Approximation Theory Based Methods for RKHS Bandits

The RKHS bandit problem (also called kernelized multi-armed bandit problem) is an online optimization problem of non-linear functions with noisy feedback. Srinivas et al. (2010) studied a multi-armed bandit problem where the reward function belongs to the reproducing kernel Hilbert space (RKHS) associated with a kernel. In this paper, we call this problem the (stochastic) RKHS bandit problem. Although the problem has been studied extensively, some issues are not completely solved yet. In this paper, we focus on mainly two issues: non-existence of general algorithms for the adversarial RKHS bandit problem and high computational complexity for the stochastic RKHS bandit algorithms.

First, as a non-linear generalization of the classical adversarial linear bandit problem, Chatterji et al. (2019) proposed the adversarial RKHS bandit problem, where a learner interacts with a sequence of any functions from the RKHS with bounded norms. However, they only consider the kernel loss, i.e., a loss function of the form $x\mapsto K(x,x_{0})$, where $x_{0}$ is a fixed point. Considering functions in the RKHS can be represented as infinite linear combinations of such functions, kernel loss is a very special function in the RKHS. Therefore, there are no algorithms for the adversarial RKHS bandits with general loss (or reward) functions.

Next, we discuss the efficiency of existing methods for the stochastic RKHS bandit problem. We note that most of the existing methods have regret guarantees at the cost of high computational complexity. For example, IGP-UCB Chowdhury & Gopalan (2017) requires matrix-vector multiplication of size $t$ for each arm at each round $t=1,\dots,T$. Therefore, the total computational complexity up to round $T$ is given as $O(|\mathcal{A}|T^{3})$, where $\mathcal{A}$ is the set of arms. To address the issue, Calandriello et al. (2020) proposed BBKB and proved its total computational complexity is given as $\widetilde{O}(|\mathcal{A}|T\gamma_{T}^{2}+\gamma_{T}^{4})$, where $\mathcal{A}\subset\Omega$ is the set of arms, $\Omega$ is a subset of a Euclidean space $\mathbb{R}^{d}$, and $\gamma_{T}$ is the maximum information gain (Srinivas et al., 2010). If the kernel is a squared exponential kernel, then since $\gamma_{T}=\widetilde{O}(\log^{d}(T))$¹¹1In this paper, we use $\widetilde{O}$ notation to ignore $\log^{c}(T)$ factor, where $c$ is a universal constant. (Srinivas et al., 2010), ignoring the polylogarithmic factor, BBKB’s computational complexity is nearly linear in $T$. However, the coefficient $|\mathcal{A}|$ in the term is large in general.

In this paper, we address these two issues by considering a novel amalgamation of approximation theory (Wendland, 2004) and the misspecified linear bandit problem Lattimore et al. (2020). That is, we approximately reduce the RKHS bandit problem to the well-studied linear bandit problem. Here, because of an approximation error, the model is a misspecified linear model. Ordinary approximation methods (such as Random Fourier Features or Nyström embedding) basically aim to approximate kernel $K(x,y)$ by an inner product of finite dimensional vectors. However, to reduce the RKHS bandits to the linear bandits, we want to approximate a function $f$ in the RKHS $\mathcal{H}_{K}\left(\Omega\right)$ by a function $\phi$ in a finite dimensional subspace so that $\|f-\phi\|_{L^{\infty}(\Omega)}$ is small. Since the usual approximation methods are not appropriate for the purpose, in this paper, we utilize a method developed in the approximation theory literature called the $P$-greedy algorithm (De Marchi et al., 2005) to minimize the $L^{\infty}$ error. More precisely, we shall introduce that any function $f$ in the RKHS is approximately equal (in terms of the $L^{\infty}$ norm) to a linear combination of $D_{q,\alpha}(T)$ functions, where $q,\alpha>0$ are parameters and $D_{q,\alpha}(T)$ is the number of functions (or equivalently points) returned by the $P$-greedy algorithm (Algorithm 1) with admissible error $\mathfrak{e}=\frac{\alpha}{T^{q}}$. If $K$ is sufficiently smooth, $D_{q,\alpha}(T)$ is much smaller than $T$ and $|\mathcal{A}|$. By this approximation, we can tackle the original RKHS bandit problem by applying an algorithm for the misspecified linear bandit problem.

First, we review previous works on the adversarial RKHS bandit problem. There are almost no existing results concerning the adversarial RKHS bandit problem except for (Chatterji et al., 2019). They also used an approximation method to solve the problem, but their approximation method can handle only a limited case. Therefore, there are no existing algorithms for the adversarial RKHS bandit problem with general reward functions. Next, we review existing results for the stochastic RKHS bandit problem. Srinivas et al. (2010) studied a multi-armed bandit problem, where the reward function is assumed to be sampled from a Gaussian process or belongs to an RKHS. Chowdhury & Gopalan (2017) improved the result of Srinivas et al. (2010) in the RKHS setting and proposed two methods called IGP-UCB and GP-TS. Valko et al. (2013) considered a stochastic RKHS bandit problem, where the arm set $\mathcal{A}$ is finite and fixed, prpoposed a method called SupKernelUCB, and proved a regret upper bound $\widetilde{O}(\sqrt{T\gamma_{T}\log^{3}(|\mathcal{A}|T/\delta)})$. To address the computational inefficiency in the stochastic RKHS bandit problem, Mutny & Krause (2018) proposed Thompson Sampling and UCB-type algorithms using an approximation method called Quadrature Fourier Features which is an improved method of Random Fourier Features Rahimi & Recht (2008). They proved that the total computational complexity of their methods is given as $\widetilde{O}(|\mathcal{A}|T\gamma_{T}^{2})$. However, their methods can be applied to only a very special class of kernels. For example, among three examples introduced in §3, only SE kernels satisfy their assumption unless $d=1$. Our methods work for general symmetric positive definite kernels with enough smoothness. Calandriello et al. (2020) proposed a method called BBKB and proved its regret is upper bounded by $55\tilde{C}^{3}R_{\text{GP-UCB}}(T)$ with $\tilde{C}>1$ and its total computational complexity is given as $\widetilde{O}(|\mathcal{A}|T\gamma^{2}(T)+\gamma^{4}(T))$. Here we use the maximum information gain instead of the effective dimension since they have the same order up to polylogarithmic factors (Calandriello et al., 2019). If the kernel is an SE kernel, ignoring polylogarithmic factors, their computational complexity is linear in $T$. However, the term incurs generally large coefficient $|\mathcal{A}|$ in the term unlike APG-PE. Finally, we note that we construct APG-PE from PHASED ELIMINATION (Lattimore et al., 2020), which is an algorithm for the stochastic misspecified linear bandit problem, where PE stands for PHASED ELIMINATION.

Let $\Omega$ be a non-empty subset of a Euclidean space $\mathbb{R}^{d}$ and $K:\Omega\times\Omega\rightarrow\mathbb{R}$ be a symmetric, positive definite kernel on $\Omega$, i.e., $K(x,y)=K(y,x)$ for all $x,y\in\Omega$ and for a pairwise distinct points $\{x_{1},\dots,x_{n}\}\subseteq\Omega$, the kernel matrix $(K(x_{i},x_{j}))_{1\leq i,j\leq n}$ is positive definite. Examples of such kernels are Rational Quadratic (RQ), Squared Exponential (SE), and Matérn kernels defined as $K_{\mathrm{RQ}}(x,y):=\left(1+\frac{s^{2}}{2\mu l^{2}}\right)^{-\mu},$ $K_{\mathrm{SE}}(x,y):=\exp\left(-\frac{s^{2}}{2l^{2}}\right),$ and $K_{\mathrm{Mat\acute{e}rn}}^{(\nu)}(x,y):=\frac{2^{1-\nu}}{\Gamma(\nu)}\left(\frac{s\sqrt{2\nu}}{l}\right)^{\nu}K_{\nu}\left(\frac{s\sqrt{2\nu}}{l}\right)$. where $s=\|x-y\|_{2}$ and $l>0,\mu>d/2$, $\nu>0$ are parameters, and $K_{\nu}$ is the modified Bessel function of the second kind. As in the previous work Chowdhury & Gopalan (2017), we normalize kernel $K$ so that $K(x,x)\leq 1$ for all $x\in\Omega$. We note that the above three examples satisfy $K(x,x)=1$ for any $x$. We denote by $\mathcal{H}_{K}\left(\Omega\right)$ the RKHS corresponding to the kernel $K$, which we shall review briefly in §4 and assume that $f\in\mathcal{H}_{K}\left(\Omega\right)$ has bounded norm, i.e., $\|f\|_{\mathcal{H}_{K}\left(\Omega\right)}\leq B$. In this paper, we consider the following multi-armed bandit problem with time interval $T$ and arm set $\mathcal{A}\subseteq\Omega$. First, we formulate the stochastic RKHS bandit problem. In each round $t=1,2,\dots,T$, a learner selects an arm $x_{t}\in\mathcal{A}$ and observes noisy reward $y_{t}=f(x_{t})+\varepsilon_{t}$. Here we assume that noise stochastic process is conditionally $R$-sub-Gaussian with respect to a filtration $\{\mathcal{F}_{t}\}_{t=1,2,\dots}$, i.e., $\mathbf{E}\left[\exp(\xi\varepsilon_{t})\mid\mathcal{F}_{t}\right]\leq\exp(\xi^{2}R^{2}/2)$ for all $t\geq 1$ and $\xi\in\mathbb{R}$. We also assume that $x_{t}$ is $\mathcal{F}_{t}$-measurable and $y_{t}$ is $\mathcal{F}_{t+1}$-measurable. The objective of the learner is to maximize the cumulative reward $\sum_{t=1}^{T}f(x_{t})$ and regret is defined by $R(T):=\sup_{x\in\mathcal{A}}\sum_{t=1}^{T}\left(f(x)-f(x_{t})\right)$. In the adversarial (or non-stochastic) bandit RKHS problem, we assume a sequence $f_{t}\in\mathcal{H}_{K}\left(\Omega\right)$ with $\|f_{t}\|_{\mathcal{H}_{K}\left(\Omega\right)}\leq B$ for $t=1,\dots,T$ is given. In each round $t=1,\dots,T$, a learner selects an arm $x_{t}\in\mathcal{A}$ and observes a reward $f_{t}(x_{t})$. The learner’s objective is to minimize the cumulative regret $R(T):=\sup_{x\in\mathcal{A}}\sum_{t=1}^{T}f_{t}(x)-\sum_{t=1}^{T}f_{t}(x_{t})$. In this paper we only consider oblivious adversary, i.e., we assume the adversary chooses a sequence $\{f_{t}\}_{t=1}^{T}$ before the game starts.

In this section, we introduce important results provided by approximation theory. For introduction to this subject, we refer to the monograph Wendland (2004). We first briefly review basic properties of the RKHS and introduce classical results regarding the convergence rate of the power function, which are required for the proof of Theorem 6. Then, we introduce the $P$-greedy algorithm and its convergence rate in Theorem 6, which generalizes the existing result Santin & Haasdonk (2017).

Since we can approximate $f\in\mathcal{H}_{K}\left(\Omega\right)$ by an element of $V(X)$, where $V(X)$ is a finite dimensional subspace of the RKHS, we study a linear bandit problem where the linear model is misspecified, i.e, the misspecified linear bandit problem (Lattimore et al., 2020; Lattimore & Szepesvári, 2020). In this section, we introduce several algorithms for the stochastic and adversarial misspecified linear bandit problems. It turns out that such algorithms can be constructed by modifying (or even without modification) algorithms for the linear bandit problem. We provide proofs in this section in the supplementary material.

First, we provide a formulation of the stochastic misspecified linear bandit problem suitable for our purpose. Let $\mathcal{A}$ be a set and suppose that there exists a map $x\mapsto\widetilde{x}$ from $\mathcal{A}$ to the unit ball $\{\xi\in\mathbb{R}^{D}:\|\xi\|_{2}\leq 1\}$ of a Euclidean space. In each round $t=1,2,\dots,T$, a learner selects an action $x_{t}\in\mathcal{A}$ and the environment reveals a noisy reward $y_{t}=g(x_{t})+\varepsilon_{t}$, where $g(x):=\langle\theta,\widetilde{x}\rangle+\omega(x)$, $\theta\in\mathbb{R}^{D}$, and $\omega(x)$ is a biased noise and satisfies $\sup_{x\in\mathcal{A}}\|\omega(x)\|\leq\epsilon$ and $\epsilon>0$ is known to the learner. We also assume that there exists $B>0$ such that $\sup_{x\in\mathcal{A}}\|g(x)\|\leq B$ and $\|\theta\|_{2}\leq B$. As before, $\{\varepsilon_{t}\}_{t\geq 1}$ is conditionally $R$-sub-Gaussian w.r.t a filtration $\{\mathcal{F}_{t}\}_{t\geq 1}$ and we assume that $\widetilde{x}_{t}$ is $\mathcal{F}_{t}$-measurable and $y_{t}$ is $\mathcal{F}_{t+1}$-measurable. The regret is defined as $R(T):=\sum_{t=1}^{T}\left(\sup_{x\in\mathcal{A}}g(x)-g(x_{t})\right).$ We can formulate the adversarial misspecified linear bandit problem in a similar way. Let $\{g_{t}\}_{t=1}^{T}$ be a sequence of functions on $\mathcal{A}$ with $g_{t}(x)=\langle\theta_{t},\widetilde{x}\rangle+\omega_{t}(x)$, $\theta_{t}\in\mathbb{R}^{D}$, and $\sup_{x\in\mathcal{A}}\|\omega_{t}(x)\|\leq\epsilon$, where the map $x\mapsto\widetilde{x}$ is as before. We also assume that there exists $B>0$ such that $\sup_{x\in\mathcal{A}}|g(x)|\leq B$ and $\|\theta_{t}\|_{2}\leq B$. In each round, $t=1,\dots,T$, the learner selects an arm $x_{t}\in\mathcal{A}$ and observes a reward $g_{t}(x_{t})$. The cumulative regret is defined as $R_{T}=\sup_{x\in\mathcal{A}}\sum_{t=1}^{T}g_{t}(x)-\sum_{t=1}^{T}g_{t}(x_{t})$.

First, we introduce a modification of LinUCB Abbasi-Yadkori et al. (2011). To do this, we prepare notation for the stochastic linear bandit problem. Let $\lambda>0$ and $\delta>0$ be parameters. We define $A_{t}:=\lambda 1_{D}+\sum_{s=1}^{t}\widetilde{x}_{s}\widetilde{x}_{s}^{\mathrm{T}}$, $b_{t}:=\sum_{s=1}^{t}y_{s}\widetilde{x}_{s}$, and $\hat{\theta}_{t}:=A_{t}^{-1}b_{t}$. Here, $1_{D}$ is the identity matrix of size $D$. For $x\in\mathbb{R}^{D}$, we define the Mahalanobis norm as $\|x\|_{A_{t}^{-1}}:=\sqrt{x^{\mathrm{T}}A_{t}^{-1}x}$ and define $\beta_{t}$ as

We note that by the proof of (Abbasi-Yadkori et al., 2011, Lemma 11), computational complexity for updating $\beta_{t}$ is $O(D^{2})$ at each round.

Lattimore et al. (2020) (see appendix of its arXiv version) considered a modification of LinUCB which selects $x\in\mathcal{A}$ maximizing (modified) UCB $\langle\hat{\theta}_{t},\widetilde{x}\rangle+\beta_{t}\|\widetilde{x}\|_{A_{t}^{-1}}+\epsilon\sum_{s=1}^{t}|\widetilde{x}^{\mathrm{T}}A_{t}^{-1}\widetilde{x}_{s}|$ in $(t+1)$th round and proved the regret of the algorithm is upper bounded by $O(D\sqrt{T}\log(T)+\epsilon T\sqrt{D\log(T)})$. However, computing the above value requires $O(t)$ time for each arm $x\in\mathcal{A}$. Therefore, instead of incurring additional $\sqrt{D}$ factor in the second term in the regret upper bound above, we consider another upper confidence bound which can be easily computed. In $(t+1)$th round, our modification of UCB type algorithm selects arm $x\in\mathcal{A}$ maximizing the modified UCB $\langle\hat{\theta}_{t},\widetilde{x}\rangle+\|\widetilde{x}\|_{A_{t}^{-1}}\left(\beta_{t}+\epsilon\psi_{t}\right),$ where $\psi_{t}$ is defined as $\sum_{s=1}^{t}\|\widetilde{x}_{s}\|_{A_{s-1}^{-1}}$. Then by storing $\psi_{t}$ in each round, the complexity for computing this value is given as $O(D^{2})$ for each $x\in\mathcal{A}$ and as is well-known, one can update $A_{t}^{-1}$ in $O(D^{2})$ time using the Sherman–Morrison formula. By the standard argument, we can prove the following regret bound.

In the supplementary material, we also introduce a modification of Thompson Sampling.

The regret upper bound provided above does not depend on the arm set $\mathcal{A}$. Moreover, the same results hold even if the arm set changes over time step $t$ (with minor modification of the definition of regret). On the other hand, several authors Lattimore et al. (2020); Auer (2002); Valko et al. (2013) studied algorithm whose regret depends on the cardinality of the arm set in the stochastic linear or RKHS setting. In some rounds, such algorithms eliminate arms that are supposed to be non-optimal with a high probability and therefore the arm set should be the same over time. Generally, these algorithms are more complicated than LinUCB or Thompson Sampling. However, recently, Lattimore et al. (2020) proposed a simpler and sophisticated algorithm called PHASED ELIMINATION using Kiefer–Wolfowitz theorem. Furthermore, they showed that it works well for the stochastic misspecified linear bandit problem without modification. More precisely, they proved the following result.

Next, we show that EXP3 for adversarial linear bandits (c.f. Lattimore & Szepesvári (2020)) works for the adversarial misspecified linear bandits without modification. We introduce notations for EXP3. Let $\eta>0$ be a learning rate, $\gamma$ an exploration parameter, and $\pi_{\mathrm{exp}}$ be an exploration distribution over $\mathcal{A}$. For a distribution $\pi$ on $\mathcal{A}$, we define a matrix $Q(\pi):=\sum_{x\in\mathcal{A}}\pi(x)\widetilde{x}\widetilde{x}^{\mathrm{T}}$. We also put $\phi_{t}:=g_{t}(x_{t})Q_{t}^{-1}\widetilde{x}_{t}$ and $\phi_{t}(x):=\langle\phi_{t},\widetilde{x}\rangle$ for $x\in\mathcal{A}$, where the matrix $Q_{t}$ is defined later. We define a distribution $q_{t}$ over $\mathcal{A}$ by $q_{t}(x)\sim\exp(\eta\sum_{s=1}^{t-1}\phi_{s}(x))$ and a distribution $p_{t}$ by $p_{t}(x)=\gamma\pi_{\mathrm{exp}}(x)+(1-\gamma)q_{t}(x)$ for $x\in\mathcal{A}$. The matrix $Q_{t}$ is defined as $Q(p_{t})$. We put $\Gamma(\pi_{\mathrm{exp}}):=\sup_{x\in\mathcal{A}}\widetilde{x}^{\mathrm{T}}Q(\pi_{\mathrm{exp}})^{-1}\widetilde{x}$.

Using results from approximation theory explained in §4 and algorithms for the misspecified bandit problem, we provide several algorithms for the stochastic and adversarial RKHS bandit problems. We provide proofs of the results in this section in the supplementary material.

Let $N_{1},\dots,N_{D}$ be the Newton basis returned by Algorithm 1 with $\mathfrak{e}=\frac{\alpha}{T^{q}}$ with $q,\alpha>0$, and $\widehat{\Omega}=\mathcal{A}$. Then, by orthonormality of the Newton basis and the definition of the power function, for any $f\in\mathcal{H}_{K}\left(\Omega\right)$ and $x\in\Omega$, we have

where $\theta_{f}=\left(\langle f,N_{i}\rangle\right)_{1\leq i\leq D}\in\mathbb{R}^{D}$ and $\widetilde{x}=\left(N_{i}(x)\right)_{1\leq i\leq D}\in\mathbb{R}^{D}$. Therefore, if $f$ is an objective function of a RKHS bandit problem, we can regard $f$ as a linearly misspecified model and apply algorithms for misspecified linear bandit problems to solve the original RKHS bandit problems.

In this section, we reduce the RKHS bandit problem to the the misspecified linear bandit problem by the map $x\mapsto\widetilde{x}$ and apply modified LinUCB, PHASED ELIMINATION, and EXP3 to the problem. We call these algorithms APG-UCB, APG-PE and APG-EXP3 respectively and APG-UCB is displayed in Algorithm 2. We denote by $D_{q,\alpha}(T)=D$ the number of points returned by Algorithm 1 with $\mathfrak{e}=\frac{\alpha}{T^{q}}$. By the results in §4, we have an upper bound of $D_{q,\alpha}(T)$ (Corollary 7).

First, we state the results for APG-UCB.

The admissible error $\mathfrak{e}$ balances the computational complexity and regret minimization. However, this is not clear from Theorem 13. The following theorem provides another upper bound of APG-UCB and it states that if we take smaller error $\mathfrak{e}$, then an upper bound of APG-UCB is almost the same as that of IGP-UCB.