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A Short Note on the Relationship of Information Gain and Eluder Dimension
Kaixuan Huang · Sham Kakade · Jason Lee · Qi Lei

Eluder dimension and information gain are two widely used methods of complexity measures in bandit and reinforcement learning. Eluder dimension was originally proposed as a general complexity measure of function classes, but the common examples of where it is known to be small are function spaces (vector spaces). In these cases, the primary tool to upper bound the eluder dimension is the elliptic potential lemma. Interestingly, the elliptic potential lemma also features prominently in the analysis of linear bandits/reinforcement learning and their nonparametric generalization, the information gain. We show that this is not a coincidence -- eluder dimension and information gain are equivalent in a precise sense for reproducing kernel Hilbert spaces.

Author Information

Kaixuan Huang (Princeton University)
Sham Kakade (University of Washington)

Sham Kakade is a Washington Research Foundation Data Science Chair, with a joint appointment in the Department of Computer Science and the Department of Statistics at the University of Washington, and is a co-director for the Algorithmic Foundations of Data Science Institute. He works on the mathematical foundations of machine learning and AI. Sham's thesis helped in laying the foundations of the PAC-MDP framework for reinforcement learning. With his collaborators, his additional contributions include: one of the first provably efficient policy search methods, Conservative Policy Iteration, for reinforcement learning; developing the mathematical foundations for the widely used linear bandit models and the Gaussian process bandit models; the tensor and spectral methodologies for provable estimation of latent variable models (applicable to mixture of Gaussians, HMMs, and LDA); the first sharp analysis of the perturbed gradient descent algorithm, along with the design and analysis of numerous other convex and non-convex algorithms. He is the recipient of the IBM Goldberg best paper award (in 2007) for contributions to fast nearest neighbor search and the best paper, INFORMS Revenue Management and Pricing Section Prize (2014). He has been program chair for COLT 2011. Sham was an undergraduate at Caltech, where he studied physics and worked under the guidance of John Preskill in quantum computing. He then completed his Ph.D. in computational neuroscience at the Gatsby Unit at University College London, under the supervision of Peter Dayan. He was a postdoc at the Dept. of Computer Science, University of Pennsylvania , where he broadened his studies to include computational game theory and economics from the guidance of Michael Kearns. Sham has been a Principal Research Scientist at Microsoft Research, New England, an associate professor at the Department of Statistics, Wharton, UPenn, and an assistant professor at the Toyota Technological Institute at Chicago.

Jason Lee (Princeton)
Qi Lei (Princeton University)

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