Kearns, M. and Ron, D. (1997). Algorithmic stability and sanitycheck bounds for leave-one-out cross validation. In Conference on Computational Learning Theory (COLT97), pages 152-162. Home/Search Document Details and Download
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A Leave-one-out Cross Validation Bound for Kernel Methods with.. - Zhang (2001) (Correct)
....if the training data are independently drawn from a xed underlying distribution, then the expected leave out out error equals the expected test error, which measures the generalization ability of the learning method. Leave one out bounds have received much attention recently. For example, see [3, 5, 6, 10] and references therein. Also in [5, 10] the leave one out analysis has already been employed to study the generalization ability of support vector classi cation. In this paper, we extend their results by deriving a general leave one out bound for a class of convex dual kernel learning machines ....
.... For example, techniques related to what we use here have also been applied in [12, 13] In fact, the proof of Lemma 1 is essentially the dual version of the technique which was used in [12] The convergence of the estimated parameter is also related to the algorithmic stability concept in [1, 6]. The former implies the latter but not conversely. Consequently, better bounds can usually be obtained if we can show the convergence of the estimated parameter. Bounds given in [1, 6] are in the style of the convergence of the empirical loss of the estimated parameter to the true loss of the ....
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Michael Kearns and Dana Ron. Algorithmic stability and sanity-check bounds for leave-one-out cross-validation. Neural Computation, 11(6):1427-1453, 1999.
Almost-Everywhere Algorithmic Stability and Generalization Error - Kutin, Niyogi (2002) (2 citations) (Correct)
....notion as weak hypothesis stability. Breiman [Bre96b] argues that unstable weak learners bene t from randomization algorithms such as bagging. He nds that, when the weak learner is stable, bagging does not help, and suggests that AdaBoost [FS97] is more e ective in this case. Kearns and Ron [KR99] consider both hypothesis stability and the weaker, related notion of error stability. They prove bounds on the error of leave one out estimates of error rates, but their arguments rely on the traditional notion of VC dimension [VC71] Bousquet and Elissee [BE01, BE02] prove that algorithms which ....
....awkward to analyze within the VC framework. Devroye and Wagner [DW79] were the rst to observe a connection between the stability of an algorithm and its generalization error. Bousquet and Elissee [BE01] proved that uniform hypothesis stability implies low generalization error. Kearns and Ron [KR99] introduced the notion of error stability, and used it to get bounds on the error of leave one out estimates. We discuss several notions of almost everywhere stability, including: strong hypothesis stability , where an algorithm is stable at most training sets, weak hypothesis stability , ....
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M. Kearns and D. Ron. Algorithmic stability and sanity-check bounds for leave-one-out cross-validation. Neural Computation, 11:1427-1453, 1999.
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....This definition of stability is known as uniform stability. It is a restrictive condition, as it needs to hold on all possible training sets, even training sets that can only occurwith probability 0. This motivates the weaker notion of (#,#) stability. Definition 2. 2 (Kearns and Ron, 1999) [11] An algorithm is # stable at S with respect to i,u,z ) ##. Definition 2.3 (Kutin and Niyogi, 2001) 5] An algorithm A is (#,#) stable with respect to IP S#Z # (A is # stable at S) #. It is obvious that a # stable algorithm is also (#,#) stable for all # 0. The following theorems ....
D. Ron and M. Kearns. Algorithmic stability and sanity-check bounds for leave-one-out crossvaildation. Neural Computation, 11(6):1427--1453, 1999.
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....of the cross validation and holdout estimates in the case where real valued functions are used as classifiers. We then devise and analyse two new error estimates, which we call the adaptive holdout estimate and the adaptive cross validation estimate. The results we obtain are what Kearns and Ron [20] (in the binary valued context) have described as sanity check bounds. This name reflects the fact that the bounds are not better than those one has for the corresponding resubstitution error estimate. Nonetheless, we believe that such sanity check bounds are worthwhile, first, as they show that ....
....no less interesting. Cheng and Titterington [11] have raised the interesting question of what happens to the result stated in Theorem 1 if a cross validation estimate is used in place of a resubstitution estimate. This problem has recently been considered by Holden [15, 16, 17] and Kearns and Ron [20], and a related problem has been studied by Blum et al. 9] There are different types of cross validation estimate. These operate by splitting the sample z into m sections. A section is removed and the remaining examples used in conjunction with L to select a hypothesis h from H. The ....
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M. J. Kearns and D. Ron. Algorithmic stability and sanity-check bounds for leave-one-out crossvalidation. In Proceedings of the 10th Annual Workshop on Computational Learning Theory, pages 152--162. ACM Press, New York, NY, 1997.
The interaction of stability and weakness in AdaBoost - Kutin, Niyogi (2001) (Correct)
....only a small change in the output hypothesis. Breiman [3] argues that unstable weak learners benefit from randomization algorithms such as bagging [2] He finds that, when the weak learner is stable, bagging does not help, and suggests that AdaBoost is more e#ective in this case. Kearns and Ron [6] consider both algorithmic stability and the weaker, related notion of error stability. They prove bounds on the error of crossvalidation estimates of generalization error, but their arguments rely on VC theory. Bousquet and Elissee# [1] prove that an algorithm which is stable everywhere has low ....
.... q# = 2T K. Hence we have shown that B is # L 1 stable. # Note 2.13 We prove Lemma 2.12 only in the case where the weights p(z) q(z) are rational. As we remark in Note 2.6, this is the only case in which we are interested. We now introduce the notion of stability at a point. Kearns and Ron [6] use leave one out stability, but for consistency we phrase our definition in terms of change one stability. Definition 2.14 (Kearns and Ron [6] We say that a learning algorithm A is # stable at S, if #i # 1, m , #u, z # Z, c(f S , z) c(f S i,u , z) # #. 8 We say that A ....
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M. Kearns and D. Ron. Algorithmic stability and sanity-check bounds for leave-one-out cross-validation. Neural Computation, 11:1427--1453, 1999.
The interaction of stability and weakness in AdaBoost - Kutin, Niyogi (2001) (Correct)
....only a small change in the output hypothesis. Breiman [3] argues that unstable weak learners bene t from randomization algorithms such as bagging [2] He nds that, when the weak learner is stable, bagging does not help, and suggests that AdaBoost is more e ective in this case. Kearns and Ron [6] consider both algorithmic stability and the weaker, related notion of error stability. They prove bounds on the error of crossvalidation estimates of generalization error, but their arguments rely on VC theory. Bousquet and Elissee [1] prove that an algorithm which is stable everywhere has low ....
....kp qk = 2T=K. Hence we have shown that B is L 1 stable. Note 2.13 We prove Lemma 2.12 only in the case where the weights p(z) q(z) are rational. As we remark in Note 2.6, this is the only case in which we are interested. We now introduce the notion of stability at a point. Kearns and Ron [6] use leave one out stability, but for consistency we phrase our de nition in terms of change one stability. De nition 2.14 (Kearns and Ron [6] We say that a learning algorithm A is stable at S, if 8i 2 f1; mg; 8u; z 2 Z; jc(f S ; z) c(f S i;u ; z)j : 8 We say that A is ( ....
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