We study the kernel instrumental variable algorithm of \citet{singh2019kernel}, a nonparametric two-stage least squares (2SLS) procedure which has demonstrated strong empirical performance. We provide a convergence analysis that covers both the identified and unidentified settings: when the structural function cannot be identified, we show that the kernel NPIV estimator converges to the IV solution with minimum norm. Crucially, our convergence is with respect to the strong $L_2$-norm, rather than a pseudo-norm. Additionally, we characterize the smoothness of the target function without relying on the instrument, instead leveraging a new description of the projected subspace size (this being closely related to the link condition in inverse learning literature). With the subspace size description and under standard kernel learning assumptions, we derive, for the first time, the minimax optimal learning rate for kernel NPIV in the strong $L_2$-norm. Our result demonstrates that the strength of the instrument is essential to achieve efficient learning. We also improve the original kernel NPIV algorithm by adopting a general spectral regularization in stage 1 regression. The modified regularization can overcome the saturation effect of Tikhonov regularization.

We study theoretical properties of a broad class of regularized algorithms with vector-valued output. These spectral algorithms include kernel ridge regression, kernel principal component regression, various implementations of gradient descent and many more. Our contributions are twofold. First, we rigorously confirm the so-called saturation effect for ridge regression with vector-valued output by deriving a novel lower bound on learning rates; this bound is shown to be suboptimal when the smoothness of the regression function exceeds a certain level. Second, we present the upper bound for the finite sample risk general vector-valued spectral algorithms, applicable to both well-specified and misspecified scenarios (where the true regression function lies outside of the hypothesis space) which is minimax optimal in various regimes. All of our results explicitly allow the case of infinite-dimensional output variables, proving consistency of recent practical applications.

We present the first optimal rates for infinite-dimensional vector-valued ridge regression on a continuous scale of norms that interpolate between $L_2$ and the hypothesis space, which we consider as a vector-valued reproducing kernel Hilbert space. These rates allow to treat the misspecified case in which the true regression function is not contained in the hypothesis space. We combine standard assumptions on the capacity of the hypothesis space with a novel tensor product construction of vector-valued interpolation spaces in order to characterize the smoothness of the regression function. Our upper bound not only attains the same rate as real-valued kernel ridge regression, but also removes the assumption that the target regression function is bounded. For the lower bound, we reduce the problem to the scalar setting using a projection argument. We show that these rates are optimal in most cases and independent of the dimension of the output space. We illustrate our results for the special case of vector-valued Sobolev spaces.

In the misspecified spectral algorithms problem, researchers usually assume the underground true function $f_{\rho}^{*} \in [\mathcal{H}]^{s}$, a less-smooth interpolation space of a reproducing kernel Hilbert space (RKHS) $\mathcal{H}$ for some $s\in (0,1)$. The existing minimax optimal results require $\|f_{\rho}^{*}\|_{L^{\infty}}<\infty$ which implicitly requires $s > \alpha_{0}$ where $\alpha_{0}\in (0,1)$ is the embedding index, a constant depending on $\mathcal{H}$. Whether the spectral algorithms are optimal for all $s\in (0,1)$ is an outstanding problem lasting for years. In this paper, we show that spectral algorithms are minimax optimal for any $\alpha_{0}-\frac{1}{\beta} < s < 1$, where $\beta$ is the eigenvalue decay rate of $\mathcal{H}$. We also give several classes of RKHSs whose embedding index satisfies $ \alpha_0 = \frac{1}{\beta} $. Thus, the spectral algorithms are minimax optimal for all $s\in (0,1)$ on these RKHSs.