Stochastic Algorithms: Foundations and Applications: 5th by Werner Römisch (auth.), Osamu Watanabe, Thomas Zeugmann

By Werner Römisch (auth.), Osamu Watanabe, Thomas Zeugmann (eds.)

This e-book constitutes the refereed court cases of the fifth overseas Symposium on Stochastic Algorithms, Foundations and functions, SAGA 2009, held in Sapporo, Japan, in October 2009.

The 15 revised complete papers offered including 2 invited papers have been conscientiously reviewed and chosen from 22 submissions. The papers are prepared in topical sections on studying, graphs, trying out, optimization and caching, in addition to stochastic algorithms in bioinformatics.

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Additional resources for Stochastic Algorithms: Foundations and Applications: 5th International Symposium, SAGA 2009, Sapporo, Japan, October 26-28, 2009. Proceedings

Example text

By definition of mj 1 cj 1 = t 1 cj1 + − 1 mj + 1 − cj1 = t t t 1 t cj1 . (17) We compare conditional probabilities P {It = j|ξ i = ci , i = j} and P {Jt = j|ξ i = ci , i = j}. The following chain of equalities and inequalities is valid: P {It = j|ξ i = ci , i = j} = 1 P {sj1:t−1 + ξ j ≥ mj |ξ i = ci , i = j} = t P {ξ j ≥ P {ξ j ≥ − sj1:t−1 )|ξ i = ci , i = j} = 1 1 P {ξ j ≥ t (mj − sj1:t−1 ) + ( t − t )(sj1:t−1 − sj1:t−1 + cj1 )|ξ i = ci , i = j} = t (mj − sj1:t−1 ) + ( j i t (mj − s1:t−1 )|ξ = ci , i = j} = t − t )(mj t exp{−( j P {ξ ≥ t (mj − sj1:t−1 ) +( t exp{−( P {ξ j ≥ t (mj − sj1:t + sjt − 1 t − 1 t cj1 ) + ( t t − − t − − j1 t )(s1:t−1 t) 1 t i cj1 |ξ = ci , i = j} ≥ j1 t )(s1:t−1 t) 1 t − sj1:t−1 )} × (18) (19) (20) − sj1:t−1 )} × cj1 |ξ i = ci , i = j} = (21) Learning Volatility of Discrete Time Series exp{−( t − j P {ξ ≥ exp − exp − Δvt μt vt 1+ sj1:t )|ξ i j t st } × = ci , i = j} = (22) (23) × 1 (sj ) − mj |ξ i = ci , i = j} ≥ μt vt 1:t j j1 ) Δvt (s1:t−1 − s1:t−1 Δvt − μt vt vt−1 μt vt P {ξ j > exp − − t (mj − sj1:t−1 ) − sj 1 (sj1:t−1 − sj1:t−1 ) − t μt vt 1 1 − μt vt−1 μt vt P {ξ j > j1 t )(s1:t−1 25 × (24) 1 (mj − sj1:t )|ξ i = ci , i = j} = μt vt 1 sj1:t−1 − sj1:t−1 vt−1 P {Jt = 1|ξ i = ci , i = j}.

Define s˜1:t = s1:t + 1 ξ for t = 1, 2, . .. Consider the one-step gains s˜t = t st + ξ 1 t − 1 t−1 for the moment. For any vector s and a unit vector d denote M (s) = argmaxd∈D {d · s}, where D = {(0, . . 1), . . (1, . . 0)} is the set of N unit vectors of dimension N and “·” is the inner product of two vectors. We first show that T M (˜ s1:t ) · s˜t ≥ M (˜ s1:T ) · s˜1:T . (32) t=1 For T = 1 this is obvious. For the induction step from T − 1 to T we need to show that M (˜ s1:T ) · s˜T ≥ M (˜ s1:T ) · s˜1:T − M (˜ s1:T −1 ) · s˜1:T −1 .

2 We consider an FPL algorithm with a variable learning rate t = 1 , μt vt−1 (11) where μt is defined by (10) and the volume vt−1 depends on experts actions on steps < t. By definition vt ≥ vt−1 and μt ≤ μt−1 for t = 1, 2, . .. Also, if γ(t) → 0 as t → ∞ then μt → 0 as t → ∞. We suppose without loss of generality that si0 = v0 = 0 for all i and 0 = ∞. The FPL algorithm is defined as follows: FPL algorithm. FOR t = 1, . . T Define It = argmaxi {si1:t−1 + 1t ξ i }, where t is defined by (11) and i = 1, 2, .

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