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Distributed Computing with Adaptive Heuristics

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Distributed Computingwith Adaptive Heuristics

Aaron D. Jaggard

Rutgers/Colgate

Michael Schapira

Princeton

Innovations in Computer Science

09 January 2011

Partially supported by NSF

Rebecca N. Wright

Rutgers

- Identify new aspects of the boundary between game theory and distributed computing
- Look at some initial results
- Identify various avenues for future work

Game Dynamics

(Natural Behaviors)

DistributedComputing

simple, myopic rules of behavior;

convergence in synchronized environment

nodes interacting in an asynchronous environment

Many real-world settings involve both simple, myopic behavior and asynchronicity

- Network protocols
- Routing
- Congestion control
- ...

- Asynchronous circuits

Distributed Computing with Adaptive Heuristics(BGP, TCP, … )

new questions

in game theory and economics

(fictitious play, regret minimization,…)

novel applications in distributed computing

(congestion control, asynchronous circuits,…)

Goal: Explore dynamics of adaptive heuristics when asynchrony is allowed

- Two stable states (pure Nash equilibria)

Bach

Stravinsky

(5,4)

(1,1)

Bach

(0,0)

(4,5)

Stravinsky

- If either player is activated alone, the system converge

Bach

Stravinsky

(5,4)

(1,1)

Bach

(0,0)

(4,5)

Stravinsky

- Without control over who is activated, the system might not converge

Bach

Stravinsky

(5,4)

(1,1)

Bach

(0,0)

(4,5)

Stravinsky

- n nodes (the players)
- Node i has action space Ai
- Each node i has a reaction function
fi: A1 x A2 x ...xAn→Ai that determines i’s next

action based on other current actions

- No dependence on own action

- Infinite sequence of discrete timestepst = 1, …
- Schedule s:{1,…} → 2[n] determines which set of players is activated at time t.
- Fair schedules

- Start at an initial state; at each time step t, let the nodes in s(t) react using their reaction functions

- The players’ action profile a=(a1,…, an) is a stable state if fi(a) = ai for every i.
- The system is convergent if the dynamics always converge (for all initial states and all fair schedules)

- What classes of systems are guaranteed (or cannot be guaranteed) to always converge to a stable state?
- How hard is it to determine whether a system always converges to a stable state?

Theorem: If the system has multiple stable states, then the system is not convergent. (I.e., there is some initial state and some schedule that diverge.)

- Actually, can strengthen this:
- Allow some history-dependence
- Allow randomness in reaction functions

Bach

Stravinsky

(5,4)

(1,1)

Bach

(0,0)

(4,5)

Stravinsky

- Inspired by approach to FLP result on impossibility of resilient consensus

- Interdomain routing
- Congestion control
- Best-reply dynamics in general games
- Diffusion of technologies in social networks
- Asynchronous circuits
- …

Theorem: Determining whether a system of n nodes, each with two actions, is convergent may require Ω(2n) bits.

- Even if all reaction functions are deterministic, and do not depend on history or own action

- Constructed system has a unique stable state

Theorem: Determining whether a system of n nodes, each with deterministic and historyless reaction function, is convergent is PSPACE-complete.

- So, difficult even if the reaction functions are succinctly represented (so that they could be transmitted quickly)
- Under complexity assumptions, no short witnesses (in general) of being convergent

Question: Does randomness help?

Bach

Stravinsky

(5,4)

(1,1)

Bach

(0,0)

(4,5)

Stravinsky

Question: Does randomness help?

Answer: No. Divergence may not only be possible, but overwhelmingly likely.

- Issues of r-fairness

- What are the convergence guarantees and impossibility results
- For other heuristics
- For other notions of convergence
- For other notions of equilibrium

- We’ve taken first steps in the context of regret-minimizing dynamics

- Variations in information
- Outdated information
- Knowledge only of own utility function (uncoupled dynamics)

- Lots of others

- Simple behaviors show up in lots of settings
- Important to understand dynamic behavior when asynchrony is allowed

- Initial results on the impossibility of guaranteeing convergence
- Lots of open questions
- What can we say about the dynamic behavior in other natural asynchronous settings?