Average Case Settings Research Articles

Nonlinear approximation of functions by sets of finite pseudo-dimension in the probabilistic and average case settings

This paper aims to explore the optimal nonlinear approximation on the class of functions in Sobolev space in the probabilistic and average case settings. Specifically, we investigate the optimal approximation through the utilization of a set with finite pseudo-dimension, measured by the Kolmogorov probabilistic nonlinear [Formula: see text]-width (or say, the Kolmogorov probabilistic pseudo-[Formula: see text]-width). Furthermore, we provide an estimation of the exact asymptotic order of the Kolmogorov probabilistic nonlinear [Formula: see text]-width and [Formula: see text]-average nonlinear [Formula: see text]-width on the class of functions in Sobolev space.

Robust Adaptive Submodular Maximization

The goal of a sequential decision-making problem is to design an interactive policy that adaptively selects a group of items, each selection is based on the feedback from the past, to maximize the expected utility of selected items. It has been shown that the utility functions of many real-world applications are adaptive submodular. However, most of existing studies on adaptive submodular optimization focus on the average-case, that is, their objective is to find a policy that maximizes the expected utility over a known distribution of realizations. Unfortunately, a policy that has a good average-case performance may have very poor performance under the worst-case realization. In this study, we propose to study two variants of adaptive submodular optimization problems, namely, worst-case adaptive submodular maximization and robust submodular maximization. The first problem aims to find a policy that maximizes the worst-case utility and the latter one aims to find a policy, if any, that achieves both near optimal average-case utility and worst-case utility simultaneously. We introduce a new class of stochastic functions, called worst-case submodular function. For the worst-case adaptive submodular maximization problem subject to a p-system constraint, we develop an adaptive worst-case greedy policy that achieves a [Formula: see text] approximation ratio against the optimal worst-case utility if the utility function is worst-case submodular. For the robust adaptive submodular maximization problem subject to cardinality constraints (respectively, partition matroid constraints), if the utility function is both worst-case submodular and adaptive submodular, we develop a hybrid adaptive policy that achieves an approximation close to [Formula: see text] (respectively, 1/3) under both worst- and average-case settings simultaneously. We also describe several applications of our theoretical results, including pool-base active learning, stochastic submodular set cover, and adaptive viral marketing. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.1239 .

Formula omitted]-superposition and truncation dimensions in average and probabilistic settings for [formula omitted]-variate linear problems

The paper deals with linear problems defined on γ-weighted Hilbert spaces of functions with infinitely many variables. The spaces are endowed with zero-mean Gaussian measures which allows to define and study ε-truncation and ε-superposition dimensions in the average case and probabilistic settings. Roughly speaking, these ε-dimensions quantify the smallest number k=k(ε) of variables that allow to approximate the ∞-variate functions by special ones that depend on at most k-variables with the average error bounded by ε. In the probabilistic setting, given δ∈(0,1), we want the error ≤ε with probability ≥1−δ. We show that the ε-dimensions are surprisingly small which, for anchored spaces, leads to very efficient algorithms, including the Multivariate Decomposition Methods.

A note about EC-[formula omitted]-weak tractability of multivariate approximation with analytic Korobov kernels

This paper is devoted to discussing multivariate approximation problems with analytic Korobov kernels in the worst and average case settings. We consider algorithms that use finitely many evaluations of arbitrary continuous linear functionals. We investigate exponential convergence-(s,t)-weak tractability (EC-(s,t)-WT) under the absolute or normalized error criterion. We completely solve the problem by filling the remaining gaps left open on EC-(s,t)-WT. That is, we obtain necessary and sufficient conditions for EC-(s,t)-WT for 0<min(s,t)<1 and max(s,t)≤1 in the worst case setting and for s,t>0 except s=t=1 in the average case setting.

Truncation in average and worst case settings for special classes of [formula omitted]-variate functions

The paper considers truncation errors for functions of the form f(x1,x2,…)=g(∑j=1∞xjξj), i.e., errors of approximating f by fk(x1,…,xk)=g(∑j=1kxjξj), where the numbers ξj converge to zero sufficiently fast and xj’s are i.i.d. random variables. As explained in the introduction, functions f of the form above appear in a number of important applications. To have positive results for possibly large classes of such functions, the paper provides bounds on truncation errors in both the average and worst case settings. The bounds are sharp in two out of three cases that we consider. In the former case, the functions g are from a Hilbert space G endowed with a zero mean probability measure with a given covariance kernel. In the latter case, the functions g are from a reproducing kernel Hilbert space, or a space of functions satisfying a Hölder condition.

How Good is the Chord Algorithm?

The Chord algorithm is a popular, simple method for the succinct approximation of curves, which is widely used, under different names, in a variety of areas, such as multiobjective and parametric optimization, computational geometry, and graphics. We analyze the performance of the Chord algorithm, as compared to the optimal approximation that achieves a desired accuracy with the minimum number of points. We prove sharp upper and lower bounds, both in the worst case and average case settings.

Approximation of additive random fields based on standard information: Average case and probabilistic settings

We consider approximation problems for tensor product and additive random fields based on standard information in the average case setting. We also study the probabilistic setting of the mentioned problem for tensor products. The main question we are concerned with in this paper is “How much do we loose by considering standard information algorithms against those using general linear information?” For both types of the fields, the error of linear algorithms has been studied in great detail; however, the power of standard information was not addressed so far, which we do here. Our main result is that in most interesting cases there is no more than a logarithmic loss in approximation error when information is being restricted to the standard one. The results are obtained by randomization techniques.

Tractability of linear problems defined over Hilbert spaces

We study d-variate approximation problems in the worst and average case settings. We consider algorithms that use finitely many evaluations of arbitrary linear functionals. In the worst case setting, we obtain necessary and sufficient conditions for quasi-polynomial tractability and uniform weak tractability. Furthermore, we give an estimate of the exponent of quasi-polynomial tractability which cannot be improved in general. In the average case setting, we obtain necessary and sufficient conditions for uniform weak tractability. As applications we discuss some examples.

最坏框架与平均框架下区间[-1, 1]上带Jacobi权的函数逼近

We study the weighted approximation of functions on the interval [-1,1] with Jocobi weights (1- x ) α (1+ x ) β , α , β > -1/2 in the worst and average case settings. In the worst case setting, we discuss the Kolmogorov n -widths d n ( BW p,α,β r , L q,α,β ) and linear n -widths δ n ( BW p,α,β r , L q,α,β ) of the weighted Sobolev classes BW p,α,β on [-1,1], where L q,α,β ,1 ≤ q ≤∞ denotes the L q space on [-1,1] with respect to Jocobi weights. Optimal asymptotic orders of d n ( BW p,α,β r , L q,α,β ) and δ n ( BW p,α,β r , L q,α,β ) as n →∞ are obtained for all 1 ≤ p,q ≤ ∞. In the average case setting, we investigate the best approximation of functions on the weighted Sobolev class B W 2, α,β r equipped with a centered Gaussian measure by polynomial subspaces in the L q,α,β metric for 1 ≤ q L q,α,β metric only for 1 q α,β }+ 1).

A splay tree-based approach for efficient resource location in P2P networks.

Resource location in structured P2P system has a critical influence on the system performance. Existing analytical studies of Chord protocol have shown some potential improvements in performance. In this paper a splay tree-based new Chord structure called SChord is proposed to improve the efficiency of locating resources. We consider a novel implementation of the Chord finger table (routing table) based on the splay tree. This approach extends the Chord finger table with additional routing entries. Adaptive routing algorithm is proposed for implementation, and it can be shown that hop count is significantly minimized without introducing any other protocol overheads. We analyze the hop count of the adaptive routing algorithm, as compared to Chord variants, and demonstrate sharp upper and lower bounds for both worst-case and average case settings. In addition, we theoretically analyze the hop reducing in SChord and derive the fact that SChord can significantly reduce the routing hops as compared to Chord. Several simulations are presented to evaluate the performance of the algorithm and support our analytical findings. The simulation results show the efficiency of SChord.

Complexity of Sobolev imbeddings and integration in the deterministic and randomized settings

This paper reformulates the problems of approximation of imbedding and integration from anisotropic Sobolev classes B ( W r p ([0, 1] d )) in the deterministic, randomized and average case settings, and obtains the exact orders of the n-th minimal error of these problems in all three settings. The results show that in the case of B ( W r p ([0, 1] d )) being not imbedded into the space of continuous functions C ([0, 1] d ), the randomized and average case error are essential smaller than the deterministic ones. Quantitatively, this maximal gain for imbedding problem amounts to the factor n -1+e for any e > 0, and for integration problem the gain is up to factor n -1 which is also the maximal speedup over the deterministic algorithms observed so far in natural numerical problems.

The approximation characteristic of diagonal matrix in probabilistic setting

In this paper, the approximation characteristic of a diagonal matrix in probabilistic and average case settings is investigated. And the asymptotic degree of the probabilistic linear ( n , δ ) -width and p -average linear n -width of diagonal matrix M are determined.

Lattice rule algorithms for multivariate approximation in the average case setting

We study multivariate approximation for continuous functions in the average case setting. The space of d variate continuous functions is equipped with the zero mean Gaussian measure whose covariance function is the reproducing kernel of a weighted Korobov space with the smoothness parameter α > 1 and weights γ d , j for j = 1 , 2 , … , d . The weight γ d , j moderates the behavior of functions with respect to the j th variable, and small γ d , j means that functions depend weakly on the j th variable. We study lattice rule algorithms which approximate the Fourier coefficients of a function based on function values at lattice sample points. The generating vector for these lattice points is constructed by the component-by-component algorithm, and it is tailored for the approximation problem. Our main interest is when d is large, and we study tractability and strong tractability of multivariate approximation. That is, we want to reduce the initial average case error by a factor ε by using a polynomial number of function values in ε - 1 and d in the case of tractability, and only polynomial in ε - 1 in the case of strong tractability. Necessary and sufficient conditions on tractability and strong tractability are obtained by applying known general tractability results for the class of arbitrary linear functionals and for the class of function values. Strong tractability holds for the two classes in the average case setting iff sup d ⩾ 1 ∑ j = 1 d γ d , j s < ∞ for some positive s < 1 , and tractability holds iff sup d ⩾ 1 ∑ j = 1 d γ d , j t / log ( d + 1 ) < ∞ for some positive t < 1 . . The previous results for the class of function values have been non-constructive. We provide a construction in this paper and prove tractability and strong tractability error bounds for lattice rule algorithms. This paper can be viewed as a continuation of our previous paper where we studied multivariate approximation for weighted Korobov spaces in the worst case setting. Many technical results from that paper are also useful for the average case setting. The exponents of ε - 1 and d corresponding to our error bounds are not sharp. However, for α close to 1 and for slow decaying weights, we obtain almost the minimal exponent of ε - 1 . We also compare the results from the worst case and the average case settings in weighted Korobov spaces.

Optimal and suboptimal algorithms in set membership identification

We discuss in this paper optimality properties of identification algorithms in a set membership framework. We deal with restricted-complexity (conditional) identification, where approximations (models) to a possibly complex system are selected from a low dimensional space. We discuss the worst- and average-case settings. In the worst-case setting, we present results on optimality, or suboptimality, of algorithms based on computing the unconditional or conditional Chebyshev centres of an uncertainty set. In the average-case setting, we show that the optimal algorithm is given by the projection of the unconditional Chebyshev centre. We show explicit formulas for its average errors, allowing us to see the contribution of all problem parameters to the minimal error. We discuss the case of weighted average errors corresponding to non-uniform distributions over uncertainty sets, and show how the weights influence the minimal identification error.

Optimal quadrature for Haar wavelet spaces

This article considers the error of the scrambled equidistribution quadrature rules in the worst-case, random-case, and average-case settings. The underlying space of integrands is a Hilbert space of multidimensional Haar wavelet series, Hwav. The asymptotic orders of the errors are derived for the case of the scrambled (λ, t, m, s)-nets and (t, s)-sequences. These rules are shown to have the best asymptotic convergence rates for any random quadrature rule for the space of integrands Hwav.

A Probabilistic Analysis of Linear Operator Testing

We test for β-conformance of an implementation linear operator A to a specification linear operator S where the operator domain and range are separable Hilbert spaces and the domain space F is equipped with a Gaussian measure μ. Given an error bound ε>0 and a tolerance parameter β∈(0, 1), we want to determine either that there is an element f in a ball Bq of radius q in domain F such that ‖Sf−Af‖>ε or that Aβ-conforms to S on a set of measure at least 1−β in the ball Bq; i.e., μq(f:‖Sf−Af‖⩽ε)⩾1−β where μq is the truncated Gaussian measure to Bq. We present a deterministic algorithm that solves this problem and uses almost a minimal number of tests where each test is an evaluation of operators S and A at an element of F. We prove that optimal tests are conducted on the eigenvectors of the covariance operator of μ. They are universal; they are independent of the operators under consideration and other problem parameters. We show that finite testing is conclusive in this probabilistic setting. In contrast, finite testing is inconclusive in the worst and average case settings; see [5, 7]. We discuss the upper and lower bounds on the minimal number of tests. For q=∞ we derive the exact bounds on the minimal number of tests, which depend on β very weakly. On the other hand, for a finite q, the bounds on the minimal number of tests depend on β more significantly. We explain our approach by an example with the Wiener measure.

Approximate evaluations of characteristic polynomials of Boolean functions

Motivated by combinational circuit verification and testing, we study the approximate evaluation of characteristic polynomials of Boolean functions. We consider an oracle model in which the values of the characteristic polynomials are approximated using the evaluations of the corresponding Boolean functions. The approximation error is defined in the worst case, average case and randomized settings. We derive lower bounds on the approximation errors in terms of the number of Boolean function evaluations. We design algorithms with an error that matches the lower bound. Let k(ε,n) denote the minimal number of Boolean function evaluations needed to reduce the initial error by a factor of ε where n is the number of Boolean variables. We show that k(ε,n) is exponential in n in the worst and average case settings, and that it is independent of n and of order ε−2 in the randomized setting.

On the Power of Standard Information for Weighted Approximation

We study weighted approximation of multivariate functions for classes of standard and linear information in the worst case and average case settings. Under natural assumptions, we show a relation between n th minimal errors for these two classes of information. This relation enables us to infer convergence and error bounds for standard information, as well as the equivalence of tractability and strong tractability for the two classes.

A Comparison of Policy Iteration Methods for Solving Continuous-State, Infinite-Horizon Markovian Decision Problems Using Random, Quasi-random, and Deterministic Discretizations

This paper compares the performance of the Howard (1960) policy iteration for infinite-horizon continuous-state Markovian decision processes (MDP's) using alternative random, quasi-random, and deterministic discretizations of the state space, or grids. Each grid corresponds to an embedded finite state Markovian decision process whose solution is used to approximate the solution to the original continuous-state Markovian decision process. Extending a result of Rust (1997), I show that policy iteration using succeeds in breaking the of dimensionality involved in approximating the solution to a class of continuous-state discrete-action MDP's known as discrete decision processes (DDP's). I compare this random policy iteration algorithm (RPI) with policy iteration algorithms using deterministically chosen including uniform and grids both of which are subject to the curse of dimensionality. I also compare the RPI to deterministic policy iteration algorithms based on quasi-random or discrepancy grids such as the Sobol and Tezuka sequences. While an analysis of the worst case computational complexity of the DDP problem shows that any deterministic solution method is subject to an inherent of dimensionality, my numerical comparisons reveal that in the test problems considered, policy iteration using the deterministic, low discrepancy were superior to the RPI algorithm. The RPI in turn, outperformed deterministic policy iteration using methods based on uniform and quadrature even in one and two dimensional test problems when the transition density in the MDP problem is sufficiently smooth, but can be inferior to the latter methods in problems where the transition density has large discontinuities or spikes, violating regularity conditions needed to establish the uniform convergence of the RPI algorithm. This finding suggests that policy iteration algorithms that use low discrepancy may succeed in breaking the of dimensionality in average case settings, since in multivariate problems the rate of convergence of these methods exceeds the rate of convergence of methods based on and other deterministically chosen grids, and thus tends to outperform these methods in problems where there are no large spikes or discontinuities in the transition density of the MDP problem.

Explicit Cost Bounds of Algorithms for Multivariate Tensor Product Problems

We study multivariate tenser product problems in the worst case and average case settings. They are defined on functions of d variables. For arbitrary d, we provide explicit upper bounds on the costs of algorithms which compute an ϵ-approximation to the solution. The cost bounds are of the form (c(d) + 2)β1(β2 + β3(ln 1/ϵ)/(d − 1))β4(d − 1)(1/ϵ)β5. Here c(d) is the cost of one function evaluation (or one linear functional evaluation), and βi′s do not depend on d; they are determined by the properties of the problem for d = 1. For certain tensor product problems, these cost bounds do not exceed c(d)Kϵ−p for some numbers K and p, both independent of d. However, the exponents p which we obtain are too large. We apply these general estimates to certain integration and approximation problems in the worst and average case settings. We also obtain an upper bound, which is independent of d, for the number, n(ϵ, d), of points for which discrepancy (with unequal weights) is at most ϵ, n(ϵ, d) ≤ 7.26ϵ−2.454, ∀d, ϵ ≤ 1.