Download PDF by Katharina Morik (auth.), Osamu Watanabe, Takashi Yokomori: Algorithmic Learning Theory: 10th International Conference,

By Katharina Morik (auth.), Osamu Watanabe, Takashi Yokomori (eds.)

ISBN-10: 3540467696

ISBN-13: 9783540467694

ISBN-10: 3540667482

ISBN-13: 9783540667483

This e-book constitutes the refereed court cases of the tenth overseas convention on Algorithmic studying conception, ALT'99, held in Tokyo, Japan, in December 1999.
The 26 complete papers awarded have been conscientiously reviewed and chosen from a complete of fifty one submissions. additionally incorporated are 3 invited papers. The papers are geared up in sections on studying measurement, Inductive Inference, Inductive good judgment Programming, PAC studying, Mathematical instruments for studying, studying Recursive capabilities, question studying and online studying.

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Extra info for Algorithmic Learning Theory: 10th International Conference, ALT’99 Tokyo, Japan, December 6–8, 1999 Proceedings

Example text

By definition, if t > or t < 0 then I(t) = 0. Lemma 3 Assume that ϕ(w) is a C0∞ -class function. Then I(t) has an asymptotic expansion for t → 0. j I(t) ∼ = ∞ mk −1 ck,m+1 tλk −1 (− log t)m (3) k=1 m=0 where m! · ck,m+1 is the coefficient of the (m + 1)-th order in the Laurent expansion of J(λ) at λ = −λk . [Proof of Lemma 3] The special case of this lemma is shown in [10]. Let IK (t) be the restricted sum in I(t) from k = 1 to k = K. It is sufficient to show that, for an arbitrary fixed K, lim (I(t) − IK (t))tλ = 0 (∀ λ > −λK+1 + 1).

Lemma 3 Assume that ϕ(w) is a C0∞ -class function. Then I(t) has an asymptotic expansion for t → 0. j I(t) ∼ = ∞ mk −1 ck,m+1 tλk −1 (− log t)m (3) k=1 m=0 where m! · ck,m+1 is the coefficient of the (m + 1)-th order in the Laurent expansion of J(λ) at λ = −λk . [Proof of Lemma 3] The special case of this lemma is shown in [10]. Let IK (t) be the restricted sum in I(t) from k = 1 to k = K. It is sufficient to show that, for an arbitrary fixed K, lim (I(t) − IK (t))tλ = 0 (∀ λ > −λK+1 + 1). t→0 (4) 1 I(t)tλ dt.

6) Then 0 < λ∗ < ∞. 3) Let G(y, z, w) = λ∗ (L(y, z) − L(y, w)). For any y, z, w ∈ [0, 1], ∂ 2 G(y, z, w)/ ∂y 2 + (∂G(y, z, w)/∂y)2 ≥ 0. For√example, λ∗ = 1 for the entropic loss, λ∗ = 2 for the square loss, and λ = 2 for the Hellinger loss. In the case of Y = {0, 1} instead of Y = [0, 1], Condition 3) is not necessarily required. ∗ 3 Asymptotical Results According to [12], we introduce the notion of ESC in order to derive upper bounds on the minimax RCL. Definition 2. Let µ be a probability measure on a hypothesis class H.

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Algorithmic Learning Theory: 10th International Conference, ALT’99 Tokyo, Japan, December 6–8, 1999 Proceedings by Katharina Morik (auth.), Osamu Watanabe, Takashi Yokomori (eds.)


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