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.
1. 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+={xi+}i=1m and negative samples S-={xi-}i=1n, 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.
The expected convex risk is
(1.1)E(f)=∬Xϕ(f(x+)-f(x-))dρ+(x+)dρ-(x-).
The corresponding empirical risk is
(1.2)ES(f)=1mn∑i=1m∑j=1nϕ(f(xi+)-f(xj-)).
Let ϑ={f∈𝔉:f=argminf∈𝔉ℰ(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 ℋK associated with the kernel K is defined (see [1]) to be the closure of the linear span of the set of functions {Kx:=K(x,·):x∈𝒳} with the inner product 〈·〉K given by 〈Kx,Kx′〉K=K(x,x′). The reproducing property takes the form f(x)=〈f,Kx〉K, for all x∈𝒳,f∈ℋK. The reproducing property with the Schwartz inequality yields that |f(x)|≤K(x,x)∥f∥ℋK. Then, ∥f∥∞≤κ∥f∥ℋK, where κ:=supx∈𝒳K(x,x).
The regularized ranking algorithm is implemented by an offline regularization scheme [2] in ℋK(1.3)fz,λ=argminf∈HK{ES(f)+λ‖f‖HK2},
where λ>0 is the regularization parameter. A data free-limit of (1.3) is
(1.4)fλ=argminf∈HK{E(f)+λ‖f‖HK2}.
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 f1S=0 and
(1.5)ft+1S=(1-ηtλ)ftS-ηtmn∑i=1m∑j=1nϕ-′(ftS(xi+)-ftS(xj-))(Kxi+-Kxj-),
where t∈ℕ and {ηt} 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 ∥ftS-fλ∥ℋKand inff∈ϑ∥ftS-f∥ℋK, 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 ℋK 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 ℋK 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.
2. 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∈C1(ℝ) satisfies
(2.1)|ϕ(u)|≤C0(1+|u|)q,|ϕ′(u)|≤C0(1+|u|)q-1,∀u∈R
for some C0>0 and q≥1. Denote the constant
(2.2)Δ*=1+4κ2(sup|u|≤1|ϕ-′(u)-ϕ′(0)||u|+C0ϕ′(0)+(8κ2ϕ′(0))q-1).
Theorem 2.1.
Assume ϕ satisfies (2.1), and choose the step size as
(2.3)ηt=η*λmax{q-2,0}t-θfor some0<θ<1,0<η*≤1Δ*.
For 0<γ<(1-θ)/min{q-1,1},s>0, one takes λ=t-γ with (mn/(m+n)3/2)s≤t≤2(mn/(m+n)3/2)s. Then, for any 0<δ<1, with confidence at least 1-δ, one has
(2.4)‖ftS-fλ‖HK2≤C(mn(m+n)3/2)-α,
where C is a constant independent of m,n, and
(2.5)α=min{sθ-sγmin{q+1,2q-1},1-sγ(1+q)}.
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 ηt→0.
The values of parameters for different convex losses.
Loss function
C0
q
Δ*
α
ϕ(t)=t2
1
2
1+12κ2
θmin{(1/2)(θ-3γ),γβ}
ϕ(t)=log(1+e-t)
2
1
1+11κ2
(γ+θ)min{θ-γ,γ-β}
ϕ(t)=(1-t)+
1
1
1+8κ2
θmin{(1/2)(θ-3γ),γβ}
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 ∥ftS-fλ∥ℋK2→0 with the order O(m(-1/2)).
Now we present the estimates of inff∈ϑ∥ftS-f∥ℋK under some approximation conditions.
Corollary 2.2.
Assume that there is f*∈ϑ such that ∥fλ-f*∥ℋK2≤Cβλβ for some 0<β<1. Under the condition in Theorem 2.1, for any 0<δ<1, with confidence at least 1-δ, one has
(2.6)inff∈ϑ‖ftS-f‖HK2≤C~(mn(m+n)3/2)-α~,
where C~ is a constant independent of m,n, and
(2.7)α~=min{sθ-sγmin{q+1,2q-1},1-sγ(1+q),sγβ}.
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 ftS 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)-ℰS(f)|. Compared with the previous theoretical studies, the approximation analysis in ℋK-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.
3. Proof of Main Result
We introduce a special property of ℰ(f)+(λ/2)∥f∥ℋK2. Since the proof is the same as that in [5], we will omit it here.
Lemma 3.1.
Let λ>0. For any f∈ℋK, there holds
(3.1)λ2‖f-fλ‖HK≤{E(f)+λ2‖f‖HK2}-{E(fλ)+λ2‖fλ‖HK2}.
For t≥0, one has
(3.3)‖ft+1S-fλ‖HK2≤(1-ηtλ)‖ftS-fλ‖HK2+ηt2‖ftλ‖HK2+2ηtφ(S,t),
where φ(S,t)=ℰS(fλ)-ℰ(fλ)+ℰ(ftS)-ℰS(ftS).
Proof.
Observe that
(3.4)‖ft+1S-fλ‖HK2=‖ftS-fλ‖HK2+ηt2‖ftλ‖HK2+2ηt〈fλ-ftS,ftλ〉K.
Note that
(3.5)〈fλ-ftS,ftλ〉K=1mn∑i=1m∑j=1nϕ′-(ftS(xi+)-ftS(xj-))(fλ(xi+)-fλ(xj-)-ftS(xi+)+ftS(xj-))+λ〈fλ-ftS,ftS〉K≤ES(fλ)-ES(ftS)-λ‖ftS‖HK2+λ〈fλ,ftS〉K≤{ES(fλ)+λ2‖fλ‖HK2}-{ES(ftS)+λ2‖ftS‖HK2},
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){ES(fλ)+λ2‖fλ‖HK2}-{ES(ftS)+λ2‖ftS‖HK2}≤{ES(fλ)-E(fλ)+E(ftS)-ES(ftS)}-λ2‖ftS-fλ‖HK2.
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 ftS is required.
Definition 3.3.
One says that ϕ-′ is locally Lipschitz at the origin if the local Lipschitz constant
(3.7)M(λ)=sup{|ϕ-′(u)-ϕ-′(0)||u|:|u|≤4κ2|ϕ-′(0)|λ}
is finite for any λ>0.
Now we estimate the bound of ftS from the ideas given in [5].
Lemma 3.4.
Assume that ϕ-′ is locally Lipschitz at the origin. If the step size ηt satisfies ηt(4κ2M(λ)+λ)≤1 for each t, then ∥ftS∥ℋK≤2κ|ϕ-′(0)|/λ.
Proof.
We prove by induction. It is trivial that f1S=0 satisfies the bound.
Suppose that this bound holds true for ftS, ∥ftS∥ℋK≤2κ|ϕ-′(0)|/λ. Consider(3.8)ft+1S=(1-ηtλ)ftS-ηtmn∑i=1m∑j=1nϕ-′(ftS(xi+)-ftS(xj-))(Kxi+-Kxj-)=(1-ηtλ)ftS-ηtmn∑i=1m∑j=1nϕ′-(ftS(xi+)-ftS(xj-))-ϕ-′(0)ftS(xi+)-ftS(xj-)⋅(ftS(xi+)-ftS(xj-))(Kxi+-Kxj-)-ηtmn∑i=1m∑j=1nϕ-′(0)(Kxi+-Kxj-).
Let Lijf=〈f,Kxi+-Kxj-〉K(Kxi+-Kxj-). Since
(3.9)0≤〈Lijf,f〉K=|〈f,Kxi+-Kxj-〉K|2≤4κ2‖f‖HK2,
we have ∥Lij∥≤4κ2.
Meanwhile, (ϕ-′ftS(xi+)-ftS(xj-)-ϕ-′(0)/ftS(xi+)-ftS(xj-)≤M(λ)). Then,
(3.10)ηtmn∑i=1m∑j=1nϕ-′(ftS(xi+)-ftS(xj-))-ϕ-′(0)ftS(xi+)-ftS(xj-)Lij
is a positive linear operator on ℋK and its norm is bounded by 4κ2M(λ).
Since ηt(4κ2M(λ)+λ)≤1, the operator
(3.11)A:=(1-ηtλ)I-ηtmn∑i=1m∑j=1nϕ-′(ftS(xi+)-ftS(xj-))-ϕ-′(0)ftS(xi+)-ftS(xj-)Lij
on ℋK is positive and A≤(1-ηtλ)I.
Thus,
(3.12)‖ft+1S‖HK≤(1-ηtλ)‖ftS‖HK+ηtmn∑i=1m∑j=1n|ϕ-′(0)|‖Kxi+-Kxj-‖HK=2κ|ϕ-′(0)|λ.
This proves the lemma.
For r>0, denote 𝔉r={f∈ℋK:∥f∥ℋK≤r}. Meanwhile, denote Lr=max{|ϕ-′(2κr)|,|ϕ-′(-2κr)|} and Mr=max{|ϕ(2κr)|,|ϕ(-2κr)|}.
Based on analysis techniques in [21, 23], we derive the capacity-independent bounds for W(S,r):=supf∈𝔉r|ℰS(f)-ℰ(f)|.
Lemma 3.5.
For every r>0 and ε>0, one has
(3.13)ProbS{|W(S,r)-EW(S,r)|>ε}≤exp{-2m2n2ε2(m+n)3Mr2},EW(S,r)≤(4Lrκr+2ϕ(0))m+nmn.
Proof.
Because of the feature of S, four cases of samples change should be taken into account to use McDiarmid's inequality. Denote by Sk the sample coinciding with S except for xk+ (or xk-) replaced by x~k+ (or x~k-). It is easy to verify that
(3.14)|W(S,r)-W(Sk,r)|=|supf∈Fr|ES(f)-E(f)|-supf∈Fr|ESk(f)-E(f)||≤supf∈Fr|ES(f)-ESk(f)|≤m+nmnMr.
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-)=ϕ(f(x+)-f(u-)). Then, ℰ(f)=Ex+Ex-ξ(x+,x-) and ℰS(f)=1/mn∑i=1m∑j=1nξ(xi+,xj-). Observe that
(3.15)W(S,r)≤supf∈Fr|E(f)-1n∑j=1nEx+ξ(x+,xj-)|+supf∈Fr|1n∑j=1nEx+ξ(x+,xj-)-ES(f)|≤Ex+supf∈Fr|Ex-ξ(x+,x-)-1n∑j=1nξ(x+,xj-)|+1n∑j=1nsupf∈Frsupx-|Ex+ξ(x+,x-)-1m∑i=1mξ(xi+,x-)|=W1+W2.
Since ϕ(h(xj-))-ϕ(0)≤Lr(h(xj-)-0), we have
(3.17)EW1≤2Lrsupx+Esuph∈Gx+|1n∑j=1nεjh(xj-)|+2ϕ(0)nE|∑j=1nεj|≤4Lrκrn+2ϕ(0)nE|∑j=1nεj|≤4Lrκr+2ϕ(0)n(E|∑j=1nεj|2)1/2≤4Lrκr+2ϕ(0)n(E∑j,j′=1nεjεj′)1/2=4Lrκr+2ϕ(0)n.
With the same fashion, we can also derive
(3.18)EW2≤4Lrκr+2ϕ(0)m.
Thus, the second desired result follows by combining (3.17) and (3.18).
Now we can derive the estimate of φ(S,t).
Lemma 3.6.
If ηt satisfies (1) for each t and r=2κ|ϕ-′(0)|/λ+2ϕ(0)/λ, then with confidence at least 1-δ one has
(3.19)φ(S,t)≤Bλ:=(4Lrκr+2ϕ(0)+Mr2log(2δ))(m+n)3/2mn.
Proof.
By Lemma 3.5, we have, with confidence at least 1-δ,
(3.20)W(S,r)≤(4Lrκr+2ϕ(0))m+nmn+(m+n)3/2Mr2mn2log(2δ).
By taking f=0 in the definition of fλ, we see that
(3.21)λ2‖fλ‖HK2≤E(0)+0≤ϕ(0).
Then, for any λ>0, we have ∥fλ∥ℋK≤2ϕ(0)/λ. Thus, ftS,fλ∈𝔉r for r=2κ|ϕ-′(0)|/λ+2ϕ(0)/λ. So, φ(S,t)≤2W(S,r) for each t. This completes the proof.
We are in a position to give bounds for the sample error. We need the following elementary inequalities that can be found in [3, 5].
Lemma 3.7.
(1) For α∈(0,1] and θ∈[0,1],
(3.22)∑i=1t1iθ∏j=i+1t(1-αjθ)≤3α.
(2) Let v∈(0,1] and θ∈(0,1]. Then
(3.23)∑t=1T-11t2θexp{-v∑j=t+1Tj-θ}≤{18vTθ+9T1-θ(1-θ)21-θexp{-v(1-2θ-1)1-θ(T+1)1-θ}ifθ<1,81-v(T+1)-vifθ=1.
(3) For any t<T and θ∈(0,1], there holds
(3.24)∑j=t+1Tj-θ≤{11-θ[(T+1)1-θ-(t+1)1-θ]ifθ<1,log(T+1)-log(t+1)ifθ=1.
Proposition 3.8.
Let ηt=η1t-θ for some θ∈[0,1], and let η1 satisfy η1(4M(λ)+λ)≤1. Set r and Bλ as in Lemma 3.6. Denote B~λ=2κLr+κ|ϕ-′(0)|. Then, with confidence at least 1-δ, the following bound holds for t≥1: when θ<1,
(3.25)‖ftS-fλ‖HK2≤‖fλ‖HK2exp{-η1λ1-θ(t1-θ-1)}+18B~λ2η1λtθ+2B~λ2η12t1-θ(1-θ)21-θexp{-η1λ(1-2θ-1)1-θ(t+1)1-θ}+6Bλλ,
when θ=1,
(3.26)‖ftS-fλ‖HK2≤‖fλ‖HK2t-η1λ+8B~λ2η121-η1λ(t+1)-η1λ+6Bλλ.
Proof.
Since ∥ftS∥ℋK≤2κ|ϕ-′(0)|/λ, we have |ftS(xi+)-ftS(xj-)|≤2κ∥ftS∥ℋK≤2κr. From the definition of ftλ, we know that ∥ftλ∥ℋK≤2κLr+κ|ϕ-′(0)|=B~λ. Thus, when φ(S,t)≤Bλ, we have from Lemma 3.2(3.27)‖ft+1S-fλ‖HK2≤(1-ηtλ)‖ftS-fλ‖HK2+ηt2B~λ2+2ηtBλ.
Applying this relation iteratively, we have
(3.28)‖ftS-fλ‖HK2≤∏i=1t-1(1-ηiλ)‖fλ‖HK2+∑i=1t-1∏j=i+1t-1(1-ηiλ)(ηi2B~λ2+2ηiBλ).
Since ηi=η1i-θ, by Lemma 3.7(2), we have for θ<1(3.29)∑i=1t-1∏j=i+1t-1(1-ηiλ)ηi2≤η12∑i=1t-112θexp{-η1λ∑j=i+1t-1j-θ}≤18η1λt+9η12t1-θ(1-θ)21-θexp{-η1λ(1-2θ-1)1-θ(t+1)1-θ}
and for θ=1(3.30)∑i=1t-1∏j=i+1t-1(1-ηiλ)ηi2≤8η121-η1λ(t+1)-η1λ.
Lemma 3.7(1) yields
(3.31)∑i=1t-1∏j=i+1t-1(1-ηiλ)ηi≤η1∑i=1t-11iθ∏j=i+1t-1(1-η1λjθ)≤3λ.
By Lemma 3.7(3), we also have for θ<1(3.32)∑i=1t-1(1-ηiλ)≤exp{-∑i=1t-1ηiλ}≤exp{η1λ1-θ(1-t1-θ)}
and for θ=1(3.33)∑i=1t-1(1-ηiλ)≤t-η1λ.
Combining the above estimations with Lemma 3.6, we derive the desired results.
Now we present the proof of Theorem 2.1.
Proof of Theorem <xref ref-type="statement" rid="thm1">2.1</xref>.
First we derive explicit expressions for the quantities in Proposition 3.8. Since λ=t-γ, we have r≤C3tγ, where C3=2κ|ϕ-′(0)|+2ϕ(0). By (2.1), we find that
(3.34)Lr≤C0(1+2κr)q-1≤C0(1+2κ)q-1C3q-1t(q-1)γ,Mr≤C0(1+2κ)qC3qtqγ.
Then,
(3.35)Bλ≤C4log(2δ)tqγ(m+n)3/2mn,B~λ≤C5t(q-1)γ,
where C4=4C0κ(1+2κ)q-1C3q-1+C0(1+2κ)qC3q+2ϕ(0) and C5=2κC0(1+2κ)q-1C3q-1+2ϕ(0).
Next, we bound M(λ). When 1≤|u|≤4κ2|ϕ-′(0)|/λ, we have
(3.36)|ϕ-′(u)-ϕ-′(0)||u|≤|ϕ-′(0)|+2q-1C0(4κ2ϕ-′(0))q-1λ2-q.
Hence,
(3.37)M(λ)≤(sup|u|≤1|ϕ-′(u)-ϕ-′(0)||u|+|ϕ-′(0)|+2q-1C0(4κ2ϕ-′(0))q-1)λmin{2-q,0}.
It follows that the condition η1(4κ2M(λ)+λ)≤1 in Proposition 3.8 holds true when ηt=η1t-θ and η1=η*λmax{q-2,0}. Based on Proposition 3.8 and (mn/m+n3/2)s≤t≤2(mn/m+n3/2)s, we have, with confidence at least 1-δ,
(3.38)‖ftS-fλ‖HK2≤(c~1‖fλ‖HK2exp{η*1-θ+c~2η*2(1-θ)21-θ(mn(m+n)3/2)s(1-θ-2(q-1)γ)})×exp{-η*(1-2θ-1)1-θt1-θ-γmax{q-1,1}}+c~3(mn(m+n)3/2)sγ(1+q)-1+c~4η*(mn(m+n)3/2)sγmin{q+1,2q-1}-sθ,
where c~1,c~2,c~3, and c~4 are constants independent of m,n,and t.
Thus, when 1-θ-γmax{q-1,1}>0, we can derive the desired result in Theorem 2.1.
Acknowledgments
This work was supported partially by the National Natural Science Foundation of China (NSFC) under Grant no. 11001092 and the Fundamental Research Funds for the Central Universities (Program no. 2011PY130, 2011QC022). The authors are indebted to the anonymous reviewers for their constructive comments.
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