Stability Analysis of Learning Algorithms for Ontology Similarity Computation

Abstract

Ontology, as a useful tool, is widely applied in lots of areas such as social science, computer science, and medical science. Ontology concept similarity calculation is the key part of the algorithms in these applications. A recent approach is to make use of similarity between vertices on ontology graphs. It is, instead of pairwise computations, based on a function that maps the vertex set of an ontology graph to real numbers. In order to obtain this, the ranking learning problem plays an important and essential role, especially k-partite ranking algorithm, which is suitable for solving some ontology problems. A ranking function is usually used to map the vertices of an ontology graph to numbers and assign ranks of the vertices through their scores. Through studying a training sample, such a function can be learned. It contains a subset of vertices of the ontology graph. A good ranking function means small ranking mistakes and good stability. For ranking algorithms, which are in a well-stable state, we study generalization bounds via some concepts of algorithmic stability. We also find that kernel-based ranking algorithms stated as regularization schemes in reproducing kernel Hilbert spaces satisfy stability conditions and have great generalization abilities.

1. Introduction and Motivations

The study of ontology deals with questions concerning what entities exist and how such entities can be grouped, related within a hierarchy, and subdivided according to similarities and differences. The developed tools have been widely applied in medicine, biology, and social science. In computer science, ontology is defined as a model for sharing formal concepts and has been applied in intelligent information integration, cooperative information systems, information retrieval, electronic commerce, and knowledge management. After a decade’s development, ontology technology has matured as an effective model of hierarchical structure and semantics for concepts, supported by systematic and comprehensive engineering theory, representation, and construction tools.

Ontology similarity computation is an essential part in practical applications. In information retrieval, it has been used to compute semantic similarity and search for concepts. We take a graph-theory approach and represent an ontology by a weighted graph 𝐺=(𝑉,𝐸,𝑤). In this setting, 𝑉={𝑣₁,…,𝑣_𝑛} is the (finite) set of vertices corresponding to concepts or objects of the ontology, 𝐸⊂𝑉×𝑉 is a set of edges, and 𝑤∶𝐸→ℝ₊ is a weight function. For two vertices 𝑣_𝑖 and 𝑣_𝑗 representing two concepts, the weight 𝑤(𝑣_𝑖,𝑣_𝑗) measures their similarity in the ontology.

Example 1. In some applications of ontology similarity computation, the weight function 𝑤 takes values on [0,1]. Then the case 𝑤(𝑣_𝑖,𝑣_𝑗)=1 means that 𝑣_𝑖 and 𝑣_𝑗 represent the same concept while 𝑤(𝑣_𝑖,𝑣_𝑗)=0 means that these two concepts have no similarity. In information retrieval, with a threshold parameter 0<𝜖<1, when one tries to find related information of the concept 𝑣_𝑖, all concepts 𝑣_𝑗 satisfying 𝑤(𝑣_𝑖,𝑣_𝑗)>𝜖 are returned, which means that 𝑣_𝑗 and 𝑣_𝑖 have a high similarity.

Traditional methods for ontology similarity computation are based on pairwise similarity calculation. Their computational complexity is high, and they required selection of many parameters, which are not so intuitive. In this paper, we use a learning theory approach. The idea is to learn a scoring function 𝑓∶𝑉→ℝ and then to determine the similarity between vertices (concepts) 𝑣_𝑖 and 𝑣_𝑗 by their value difference |𝑓(𝑣_𝑖)−𝑓(𝑣_𝑗)|: the smaller the difference the higher the similarity. Formally, Sim(𝑣_𝑖,𝑣_𝑗1)≥Sim(𝑣_𝑖,𝑣_𝑗2) if and only if |𝑓(𝑣_𝑖)−𝑓(𝑣_𝑗1)|≤|𝑓(𝑣_𝑖)−𝑓(𝑣_𝑗2)|. Such an inspiring approach was introduced from the viewpoint of ranking in [1] where a ranking algorithm is used for learning from samples a scoring function 𝑓 with small ranking error. The method was employed in the ontology setting in [2] which demonstrates accuracy and efficiency. Another possible way to learn such a function 𝑓 is by a graph Laplacian and taking an eigenvector associated with its second smallest eigenvalue. See [3–6] for details. This method requires a positive definiteness condition for a similarity matrix which is hard to check in our setting. Also, when the size of the graph is large, the computational complexity is high.

In this paper, we explore the learning theory approach for ontology similarity computations in a setting when the ontology graph is a tree. It is a connected graph without cycle. Thus, there is a unique path between any two vertices. The tree structure gives restrictions on similarity of vertices (concepts). For example, we assign a top vertex 𝑣_top and let it be the root, then denote 𝑘 the degree (the number of edges that link to a vertex) of the top vertex. Let 𝑁_𝐺(𝑣_top)={𝑣₁,𝑣₂,…,𝑣_𝑘} be the neighbor set of 𝑣_top. If there is a path from one vertex to 𝑣_top through 𝑣_𝑖, then it belongs to branch 𝑖. Thus, we have 𝑘 branches in the tree and any two vertices belonging to different branches have no edge between them. The concepts in the same branch of the tree should have higher similarity, compared with concepts in different branches. This observation motivates us to apply the 𝑘-partite ranking algorithm [7] in which the 𝑘 parts correspond to the 𝑘 classes of vertices of 𝑘 rates. The rate values of all classes are decided by experts. Intuitively, a vertex of a higher rate 𝑏 is ranked higher than any vertex of rate 𝑎 if 1≤𝑎<𝑏≤𝑘. Thus, the 𝑘-partite ranking algorithm is reasonable to learn a similarity function for an ontology graph with a tree structure.

The main contribution of this paper is to state some ontology computations as a 𝑘-partite ranking problem and to conduct stability analysis of the algorithms with mild conditions, which leads to useful error bounds for ontology applications.

The organization of the rest part of this paper is as follows. The setting and main results are given in the next section. The generalization bounds for learning algorithms will be shown in Section 4. The stability and generalization bounds for the learning algorithms stated as regularization schemes in reproducing kernel Hilbert spaces will be discussed in Section 5.

2. Formal Setting and Main Results

Now, we state our learning algorithm for ontology similarity computation.

Let 𝑉={𝑣₁,…,𝑣_𝑛} be the finite set of vertices of an ontology graph. It is divided into 𝑘 disjoint subsets {𝑉_𝑖}^𝑘_𝑖=1 corresponding to 𝑘 rates. Let 𝒟 be a probability measure on 𝑉.

The performance of a ranking function 𝑓∶𝑉→ℝ can be measured by the following concept.

Definition 2. A ranking loss function is a function 𝑙∶ℝ^𝑉×𝑉×𝑉→ℝ₊∪{0} that assigns, for 𝑓∶𝑉→ℝ and 𝑣,𝑣^′∈𝑉, a nonnegative real number 𝑙(𝑓,𝑣,𝑣^′) interpreted as the loss of 𝑓 in its relative ranking of 𝑣 and 𝑣^′. The expected 𝑘-partite ranking error on the ontology graph for a ranking function 𝑓∶𝑉→ℝ associated with the ranking loss function 𝑙 is defined as 𝑒𝑟_𝑙𝑓=^𝑘−1𝑘_{𝑎=1𝑏=𝑎+1}𝐄_{𝑣∼𝒟𝑎,𝑣′∼𝒟𝑏}𝑙𝑓,𝑣,𝑣^′, (1) where 𝒟_𝑎 is the conditional distribution of 𝒟 on 𝑉_𝑎.

Example 3. One commonly used ranking loss function is the hinge ranking loss defined as 𝑙_ℎ𝑓,𝑣,𝑣^′=1−𝑓(𝑣)−𝑓𝑣^′₊, (2) where 𝑥₊=max{𝑥,0}. Another ranking loss function is the 𝛾-ranking loss with a smoothing parameter 𝛾>0 defined as 𝑙_𝛾𝑓,𝑣,𝑣^′=1,if𝑓(𝑣)−𝑓𝑣^′1<0,1−𝛾𝑓(𝑣)−𝑓𝑣^′,if0≤𝑓(𝑣)−𝑓𝑣^′≤𝛾𝜏𝑣,𝑣^′,0,otherwise. (3)

Learning algorithms are implemented with a sample 𝒯={𝑡_𝑖}^𝑀_𝑖=1 of size 𝑀, called a preference graph, which is assumed here to be independently drawn according to 𝒟. It can also be divided into 𝑘 parts {𝒯₁,…,𝒯_𝑘}, where 𝒯_𝑎={𝑡_𝑖∶𝑡_𝑖∈𝑉_𝑎} consists of those sampling points of rate 𝑎.

In [8], Agarwal and Niyogi have studied the algorithmic stability in a general setting, where the training examples take labels 𝑦∈[0,𝑀₁] for some 𝑀₁>0. A goal ranking function ranks future instances with larger labels higher than those with smaller labels. Here, our setting is more specific. The learner is given a preference graph 𝒯 consisting of 𝑘 disjoint parts corresponding to the 𝑘 classes of vertices. Every part has a rate value. The target ranking function ranks future instances in higher-rate parts higher than those in lower-rate parts.

A large class of learning algorithms is generated by regularization schemes. They penalize an empirical error which is chosen here to be the empirical 𝑘-partite ranking error on the ontology graph defined for a 𝑓∶𝑉→ℝ associated with the sample 𝒯 as 𝑒𝑟_𝒯,𝑙𝑓=1𝑀^{2𝑘−1𝑘}_{𝑎=1𝑏=𝑎+1𝑖∶𝑡𝑖∈𝒯𝑎𝑗∶𝑡𝑗∈𝒯𝑏}𝑙𝑓,𝑡_𝑖,𝑡_𝑗. (4)

In this paper, we study a learning algorithm generated by a regularization scheme in a reproducing kernel Hilbert space (RKHS) (ℋ_𝐾,‖⋅‖_𝐾) associated with a Mercer kernel 𝐾∶𝑉×𝑉→ℝ. Now, the regularization scheme is defined by 𝑓_𝒯=argmin_{𝑓∈ℋ𝐾}𝑒𝑟_𝒯,𝑙𝑓+𝜆𝑓²_𝐾, (5) where 𝜆>0 is a regularization parameter. On the selection of the regularized parameter, readers are referred to [9, 10] for more details about the method of cross-validation.

One point we need to emphasize that we abuse terminology for the sake of better readability. If the ranking function 𝑓 does not associate with RKHS (for instance, in Lemma 9, Theorems 10 and 12), then the second term in the right-hand side of (5) vanishes.

Our error analysis provides a learning rate of algorithm (5) when the ranking loss is 𝜎-admissible.

Definition 4. Let 𝑙 be a ranking loss, 𝜎>0, and ℱ a class of real-valued functions on 𝑉. We say that 𝑙 is 𝜎-admissible with respect to ℱ if for any 𝑓₁, 𝑓₂∈ℱ and 𝑣,𝑣^′∈𝑉, 𝑙𝑓₁,𝑣,𝑣^′−𝑙𝑓₂,𝑣,𝑣^′≤𝜎𝑓₁(𝑣)−𝑓₂(𝑣)+𝑓₁𝑣^′−𝑓₂𝑣^′. (6)

Let us state the estimate of learning rates which will be proved in Section 5.

Theorem 5. Let ℋ_𝐾 be a RKHS such that 𝐾(𝑣,𝑣)≤𝜅²<∞ for all 𝑣∈𝑉. Let 𝑙 be a ranking loss, 𝜎-admissible with respect to ℋ_𝐾, and bounded by some 𝐵>0, such that 𝑙(𝑓,𝑣,𝑣^′) is convex with respect to 𝑓. Let 𝑓^∗_𝑙 be a fixed function in ℋ_𝐾 satisfying 𝑒𝑟_𝑙𝑓^∗_𝑙≤inf_{𝑓∈ℋ𝐾}er_𝑙𝑓+1𝑀^′ (7) for some 𝑀^′>0. Then, for any 0<𝛿<1, with confidence at least 1−𝛿, one has er_𝑙𝑓_𝒯≤inf_{𝑓∈ℋ𝐾}𝑒𝑟_𝑙𝑓+𝜆𝑓^∗_𝑙²_𝐾+1𝑀^′+16𝜎²𝜅²+𝜆𝑀24𝜎²𝜅²𝜆+𝐵ln(2/𝛿).2𝑀 (8)

Form Theorem 5, we see that if 𝜆→0 and 𝜆√𝑀→∞ (e.g., 𝜆=𝑀^−1/4), then 𝑒𝑟_𝑙(𝑓_𝒯) converges with confidence to inf_{𝑓∈ℋ𝐾}𝑒𝑟_𝑙(𝑓). The quantity inf_{𝑓∈ℋ𝐾}𝑒𝑟_𝑙(𝑓) is well understood in the literature related to learning theory (e.g., [11–14]).

3. Stability Analysis

An algorithm is stable if any change of a single point in a training set yields only a small change in the output. It is natural to consider that a good ranking algorithm is one with good stability; that is, a mild change of samples does not necessarily lead to too much change in the ranking function. Some analysis of the stability of ranking algorithms is given in [1, 8, 15].

Let 𝑛_𝑎=|𝒯_𝑎| and 𝑡^𝑖𝑎_𝑎 be the 𝑖_𝑎th element in 𝒯_𝑎 for 1≤𝑎≤𝑘,1≤𝑖_𝑎≤𝑛_𝑎. Let 𝒯^{(𝑡𝑖𝑎𝑎,𝑡𝑎)} be the sequence obtained by replacing 𝑡^𝑖𝑎_𝑎 in 𝒯 by a new sampling point 𝑡^𝑎 of rate 𝑎. We define some notions of stability for 𝑘-partite ranking algorithms.

Definition 6 (uniform loss stability for a 𝑘-partite ranking algorithm on ontology graph). Let A be a 𝑘-partite ranking algorithm for ontology whose output on a preference graph 𝒯=(𝒯₁,…,𝒯_𝑘) is denoted by 𝑓_𝒯. Let 𝑙 be a ranking loss function and 𝛼_𝑎∶ℕ^𝑘→ℝ for 1≤𝑎≤𝑘. We say that 𝐴 has uniform loss stability (𝛼₁,…,𝛼_𝑘) with respect to 𝑙 if for all 1≤𝑎≤𝑘, 𝑛_𝑎∈ℕ, 𝑡^𝑖𝑎_𝑎∈𝒯_𝑎 and 𝑖_𝑎∈{1,…,𝑛_𝑎}, we have for all 𝑡^𝑎∈𝑉_𝑎, 𝑣,𝑣^′∈𝑉 but belong to different rate, 𝑙𝑓_𝒯,𝑣,𝑣^′−𝑙𝑓_𝒯^𝑖_{𝑗𝑗𝑗)(𝑡,𝑡},𝑣,𝑣^′≤𝛼_𝑗𝑛₁,…,𝑛_𝑘,𝑗=1,…,𝑘. (9)

Definition 7 (uniform score stability for a 𝑘-partite ranking algorithm on ontology graph). Let A be a 𝑘-partite ranking algorithm for ontology whose output on a preference graph 𝒯=(𝒯₁,…,𝒯_𝑘) is denoted by 𝑓_𝒯. Let 𝑙 be a ranking loss function and 𝜇_𝑎∶ℕ^𝑘→ℝ for 1≤𝑎≤𝑘. We say that 𝐴 has uniform score stability (𝜇₁,…,𝜇_𝑘) with respect to 𝑙 if for all 1≤𝑎≤𝑘, 𝑛_𝑎∈ℕ, 𝑡^𝑖𝑎_𝑎∈𝒯_𝑎, 𝑖_𝑎∈{1,…,𝑛_𝑎}, and 𝑡^𝑎∈𝑉_𝑎, and for all 𝑣∈𝑉, 𝑓_𝒯(𝑣)−𝑓_𝒯^𝑖_{𝑗𝑗𝑗)(𝑡,𝑡}(𝑣)≤𝜇_𝑗𝑛₁,…,𝑛_𝑘,𝑗=1,…,𝑘. (10)

The main tool used here is McDiarmid’s inequality, which bounds the deviation of any function of a sample on which a single change in the sample has limited effect.

Theorem 8 (see [16]). Let 𝑋₁,…,𝑋_𝑁 be independent random variables, each taking values in a set 𝐴. Let 𝜙∶𝐴^𝑁→ℝ such that for each 𝑖∈{1,…,𝑁}, there exists a constant 𝑐_𝑖>0 such that sup_{𝑥1,…,𝑥𝑁∈𝐴,𝑥′𝑖∈𝐴}𝜙𝑥₁,…,𝑥_𝑁−𝜙𝑥₁,…,𝑥_𝑖−1,𝑥^′_𝑖,𝑥_𝑖+1,…,𝑥_𝑁≤𝑐_𝑖. (11) Then, for any 𝜀>0, 𝐏𝜙𝑋₁,…,𝑋_𝑁−𝐄𝜙𝑋₁,⋯,𝑋_𝑁≥𝜀≤𝑒^{−2𝜀2/∑𝑁𝑖=1𝑐2𝑖}. (12)

In what follows, 𝒯^{(𝑡𝑖11,𝑡1),…,(𝑡𝑖𝑘𝑘,𝑡𝑘)} denotes a training sample set obtained by replacing 𝑡^𝑖1₁ in 𝒯 by 𝑡¹,…,𝑡^𝑖𝑘_𝑘 by 𝑡^𝑘 for 𝑡^𝑖𝑎_𝑎∈𝒯_𝑎, and 𝑖_𝑎∈{1,…,𝑛_𝑎}. Also, 𝛼_𝑎(𝑛₁,…,𝑛_𝑘) and 𝜇_𝑎(𝑛₁,…,𝑛_𝑘) are simply denoted by 𝛼_𝑎 and 𝜇_𝑎, respectively. We only consider the case of sample replacements with the same rate: for some ontology graphs, the graph structure is fixed; hence the members of vertices and edges in each branch are fixed.

4. Generalization Bounds for Stable 𝑘-Partite Ranking Algorithms on Ontology Graph

From this section, our analysis for stability of 𝑘-partite ranking algorithms is stated on an ontology graph and our organization follows [8]. In this section, generalization bounds for ranking algorithms that exhibit good stability properties will be derived. Our tricks are based on those of [17]. We start with the following technical lemma.

Lemma 9. Let A be a symmetric 𝑘-partite ranking algorithm for ontology whose output on a preference graph 𝒯=(𝒯₁,…,𝒯_𝑘) is denoted by 𝑓_𝒯, and let 𝑙 be a ranking loss function. Then, for all 𝑡^𝑖𝑎_𝑎∈𝒯_𝑎, 𝑖_𝑎∈{1,…,𝑛_𝑎}, and 𝑡^𝑎∈𝑉_𝑎, 𝑡^𝑏∈𝑉_𝑏, one has 𝐄_𝒯er_𝑙𝑓_𝒯−𝑒𝑟_𝒯,𝑙𝑓_𝒯=1𝑀^{2𝑘−1𝑘}_{𝑎=1𝑏=𝑎+1}𝐄_{𝒯,𝑡𝑎,𝑡𝑏}𝑛_𝑎𝑛_𝑏𝑙𝑓_𝒯,𝑡^𝑎,𝑡^𝑏−𝑙𝑓_{𝒯𝑖𝑎𝑎𝑎}^𝑖_{𝑏𝑏𝑏)(𝑡,𝑡),(𝑡,𝑡},𝑡^𝑎,𝑡^𝑏. (13)

Proof. We have 𝐄_𝒯𝑒𝑟_𝒯,𝑙𝑓_𝒯=1𝑀^{2𝑘−1𝑘}_{𝑎=1𝑏=𝑎+1}𝐄_{𝒯𝑖∶𝑡𝑖∈𝒯𝑎𝑗∶𝑡𝑗∈𝒯𝑏}𝑙𝑓_𝒯,𝑡_𝑖,𝑡_𝑗. (14) By symmetry, the term in the summation is the same for all 𝑖,𝑗. Therefore, we get 𝐄_𝒯𝑒𝑟_𝒯,𝑙𝑓_𝒯=1𝑀^{2𝑘−1𝑘}_{𝑎=1𝑏=𝑎+1}𝐄_{𝒯,𝑡𝑎,𝑡𝑏𝑖∶𝑡𝑖∈𝒯𝑎𝑗∶𝑡𝑗∈𝒯𝑏}𝑙𝑓_𝒯,𝑡_𝑖,𝑡_𝑗. (15) Interchanging the roles of 𝑡_𝑖 with 𝑡^𝑎 and 𝑡_𝑗 with 𝑡^𝑏, we get 𝐄_𝒯𝑒𝑟_𝒯,𝑙𝑓_𝒯=1𝑀^{2𝑘−1𝑘}_{𝑎=1𝑏=𝑎+1}𝐄_{𝒯,𝑡𝑎,𝑡𝑏}𝑛_𝑎𝑛_𝑏𝑙𝑓_{𝒯𝑖𝑎𝑎𝑎}^𝑖_{𝑏𝑏𝑏)(𝑡,𝑡),(𝑡,𝑡},𝑡^𝑎,𝑡^𝑏. (16) Since 𝐄_𝒯𝑒𝑟_𝑙𝑓_𝒯=1𝑀^{2𝑘−1𝑘}_{𝑎=1𝑏=𝑎+1}𝐄_{𝒯,𝑡𝑎,𝑡𝑏}𝑛_𝑎𝑛_𝑏𝑙𝑓_𝒯, (17) the results follow.

We are now ready to give our main result of this section, which bounds the expected 𝑙-error of a ranking function learned by a 𝑘-partite ranking algorithm with good uniform loss stability in terms of its empirical 𝑙-error on the training sample. The proof follows [18].

Theorem 10. Let A be a symmetric 𝑘-partite ranking algorithm for ontology whose output on a preference graph 𝒯=(𝒯₁,…,𝒯_𝑘) is denoted by 𝑓_𝒯, and let 𝑙 be ranking loss function such that 0≤𝑙(𝑓,𝑣,𝑣^′)≤𝐵 for all 𝑓∶𝑉→ℝ and 𝑣,𝑣^′∈𝒯. Let 𝛼_𝑎∶ℕ^𝑘→ℝ for 1≤𝑎≤𝑘 such that A has uniform loss stability (𝛼₁,…,𝛼_𝑘) with respect to 𝑙. Let 𝛼_max=max{𝛼₁,…,𝛼_𝑘}. Then, for any 0<𝛿<1, with confidence at least 1−𝛿, one has 𝑒𝑟_𝑙𝑓_𝒯≤𝑒𝑟_𝒯,𝑙𝑓_𝒯+2𝛼_max+2𝛼_max+𝐵𝑀𝑀ln(1/𝛿)2. (18)

Proof. Let 𝜙∶𝑉^𝑀→ℝ be defined by 𝜙(𝒯)=𝑒𝑟_𝑙𝑓_𝒯−𝑒𝑟_𝒯,𝑙𝑓_𝒯. (19) We show that 𝜙 satisfies the condition for McDiarmid’s inequality. To this end, let 𝑡^𝑎∈𝑉_𝑎. For each 𝑖₁∈{1,…,𝑛₁}, we have 𝑒𝑟_𝑙𝑓_𝒯−𝑒𝑟_𝑙𝑓_𝒯^𝑖_{111)(𝑡,𝑡}=^𝑘−1𝑘_{𝑎=1𝑏=𝑎+1}𝐄_{𝑣∼𝒟𝑎,𝑣′∼𝒟𝑏}𝑙𝑓_𝒯,𝑣,𝑣^′−𝑙𝑓_𝒯^𝑖_{111)(𝑡,𝑡},𝑣,𝑣^′≤𝛼₁,𝑒𝑟_𝒯,𝑙𝑓_𝒯−𝑒𝑟_𝒯,𝑙𝑓_𝒯^𝑖_{111)(𝑡,𝑡}≤1𝑀^{2𝑘−1𝑘}_{𝑎=1𝑏=𝑎+1𝑡𝑖∶𝑖∈𝒯𝑎,𝑡𝑖≠𝑡𝑖11𝑗∶𝑡𝑗∈𝒯𝑏}𝑙𝑓_𝒯,𝑡_𝑖,𝑡_𝑗−𝑙𝑓_𝒯^𝑖_{111)(𝑡,𝑡},𝑡_𝑖,𝑡_𝑗+1𝑀^2𝑘_{𝑏=2𝑗∶𝑡𝑗∈𝒯𝑏}𝑙𝑓_𝒯,𝑡^𝑖1₁,𝑡_𝑗,−𝑙𝑓_𝒯^𝑖_{111)(𝑡,𝑡},𝑡^𝑖1₁,𝑡_𝑗≤𝛼₁+𝐵∑^𝑘_𝑏=2𝑛_𝑏𝑀²≤𝛼₁+𝐵𝑀. (20) These give 𝜙(𝒯)−𝜙𝒯^{(𝑡𝑖11,𝑡1)}≤2𝛼₁+𝐵𝑀. (21) Similarly, it can be shown that for any 𝑖_𝑎∈{1,…,𝑛_𝑎}, 1≤𝑎≤𝑘, 𝜙(𝒯)−𝜙𝒯^{(𝑡𝑖𝑎𝑎,𝑡𝑎)}≤2𝛼_𝑎+𝐵𝑀. (22) Thus, applying McDiarmid’s inequality to 𝜙, we get for any 𝜀>0, 𝐏_𝒯𝑒𝑟_𝑙𝑓_𝒯−𝑒𝑟_𝒯,𝑙𝑓_𝒯−𝐄_𝒯𝑒𝑟_𝑙𝑓_𝒯−𝑒𝑟_𝒯,𝑙𝑓_𝒯−≥𝜀≤exp2𝜀²𝑀2𝛼_max+𝐵/𝑀². (23) Now, by Lemma 9, we know that the expectation 𝐄_𝒯[𝑒𝑟_𝑙(𝑓_𝒯)−𝑒𝑟_𝒯,𝑙(𝑓_𝒯)] can be bounded as 1𝑀^{2𝑘−1𝑘}_{𝑎=1𝑏=𝑎+1}𝐄_{𝒯,𝑡𝑎,𝑡𝑏}𝑛_𝑎𝑛_𝑏𝑙𝑓_𝒯,𝑡^𝑎,𝑡^𝑏−𝑙𝑓_{𝒯𝑖𝑎𝑎𝑎}^𝑖_{𝑏𝑏𝑏)(𝑡,𝑡),(𝑡,𝑡},𝑡^𝑎,𝑡^𝑏=1𝑀^{2𝑘−1𝑘}_{𝑎=1𝑏=𝑎+1}𝐄_{𝒯,𝑡𝑎,𝑡𝑏}𝑛_𝑎𝑛_𝑏𝑙𝑓_𝒯,𝑡^𝑎,𝑡^𝑏−𝑙𝑓_𝒯^𝑖_{𝑎𝑎𝑎)(𝑡,𝑡},𝑡^𝑎,𝑡^𝑏+1𝑀^{2𝑘−1𝑘}_{𝑎=1𝑏=𝑎+1}𝐄_{𝒯,𝑡𝑎,𝑡𝑏}𝑛_𝑎𝑛_𝑏𝑙𝑓_𝒯^𝑖_{𝑎𝑎𝑎)(𝑡,𝑡},𝑡^𝑎,𝑡^𝑏−𝑙𝑓_{𝒯𝑖𝑎𝑎𝑎}^𝑖_{𝑏𝑏𝑏)(𝑡,𝑡),(𝑡,𝑡},𝑡^𝑎,𝑡^𝑏≤1𝑀^{2𝑘−1𝑘}_{𝑎=1𝑏=𝑎+1}𝐄_{𝒯,𝑡𝑎,𝑡𝑏}𝑛_𝑎𝑛_𝑏𝛼_𝑎+1𝑀^{2𝑘−1𝑘}_{𝑎=1𝑏=𝑎+1}𝐄_{𝒯,𝑡𝑎,𝑡𝑏}𝑛_𝑎𝑛_𝑏𝛼_𝑏≤2𝛼_max. (24)
Thus, for any 𝜀>0, 𝐏_𝒯𝑒𝑟_𝑙𝑓_𝒯−𝑒𝑟_𝒯,𝑙𝑓_𝒯−2𝛼_max−≥𝜀≤exp2𝜀²𝑀2𝛼_max+𝐵/𝑀². (25) The result follows by setting the right-hand side equal to 𝛿 and solving it for 𝜀.

For any 𝛾>0, and any 𝑘-partite ranking algorithm with good uniform loss stability with respect to 𝑙_𝛾, Theorem 10 can be applied to bound the expected ranking error of a learned ranking function in terms of its empirical 𝑙_𝛾-error on the training sample. The following lemma shows that, for every 𝛾>0, a ranking algorithm with good uniform score stability also has good uniform loss stability with respect to 𝑙_𝛾. Using the techniques of Lemma 2 in [17], and taking 𝜏(𝑣_𝑖,𝑣_𝑗) in Example 3 as 1, the following lemma can be obtained immediately.

Lemma 11. Let A be a 𝑘-partite ranking algorithm for ontology whose output on a preference graph 𝒯=(𝒯₁,…,𝒯_𝑘) is denoted by 𝑓_𝒯. Let 𝜇_𝑎∶ℕ^𝑘→ℝ for 1≤𝑎≤𝑘 such that A has uniform score stability (𝜇₁,…,𝜇_𝑘). Then, for every 𝛾>0, A has uniform loss stability (𝛼^𝛾₁,…,𝛼^𝛾_𝑘) with respect to the 𝛾 ranking loss 𝑙_𝛾, where for all 𝑛₁,…,𝑛_𝑘∈ℕ, 𝛼^𝛾_𝑗𝑛₁,…,𝑛_𝑘=2𝜇_𝑗𝛾,𝑗=1,…,𝑘. (26)

Combining Theorem 10 and Lemma 11, we get the following result which bounds the expected ranking error of a learned ranking function in terms of its empirical 𝑙_𝛾-error for any ranking algorithm with good uniform score stability.

Theorem 12. Let A be a 𝑘-partite ranking algorithm for ontology whose output on a preference graph 𝒯=(𝒯₁,…,𝒯_𝑘) is denoted by 𝑓_𝒯. Let 𝜇_𝑎∶ℕ^𝑘→ℝ for 1≤𝑎≤𝑘 such that A has uniform score stability (𝜇₁,…,𝜇_𝑘), and 𝛾>0. Denote 𝜇_max=max{𝜇₁,…,𝜇_𝑘}. If 𝑙 is a ranking loss satisfying 0≤𝑙(𝑓,𝑣,𝑣^′)≤𝐵 for all 𝑓∶𝑉→ℝ and 𝑣,𝑣^′∈𝒯, then for any 0<𝛿<1, with probability of at least 1−𝛿, 𝑒𝑟_𝑙𝑓_𝒯≤𝑒𝑟_𝒯,𝑙𝑓_𝒯+4𝛾𝜇_max+4𝛾𝜇_max+𝐵𝑀𝑀ln(1/𝛿)2. (27)

Proof. One applies Theorem 10 to 𝐴 with the ranking loss 𝑙_𝛾 (using Lemma 11), which satisfies 0≤𝑙_𝛾≤𝐵. One finishes the proof thanks to the fact that 𝑒𝑟≤𝑒𝑟_𝑙𝛾

5. Stable Ranking Algorithms

In this section, we will demonstrate stability of some ranking algorithms in which a ranking function is selected by minimizing a regularized objective function. A general result for regularization-based 𝑘-partite ranking algorithms will be derived in Section 5.1. In Section 5.2, this result is used to illustrate stability of kernel-based 𝑘-partite ranking algorithms that perform regularization in a reproducing kernel Hilbert space. These stability results are also used to achieve consistency theorem for kernel-based 𝑘-partite ranking algorithms in Section 5.3.

5.1. General Regularizers

Let 𝑙 be given a ranking loss function, ℱ a class of real-valued functions on 𝑉, and 𝑁∶ℱ→ℝ₊∪{0} a regularization functional. Consider the following regularized empirical 𝑙-error of a ranking function 𝑓∈ℱ (with respect to a preference graph 𝒯) with regularization parameter 𝜆>0, 𝑒𝑟^𝜆_𝒯,𝑙=1𝑀^{2𝑘−1𝑘}_{𝑎=1𝑏=𝑎+1𝑖∶𝑡𝑖∈𝒯𝑎𝑗∶𝑡𝑗∈𝒯𝑏}𝑙𝑓,𝑡_𝑖,𝑡_𝑗+𝜆𝑁𝑓. (28) We consider 𝑘-partite ranking algorithms that minimize such a regularized objective function; that is, ranking algorithms that, given a preference graph 𝒯, output a ranking function 𝑓_𝒯∈ℱ that satisfies 𝑓_𝒯=argmin_𝑓∈ℱ𝑒𝑟^𝜆_𝒯,𝑙=argmin_𝑓∈ℱ𝑒𝑟_𝒯,𝑙+𝜆𝑁𝑓, (29) for some fixed choice of ranking loss 𝑙, function class ℱ, regularized 𝑁, and regularization parameter 𝜆. We derive a general result below that will be useful for showing stability of such regularization-based algorithms.

Lemma 13. Let 𝑙 be a ranking loss such that 𝑙(𝑓,𝑣,𝑣^′) is convex in 𝑓. Let ℱ be a convex set of real-valued functions on 𝑉, and let 𝜎>0 such that 𝑙 is 𝜎-admissible with respect to ℱ. Let 𝜆>0, and let 𝑁∶ℱ→ℝ₊∪{0} be a functional defined on ℱ such that for preference graph 𝒯=(𝒯₁,…,𝒯_𝑘), the regularized empirical 𝑙-error 𝑒𝑟^𝜆_𝒯,𝑙 has a minimum (not necessarily unique) in ℱ. Let A be a 𝑘-partite ranking algorithm for ontology defined by (29). Let 𝑡^𝑎∈𝑉_𝑎, 𝑡^𝑖𝑎_𝑎∈𝒯_𝑎, and 𝑖_𝑎∈{1,…,𝑛_𝑎}. For brevity, denote 𝑓≡𝑓_𝒯,𝑓_1,𝑖1≡𝑓_𝒯^𝑖_{111((𝑡,𝑡))},…,𝑓_{𝑘,𝑖𝑘}≡𝑓_𝒯^𝑖_{𝑘𝑘𝑘((𝑡,𝑡))}, (30) and let Δ𝑓_1,𝑖1=𝑓_1,𝑖1−𝑓,…,Δ𝑓_{𝑘,𝑖𝑘}=𝑓_{𝑘,𝑖𝑘}−𝑓. (31) Then for any 𝑡∈[0,1] and 𝑞=1,…,𝑘, one has 𝑁𝑓−𝑁𝑓+𝑡Δ𝑓_{𝑞,𝑖𝑞}+𝑁𝑓_{𝑞,𝑖𝑞}−𝑁𝑓_{𝑞,𝑖𝑞}−𝑡Δ𝑓_{𝑞,𝑖𝑞}≤𝑡𝜎𝜆𝑀²_{𝑏≠𝑞𝑗∶𝑡𝑗∈𝒯𝑏}Δ𝑓_{𝑞,𝑖𝑞}𝑡^𝑖𝑞_𝑞+2Δ𝑓_{𝑞,𝑖𝑞}𝑡_𝑗+Δ𝑓_{𝑞,𝑖𝑞}𝑡^𝑞. (32)

Proof. Recall that a convex function 𝑔 satisfies 𝑔𝑥+𝑡𝑦−𝑥−𝑔(𝑥)≤𝑡𝑔𝑦−𝑔(𝑥),∀𝑡∈[0,1]. (33) Since 𝑙(𝑓,𝑣,𝑣^′) is convex in 𝑓, 𝑒𝑟_𝒯,𝑙(𝑓) is convex in 𝑓. Therefore, for any 𝑡∈[0,1], we have 𝑒𝑟_𝒯,𝑙𝑓+𝑡Δ𝑓_1,𝑖1−𝑒𝑟_𝒯,𝑙𝑓≤𝑡𝑒𝑟_𝒯,𝑙𝑓_1,𝑖1−𝑒𝑟_𝒯,𝑙𝑓, (34) and also (interchanging the roles of 𝑓 and 𝑓_1,𝑖1), 𝑒𝑟_𝒯,𝑙𝑓_1,𝑖1−𝑡Δ𝑓_1,𝑖1−𝑒𝑟_𝒯,𝑙𝑓_1,𝑖1≤𝑡𝑒𝑟_𝒯,𝑙𝑓−𝑒𝑟_𝒯,𝑙𝑓_1,𝑖1. (35) Adding the above two inequalities yields 𝑒𝑟_𝒯,𝑙𝑓+𝑡Δ𝑓_1,𝑖1−𝑒𝑟_𝒯,𝑙𝑓+𝑒𝑟_𝒯,𝑙𝑓_1,𝑖1−𝑡Δ𝑓_1,𝑖1−𝑒𝑟_𝒯,𝑙𝑓_1,𝑖1≤0. (36) Now, since ℱ is convex, (𝑓+𝑡Δ𝑓_1,𝑖1)∈ℱ and (𝑓_1,𝑖1−𝑡Δ𝑓_1,𝑖1)∈ℱ. Since 𝑓 minimizes 𝑒𝑟^𝜆_𝒯,𝑙(𝑓) in ℱ and 𝑓_1,𝑖1 minimizes 𝑒𝑟^𝜆_{𝒯𝑖111)(𝑡,𝑡,𝑙}(𝑓) in ℱ, we have 𝑒𝑟^𝜆_𝒯,𝑙𝑓−𝑒𝑟^𝜆_𝒯,𝑙𝑓+𝑡Δ𝑓_1,𝑖1≤0,𝑒𝑟^𝜆_{𝒯𝑖111)(𝑡,𝑡,𝑙}𝑓_1,𝑖1−𝑒𝑟^𝜆_{𝒯𝑖111)(𝑡,𝑡,𝑙}𝑓_1,𝑖1−𝑡Δ𝑓_1,𝑖1≤0. (37) Adding these two inequalities and applying (36), we get 𝜆𝑁𝑓−𝑁𝑓+𝑡Δ𝑓_1,𝑖1+𝑁𝑓_1,𝑖1−𝑁𝑓_1,𝑖1−𝑡Δ𝑓_1,𝑖1≤𝑒𝑟_𝒯,𝑙𝑓_1,𝑖1−𝑒𝑟_𝒯^𝑖_{111)(𝑡,𝑡,𝑙}𝑓_1,𝑖1+𝑒𝑟_𝒯^𝑖_{111)(𝑡,𝑡,𝑙}𝑓_1,𝑖1−𝑡Δ𝑓_1,𝑖1−𝑒𝑟_𝒯,𝑙𝑓_1,𝑖1−𝑡Δ𝑓_1,𝑖1=1𝑀^2𝑘_{𝑏=2𝑗∶𝑡𝑗∈𝒯𝑏}𝑙𝑓_1,𝑖1,𝑡^𝑖1₁,𝑡_𝑗−𝑙𝑓_1,𝑖1−𝑡Δ𝑓_1,𝑖1,𝑡^𝑖1₁,𝑡_𝑗+𝑙𝑓_1,𝑖1−𝑡Δ𝑓_1,𝑖1,𝑡¹,𝑡_𝑗−𝑙𝑓_1,𝑖1,𝑡¹,𝑡_𝑗≤𝑡𝜎𝑀^2𝑘_{𝑏=2𝑗∶𝑡𝑗∈𝒯𝑏}Δ𝑓_1,𝑖1𝑡^𝑖1₁+2Δ𝑓_1,𝑖1𝑡_𝑗+Δ𝑓_1,𝑖1𝑡¹. (38) Similarly, for 𝑞=2,…,𝑘, we have 𝜆𝑁𝑓−𝑁𝑓+𝑡Δ𝑓_{𝑞,𝑖𝑞}+𝑁𝑓_{𝑞,𝑖𝑞}−𝑁𝑓_{𝑞,𝑖𝑞}−𝑡Δ𝑓_{𝑞,𝑖𝑞}≤𝑡𝜎𝑀²_{𝑏≠𝑞𝑗∶𝑡𝑗∈𝒯𝑏}Δ𝑓_{𝑞,𝑖𝑞}𝑡^𝑖𝑞_𝑞+2Δ𝑓_{𝑞,𝑖𝑞}𝑡_𝑗+Δ𝑓_{𝑞,𝑖𝑞}𝑡^𝑞. (39) The results follow.

As we will see below, the above result can be used to establish stability of some regularization-based ranking algorithms.

5.2. Regularization in Reproducing Kernel Hilbert Spaces

Let ℱ be a reproducing kernel Hilbert space (RKHS) of real-valued functions on 𝑉 associated with a Mercer kernel 𝐾∶𝑉×𝑉→ℝ. Here, 𝐾_𝑣∶𝑉→ℝ is defined as 𝐾_𝑣(𝑣^′)=𝐾(𝑣,𝑣^′), and the reproducing property of ℱ gives that for all 𝑓∈ℱ and all 𝑣∈𝑉, 𝑓(𝑣)=𝑓,𝐾_𝑣𝐾, (40) where ⟨⋅,⋅⟩_𝐾 denotes the RKHS inner product in ℱ. By the Schwartz inequality, it is easy to show that for all 𝑓∈ℱ and all 𝑣∈𝑉, 𝑓(𝑣)≤𝑓_𝐾𝐾_𝑣𝐾=𝑓_𝐾𝐾(𝑣,𝑣), (41) where ‖⋅‖_𝐾 denotes the RKHS norm in ℱ. We consider ranking algorithms that perform regularization in the RKHS ℱ using the squared norm in ℱ as regularizers. Specifically, let 𝑁∶ℱ→ℝ₊∪{0} be the regularizer defined by 𝑁𝑓=𝑓²_𝐾. (42) It will be demonstrated below that if for some 0≤𝜅<∞, 𝐾(𝑣,𝑣)≤𝜅 for any 𝑣∈𝑉, then a ranking algorithm that minimizes an appropriate regularized error over ℱ, with regularizer 𝑁 defined as above, has good uniform score stability.

Theorem 14. Let ℱ be an RKHS with kernel 𝐾 such that for all 𝑣∈𝑉,𝐾(𝑣,𝑣)≤𝜅²<∞. Let 𝑙 be a ranking loss such that 𝑙(𝑓,𝑣,𝑣^′) is convex in 𝑓 and 𝑙 is 𝜎-admissible with respect to ℱ. Let 𝜆>0, and let 𝑁 be given by (42). Let A be the 𝑘-partite ranking algorithm for ontology that, given a preference graph 𝒯, outputs a ranking function 𝑓_𝒯∈ℱ defined by (29). Then, A has uniform score stability (𝜇₁,…,𝜇_𝑘) with 𝜇_𝑞𝑛₁,…,𝑛_𝑘=4𝜎𝜅²𝜆𝑀,𝑞=1,…,𝑘. (43)

Proof. Let 𝑣^𝑎∈𝑉_𝑎 and 𝑖_𝑎∈{1,…,𝑛_𝑎}.
Applying Lemma 13 with 𝑡=1/2, we get (using the notation in the proof of Lemma 13) that 𝑓²_𝐾−1𝑓+2Δ𝑓_1,𝑖1²_𝐾+𝑓_1,𝑖1²_𝐾−𝑓_1,𝑖1−12Δ𝑓_1,𝑖1²_𝐾≤𝜎2𝜆𝑀^2𝑘_{𝑏=2𝑗∶𝑡𝑗∈𝒯𝑏}Δ𝑓_1,𝑖1𝑡^𝑖1₁+2Δ𝑓_1,𝑖1𝑡_𝑗+Δ𝑓_1,𝑖1𝑡¹. (44) Since ℱ is a vector space, Δ𝑓_1,𝑖1∈ℱ, (𝑓+(1/2)Δ𝑓_1,𝑖1)∈ℱ, and (𝑓_1,𝑖1−(1/2)Δ𝑓_1,𝑖1)∈ℱ, so ‖𝑓+(1/2)Δ𝑓_1,𝑖1‖²_𝑘 and ‖𝑓_1,𝑖1−(1/2)Δ𝑓_1,𝑖1‖²_𝑘 are well defined. It is easy to check that 𝑓²_𝐾−1𝑓+2Δ𝑓_1,𝑖1²_𝐾+𝑓_1,𝑖1²_𝐾−𝑓_1,𝑖1−12Δ𝑓_1,𝑖1²_𝐾=12Δ𝑓_1,𝑖1²_𝐾. (45) Combined with (44), this gives Δ𝑓_1,𝑖1²_𝐾≤𝜎𝜆𝑀^2𝑘_{𝑏=2𝑗∶𝑡𝑗∈𝒯𝑏}Δ𝑓_1,𝑖1𝑡^𝑖1₁+2Δ𝑓_1,𝑖1𝑡_𝑗+Δ𝑓_1,𝑖1𝑡¹. (46) Since (as noted above) Δ𝑓_1,𝑖1∈ℱ, this together with (41) gives Δ𝑓_1,𝑖1²_𝐾≤𝜎𝜆𝑀²Δ𝑓_1,𝑖1𝐾^𝑘_{𝑏=2𝑗∶𝑡𝑗∈𝒯𝑏}𝐾𝑡^𝑖1₁,𝑡^𝑖1₁+2𝐾𝑡_𝑗,𝑡_𝑗+𝐾𝑡¹,𝑡¹≤4𝜎𝜅𝜆𝑀Δ𝑓_1,𝑖1𝐾. (47) It follows that Δ𝑓_1,𝑖1𝐾≤4𝜎𝜅.𝜆𝑀 (48) This together with (41) tells us that for any 𝑣∈𝑉, 𝑓_𝒯(𝑣)−𝑓_𝒯𝑡^𝑖_111,𝑡(𝑣)≤4𝜎𝜅².𝜆𝑀 (49) Similarly, we can also obtain 𝑓_𝒯(𝑣)−𝑓_𝒯𝑡^𝑖_{𝑞𝑞𝑞,𝑡}(𝑣)≤4𝜎𝜅²𝜆𝑀,𝑞=2,…,𝑘. (50) The conclusion follows.

Theorems 12 and 14 give the following generalization bound for kernel-based ranking algorithms.

Corollary 15. Under the conditions of Theorem 14, one has that for any 0<𝛿<1, with probability of at least 1−𝛿 over the draw of 𝒯, the expected ranking error of the ranking function 𝑓_𝒯 learned by the regularized algorithm associated with the 𝑙₁ ranking loss is bounded by 𝑒𝑟_𝑙1𝑓_𝒯≤𝑒𝑟_𝒯,𝑙1𝑓_𝒯+16𝜎𝜅²+𝜆𝑀16𝜎𝜅²+1𝜆𝑀𝑀𝑀ln(1/𝛿)2. (51)

The result of Corollary 15 shows that a larger regularization parameter 𝜆 leads to better stability and, therefore, a tighter confidence interval in the resulting generalization bound.

Under the conditions of the above results, a kernel-based ranking algorithm minimizing a regularized empirical 𝑙-error also has good uniform loss stability with respect to 𝑙; this follows from the following simple lemma.

Lemma 16. Let ℱ be a class of real-valued functions on 𝑉, and let A be a 𝑘-partite ranking algorithm for ontology that, given a preference graph 𝒯, outputs a ranking function 𝑓_𝒯∈ℱ. If A has uniform score stability (𝜇₁,…,𝜇_𝑘) and 𝑙 is a ranking loss that is 𝜎-admissible with respect to ℱ, then A has uniform loss stability (𝛼₁,…,𝛼_𝑘) with respect to 𝑙, where for all 𝑚∈ℕ, 𝛼_𝑗𝑛₁,…,𝑛_𝑘=2𝜎𝜇_𝑗𝑛₁,…,𝑛_𝑘,𝑗=1,…,𝑘. (52)

The proof of this result can follow the proof of Lemma 13 in [8]. Using Theorem 14 and Lemma 16, we can immediately get the following corollary.

Corollary 17. Under the conditions of Theorem 14, A has uniform loss stability (𝛼₁,…,𝛼_𝑘) with respect to 𝑙, where for all 𝑛_𝑎∈ℕ, 𝛼_𝑗𝑛₁,…,𝑛_𝑘=8𝜎²𝜅²𝜆𝑀,𝑗=1,…,𝑘. (53)

5.3. Consistency

We can also use the above results to show consistency of kernel-based ranking algorithms. In particular, let 𝑒𝑟^∗_𝑙(ℱ) denote the optimal expected 𝑙-error in an RKHS ℱ (for a given distribution): 𝑒𝑟^∗_𝑙(ℱ)=inf_𝑓∈ℱ𝑒𝑟_𝑙𝑓. (54) Then, for a bounded loss function 𝑙, we can show that with an appropriate choice of the regularization parameter 𝜆, the expected 𝑙-error 𝑒𝑟_𝑙(𝑓_𝒯) of the ranking function 𝑓_𝒯 learned by a kernel-based ranking algorithm that minimizes a regularized empirical 𝑙-error in ℱ converges (in probability) to this optimal value. We first show the following simple lemma.

Lemma 18. Let 𝑓∶𝑉→ℝ be a fixed ranking function, and let 𝑙 be a bounded ranking loss function such that 0≤𝑙(𝑓,𝑣,𝑣^′)≤𝐵 for all 𝑓∶𝑉→ℝ and 𝑣,𝑣^′∈𝑉. Then, for any 0<𝛿<1, with probability of at least 1−𝛿, 𝑒𝑟_𝒯,𝑙𝑓≤𝑒𝑟_𝑙𝑓+8𝜎²𝜅²𝜆ln(1/𝛿).2𝑀 (55)

Proof. Define 𝜙 as 𝜙(𝒯)=𝑒𝑟_𝒯,𝑙𝑓. (56) Then, 𝐄_𝒯[𝜙(𝒯)]=𝑒𝑟_𝑙(𝑓). We show that 𝜙 satisfies the condition of McDiarmid’s inequality. For each 𝑖_𝑎∈{1,…,𝑛_𝑎} and 𝑣^𝑎∈𝑉_𝑎, we have 𝜙(𝒯)−𝜙𝒯^{𝑣𝑖𝑎𝑎,𝑣𝑎}=𝑒𝑟_𝒯,𝑙𝑓−𝑒𝑟_𝒯^𝑖_{𝑎𝑎𝑎)(𝑣,𝑣,𝑙}𝑓≤1𝑀^2𝑘_{𝑏=𝑎+1𝑗∶𝑡𝑗∈𝒯𝑏}𝑙𝑓,𝑣^𝑖𝑎_𝑎,𝑡_𝑗−𝑙𝑓,𝑣^𝑎,𝑡_𝑗+1𝑀^2𝑎−1_{𝑚=1𝑖∶𝑡𝑖∈𝒯𝑚}𝑙𝑓,𝑡_𝑖,𝑣^𝑖𝑎_𝑎−𝑙𝑓,𝑡_𝑖,𝑣^𝑎≤𝑘𝐵𝑀. (57) Therefore, applying McDiarmid’s inequality, we know that for any 𝜀>0, 𝐏_𝒯𝑒𝑟_𝒯,𝑙𝑓−𝑒𝑟_𝑙𝑓−𝜀≤𝜀≤exp²𝜆²𝑀32𝜎⁴𝜅⁴. (58) The result follows by setting the right-hand side equal to 𝛿 and solving it for 𝜀.

We are now in a position to prove our main result (Theorem 5).

Proof of Theorem 5. We use Corollary 17 and apply Theorem 10 with 𝛿/2 to get that with probability of at least 1−𝛿/2, 𝑒𝑟_𝑙𝑓_𝒯≤𝑒𝑟_𝒯,𝑙𝑓_𝒯+16𝜎²𝜅²+𝜆𝑀16𝜎²𝜅²+𝐵𝜆𝑀𝑀𝑀ln(2/𝛿)2. (59) Clearly, 𝑒𝑟_𝒯,𝑙𝑓_𝒯≤𝑒𝑟_𝒯,𝑙𝑓_𝒯+𝜆𝑓_𝒯²_𝐾≤𝑒𝑟_𝒯,𝑙𝑓^∗_𝑙+𝜆𝑓^∗_𝑙²_𝐾. (60) Applying Lemma 18 to 𝑓^∗_𝑙 with 𝛿/2, we, thus, get that with probability of at least 1−𝛿/2: 𝑒𝑟_𝒯,𝑙𝑓_𝒯≤𝑒𝑟_𝑙𝑓^∗_𝑙+8𝜎²𝜅²𝜆ln(2/𝛿)𝑓2𝑀+𝜆^∗_𝑙²_𝐾. (61) One finishes the proof by combining the inequalities in (59) and (61), each of which holds with probability at least 1−𝛿/2, together with the condition in (7).

6. Conclusion

The main focus of this paper is on studying the stability and generalization properties of 𝑘-partite ranking algorithm used for ontology computation. This algorithm shows good intuition about the vertex in ontology graph mapping to a vertex in a line. The representation of vertices in ontology graph does not take real-valued labels, and the samples are given by preference graph (pairwise vertices in different ranking rates). This setting is suitable for ontology. We have derived generalization bounds for 𝑘-partite ranking algorithms in this setting using the notion of algorithmic stability. It is also shown that 𝑘-partite ranking algorithms revealing good stability properties have good generalization properties. Our results are applied to obtain generalization bounds for kernel-based 𝑘-partite ranking algorithms that perform regularization in a reproducing kernel Hilbert space.

Acknowledgments

The authors thank the reviewers for their constructive comments and detailed suggestions for improving the quality of this paper. This work was supported in part by the Key Laboratory of Educational Informatization for Nationalities, Ministry of Education, the National Natural Science Foundation of China (60903131) and Key Science and Technology Research Project of Education Ministry (210210).