We study the convergence behavior of regularized regression based on reproducing kernel Banach spaces (RKBSs). The convex inequality of uniform convex Banach spaces is used to show the robustness of the optimal solution with respect to the distributions. The learning rates are derived in terms of the covering number and K-functional.

1. Introduction

Recently, there is an increasing research interest in learning with abstract functional spaces, and considerable work has been done in [1–3] and so on.

Let (ℬ, ∥·∥_ℬ) be a normed vector space consisting of real functions on a compact distance space (X, d(·, ·)), and let M > 0 be a given positive number. Let

be a finite set of samples drawn independently and identically (i.i.d.) according to a distribution ρ(x, y) on Z. Then, the regularized learning scheme associating with a given hypothesis space ℬ and the least square loss is

()

where q ≥ 1 is a given real number. The unknown Borel probability distribution ρ(x, y) can be decomposed into ρ(y∣x) and ρ_X(x), where ρ(y∣x) is the conditional probability of ρ at x ∈ X and ρ_X(x) is the marginal probability on X.

The regression function corresponding to the least square loss is

()

which satisfies

()

When the hypothesis spaces ℬ in (1) are reproducing kernel Banach spaces, we call it the RKBSs based on regularized regression learning defined by [4, 5] recently. The represented theorem related closely to regularized learning is studied in case that ℬ is an RKBS, and the discussions are extended to the generalized semi-inner-product RKBSs in [6].

In the present paper, we will provide an investigation on the learning rates of scheme (1) when ℬ is an RKBS with uniform convexity. The paper is organized as follows. In Section 2, we show the main results of the present paper. The robustness is studied in Section 3, and the sample errors are bounded in Section 4. The approximation error boils down to a K-functional. The learning rates are bounded in Section 5.

For a given real number p ≥ 1, we denote by L^p(ρ_X) the class of ρ_X-measurable functions f satisfying .

We say A = O(B) if there is a constant C > 0 such that A/B ≤ C. We say A ~ B if both A = O(B) and B = O(A).

2. Notions and Results

To state the results of the present paper, we first introduce some notions as follows.

2.1. The RKBSs

We denote by ℬ the Banach space with dual space ℬ^* and norm ∥·∥_ℬ. For f ∈ ℬ and f^* ∈ ℬ^*, we write.

A reproducing kernel Banach space (RKBS) on X is a reflexive Banach space of real functions on X whose dual space ℬ^* is isometric to a Banach space ℬ^# of functions on X, and the point evaluations are continuous functions on both ℬ and ℬ^#. It was shown by Theorem 2 of [4] that if B is an RKBS on X, then, there exists uniquely a function K : X × X → ℜ called the reproducing kernel of ℬ satisfying the following:

(i)
K(x, ·) ∈ ℬ, K(·, x) ∈ ℬ^*, x ∈ X;
(ii)
f(x) = 〈f, K(·, x)〉 _ℬ, f^*(x) = 〈K(x, ·), f^*〉 _ℬ, f^* ∈ ℬ^*, x ∈ X.
(iii)
The linear span of {K(x, ·) : x ∈ X} is dense in ℬ, namely,
()

(iv)
The linear span of {K(·, x) : x ∈ X} is dense in ℬ^*, namely,

()

(v)
For all x, y ∈ X there holds K(x, y) = 〈K(x, ·), K(·, y)〉 _ℬ.

When ℬ is an RKHS, K is indeed the reproducing kernel in the usual sense (see [7]).

Since ℬ is a reflective Banach space, we have

()

A way of producing reproducing kernel spaces in L^p spaces by the idempotent integral operators was provided in [8]. In the present paper, we provide a method to construct RKBSs by orthogonal function series.

Example 1. Let X = [a, b] be a given closed interval and let,be a sequence of continuous functions on [a, b] satisfying the following:

(i)
φ_k ∈ L^p(ρ) for k = 0,1, 2, …;
(ii)
φ_k and φ_l are orthonormal (in L²(ρ)) when l ≠ k;
(iii)
is dense in L^p(ρ) for 1 < p < +∞.

Letbe a given positive real number sequence satisfying . Define

()

and the functional classon X by

()

where a_k(f) = ∫_[a,b] f(y)φ_k(y)dρ(y), k ∈ N. We define the spacefor p^′ = p/(p − 1) in an analogous way.

We have the following proposition.

Proposition 2. Define a bivariate operation onandby

()

Then, is a reproducing kernel Banach space with reproducing kernel K(x, y).

Proof. Let andbe defined in an analogous way. Then, bothandare Banach spaces andand.

By (9) we knowandare isometric isomorphisms. Therefore,are Banach spaces.

Since a_l(K(·, y)) = λ_lφ_l(y), we have for

that

()

By the same way, we have for any that 〈K(x, ·), g〉 = g(x); that is, the reproducing property holds.

2.2. The Uniform Convexity

In this subsection, we focus on some notions in convex analysis and Banach geometry theory.

Let F(f) : ℬ → ℜ be a convex function. Then,

()

We call ∂F(f) the subdifferential of F(f) at f ∈ ℬ. If ξ ∈ ∂F(f), then, we call ξ a subgradient of F at f.

A well-known result is that f₀ is a minimal value point of a convex function F(f) on ℬ if and only if 0 ∈ ∂F(f₀) (see [9]).

A Banach space ℬ is called q-uniform convex if there are constants q > 0, c > 0 such that the modulus defined by

()

satisfies δ_ℬ(ɛ) ≥ cɛ^q. In particular, any Hilbert spaces are 2-uniform convex Banach spaces.

Define

Then, by (28) in Corollary 1 of [10] we know ℬ is q-uniform convex if and only if there is a positive constant c_q > 0 such that for all f, g ∈ ℬ and all j_q(f) ∈ J_q(f)_ℬ there holds

()

In [11–14] we know that, for a given 1 < p < +∞, the space l_p, the Lebesgue spaces L_p and the Sobolev spaceare max {2, p}-uniform convex. Also, letandbe defined as in Section 2.1. Then, by the fact thatand are isometric isomorphisms, we knowis 2-uniform convex if p > 2 and p^′-uniform convex if 1 < p ≤ 2. Therefore, we knowis a q-uniform convex Banach space, where q is 2 if p > 2 and its value is p/(p − 1) if 1 < p ≤ 2.

2.3. Main Results

Let S be a distance space and η > 0. The covering number 𝒩(S, η) is defined to be the minimal positive integer number l such that there exists l disk in S with radius η covering S.

We say a compact subset E in a distance space (ℬ, ∥·∥_ℬ) has logarithmic complexity exponent s ≥ 0 if there is a constant c_s > 0 such that the closed ball of radius R centered at origin, that is, ℬ_R = {f ∈ E : ∥f∥_ℬ ≤ R}, satisfies

()

Now we are in a position to present the main results of this paper.

Theorem 3. Let ℬ be an RKBS with q-uniform convexity and a reproducing kernel K(·, x) which is uniform continuous on X in terms of the norm, that is, . is a uniform continuous function on X, and there is a constant k > 0 such thatholds for all x ∈ X. Let f_z be the unique minimizer of scheme (1). If f_ρ ∈ L²(ρ_X), then for any ϵ > 0 there holds

()

where

()

is a K-functional,and

()

The covering number involved in (16) has been studied widely (see [15–19]). In this paper, we assume 𝒩(ℬ_R, η) has the logarithmic complexity.

Theorem 4. Under the conditions of Theorem 3, if f_ρ ∈ L²(ρ_X) and (ℬ, ∥·∥_ℬ) has logarithmic complexity with exponent s ≥ 0, then for any δ ∈ (0,1), with confidence 1 − δ, there holds

()

where c_s is defined in (15).

We now give some remarks on Theorems 3 and 4.

(i)
In Theorem 3, we require that the kernel K(x, y) is uniform continuous and uniform bounded on X. In fact, a large class of real bivariate functions satisfies these conditions. For example, if the function sequencedefined in Example 1 is uniformly bounded, that is, |φ_l(x)| ≤ 1 holds for all l and all x ∈ [a, b], then, kernel K(x, y) is continuous on [a, b]×[a, b] which turns out that K(x, y) is uniform continuous on [a, b]×[a, b]. Therefore, shows that K(x, y) is uniform continuous and bounded with norm .
(ii)
By the definition of γ_q, we know that if D_q(f_ρ, λ) = O(λ^β), 0 < β ≤ 1, then, . It is bounded if β = 1.
(iii)
If ℬ is a reproducing kernel Hilbert space, then, q = 2, c_q = 1. Moreover, if D_q(f_ρ, λ) = O(λ^β), 0 < β ≤ 1, then, we have by (19) that
()

(iv)
We can show a way of bounding the decay rates of D_q(f_ρ, λ) for 1 < p ≤ 2. Let f ∈ L^p(ρ). Then, we have the following Fourier expansion:

()

Define an operator sequence by
()
Then, for a given positive integer n we have a_l(V_n(f)) = λ_la_l(f) and
()
where we have used the generalized Bessel inequality (see [20]):
()
Also,
()
By (25) and (23) we knowholds for all positive integers n and, in this case,
()
One can choose suitable n such that it depends upon the sample number m and obtain the decay rates when m → +∞. There are many choices for the type of operator (22). For example, the Bernstein-Durrmeyer operators (see, e.g., [21–23]) and the de la Valle-Poussin sum operators are such types (see [24]). This method was first provided by [25] and was extended in [26, 27].
(v)
We know from [19] that the RKHSs with logarithmic complexity with exponent s ≥ 0 exist. By Corollary 4.1 and Theorem 2.1 of [16] we know that if λ_l satisfy λ_l ~ 1/(1+l)^α, α > 1, then, the covering number ofmay attain the decay of complexity exponent. In a recent paper (see [28]), Guntuboyina and Sen showed that the set of all convex functions defined on [a, b] ^d that are uniform bounded has the logarithmic complexity exponent d/2 in the L_p-metric.

3. Robustness

Robustness is a quantitative description of the solutions on the distributions.

Define the ρ-control integral regularized model corresponding to (1) by

()

where ℰ_ρ(f) is defined in (3). Then, f^(ρ) is influenced by the distributions ρ. For any bounded ρ-measurable function f(x, y) on Z, we define the empirical measure γ_z(x, y) as follows:

()

Then,We give the following theorem.

Theorem 5. Let ℬ be an RKBS with q-uniform convexity and the reproducing kernel K(x, y), and let f^(ρ) and f^(γ) be the solutions of scheme (27) with respect to distributions ρ and γ, respectively. Then,

()

where c_q is the constant defined in (14).

Theorem 5 shows how ρ influences the unique solution f^(ρ).

To prove Theorem 5, we need the following lemmas.

Lemma 6. Under the conditions of Theorem 5, there holds

()

where the point · in K(·, x) means K(·, x) ∈ ℬ for any x ∈ X.

Proof. We restate the following statement.

Let (ℬ, ∥·∥_ℬ) be a Banach space, F(f) : ℬ → ℜ⋃ {∓∞} be a real function. We say F is Gateaux differentiable at f₀ ∈ ℬ if there is an ξ ∈ ℬ^* such that for any g ∈ ℬ there holds

()

and writeBy [29] we know that if F is convex on ℬ and is Gateaux differentiable at f₀ ∈ ℬ, then,

By equality

()

we have for any g(x) = 〈g, K(·, x)〉 _ℬ ∈ ℬ that

()

Since ℰ_ρ(f) is a convex function on ℬ, we know (30) holds.

Lemma 7. Take. Then, under the conditions of Theorem 5, there hold the following.

(i)
There exists uniquely a minimizer f^(ρ) of the problem (27) and
()
(ii)
There is a j_q(f^(ρ)) ∈ J_q(f^(ρ)) such that
()

Proof. The uniqueness of the minimizer can be obtained by the fact that (27) is a strict convex optimization problem. By the definition of f^(ρ), we have

()

We then have (34).

Proof of (35). Since f^(ρ) is the unique solution of (27), we have

()

Notice that both ℰ_ρ(f) andare convex functions about f on ℬ. We have

()

By (30), we know that (37) leads to

()

Therefore, there is j_q(f^(ρ)) ∈ J_q(f^(ρ)) such that (35) holds.

Lemma 8. Let ℬ be an RKBS satisfying the conditions of Theorem 3. Then,

()

Proof. The reproducing property and (16) give

()

Then, the factgives (40).

Lemma 9. Let K(x, y) be the reproducing kernel of ℬ, and K(·, x) is uniform continuous about x on X in norm, R > 0 be a given real number. Then, the ball ℬ_R = {f ∈ ℬ : ∥f∥_ℬ ≤ R} is a compact subset of C(X).

Proof. Since X is a compact distance space, so is X × X. Since K(·, x) is uniform continuous about x in norm, we know that for any ϵ > 0 there is a δ > 0 such that for all x, x^′ ∈ X with d(x, x^′) < δ, we have

()

and for any f ∈ ℬ_R holds

()

By (43), we know that ℬ_R is a closed, bounded, and equicontinuous set. Therefore, ℬ_R is a compact set of C(X).

Proof of Theorem 5. By the definition of ∂ℰ_γ(f^(ρ)) and (30) we know

()

Also, by (44) and the definitions of f^(ρ) and f^(γ) we have

()

Since ℬ is q-uniform convex, we have by (14) and the definition of j_q(f^(ρ)) that

()

Combining (46) with (45), we have

()

It follows that

()

We then have (29).

4. Sample Error

We give the following sample error bounds.

Theorem 10. Let ℬ be an RKBS satisfying the conditions of Theorem 3. f^(ρ) is the solution of scheme (27) with respect to ρ and f_z is the solution of (1). Then, for all ϵ > 0 there hold

()

where

()

To show Theorem 10, we first give a lemma.

Lemma 11 (see [15].)Let ℱ be a family of functions from a probability space Z to ℜ and d(·, ·) a distance on ℱ. Let 𝒰 ⊂ Z be of full measure and constants B, L > 0 such that

(i)
|ξ(z)| ≤ B for all ξ ∈ ℱ and all z ∈ 𝒰,
(ii)
|L_z(ξ₁) − L_z(ξ₂)| ≤ Ld(ξ₁, ξ₂) for all ξ₁, ξ₂ ∈ ℱ and all z ∈ 𝒰^m, where
()

Then, for all ϵ > 0,

()

Proof of Theorem 10. Take γ = γ_z into (29). Then,

()

By (7) and the reproducing property, we have

()

Since

()

and (40), we have

()

Define

()

Then,

()

By (52), we have for all ϵ > 0 that

()

By (53), (56), and (59), we know

()

which gives

()

It follows that

()

That is,

()

We then have (49).

5. Learning Rates

Proof of Theorem 3. We know from [30] that for any f ∈ L²(ρ_X) there holds

()

Since X is a compact set, we have by (40) thatTherefore,

()

By (65) we have

()

which gives for any h > 0 that

()

By (49) and above inequality we have

()

Since ℱ ⊂ ℬ₁,we know

()

By (69) and above inequality we have (16).

To show Theorem 4, we need two lemmas.

Lemma 12 (see [31].)Let c₁ > 0, c₂ > 0 and u > t > 0. Then, the equation

()

has a unique positive zero x^*. In addition,

()

Proof of Theorem 4. Since (ℬ, ∥·∥_ℬ) has logarithmic complexity exponent s ≥ 0, we have by (15) a constant c_s > 0 such that

()

Then, by (16) we have

()

Take

()

Then,

()

By Lemma 12, we know that the unique solution ϵ^* of (75) satisfies

()

By (74) and (77), we have (19).

Acknowledgments

This work was supported partially by the National Natural Science Foundation of China under Grant nos. 10871226, 61179041, 11271199. The authors thank the reviewers for giving many valuable suggestions and comments which make the paper presented in a better form.

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The Learning Rates of Regularized Regression Based on Reproducing Kernel Banach Spaces

Abstract

1. Introduction

2. Notions and Results

2.1. The RKBSs

2.2. The Uniform Convexity

2.3. Main Results

3. Robustness

4. Sample Error

5. Learning Rates

Acknowledgments

References

Citing Literature

References

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The Learning Rates of Regularized Regression Based on Reproducing Kernel Banach Spaces

Abstract

1. Introduction

2. Notions and Results

2.1. The RKBSs

2.2. The Uniform Convexity

2.3. Main Results

3. Robustness

4. Sample Error

5. Learning Rates

Acknowledgments

References

Citing Literature

References

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