Optimality and Adaptivity of Deep Neural Features for Instrumental Variable Regression

We provide a convergence analysis of deep feature instrumental variable (DFIV) regression (Xu et al., 2021), a nonparametric approach to IV regression using data-adaptive features learned by deep neural networks in two stages. We prove that the DFIV algorithm achieves the minimax optimal learning rate when the target structural function lies in a Besov space. This is shown under standard nonparametric IV assumptions, and an additional smoothness assumption on the regularity of the conditional distribution of the covariate given the instrument, which controls the difficulty of Stage 1. We further demonstrate that DFIV, as a data-adaptive algorithm, is superior to fixed-feature (kernel or sieve) IV methods in two ways. First, when the target function possesses low spatial homogeneity (i.e., it has both smooth and spiky/discontinuous regions), DFIV still achieves the optimal rate, while fixed-feature methods are shown to be strictly suboptimal. Second, comparing with kernel-based two-stage regression estimators, DFIV is provably more data efficient in the Stage 1 samples.

Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimal

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.

Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithms

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.

Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithm

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.

Nonlinear Meta-Learning Can Guarantee Faster Rates

Many recent theoretical works on \emph{meta-learning} aim to achieve guarantees in leveraging similar representational structures from related tasks towards simplifying a target task. Importantly, the main aim in theory works on the subject is to understand the extent to which convergence rates -- in learning a common representation -- \emph{may scale with the number $N$ of tasks} (as well as the number of samples per task). First steps in this setting demonstrate this property when both the shared representation amongst tasks, and task-specific regression functions, are linear. This linear setting readily reveals the benefits of aggregating tasks, e.g., via averaging arguments. In practice, however, the representation is often highly nonlinear, introducing nontrivial biases in each task that cannot easily be averaged out as in the linear case. In the present work, we derive theoretical guarantees for meta-learning with nonlinear representations. In particular, assuming the shared nonlinearity maps to an infinite-dimensional RKHS, we show that additional biases can be mitigated with careful regularization that leverages the smoothness of task-specific regression functions,

Optimal Rates for Regularized Conditional Mean Embedding Learning

We address the consistency of a kernel ridge regression estimate of the conditional mean embedding (CME), which is an embedding of the conditional distribution of $Y$ given $X$ into a target reproducing kernel Hilbert space $\mathcal{H}_Y$. The CME allows us to take conditional expectations of target RKHS functions, and has been employed in nonparametric causal and Bayesian inference. We address the misspecified setting, where the target CME is in the space of Hilbert-Schmidt operators acting from an input interpolation space between $\mathcal{H}_X$ and $L_2$, to $\mathcal{H}_Y$. This space of operators is shown to be isomorphic to a newly defined vector-valued interpolation space. Using this isomorphism, we derive a novel and adaptive statistical learning rate for the empirical CME estimator under the misspecified setting. Our analysis reveals that our rates match the optimal $O(\log n / n)$ rates without assuming $\mathcal{H}_Y$ to be finite dimensional. We further establish a lower bound on the learning rate, which shows that the obtained upper bound is optimal.

Distribution Regression with Sliced Wasserstein Kernels

The problem of learning functions over spaces of probabilities - or distribution regression - is gaining significant interest in the machine learning community. A key challenge behind this problem is to identify a suitable representation capturing all relevant properties of the underlying functional mapping. A principled approach to distribution regression is provided by kernel mean embeddings, which lifts kernel-induced similarity on the input domain at the probability level. This strategy effectively tackles the two-stage sampling nature of the problem, enabling one to derive estimators with strong statistical guarantees, such as universal consistency and excess risk bounds. However, kernel mean embeddings implicitly hinge on the maximum mean discrepancy (MMD), a metric on probabilities, which may fail to capture key geometrical relations between distributions. In contrast, optimal transport (OT) metrics, are potentially more appealing, as documented by the recent literature on the topic. In this work, we propose the first OT-based estimator for distribution regression. We build on the Sliced Wasserstein distance to obtain an OT-based representation. We study the theoretical properties of a kernel ridge regression estimator based on such representation, for which we prove universal consistency and excess risk bounds. Preliminary experiments complement our theoretical findings by showing the effectiveness of the proposed approach and compare it with MMD-based estimators.

Meta-strategy for Learning Tuning Parameters with Guarantees

Online gradient methods, like the online gradient algorithm (OGA), often depend on tuning parameters that are difficult to set in practice. We consider an online meta-learning scenario, and we propose a meta-strategy to learn these parameters from past tasks. Our strategy is based on the minimization of a regret bound. It allows to learn the initialization and the step size in OGA with guarantees. We provide a regret analysis of the strategy in the case of convex losses. It suggests that, when there are parameters $\theta_1,\dots,\theta_T$ solving well tasks $1,\dots,T$ respectively and that are close enough one to each other, our strategy indeed improves on learning each task in isolation.