george saad university of new mexico department of computer science
TRANSCRIPT
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George SaadUniversity of New Mexico
Department of Computer Science
Selfishness and Malice in Distributed Systems
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Selfishness and Malice• Selfishness and malice have negative influence on the
performance of distributed systems.• Selfishness of players in a game can reduce social welfare.• Malicious nodes can seriously disrupt the network.
• In this dissertation, we provide algorithms to address these issues.
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Selfishness and Malice• Selfishness (El-Farol game): we characterize
BCE for game of +ve/-ve network effects. “The Power of Mediation in an Extended El Farol Game”, SAGT’13
2013
2013 2014
“Self-Healing Communication”, SSS’13 “Self-Healing Computation”, SSS’14
• Malice: we develop algorithms to recover networks from Byzantine faults.
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Part I : Selfishness
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El-Farol Game
• A set of n selfish players• Actions:• go to the bar• stay home
• The cost function:• cost to stay = 1,• cost to go: f(x)
Objective: find an equilibrium which minimizes Social Cost, where
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Our El Farol Extension
We extend the cost function:• The cost to stay can be any constant t > 0,• The cost to go, f(x):
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Positive and Negative Network Effects
“Many real situations in fact display both kinds of [positive and negative] externalities … an on-line social media site with limited infrastructure might be most enjoyable if it has a reasonably large audience, but not so large that connecting to the Web site becomes very slow due to the congestion.”
“Many real situations in fact display both kinds of [positive and negative] externalities … an on-line social media site with limited infrastructure might be most enjoyable if it has a reasonably large audience, but not so large that connecting to the Web site becomes very slow due to the congestion.”
[Easley and Kleinberg, 2010]
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Solution ConceptsHow to minimize
my own cost unilaterally?• Nash Equilibrium • Unfortunately, NE has high social cost.
• Correlated Equilibrium (CE)• Mediator implements CE.
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Mediator• A trusted coordinator that • gives recommendations to the players, • implements a correlated equilibrium.• Note that all players have free will.
• A mediator is optimal when it implements the best correlated equilibrium.
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Let mediator have a probability distribution on k ≥ 1 strategy profiles.• The players know probability distribution and strategy profiles.• Mediator selects secretly one strategy profile according to the
probability distribution. • Mediator advises each player privately and separately.• No player has incentive to deviate unilaterally from the advice.
How to design such a mediator?
( [s11,…,s1n], p1 )
( [s21,…,s2n] , p2 )
( [sk1,…,skn] , pk )
( x1 , p1 )
( x2 , p2 )
( xk , pk )
( x1 , p1 )
( x2 , p2 )
( xk , pk )
( x1 , p1 )
( x2 , p2 )
( xk , pk )
( x1 , p1 )
( x2 , p2 )
( xk , pk )
( x1 , p1 )
( x2 , p2 )
( xk , pk )
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Example for (c, s1, s2)-El Farol Game
• For a (2, 4, 4)-El Farol game:• Best Nash Equilibrium:
• ¼-fraction of players go.• Social cost = n.
• An optimal mediator:• Strategy profile 1: (x1 = 0, p1 = 1/3)
• Strategy profile 2: (x2 = ½, p2 = 2/3)• Expected social cost = ⅔ n.
• The optimal social cost (no selfishness)• ½-fraction of players go.• Social cost = ½ n.
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How efficient is our mediator?
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Our Contributions• Game of positive and negative network effects, we characterize: • Optimal Social Cost,• Best Nash Equilibrium (BNE), and• Best Correlated Equilibrium (BCE).
• Efficiency of optimal mediator for this game• When BCE = BNE?• MV and EV can be unbounded!
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Optimal Social Cost
We characterize x* as a function of parameters of our game.
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Best Nash Equilibrium
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Optimal Mediator
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- - p is a function of c, s1 and s2.- p can be 0 or 1 for some values of c, s1 and s2.
When is BCE = BNE?
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If c ≤ 1, then all players would rather stay, if f(1) ≥ 1; all players would rather go, if f(1) < 1.
If c > 1 and λ(c, s1, s2) ≥ 1, then all players would rather go, where:
When BCE = BNE?
BCE is advantageous over BNE when c > 1 and λ < 1.
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Can MV be unbounded?c s1 s2 c/s1 1
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Can EV be unbounded?c s1 s2 c/s1 1
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Related Work• Linear Congestion Games [CK’05]:• 1.577 ≤ EV ≤ 1.6 and MV ≤ 1.015.
• Ranking Games [BFHS’07]:• EV = n-1 and MV = n-1 for n>3.
• Virus Inoculation Game [DMNS’09]:• EV = and MV = .
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Conclusion
• We extended the El-Farol game to have both positive and negative network effects.
• For this extension, we have characterized:• the optimal social cost, • the BNE, and• the BCE.
• We characterized the MV and the EV for this game.• We show when BCE = BNE.• We show that MV and EV can be unbounded in this game.
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Open Problems
• Multi-Site El-Farol Game (> 2 actions): • The bar has k > 2 sites.• Each player chooses which site to go to.• How many strategy profiles required for BCE?
• If f(x) is polynomial in x, with degree > 1, then• what is the characterization of BCE? • Is # strategy profiles related to degree of
f(x)?
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Self-Healing Communication Self-Healing Computation
Part II : Malice
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Malice• We consider the presence of an adversary.
• Adversary takes over a subset of nodes to cause faults.
• Byzantine Faults vs Fail-Stop Faults
• Fault Tolerance:
• Replication
• Self-healing (automatic recovery)
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Fault Tolerance• Non-self-healing algorithms for Byzantine model: [NW’03,
HK’04, FSY’05, AS’06, AJR’06, AS’07, JY’08, GKKY’10, GKKY’13].
• Self-healing algorithms for fail-stop model: [BSAS’06, ST’06, HRST’08, HST’09, PT’11, ST’11].
• Self-healing Algorithms for Byzantine faults?
• We develop self-healing algorithms to recover from Byzantine faults.
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How to recover from Byzantine faults?
Self-Healing CommunicationMessage is sent through a path of nodes.
Self-Healing ComputationComputation is performed through circuits.
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Our Model• A network of n nodes• Static and Computationally Bounded Adversary• Adversary controls up to ¼ of the nodes.• Partially Synchronous Communication: Upper bound of time
steps between sending and receiving messages.• Rushing Adversary: Waiting until receiving all messages from
good nodes before responding.• After bad nodes selected, Quorum Graph is built up [KLST’10]• Any quorum is a set of θ(log n) nodes; and • Each node is in θ(log n) quorums.• At most ¼ of nodes in any quorum are bad.
KLST’10 : Valerie King, Steve Lonargan, Jared Saia and Amitabh Trehan, “Load balanced Scalable Byzantine Agreement through Quorum Building, with Full Information”, ICDCN 2010.
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Naïve Communication (no self-healing)
• All-to-all communication between quorums• Message cost O(l log2 n), and latency O(l)• However, we can do better by self-healing.
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Our Contribution• We developed a self-healing algorithm that detects message
corruptions and marks bad nodes.
• Each bad node causes O((log∗ n)2) corruptions, in expectation.“Fool me once, shame on you. Fool me ω((log* n)2) times,
shame on me.”
Iterated Logarithme.g. log*
1010 = 5
Naïve Communication Our Algorithm
Message cost O(l log2 n ) O(l + log n)Latency O(l) O(l)Corruptions No corruptions O(t(log∗ n)2))
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Our Algorithm (SEND)
SEND-PATH
SEND
CHECK
CHECK1 CHECK2
HEAL
HEAL is triggered O(t) times before all bad nodes are marked.
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CHECK1• SEND triggers CHECK1 with probability 1/(log log n)2.• Subquorum size is O(log log n).• Latency is O(l) and Message Cost is O(l (log log n)2).• Detects corruptions with const prob. for l = O(log2 n).
• SEND triggers CHECK1 with probability 1/(log log n)2.• Subquorum size is O(log log n).• Latency is O(l) and Message Cost is O(l (log log n)2).
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CHECK2• SEND triggers CHECK2 with probability 1/(log ∗ n)2.• CHECK2 has O(log ∗ n) rounds.• Incremental subquorum size, up to O(log∗ n).• Latency is O(l log ∗ n) and Message Cost is O(l (log ∗ n)2).
• SEND triggers CHECK2 with probability 1/(log ∗n)2.• CHECK2 has O(log ∗ n) rounds.• Incremental subquorum size, up to O(log∗ n).• Latency is O(l log ∗ n) and Message Cost is O(l (log ∗n)2).
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CHECK2 Analysis• Deception Interval : a substring of bad nodes, where a
corruption occurs.• Key Points of Detecting Corruptions:• Deception interval shrinks logarithmically with prob. ≥ ½.• O(log* n) rounds to shrink deception interval to size zero.
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CHECK2 Analysis• Deception Interval shrinks logarithmically from round to round:
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HEAL
• Inspects each node participated what it received and sent
• Marks the nodes that are in conflict* A pair of nodes is in conflict if they accuse each other
• Each pair of nodes in conflict has at least one bad node
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?HEAL
• If ½ nodes in any quorum are marked, they are set unmarked.
• HEAL is triggered O(t) times before all bad nodes are marked.
• We show that using a potential function argument.
• f(b,g) is monotonically increasing,• Δf(b,g) is at least some +ve constant.• When f(b,g) = t, we are done.
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Empirical Results• Our simulation runs:• over butterfly networks of quorums,• for different network sizes, up to
n=30k, and • for different fractions of bad nodes.
• Simulation terminates after all bad nodes are marked.
• The results are taken over 3000 experiments.
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# messages is improved by a factor of 60 for CHECK1
39,100
649
Empirical Results# Messages reduces by a factor of 60 (n~30k)
39,100
1,177
# messages is improved by a factor of 33 for CHECK2
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Empirical ResultsLatency increases by 1½ times (n~30k)
Latency increases by 1½ times for CHECK1
39,100
649
Latency increases by 2 times for CHECK2
18
13
25
13
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Empirical ResultsCorruption Probability 0
39,100
649
18
13
25
13
CHECK1 CHECK2
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Empirical Results# Messages reduces by O(log2 n) times
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Empirical ResultsLatency increases by (1) timesθ
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How to recover from Byzantine faults?
Self-Healing CommunicationMessage is sent through a path of nodes.
Self-Healing ComputationComputation is performed through circuits.
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Quorum Graph• Quorum Graph has:• n input quorums; • m quorum gates; and• one output quorum
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• No self-healing• All nodes in each quorum (gate) perform the same computation• Results are sent between quorums via all-to-all communication• Expensive resource cost
Naïve Computation
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Our Contribution
Naïve Computation Our Algorithm
Message cost O( (n+m) log2 n ) O(m + nlog n)
Computation cost O( (n+m) log2 n ) O(m + nlog n)
Latency O(l) O(l)Corruptions No corruptions O(t(log∗ n)2))
We develop a self-healing algorithm for computation networks
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Our Algorithm (COMPUTE)
COMPUTE
CHECK
EVALUATE
RECOVER
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CHECK Algorithm• CHECK has O(log* n) rounds• In each round, nodes are selected uniformly at random, and same
computation is performed
Round 1
Round 2
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CHECK Algorithm• Adversary corrupts computation in a Deception Subgraph.
• Key points of corruption detection:• We prove that deception subgraph shrinks logarithmically in each
round with constant probability.• Once deception subgraph shrinks to size zero, corruption is
detected.
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Shrinks Logarithmically
Round 1
Round 2
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Shrinks Logarithmically
Round 2
Round 3
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Shrinks Logarithmically
Round 3
Round 4
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RECOVER
• Inspects each node participated what it received and sent
• Marks the nodes that are in conflict* A pair of nodes is in conflict if they accuse each other
• Each pair of nodes in conflict has at least one bad node
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?RECOVER
• If ½ nodes in any quorum are marked, they are set unmarked.
• HEAL is triggered O(t) times before all bad nodes are marked.
• We show that using a potential function argument.
• f(b,g) is monotonically increasing, and• when it reaches t, we are done.
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Empirical Results• Our simulation runs:• over perfect binary trees of quorums,• for different network sizes, up to 8k, and • for different fractions of bad nodes.
• Simulation terminates after all leaders are good.
• The results are taken over 3000 experiments.
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Empirical Results# Messages reduces by factor of 65 (n~8k)
Reduced by afactor of 651.01M
66M
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Empirical ResultsLatency increases by 1.75 times (n~8k)
Increases 1.75 times
63 time steps
36
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Empirical ResultsCorruption Probability 0
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Empirical Results# Messages reduces by O(log2n) times!
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Empirical ResultsLatency increases by (1) timesθ
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Conclusion
• We developed self-healing algorithms to recover networks from Byzantine faults.
• Message cost is reduced polylogarithmically in n, compared to non-self-healing algorithms.
• Experiments show that message cost reduced by • Up to a factor of 60 for communication networks• Up to a factor of 65 for computation networks
• For t < n/4, the expected total number of corruptions is O(t(log∗ n)2)
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Open Problems• Can we limit the number of corruptions to O(t)?• How to self-heal networks with churn? adaptive adversary?• How to self-healing asynchronous networks?• We trigger CHECK and select the nodes in a centralized
manner. How we make CHECK decentralized?• We propose a decentralized CHECK for future work.• We implement a simulation that suggests interesting results.
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Thanks! Any Questions?