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Information Complexity: an Overview Rotem Oshman, Princeton CCI Based on work by Braverman, Barak, Chen, Rao, and others Charles River Science of Information Day 2014 Slide 2 Classical Information Theory Shannon 48, A Mathematical Theory of Communication: Slide 3 Motivation: Communication Complexity Yao 79, Some complexity questions related to distributive computing Slide 4 Motivation: Communication Complexity Yao 79, Some complexity questions related to distributive computing Slide 5 Applications: Circuit complexity Streaming algorithms Data structures Distributed computing Property testing Motivation: Communication Complexity Slide 6 Example: Streaming Lower Bounds Streaming algorithm: Reduction from communication complexity [AMS97] data algorithm How much space is required to approximate f(data)? Slide 7 Example: Streaming Lower Bounds Streaming algorithm: Reduction from communication complexity [Alon, Matias, Szegedy 99] data algorithm State of the algorithm Slide 8 Advances in Communication Complexity Slide 9 Extending Information Theory to Interactive Computation Slide 10 Information Cost Slide 11 Slide 12 Information Cost: Basic Properties Slide 13 Slide 14 Slide 15 Information vs. Communication Slide 16 Slide 17 Slide 18 Information vs. Amortized Communication Slide 19 Direct Sum Theorem [BR10] Slide 20 Slide 21 Slide 22 Slide 23 Compression Slide 24 Using Information Complexity to Prove Communication Lower Bounds Slide 25 Extending IC to Multiple Players Recent interest in multi-player number-in- hand communication complexity Motivated by big data: Streaming and sketching, e.g., [Woodruff, Zhang 11,12,13] Distributed learning, e.g., [Awasthi, Balcan, Long 14] Slide 26 Extending IC to Multiple Players Slide 27 Information Complexity on Private Channels Slide 28 Extending IC to Multiple Players Slide 29 Slide 30 Conclusion Information complexity extends classical information theory to the interactive setting Picture is much less well-understood Powerful tool for lower bounds Fascinating open problems: Compression Information complexity for multi-player computation, quantum communication,