In many domains, collecting sufficient labeled training data for supervised machine learning requires easily accessible but noisy sources, such as crowdsourcing services or tagged Web data. Noisy labels occur frequently in data sets harvested via these means, sometimes resulting in entire classes of data on which learned classifiers generalize poorly. For real world applications, we argue that it can be beneficial to avoid training on such classes entirely. In this work, we aim to explore the classes in a given data set, and guide supervised training to spend time on a class proportional to its learnability. By focusing the training process, we aim to improve model generalization on classes with a strong signal. To that end, we develop an online algorithm that works in conjunction with classifier and training algorithm, iteratively selecting training data for the classifier based on how well it appears to generalize on each class. Testing our approach on a variety of data sets, we show our algorithm learns to focus on classes for which the model has low generalization error relative to strong baselines, yielding a classifier with good performance on learnable classes.
In our approach, data is selected at each training step using a Multi-Armed Bandit algorithm, a problem first introduced in @cite_17 . The original formulation of the problem assumes that the reward for a given decision made by the algorithm is drawn from a fixed distribution every time that decision is made (i.e. a stochastic payoff function). In our setting, rewards are based on the state of a classifier at a given training step, which changes over time. Bandit algorithms developed for such scenarios are called MAB algorithms @cite_5 , and make no assumptions on the payoff structure of decisions. We additionally assume that the class set we are choosing from is large, rendering naive selection strategies ineffective. For these settings, it is common to assume a structure on the decision-space, specifically that there exists a means of measuring similarity between decisions @cite_25 . One such approach is to assume that the reward function changes smoothly over the decision space, i.e that similar decisions will have similar payoffs @cite_16 . We use a time-varying version of this algorithm presented in @cite_10 .
@cite_5 @cite_16 @cite_10 @cite_25 @cite_17
2742049221 2148434045 1761637522 1620761767
The multi-armed bandit (MAB) problem features the classical tradeoff between exploration and exploitation. The input specifies several stochastic arms which evolve with each pull, and the goal is to maximize the expected reward after a fixed budget of pulls. The celebrated work of , surveyed in [8], presumes a condition on the arms called the martingale assumption. In [9], A. obtained an LP-based 1 48-approximation for the problem with the martingale assumption removed. We improve the algorithm to a 4 27-approximation, with simpler analysis. Our algorithm also generalizes to the case of MAB superprocesses with (stochastic) multi-period actions. This generalization captures the framework introduced by Guha and Munagala in [11], and yields new results for their budgeted learning problems. Also, we obtain a (1 2 -- e)-approximation for the variant of MAB where preemption (playing an arm, switching to another arm, then coming back to the first arm) is not allowed. This contains the stochastic knapsack problem of Dean, Goemans, and Vondrak in [6] with correlated rewards, where we are given a knapsack of fixed size, and a set of jobs each with a joint distribution for its size and reward. The actual size and reward of a job can only be discovered in real-time as it is being scheduled, and the objective is to maximize expected reward before the knapsack size is exhausted. Our (1 2 -- e)-approximation improves the 1 16 and 1 8 approximations in [9] for correlated stochastic knapsack with cancellation and no cancellation, respectively, providing to our knowledge the first tight algorithm for these problems that matches the integrality gap of 2. We sample probabilities from an exponential-sized dynamic programming solution, whose existence is guaranteed by an LP projection argument. We hope this technique can also be applied to other dynamic programming problems which can be projected down onto a small LP. In a multi-armed bandit (MAB) problem, an online algorithm makes a sequence of choices. In each round it chooses from a time-invariant set of alternatives and receives the payoff associated with this alternative. While the case of small strategy sets is by now well-understood, a lot of recent work has focused on MAB problems with exponentially or infinitely large strategy sets, where one needs to assume extra structure in order to make the problem tractable. In particular, recent literature considered information on similarity between arms. We consider similarity information in the setting of contextual bandits, a natural extension of the basic MAB problem where before each round an algorithm is given the context--a hint about the payoffs in this round. Contextual bandits are directly motivated by placing advertisements on web pages, one of the crucial problems in sponsored search. A particularly simple way to represent similarity information in the contextual bandit setting is via a similarity distance between the context-arm pairs which bounds from above the difference between the respective expected payoffs. Prior work on contextual bandits with similarity uses "uniform" partitions of the similarity space, so that each context-arm pair is approximated by the closest pair in the partition. Algorithms based on "uniform" partitions disregard the structure of the payoffs and the context arrivals, which is potentially wasteful. We present algorithms that are based on adaptive partitions, and take advantage of "benign" payoffs and context arrivals without sacrificing the worst-case performance. The central idea is to maintain a finer partition in high-payoff regions of the similarity space and in popular regions of the context space. Our results apply to several other settings, e.g., MAB with constrained temporal change (Slivkins and Upfal, 2008) and sleeping bandits (, 2008a). We introduce the budget-limited multi-armed bandit (MAB), which captures situations where a learner's actions are costly and constrained by a fixed budget that is incommensurable with the rewards earned from the bandit machine, and then describe a first algorithm for solving it. Since the learner has a budget, the problem's duration is finite. Consequently an optimal exploitation policy is not to pull the optimal arm repeatedly, but to pull the combination of arms that maximises the agent's total reward within the budget. As such, the rewards for all arms must be estimated, because any of them may appear in the optimal combination. This difference from existing MABs means that new approaches to maximising the total reward are required. To this end, we propose an ∊-first algorithm, in which the first ∊ of the budget is used solely to learn the arms' rewards (exploration), while the remaining 1 - ∊ is used to maximise the received reward based on those estimates (exploitation). We derive bounds on the algorithm's loss for generic and uniform exploration methods, and compare its performance with traditional MAB algorithms under various distributions of rewards and costs, showing that it outperforms the others by up to 50 . In a multi-armed bandit problem, 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. While the performance of bandit algorithms with a small finite strategy set is quite well understood, bandit problems with large strategy sets are still a topic of very active investigation, motivated by practical applications such as online auctions and web advertisement. The goal of such research is to identify broad and natural classes of strategy sets and payoff functions which enable the design of efficient solutions. In this work we study a very general setting for the multi-armed bandit problem in which the strategies form a metric space, and the payoff function satisfies a Lipschitz condition with respect to the metric. We refer to this problem as the "Lipschitz MAB problem". We present a solution for the multi-armed bandit problem in this setting. That is, for every metric space we define an isometry invariant which bounds from below the performance of Lipschitz MAB algorithms for this metric space, and we present an algorithm which comes arbitrarily close to meeting this bound. Furthermore, our technique gives even better results for benign payoff functions. We also address the full-feedback ("best expert") version of the problem, where after every round the payoffs from all arms are revealed.