On the Optimality of Misspecified Kernel Ridge Regression

Kernel ridge regression has been studied as a special kind of spectral regularization algorithm (Rosasco et al., 2005; Caponnetto, 2006; Bauer et al., 2007; Gerfo et al., 2008; Mendelson & Neeman, 2010). In large part of the literature, the ‘hardness’ of the regression problem is determined by two parameters: 1. the source condition ($s$), which characterizes a function’s relative ‘smoothness’ with respect to the RKHS; 2. the eigenvalue decay rate ($\beta$) (or capacity, effective dimension equivalently), which characterizes the RKHS itself. These two parameters divide the convergence behavior of KRR or spectral regularization algorithm into different regimes and lead to different convergence rates (Dicker et al., 2017; Blanchard & Mücke, 2018; Lin et al., 2018; Lin & Cevher, 2020; Li et al., 2023, etc.). Caponnetto & de Vito (2007) first proves the optimality when $1\leq s\leq 2$ and Lin et al. (2018) extends the desired upper bound of the convergence rate to the regime $s+\frac{1}{\beta}>1$.

KRR in the misspecified case ($f_{\rho}^{*}\notin\mathcal{H}~{}\text{or}~{}s<1$) has also been discussed by another important line of work which considers the embedding index ($\alpha_{0}$) of the RKHS and performs refined analysis (Steinwart et al., 2009; Dicker et al., 2017; Pillaud-Vivien et al., 2018; Fischer & Steinwart, 2020; Celisse & Wahl, 2020; Li et al., 2022). The desired upper bound $n^{-\frac{s\beta}{s\beta+1}}$ is extended to the regime $s+\frac{1}{\beta}>\alpha_{0}$, and the minimax optimality is extended to the regime $s>\alpha_{0}$. It is worth pointing out that when $f_{\rho}^{*}$ falls into a less-smooth interpolation space which does not imply the boundedness of functions therein, all existing works (either considering embedding index or not) require an additional boundedness assumption, i.e., $\|f_{\rho}^{*}\|_{L^{\infty}(\mathcal{X},\mu)}\leq B_{\infty}<\infty$ (Lin & Cevher, 2020; Fischer & Steinwart, 2020; Talwai & Simchi-Levi, 2022; Li et al., 2022, etc). As discussed in the introduction, this will lead to the suboptimality in the $s\leq\alpha_{0}$ regime.

This paper follows the line of work that considers the embedding index, refines the proof by handling the additional boundedness assumption, and solves the optimality problem for Sobolev RKHS and all $s\in(0,1)$. In addition, our technical improvement also sheds light on the optimality for general RKHS. Specifically, we replace the boundedness assumption with a far weaker $L^{q}$-integrability assumption, which turns out to be reasonable for many RKHS. Note that our results focus on the most frequently used $L^{2}$ convergence rate for KRR, and it can be easily extended to $[\mathcal{H}]^{\gamma},\gamma\geq 0$ convergence rate (e.g., Fischer & Steinwart 2020) and general spectral regularization algorithms (e.g., Lin et al. 2018).

Let $\mathcal{X}\subseteq\mathbb{R}^{d}$ be the input space and $\mathcal{Y}\subseteq\mathbb{R}$ be the output space. Let $\rho$ be an unknown probability distribution on $\mathcal{X}\times\mathcal{Y}$ satisfying $\int_{\mathcal{X}\times\mathcal{Y}}y^{2}\mathrm{d}\rho(x,y)<\infty$, and denote the corresponding marginal distribution on $\mathcal{X}$ as $\mu$. We use $L^{p}(\mathcal{X},\mu)$ (in short $L^{p}$) to represent the $L^{p}$-spaces. Then the generalization error (1) can be written as

Throughout the paper, we denote $\mathcal{H}$ as a separable RKHS on $\mathcal{X}$ with respect to a continuous and bounded kernel function $k$ satisfying

Define the integral operator $T:L^{2}(\mathcal{X},\mu)\to L^{2}(\mathcal{X},\mu)$ as

It is well known that $T$ is a positive, self-adjoint, trace-class, and hence a compact operator (Steinwart & Scovel, 2012). The spectral theorem for self-adjoint compact operators and Mercer’s decomposition theorem yield that

where $N$ is an at most countable set, the eigenvalues $\{\lambda_{i}\}_{i\in N}\subseteq(0,\infty)$ is a non-increasing summable sequence, and $\{e_{i}\}_{i\in N}$ are the corresponding eigenfunctions. Denote the samples as $X=\left(x_{1},\cdots,x_{n}\right)$ and $\textbf{y}=\left(y_{1},\cdots,y_{n}\right)^{\prime}$. The representer theorem (see, e.g., Steinwart & Christmann 2008) gives an explicit expression of the KRR estimator defined by (2), i.e.,

where

and

We also need to introduce the interpolation spaces of RKHS. For any $s\geq 0$, the fractional power integral operator $T^{s}:L^{2}(\mathcal{X},\mu)\to L^{2}(\mathcal{X},\mu)$ is defined as

and the interpolation space $[\mathcal{H}]^{s},s\geq 0$ of $\mathcal{H}$ is defined as

with the inner product

It is easy to show that $[\mathcal{H}]^{s}$ is also a separable Hilbert space with orthogonal basis $\{\lambda_{i}^{\frac{s}{2}}e_{i}\}_{i\in N}$. Specially, we have $[\mathcal{H}]^{0}\subseteq L^{2}(\mathcal{X},\mu)$ and $[\mathcal{H}]^{1}\subseteq\mathcal{H}$. For $0<s_{1}<s_{2}$, the embeddings $[\mathcal{H}]^{s_{2}}\hookrightarrow[\mathcal{H}]^{s_{1}}\hookrightarrow[\mathcal{H}]^{0}$ exist and are compact (Fischer & Steinwart, 2020). For the functions in $[\mathcal{H}]^{s}$ with larger $s$, we say they have higher regularity (smoothness) with respect to the RKHS.

As an example, the Sobolev space $H^{m}(\mathcal{X})$ is an RKHS if $m>\frac{d}{2}$, and its interpolation space is still a Sobolev space given by $[H^{m}(\mathcal{X})]^{s}\cong H^{ms}(\mathcal{X}),\forall s>0$, see Section 3.1 for detailed discussions.

This subsection lists the standard assumptions that frequently appeared in related literature.

Note that the eigenvalues $\lambda_{i}$ and EDR are only determined by the marginal distribution $\mu$ and the RKHS $\mathcal{H}$. The polynomial eigenvalue decay rate assumption is standard in related literature and is also referred to as the capacity condition or effective dimension condition.

In fact, for any $\alpha>0$, we can define $M_{\alpha}$ as the smallest constant $A>0$ such that

if there is no such constant, set $M_{\alpha}=\infty$. Then Fischer & Steinwart (2020, Theorem 9) shows that for $\alpha>0$,

The larger $\alpha$ is, the weaker the embedding property is. Note that since $\sup_{x\in\mathcal{X}}k(x,x)\leq\kappa^{2}$, $M_{\alpha}\leq\kappa<\infty$ is always true for $\alpha\geq 1$. In addition, Fischer & Steinwart (2020, Lemma 10) also shows that $\alpha$ can not be less than $\frac{1}{\beta}$.

Note that the embedding property (6) holds for any $\alpha>\alpha_{0}$. This directly implies that all the functions in $[\mathcal{H}]^{\alpha}$ are $\mu\text{-a.e.}$ bounded, $\alpha>\alpha_{0}$. However, the embedding property may not hold for $\alpha=\alpha_{0}$.

Functions in $[\mathcal{H}]^{s}$ with smaller $s$ are less smooth, which will be harder for an algorithm to estimate.

This is a standard assumption to control the noise such that the tail probability decays fast (Lin & Cevher, 2020; Fischer & Steinwart, 2020). It is satisfied for, for instance, the Gaussian noise with bounded variance or sub-Gaussian noise. Some literature (e.g., Steinwart et al. 2009; Pillaud-Vivien et al. 2018; Jun et al. 2019, etc) also uses a stronger assumption $y\in[-L_{0},L_{0}]$ which directly implies both Assumption 2.4 and the boundedness of $f_{\rho}^{*}$.

For the convenience of comparing our results with previous works, we review state-of-the-art upper and lower bounds of the convergence rate of KRR (Fischer & Steinwart, 2020).

Note that in Proposition 2.7, we consider the distributions in $\mathcal{P}\cap\mathcal{P}_{\infty}$. If we consider all the distributions in $\mathcal{P}$, we have the following minimax lower bound, which is often referred to as the information-theoretic lower bound (see, e.g., Rastogi & Sampath 2017).

Let us first introduce some concepts of (fractional) Sobolev space (see, e.g., Adams & Fournier 2003). In this section, we assume that $\mathcal{X}\subseteq\mathbb{R}^{d}$ is a bounded domain with smooth boundary and the Lebesgue measure $\nu$. Denote $L^{2}(\mathcal{X})\coloneqq L^{2}(\mathcal{X},\nu)$ as the corresponding $L^{2}$ space. For $m\in\mathbb{N}$, we denote $H^{m}(\mathcal{X})$ as the Sobolev space with smoothness $m$ and $H^{0}(\mathcal{X})\coloneqq L^{2}(\mathcal{X})$. Then the (fractional) Sobolev space for any real number $r>0$ can be defined through real interpolation:

where $m:=\min\{k\in\mathbb{N}:k>r\}$. (For details of real interpolation of Banach spaces, we refer to Sawano (2018, Chapter 4.2.2)). It is well known that when $r>\frac{d}{2}$, $H^{r}(\mathcal{X})$ is a separable RKHS with respect to a bounded kernel, and the corresponding EDR is (see, e.g., Edmunds & Triebel 1996)

Furthermore, for the interpolation space of $H^{r}(\mathcal{X})$ under Lebesgue measure defined by (4), we have (see, e.g., Steinwart & Scovel 2012, Theorem 4.6), for $s>0$,

Now we begin to introduce the embedding theorem of (fractional) Sobolev space from 7.57 of Adams (1975), which is crucial when applying Theorem 3.1 to Sobolev RKHS.

On the one hand, (iii) of Proposition 3.5 shows that, for a Sobolev RKHS $\mathcal{H}=H^{r}(\mathcal{X}),r>\frac{d}{2}$ and any $\alpha>\frac{1}{\beta}=\frac{d}{2r}$,

where $\gamma>0$. So the Assumption 2.2 holds for $\alpha_{0}=\frac{1}{\beta}$, and thus $\frac{2(s\beta+1)}{2+(s-\alpha_{0})\beta}=2$. On the other hand, (i) of Proposition 3.5 shows that if $d>2rs$,

where $q=\frac{2d}{d-2rs}=\frac{2}{1-s\beta}>2$. In addition, (ii) and (iii) of Proposition 3.5 show that $q>2$ also holds if $d\leq 2rs$. Therefore, for any $0<s\leq 2$, we have

hold simultaneously.

Now we are ready to state a theorem as the corollary of Theorem 3.1 and Proposition 3.5, which implies the optimality of KRR for Sobolev RKHS and any $0<s\leq 2$.

Note that we say that the distribution $\mu$ has Lebesgue density $0<c\leq p(x)\leq C$, if $\mu$ is equivalent to the Lebesgue measure $\nu$, i.e., $\mu\ll\nu,\nu\ll\mu$, and there exist constants $c,C>0$ such that $c\leq\frac{\mathrm{d}\mu}{\mathrm{d}\nu}\leq C$.

In this subsection, we present the sketch of the proof of Theorem 3.1. The rigorous proof of Theorem 3.1, Proposition 2.9, and Theorem 3.6 will be in the appendix. We refer to Fischer & Steinwart (2020, Chapter 6) for the proof of Proposition 2.5 and 2.7.

Our proofs are based on the standard integral operator techniques dating back to Smale & Zhou (2007), and we refine the proof to handle the unbounded case. Let us first introduce some frequently used notations. Define the sample operator as

and its adjoint operator

Next we define the sample covariance operator $T_{X}:\mathcal{H}\to\mathcal{H}$ as

and the sample basis function

Then the KRR estimator (2) can be expressed by (see, e.g., Caponnetto & de Vito (2007, Chapter 5))

Define $f_{\lambda}$ as the unique minimizer given by

Note that the integral operator $T$ can also be seen as a bounded linear operator on $\mathcal{H}$, $f_{\lambda}$ can be expressed by

where $g$ is the expectation of $g_{Z}$ given by

The first step in our proof is to decompose the generalization error into two terms, which are often referred to as the approximation error and estimation error,

Then we will show that by choosing $\lambda\asymp n^{-\frac{\beta}{s\beta+1}}$,

and for any fixed $\delta\in(0,1)$, when $n$ is sufficiently large, with probability at least $1-\delta$, we have

Plugging (10) and (11) into (9) and we will finish the proof of Theorem 3.1.

Suppose that $\mathcal{X}=[0,1]$ and the marginal distribution $\mu$ is the uniform distribution on $[0,1]$. We consider the RKHS $\mathcal{H}=H^{1}(\mathcal{X})$ to be the Sobolev space with smoothness 1. Section 3.1 shows that the EDR is $\beta=2$ and embedding index is $\alpha_{0}=\frac{1}{\beta}$. We construct a function in $[\mathcal{H}]^{s}\backslash L^{\infty}$ by

for some $0<s<\frac{1}{\beta}=0.5$. We will show in Appendix F that the series in (14) converges on $(0,1)$. In addition, since $\sin 2k\pi+\cos 2k\pi\equiv 1$, we also have $f^{*}\notin L^{\infty}(\mathcal{X})$. The explicit formula of the kernel associated to $H^{1}(\mathcal{X})$ is given by Thomas-Agnan (1996, Corollary 2), i.e., $k(x,y)=\frac{1}{\sinh{1}}\cosh{(1-\max(x,y))\cosh{(1-\min(x,y))}}$.

We consider the following data generation procedure:

where $f^{*}$ is numerically approximated by the first 1000 terms in (14) with $s=0.4$, $x\sim\mathcal{U}[0,1]$ and $\epsilon\sim\mathcal{N}(0,1)$. We choose the regularization parameter as $\lambda=cn^{-\frac{\beta}{s\beta+1}}=cn^{-\frac{10}{9}}$ for a fixed $c$. The sample size $n$ is chosen from 1000 to 5000, with intervals of 100. We numerically compute the generalization error $\|\hat{f}-f^{*}\|_{L^{2}}$ by Simpson’s formula with $N\gg n$ testing points. For each $n$, we repeat the experiments 50 times and present the average generalization error as well as the region within one standard deviation. To visualize the convergence rate $r$, we perform logarithmic least-squares $\log\text{err}=r\log n+b$ to fit the generalization error with respect to the sample size and display the value of $r$.

We try different values of $c$, Figure 1 presents the convergence curve under the best choice of $c$. It can be concluded that the convergence rate of KRR’s generalization error is indeed approximately equal to $n^{-\frac{s\beta}{s\beta+1}}=n^{-\frac{4}{9}}$, without the boundedness assumption of the true function $f^{*}$. We also did another experiment that used cross validation to choose the regularization parameter. Figure 3 in Appendix F shows a similar result as Figure 1. We refer to Appendix F for more details of the experiments.

Suppose that $\mathcal{X}=[0,1]$ and the marginal distribution $\mu$ is the uniform distribution on $[0,1]$. It is well known that the following RKHS

is associated with the kernel $k(x,y)=\min(x,y)$ (Wainwright, 2019). Further, its eigenvalues and eigenfunctions can be written as

and

It is easy to see that the EDR is $\beta=2$, the eigenfunctions are uniformly bounded, and the embedding index is $\alpha_{0}=\frac{1}{\beta}$ (see Remark 3.3). We construct a function in $[\mathcal{H}]^{s}\backslash L^{\infty}$ by

for some $0<s<\frac{1}{\beta}=0.5$. We will show in Appendix F that the series in (15) converges on $(0,1)$. Since $e_{2k-1}(1)\equiv 1$, we also have $f^{*}\notin L^{\infty}(\mathcal{X})$.

We use the same data generation procedure as Section 4.1:

where $f^{*}$ is numerically approximated by the first 1000 terms in (15) with $s=0.4$, $x\sim\mathcal{U}[0,1]$ and $\epsilon\sim\mathcal{N}(0,1)$.

Figure 2 presents the convergence curve under the best choice of $c$. It can also be concluded that the convergence rate of KRR’s generalization error is indeed approximately equal to $n^{-\frac{s\beta}{s\beta+1}}=n^{-\frac{4}{9}}$. Figure 4 in Appendix F shows the result of using cross validation.