Gaussian processes with deep neural networks demonstrate to be a strong learner for few-shot learning since they combine the strength of deep learning and kernels while being able to well capture uncertainty. However, it remains an open problem to leverage the shared knowledge provided by related tasks. In this paper, we propose to learn Gaussian processes with dense inducing variables by meta-learning for few-shot learning. In contrast to sparse Gaussian processes, we define a set of dense inducing variables to be of a much larger size than the support set in each task, which collects prior knowledge from experienced tasks. The dense inducing variables specify a shared Gaussian process prior over prediction functions of all tasks, which are learned in a variational inference framework and offer a strong inductive bias for learning new tasks. To achieve task-specific prediction functions, we propose to adapt the inducing variables to each task by efficient gradient descent. We conduct extensive experiments on common benchmark datasets for a variety of few-shot learning tasks. Our dense Gaussian processes present significant improvements over vanilla Gaussian processes and comparable or even better performance with state-of-the-art methods.

In this paper, we introduce a new functional point of view on bilevel optimization problems for machine learning, where the inner objective is minimized over a function space. These types of problems are most often solved by using methods developed in the parametric setting, where the inner objective is strongly convex with respect to the parameters of the prediction function. The functional point of view does not rely on this assumption and notably allows using over-parameterized neural networks as the inner prediction function. We propose scalable and efficient algorithms for the functional bilevel optimization problem and illustrate the benefits of our approach on instrumental regression and reinforcement learning tasks.

We consider supervised learning problems in which set predictions provide explicit uncertainty estimates. Using Choquet integrals (a.k.a. Lov{á}sz extensions), we propose a convex loss function for nondecreasing subset-valued functions obtained as level sets of a real-valued function. This loss function allows optimal trade-offs between conditional probabilistic coverage and the ''size'' of the set, measured by a non-decreasing submodular function. We also propose several extensions that mimic loss functions and criteria for binary classification with asymmetric losses, and show how to naturally obtain sets with optimized conditional coverage. We derive efficient optimization algorithms, either based on stochastic gradient descent or reweighted least-squares formulations, and illustrate our findings with a series of experiments on synthetic datasets for classification and regression tasks, showing improvements over approaches that aim for marginal coverage.

We thank all reviewers for their helpful and constructive feedback. Should we address your main concerns well, we hope you improve your score accordingly. This objective has a trivial solution. For this work, we opted for the simplest approach with fewer hyperparameters. Section 3.2, both proposing a method to reduce the likelihood of spurious modes and showing that the simpler approach We actually meant negative sampling is problematic and we avoid it.

First of all, we thank the reviewers for their helpful comments and remarks. Moore-Penrose pseudo-inverse which explains why the complexity is only quadratic in K). Adaptative Neural Trees ([2]) and Deep Neural Decision Forests ([3]) are both built from decision trees. It is true that a standard regression tree with enough leaves can also approximate a smooth link function. We'll modify lines 228-230 as In that case, the distribution of x over regions is concentrated on one region.

Theorem 6 is stated in terms of Gaussian complexity. Ben-David (2014) has a full proof. M (α)null is the linear class following the depth-K neural network. The second term relies on the Lipschitz constant of DNN, which we bound with the following lemma. Similar results are given by Scaman and Virmaux (2018); Fazlyab et al. (2019).