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Information Complexity: an Overview

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. Classical Information Theory. Shannon ‘48, A Mathematical Theory of Communication :.

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Information Complexity: an Overview

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  1. 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

  2. Classical Information Theory • Shannon ‘48, A Mathematical Theory of Communication:

  3. Motivation: Communication Complexity = ? Yao ‘79, “Some complexity questions related to distributive computing”

  4. Motivation: Communication Complexity More generally: solve some task Yao ‘79, “Some complexity questions related to distributive computing”

  5. Motivation: Communication Complexity • Applications: • Circuit complexity • Streaming algorithms • Data structures • Distributed computing • Property testing • …

  6. Example: Streaming Lower Bounds • Streaming algorithm: • Reduction from communication complexity [AMS’97] How much space is required to approximate f(data)? algorithm data

  7. Example: Streaming Lower Bounds State of the algorithm • Streaming algorithm: • Reduction from communication complexity [Alon, Matias, Szegedy ’99] algorithm data

  8. Advances in Communication Complexity • Very successful in proving unconditional lower bounds, e.g., • for set disjointness[KS’92, Razborov ‘92] • for gap hamming distance [Chakrabarti, Regev ‘10] • But stuck on some hard questions • Multi-party communication complexity • Karchmer-Wigderson games • [Chakrabarty, Shi, Wirth, Yao ’01], [Bar-Yossef, Kumar, Jayram, Srivakumar ‘04]: use tools from information theory

  9. Extending Information Theory to Interactive Computation • One-way communication: • Task: send across the channel • Cost: bits • Shannon: in the limit over many instances • Huffman: bits for one instance • Interactive computation: • Task: e.g., compute • Cost?

  10. Information Cost • Reminder: mutual information • Conditional mutual information: • Basic properties: • and • Chain rule:

  11. Information Cost • Fix a protocol • Notation abuse: let also denote the transcript of the protocol • Two ways to measure information cost: • External information cost: • Internal information cost: • Cost of a task: infimum over all protocols • Which cost is “the right one”?

  12. Information Cost: Basic Properties External information: Internal information: • Internal external • Can be much smaller, e.g.: • uniform over • Alice sends to Bob • But equal if inependent

  13. Information Cost: Basic Properties External information: Internal information: • External information communication:

  14. Information Cost: Basic Properties • Internal information communication cost: • By induction: let . • : what we know after r rounds what we knew after r-1 rounds what we learn in round r, given what we already know I.H.

  15. Information vs. Communication • Want: • Suppose is sent by Alice. • What does Alice learn? • is a function of and so • What does Bob learn?

  16. Information vs. Communication • We have: Internal information communication External information communication Internal information external information

  17. Information vs. Communication • “Information cost = communication cost”? • In the limit: internal information! [Braverman, Rao ‘10] • For one instance: external information! [Braverman, Barak, Rao, Chen ‘10] Big question: can protocols be compressed down to their internal information cost? • [Ganor, Kol, Raz ’14]: no! • There is a task with internal IC=, CC=. … but: remains open for functions, small output.

  18. Information vs. Amortized Communication • Theorem [Braverman, Rao ‘10]: • The “” direction: compression • The “” direction: direct sum • We know: • We can show:

  19. Direct Sum Theorem [BR‘10] • Let be a protocol for on -copy inputs • Construct for as follows: • Alice and Bob get inputs • Choose a random coordinate , set • Bad idea: publicly sample

  20. Direct Sum Theorem [BR‘10] • Let be a protocol for on -copy inputs • Construct for as follows: • Alice and Bob get inputs • Choose a random coordinate , set • Bad idea: publicly sample Suppose in , Alice sends . In , Bob learns one bit in he should learn bit But if is public Bob learns 1 bit about !

  21. Direct Sum Theorem [BR‘10] • Let be a protocol for on -copy inputs • Construct for as follows: • Alice and Bob get inputs • Choose a random coordinate , set Publicly sample Privately sample Privately sample Publicly sample

  22. Compression • What we know: a protocol with communication , internal info and external info can be compressed to • [BBCR’10] • [BBCR’10] • [Braverman’10] • Major open question:can we compress to [GKR, partial answer: no]

  23. Using Information Complexity to Prove Communication Lower Bounds • Internal/external info communication • Essentially the most powerful technique known [Kerenidis,Laplante,Lerays,Roland,Xiao’12]: most lower bound techniques imply IC lower bounds • Disadvantage: hard to show incompressibility! • Must exhibit problem with low IC, high CC • But proving high CC usually proves high IC…

  24. 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]

  25. Extending IC to Multiple Players • Multi-player computation traditionally hard to analyze • [Braverman,Ellen,O.,Pitassi,Vaikuntanathan]: for Set Disjointness with elements, players, private channels, NIH input

  26. Information Complexity on Private Channels • First obstacle: secure multi-party computation • [Goldreich,Micali,Wigderson’87]: any function can be computed with perfect information-theoretic security against players • Solution: redefine information cost, measure both • Information a player learns, and • Information a player leaks to all the others.

  27. Extending IC to Multiple Players • Set disjointness: • Input: • Output: • Open problem: can we extend to gap set disjointness? • First step: “purely info-theoretic” 2-party analysis

  28. Extending IC to Multiple Players • In [Braverman,Ellen,O.,Pitassi,Vaikuntanathan] we show direct sum for multi-party • Solving instances = solving one instance • Does direct sum hold “across players”? • Solving with players = solving with 2 players? • Not always • Does compression work for multi-party?

  29. 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, …

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