By Robert E. Schapire (auth.), Paul Fischer, Hans Ulrich Simon (eds.)
This booklet constitutes the refereed complaints of the 4th ecu convention on Computational studying idea, EuroCOLT'99, held in Nordkirchen, Germany in March 1999. The 21 revised complete papers offered have been chosen from a complete of 35 submissions; additionally integrated are invited contributions. The e-book is split in topical sections on studying from queries and counterexamples, reinforcement studying, on-line studying and export recommendation, instructing and studying, inductive inference, and statistical concept of studying and development attractiveness.
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Additional info for Computational Learning Theory: 4th European Conference, EuroCOLT’99 Nordkirchen, Germany, March 29–31, 1999 Proceedings
1/ m) Repeat: D = g − H(g · H) S =∅ for i = 1 to m do if Di < 0 then add (xi , −yi ) to S m Di = −Di / i=1 |Di | else add (xi , yi ) to S m Di = Di / i=1 |Di | end do call weak learner with distribution D over S , obtaining hypothesis h g·(h−H(H·h)) α = g·(H(h·h)−h(H·h)) m H = (H + αh)/ (Hi + αhi )2 i=1 Fig. 3. The GeoLev Algorithm. Breiman shows that, under certain conditions on f , arcing algorithms converge to good hypotheses in the limit. Furthermore, he shows that AdaBoost is an arcing algorithm with f (x) = ex and arc-x4 is an arcing algorithm with polynomial f (x).
Define µr2n 1 , if r ≤ p = v1v+v 1 n then p ≤ µ. 2 n/2 1 nπ and v2 ≥ Proof. From corollary 1 it follows that v1 ≤ Γr(n+2/2) fore nr1 v1 ≤ n v2 r2 Let µ ≤ 1 . Substituting r1 ≤ µr2n n p= r2n π n/2 Γ (n+2/2) . There(13) we conclude that v1 v1 ≤ ≤µ v1 + v2 v2 (14) Query by Committee, Linear Separation and Random Walks 47 Lemma 4. For any sample x, let q be the probability that the QBC algorithm will query for a tag and let qˆ be the similar probability for QBC”. Then |q − qˆ| ≤ 4 (15) k Proof. e. min(r+ , r− ) ≥ µk (max(r+ , r− ))n .
We would like to bound the probability that Pr[W ] > βk /4 when QBC” didn’t query for a tag for tk at the last consecutive samples: If Pr[W ] > βk /4, then the probability that the QBC algorithm will query for a tag is greater then αβk /4. From lemma 4 we conclude that the probability that the QBC” algorithm will query for a tag is greater then αβk /4 − 4 k .
Computational Learning Theory: 4th European Conference, EuroCOLT’99 Nordkirchen, Germany, March 29–31, 1999 Proceedings by Robert E. Schapire (auth.), Paul Fischer, Hans Ulrich Simon (eds.)