ICML 2023 Papers

In the misspecified kernel ridge regression problem, researchers usually assume the underground true function $f_{\rho}^{\star} \in [\mathcal{H}]^{s}$, 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 $\left\Vert f_{\rho}^{\star} \right \Vert_{L^{\infty}} < \infty$ which implicitly requires $s > \alpha_{0}$ where $\alpha_{0} \in (0,1) $ is the embedding index, a constant depending on $\mathcal{H}$. Whether the KRR is optimal for all $s\in (0,1)$ is an outstanding problem lasting for years. In this paper, we show that KRR is minimax optimal for any $s\in (0,1)$ when the $\mathcal{H}$ is a Sobolev RKHS.

Semi-supervised semantic segmentation involves assigning pixel-wise labels to unlabeled images at training time. This is useful in a wide range of real-world applications where collecting pixel-wise labels is not feasible in time or cost. Current approaches to semi-supervised semantic segmentation work by predicting pseudo-labels for each pixel from a class-wise probability distribution output by a model. If this predicted probability distribution is incorrect, however, it leads to poor segmentation results which can have knock-on consequences in safety critical systems, like medical images or self-driving cars. It is, therefore, important to understand what a model does not know, which is mainly achieved by uncertainty quantification. Recently, neural processes (NPs) have been explored in semi-supervised image classification, and they have been a computationally efficient and effective method for uncertainty quantification. In this work, we move one step forward by adapting NPs to semi-supervised semantic segmentation, resulting in a new model called NP-SemiSeg. We experimentally evaluated NP-SemiSeg on the public benchmarks PASCAL VOC 2012 and Cityscapes, with different training settings, and the results verify its effectiveness.

Multi-channel imaging data is a prevalent data format in scientific fields such as astronomy and biology. The structured information and the high dimensionality of these 3-D tensor data makes the analysis an intriguing but challenging topic for statisticians and practitioners. The low-rank scalar-on-tensor regression model, in particular, has received widespread attention and has been re-formulated as a tensor Gaussian Process (Tensor-GP) model with multi-linear kernel in Yu et al. (2018). In this paper, we extend the Tensor-GP model by introducing an integrative dimensionality reduction technique, called tensor contraction, with a Tensor-GP for a scalar-on-tensor regression task with multi-channel imaging data. This is motivated by the solar flare forecasting problem with high dimensional multi-channel imaging data. We first estimate a latent, reduced-size tensor for each data tensor and then apply a multi-linear Tensor-GP on the latent tensor data for prediction. We introduce an anisotropic total-variation regularization when conducting the tensor contraction to obtain a sparse and smooth latent tensor. We then propose an alternating proximal gradient descent algorithm for estimation. We validate our approach via extensive simulation studies and applying it to the solar flare forecasting problem.

Despite the significant interest and progress in reinforcement learning (RL) problems with adversarial corruption, current works are either confined to the linear setting or lead to an undesired $\tilde{\mathcal O}(\sqrt{T}\zeta)$ regret bound, where $T$ is the number of rounds and $\zeta$ is the total amount of corruption. In this paper, we consider contextual bandits with general function approximation and propose a computationally efficient algorithm to achieve a regret of $\tilde{\mathcal O}(\sqrt{T}+\zeta)$. The proposed algorithm relies on the recently developed uncertainty-weighted least-squares regression from linear contextual bandits (He et al., 2022) and a new weighted estimator of uncertainty for the general function class. In contrast to the existing analysis for the sum of uncertainty that is heavily based on the linear structure, we develop a novel technique to control the sum of weighted uncertainty, thus establishing the final regret bound. We then generalize our algorithm to the episodic MDP and first achieve an additive dependence on the corruption level $\zeta$ in the scenario of general function approximation. Notably, our algorithms achieve regret bounds that either nearly match the lower bound or improve the performance of existing methods for all the corruption levels in both known and unknown $\zeta$ cases.

Partial differential equations (PDEs) are important tools to model physical systems and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant coefficients, we propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations are exact solutions of this system. We apply the Ehrenpreis-Palamodov fundamental principle, which works as a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs. Our approach can infer probable solutions of linear PDE systems from any data such as noisy measurements, or pointwise defined initial and boundary conditions. Constructing EPGP-priors is algorithmic, generally applicable, and comes with a sparse version (S-EPGP) that learns the relevant spectral frequencies and works better for big data sets. We demonstrate our approach on three families of systems of PDEs, the heat equation, wave equation, and Maxwell's equations, where we improve upon the state of the art in computation time and precision, in some experiments by several orders of magnitude.

Despite their importance for assessing reliability of predictions, uncertainty quantification (UQ) measures in machine learning models have only recently begun to be rigorously characterized. One prominent issue is the *curse of dimensionality*: it is commonly believed that the marginal likelihood should be reminiscent of cross-validation metrics and both should deteriorate with larger input dimensions. However, we prove that by tuning hyperparameters to maximize marginal likelihood (the empirical Bayes procedure), performance, as measured by the marginal likelihood, *improves monotonically* with the input dimension. On the other hand, cross-validation metrics exhibit qualitatively different behavior that is characteristic of double descent. Cold posteriors, which have recently attracted interest due to their improved performance in certain settings, appear to exacerbate these phenomena. We verify empirically that our results hold for real data, beyond our considered assumptions, and we explore consequences involving synthetic covariates.