not available as a part of the dissertation due to the copyright.Thompson Sampling (TS) is a state-of-art algorithm for bandit problems set in a Bayesian framework. Both the theoretical foundation and the empirical efficiency of TS is wellexplored for plain bandit problems. However, the Bayesian underpinning of TS means that TS could potentially be applied to other, more complex, problems as well, beyond the bandit problem, if suitable Bayesian structures can be found.
The objective of this thesis is the development and analysis of TS-based schemes for more complex optimization problems, founded on Bayesian reasoning. We address several complex optimization problems where the previous state-of-art relies on a relatively myopic perspective on the problem. These includes stochastic searching on the line, the Goore game, the knapsack problem, travel time estimation, and equipartitioning. Instead of employing Bayesian reasoning to obtain a solution, they rely on carefully engineered rules. In all brevity, we recast each of these optimization problems in a Bayesian framework, introducing dedicated TS based solution schemes. For all of the addressed problems, the results show that besides being more effective, the TS based approaches we introduce are also capable of solving more adverse versions of the problems, such as dealing with stochastic liars.publishedVersio