Concentration inequalities are indispensable tools for studying the generalization capacity of learning models. Hoeffding's and McDiarmid's inequalities are commonly used, giving bounds independent of the data distribution. Although this makes them widely applicable, a drawback is that the bounds can be too loose in some specific cases. Although efforts have been devoted to improving the bounds, we find that the bounds can be further tightened in some distribution-dependent scenarios and conditions for the inequalities can be relaxed. In particular, we propose four types of conditions for probabilistic boundedness and bounded differences, and derive several distribution-dependent extensions of Hoeffding's and McDiarmid's inequalities. These extensions provide bounds for functions not satisfying the conditions of the existing inequalities, and in some special cases, tighter bounds. Furthermore, we obtain generalization bounds for unbounded and hierarchy-bounded loss functions. Finally we discuss the potential applications of our extensions to learning theory.
@math Being similar to the boundedness conditions given by @cite_27 (See Assumptions 1 and 3 in Section 4). In this case, we will obtain a refined bound.
@cite_27
2547508527 2531021157 2000931246 2001663593
We present an algorithm for the statistical learning setting with a bounded exp-concave loss in @math dimensions that obtains excess risk @math with probability at least @math . The core technique is to boost the confidence of recent in-expectation @math excess risk bounds for empirical risk minimization (ERM), without sacrificing the rate, by leveraging a Bernstein condition which holds due to exp-concavity. We also show that with probability @math the standard ERM method obtains excess risk @math . We further show that a regret bound for any online learner in this setting translates to a high probability excess risk bound for the corresponding online-to-batch conversion of the online learner. Lastly, we present two high probability bounds for the exp-concave model selection aggregation problem that are quantile-adaptive in a certain sense. The first bound is a purely exponential weights type algorithm, obtains a nearly optimal rate, and has no explicit dependence on the Lipschitz continuity of the loss. The second bound requires Lipschitz continuity but obtains the optimal rate. In our LICS 2004 paper we introduced an approach to the study of the local structure of finite algebras and relational structures that aims at applications in the Constraint Satisfaction Problem (CSP). This approach involves a graph associated with an algebra @math or a relational structure A, whose vertices are the elements of @math (or A), the edges represent subsets of @math such that the restriction of some term operation of @math is ‘good’ on the subset, that is, act as an operation of one of the 3 types: semilattice, majority, or affine. In this paper we significantly refine and advance this approach. In particular, we prove certain connectivity and rectangularity properties of relations over algebras related to components of the graph connected by semilattice and affine edges. We also prove a result similar to 2-decomposition of relations invariant under a majority operation, only here we do not impose any restrictions on the relation. These results allow us to give a new, somewhat more intuitive proof of the bounded width theorem: the CSP over algebra @math has bounded width if and only if @math does not contain affine edges. Actually, this result shows that bounded width implies width (2,3). We also consider algebras with edges from a restricted set of types. In particular, it can be proved that type restrictions are preserved under the standard algebraic constructions. Finally, we prove that algebras without semilattice edges have few subalgebras of powers, that is, the CSP over such algebras is also polynomial time. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 .