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Deterministic Communication Amid Uncertainty - Insights on Classical Communication Framework

Explore the concept of deterministic communication amid uncertainty in classical communication settings, tackling issues and contrasting with randomized approaches. Delve into the motivation, formalism, and behavioral aspects of natural communication, offering insight into modeling, solutions, performance, and implications for human and designed communication. Investigate the challenge of deterministic compression and consider a challenging special case involving permutation functions. This work sheds light on practical implications and theoretical foundations in the realm of communication under uncertainty.

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Deterministic Communication Amid Uncertainty - Insights on Classical Communication Framework

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  1. (Deterministic) Communication amid Uncertainty MadhuSudan Microsoft, New England Based on joint works with: (1) Adam Kalai(MSR), SanjeevKhanna(U.Penn), Brendan Juba (Harvard) and (2)EladHaramaty(Technion) MSR-I: Deterministic Communication Amid Uncertainty

  2. Classical Communication • The Shannon setting • Alice gets chosen from distribution • Sends some compression to Bob. • Bob computes • (with knowledgeof ). • Hope . • Classical Uncertainty: • Today’s talk: Bob knows . MSR-I: Deterministic Communication Amid Uncertainty

  3. Outline Part 1: Motivation Part 2: Formalism Part 3: Randomized Solution Part 4: Issues with Randomized Solution Part 5: Deterministic Issues. MSR-I: Deterministic Communication Amid Uncertainty

  4. Motivation: Human Communication • Human communication vs. Designed communication: • Human comm. dictated by languages, grammars … • Grammar: Rules, often violated. • Dictionary: multiple meanings to word. • Redundant: But error-correcting code. • Theory for human communication? • Information theory? • Linguistics? (Universal grammars etc.)? MSR-I: Deterministic Communication Amid Uncertainty

  5. Behavioral aspects of natural communication • (Vast) Implicit context. • Sender sends increasingly long messages to receiver till receiver “gets” (the meaning of) the message. • Where do the options come from? • Comes from language/dictionary – but how/why? • Sender may use feedback from receiver if available; or estimates receiver’s knowledge if not. • How does estimation influence message. MSR-I: Deterministic Communication Amid Uncertainty

  6. Model: • Reason to choose short messages: Compression. • Channel is still a scarce resource; still want to use optimally. • Reason to choose long messages (when short ones are available): Reducing ambiguity. • Sender unsure of receiver’s prior (context). • Sender wishes to ensure receiver gets the message, no matter what its prior (within reason). MSR-I: Deterministic Communication Amid Uncertainty

  7. Back to Problem • Design encoding/decoding schemes () so that • Sender has distribution on • Receiver has distribution on • Sender gets • Sends to receiver. • Receiver receives • Decodes to • Want: (provided close), • While minimizing MSR-I: Deterministic Communication Amid Uncertainty

  8. Contrast with some previous models • Universal compression? • Doesn’t apply: P,Q are not finitely specified. • Don’t have a sequence of samples from P; just one! • K-L divergence? • Measures inefficiency of compressing for Q if real distribution is P. • But assumes encoding/decoding according to same distribution Q. • Semantic Communication: • Uncertainty of sender/receiver; but no special goal. MSR-I: Deterministic Communication Amid Uncertainty

  9. Closeness of distributions: is -close to if for all , -close to (symmetrized, “worst-case” KL-divergence) MSR-I: Deterministic Communication Amid Uncertainty

  10. Dictionary = Shared Randomness? • Modelling the dictionary: What should it be? • Simplifying assumption – it is shared randomness, so … • Assume sender and receiver have some shared randomness and are independent of . • Want MSR-I: Deterministic Communication Amid Uncertainty

  11. Solution (variant of Arith. Coding) • Use to define sequences • … • where chosen s.t. Either Or MSR-I: Deterministic Communication Amid Uncertainty

  12. Performance • Obviously decoding always correct. • Easy exercise: • ( “binary entropy”) • Limits: • No scheme can achieve • Can reduce randomness needed. MSR-I: Deterministic Communication Amid Uncertainty

  13. Implications • Reflects the tension between ambiguity resolution and compression. • Larger the ((estimated) gap in context), larger the encoding length. • Coding scheme reflects the nature of human process (extend messages till they feel unambiguous). • The “shared randomness’’ is a convenient starting point for discussion • Dictionaries do have more structure. • But have plenty of entropy too. • Still … should try to do without it. MSR-I: Deterministic Communication Amid Uncertainty

  14. Deterministic Compression? • Randomness fundamental to solution. • Needs independent of to work. • Can there be a deterministic solution? • Technically: Hard to come up with single scheme that compresses consistently for all (). • Conceptually: Nicer to know “dictionary” and context can be interdependent. MSR-I: Deterministic Communication Amid Uncertainty

  15. Challenging special case • Alice has permutation on • i.e., 1-1 function mapping • Bob has permutation • Know both are close: • (say ) • Alice and Bob know (say ). • Alice wishes to communicate to Bob. • Can we do this with few bits? • Say bits if . MSR-I: Deterministic Communication Amid Uncertainty

  16. Model as a graph coloring problem X • Consider family of graphs • Vertices = permutations on • Edges = close permutations with distinct messages. (two potential Alices). • Central question: What is ? MSR-I: Deterministic Communication Amid Uncertainty

  17. Main Results [w. EladHaramaty] • Claim: Compression length for toy problem • Thm 1: • ( times) • . • Thm 2: uncertain comm. schemes with (-error). • -error). • Rest of the talk: Graph coloring MSR-I: Deterministic Communication Amid Uncertainty

  18. Restricted Uncertainty Graphs X • Will look at • Vertices: restrictions of permutations to first coordinates. • Edges: MSR-I: Deterministic Communication Amid Uncertainty

  19. Homomorphisms • homomorphic to () if • Homorphisms? • is -colorable • Homomorphisms and Uncertainty graphs. • Suffices to upper bound MSR-I: Deterministic Communication Amid Uncertainty

  20. Chromatic number of For , Let Claim: is an independent set of Claim: Corollary: MSR-I: Deterministic Communication Amid Uncertainty

  21. Better upper bounds: Say Lemma: For MSR-I: Deterministic Communication Amid Uncertainty

  22. Better upper bounds: MSR-I: Deterministic Communication Amid Uncertainty • Lemma: • For • Corollary: • Aside: Can show: • Implies can’t expect simple derandomization of the randomized compression scheme.

  23. Future work? • Open Questions: • Is ? • Can we compress arbitrary distributions to ? or even • On conceptual side: • Better mathematical understanding of forces on language. • Information-theoretic • Computational • Evolutionary MSR-I: Deterministic Communication Amid Uncertainty

  24. Thank You MSR-I: Deterministic Communication Amid Uncertainty

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