Approximation Analysis of Gradient Descent Algorithm for Bipartite Ranking

We introduce a gradient descent algorithm for bipartite ranking with general convex losses. The implementation of this algorithm is simple, and its generalization performance is investigated. Explicit learning rates are presented in terms of the suitable choices of the regularization parameter and the step size. The result fills the theoretical gap in learning rates for ranking problem with general convex losses.


Introduction
In this paper we consider a gradient descent algorithm for bipartite ranking generated from Tikhonov regularization scheme with general convex losses and reproducing kernel Hilbert spaces RKHS .
Let X be a compact metric space and Y {−1, 1}.In bipartite ranking problem, the learner is given positive samples S {x i } m i 1 and negative samples S − {x − i } n i 1 , which are randomly independent drawn from ρ and ρ − , respectively.Given training set S : S , S − , the goal of bipartite ranking is to learn a real-valued ranking function f : X → R that ranks future positive samples higher than negative ones.
The expected loss incurred by a ranking function f on a pair of instances x , x − is I {f x −f x − ≤0} , where I {t} is 1 if t is true and 0 otherwise.However, due to the nonconvexity of I, the empirical minimization method based on I is NP-hard.Thus, we consider replacing I by a convex upper loss function φ f x − f x − .Typical choices of φ include the hinge loss, the least square loss, and the logistic loss.

Journal of Applied Mathematics
The expected convex risk is The corresponding empirical risk is Let ϑ {f ∈ F : f arg min f∈F E f } be the target function set, where F is the measurable function space.We can observe that the target function is not unique.In particular, for the least square loss, the regression function is one element in this set.
The ranking algorithm we investigate in this paper is based on a Tikhonov regularization scheme associated with a Mercer kernel.We usually call a symmetric and positive semidefinite continuous function K : X × X → R a Mercer kernel.The RKHS H K associated with the kernel K is defined see 1 to be the closure of the linear span of the set of functions {K x : K x, • : x ∈ X} with the inner product • K given by K x , K x K K x, x .The reproducing property takes the form f x f, K x K , for all x ∈ X, f ∈ H K .The reproducing property with the Schwartz inequality yields that The regularized ranking algorithm is implemented by an offline regularization scheme 2 in H K f z,λ arg min where λ > 0 is the regularization parameter.A data free-limit of 1.3 is Though the offline algorithm 1.3 has been well understood in 2 , it might be practically challenging when the sample size m or n is large.The same difficulty for classification and regression algorithms is overcome by reducing the computational complexity through a stochastic gradient descent method.Such algorithms have been proposed for online regression in 3, 4 , online classification in 5, 6 , and gradient learning in 7, 8 .In this paper, we use the idea of gradient descent to propose an algorithm for learning a target function in ϑ.
Since φ is convex, we know that its left derivative φ − is well defined and nondecreasing on R. By taking functional derivatives in 1.3 , we introduce the following algorithm for ranking.
Definition 1.1.The stochastic gradient descent ranking algorithm is defined for the sample S by f S 1 0 and where t ∈ N and {η t } is the sequence of step sizes.
In fact, Burges et al. 9 investigate gradient descent methods for learning ranking functions and introducing a neural network to model the underlying ranking function.From the idea of maximizing the generalized Wilcoxon-Mann-Whitney statistic, a ranking algorithm using gradient approximation has been proposed in 10 .However, these approaches are different from ours and their analysis focuses on computational complexity.Recently, for least square loss, numerical experiments by gradient descent algorithm have been presented in 11 .The aim of this paper is to provide generalization bounds for the gradient descent ranking algorithm 1.5 with general convex losses.To the best of our knowledge, there is no error analysis in this case; This is why we conduct our study in this paper.
We mainly analyze the errors f S t − f λ H K and inf f∈ϑ f S t − f H K , which is different from previous error analysis for ranking algorithms based on uniform convergence e.g., 12-16 and stability analysis in 2, 17, 18 .Though the convergence rates of H K norm for classification and regression algorithms have been elegantly investigated in 19, 20 , there is no such analysis in the ranking setting.The main difference in the formulation of the ranking problem as compared to the problems of classification and regression is that the performance or loss in ranking is measured on pairs of examples, rather than on individual examples.This means in particular that, unlike the empirical error in classification or regression, the empirical error in ranking cannot be expressed as a sum of independent random variables 17 .This makes the convergence analysis of H K norm difficult and previous techniques invalid.Fortunately, we observe that similar difficulty for gradient learning has been well overcome in 7, 21, 22 for gradient learning by introducing some novel techniques.In this paper, we will develop an elaborative analysis in terms of these analysis techniques.

Main Result
In this section we present our main results on learning rates of algorithm 1.5 for learning ranking functions.We assume that φ ∈ C 1 R satisfies for some C 0 > 0 and q ≥ 1. Denote the constant

2.2
Table 1: The values of parameters for different convex losses.

Loss function
Theorem 2.1.Assume φ satisfies 2.1 , and choose the step size as where C is a constant independent of m, n, and Theorem 2.1 will be proved in the next section where the constant C can be obtained explicitly.The explicit parameters in Theorem 2.1 are described in Table 1 for some special loss functions φ.Note that the iteration steps and iterative numbers depend on sample number m, n.When m O n and m → ∞, we have t → ∞ and η t → 0.
From the results in Theorem 2.1, we know that the balance of samples is crucial to reach fast learning rates.For m O n and the least square loss, the approximation order is O m −1/2 min{sθ−3sγ,1−3sγ} .Moreover, when sθ → 1 and sγ → 0, we have Now we present the estimates of inf f∈ϑ f S t − f H K under some approximation conditions.

Corollary 2.2. Assume that there is
where C is a constant independent of m, n, and For m O n and the least square loss, by setting s 1/γ 3 β , we can derive the learning rate O m −1/γ 6 2β min{θ−3γ,γβ} .Moreover, if β < θ −3γ /γ, we get the approximation order O m − β / 6 2β .
For the least square loss, the regression function is an optimal predictor in ϑ.Then, the bipartite ranking problem can be reduced as a regression problem.Based on the theoretical analysis in 19, 20 , we know that the approximation condition in Corollary 2.2 can be achieved when the regression function lies in the β 1 /2th power of the integral operator associated with the kernel K.
The highlight of our theoretical analysis results is to provide the estimate of the distance between f S t and the target function set ϑ in RKHS.This is different from the previous result on error analysis that focuses on establishing the estimate of |E f − E S f |.Compared with the previous theoretical studies, the approximation analysis in H K -norm is new and fills the gap on learning rates for ranking problem with general convex losses.
We also note that the techniques of previous error estimate for ranking problem mainly include stability analysis in 2, 17 , concentration estimation based on U-statistics in 14 , and uniform convergence bounds based on covering numbers 15, 16 .Our analysis presents a novel capacity-independent procedure to investigate the generalization performance of ranking algorithms.

Proof of Main Result
We introduce a special property of E f λ/2 f 2 H K .Since the proof is the same as that in 5 , we will omit it here.Lemma 3.1.Let λ > 0. For any f ∈ H K , there holds

3.2
Now we give the one-step analysis.

Journal of Applied Mathematics
Proof.Observe that Note that where the first and the second inequalities are derived by the convexity of φ and the Schwartz inequality, respectively.By Lemma 3.1, we know that

3.6
Thus, the desired result follows by combining 3.5 and 3.6 with 3.4 .
To deal with the sample error iteratively by applying 3.3 , we need to bound the quantity ϕ S, t by the theory of uniform convergence.To this end, a bound for the norm of f S t is required.
Definition 3.3.One says that φ − is locally Lipschitz at the origin if the local Lipschitz constant is finite for any λ > 0.
Now we estimate the bound of f S t from the ideas given in 5 .
Lemma 3.4.Assume that φ − is locally Lipschitz at the origin.If the step size η t satisfies η t 4κ 2 M λ λ ≤ 1 for each t, then Proof.We prove by induction.It is trivial that f S 1 0 satisfies the bound.Suppose that this bound holds true for is a positive linear operator on H K and its norm is bounded by 4κ 2 M λ .Since η t 4κ 2 M λ λ ≤ 1, the operator

3.12
This proves the lemma.
For r > 0, denote Based on analysis techniques in 21, 23 , we derive the capacity-independent bounds for W S, r : sup f∈F r |E S f − E f |.Lemma 3.5.For every r > 0 and ε > 0, one has

3.13
Proof.Because of the feature of S, four cases of samples change should be taken into account to use McDiarmid's inequality.Denote by S k the sample coinciding with S except for

3.14
Based on MicDiarmid's inequality in 24 , we can derive the first result in Lemma 3.5.To derive the second result, we denote ξ x , u

3.17
With the same fashion, we can also derive Thus, the second desired result follows by combining 3.17

3.27
Applying this relation iteratively, we have

3.31
By Lemma 3.7 3 , we also have for θ < 1 and for θ 1

3.33
Combining the above estimations with Lemma 3.6, we derive the desired results.Now we present the proof of Theorem 2.1.