Spectral Representation for Causal Estimation with Hidden Confounders

We address the problem of causal effect estimation where hidden confounders are present, with a focus on two settings: instrumental variable regression with additional observed confounders, and proxy causal learning. Our approach uses a singular value decomposition of a conditional expectation operator, followed by a saddle-point optimization problem, which, in the context of IV regression, can be thought of as a neural net generalization of the seminal approach due to Darolles et al. [2011]. Saddle-point formulations have gathered considerable attention recently, as they can avoid double sampling bias and are amenable to modern function approximation methods. We provide experimental validation in various settings, and show that our approach outperforms existing methods on common benchmarks.

Mind the Graph When Balancing Data for Fairness or Robustness

Failures of fairness or robustness in machine learning predictive settings can be due to undesired dependencies between covariates, outcomes and auxiliary factors of variation. A common strategy to mitigate these failures is data balancing, which attempts to remove those undesired dependencies. In this work, we define conditions on the training distribution for data balancing to lead to fair or robust models. Our results display that, in many cases, the balanced distribution does not correspond to selectively removing the undesired dependencies in a causal graph of the task, leading to multiple failure modes and even interference with other mitigation techniques such as regularization. Overall, our results highlight the importance of taking the causal graph into account before performing data balancing.

Conditional Bayesian Quadrature

We propose a novel approach for estimating conditional or parametric expectations in the setting where obtaining samples or evaluating integrands is costly. Through the framework of probabilistic numerical methods (such as Bayesian quadrature), our novel approach allows to incorporates prior information about the integrands especially the prior smoothness knowledge about the integrands and the conditional expectation. As a result, our approach provides a way of quantifying uncertainty and leads to a fast convergence rate, which is confirmed both theoretically and empirically on challenging tasks in Bayesian sensitivity analysis, computational finance and decision making under uncertainty.

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.

Deep MMD Gradient Flow without adversarial training

We propose a gradient flow procedure for generative modeling by transporting particles from an initial source distribution to a target distribution, where the gradient field on the particles is given by a noise-adaptive Wasserstein Gradient of the Maximum Mean Discrepancy (MMD). The noise-adaptive MMD is trained on data distributions corrupted by increasing levels of noise, obtained via a forward diffusion process, as commonly used in denoising diffusion probabilistic models. The result is a generalization of MMD Gradient Flow, which we call Diffusion-MMD-Gradient Flow or DMMD. The divergence training procedure is related to discriminator training in Generative Adversarial Networks (GAN), but does not require adversarial training. We obtain competitive empirical performance in unconditional image generation on CIFAR10, MNIST, CELEB-A (64 x64) and LSUN Church (64 x 64). Furthermore, we demonstrate the validity of the approach when MMD is replaced by a lower bound on the KL divergence.

Proxy Methods for Domain Adaptation

We study the problem of domain adaptation under distribution shift, where the shift is due to a change in the distribution of an unobserved, latent variable that confounds both the covariates and the labels. In this setting, neither the covariate shift nor the label shift assumptions apply. Our approach to adaptation employs proximal causal learning, a technique for estimating causal effects in settings where proxies of unobserved confounders are available. We demonstrate that proxy variables allow for adaptation to distribution shift without explicitly recovering or modeling latent variables. We consider two settings, (i) Concept Bottleneck: an additional ''concept'' variable is observed that mediates the relationship between the covariates and labels; (ii) Multi-domain: training data from multiple source domains is available, where each source domain exhibits a different distribution over the latent confounder. We develop a two-stage kernel estimation approach to adapt to complex distribution shifts in both settings. In our experiments, we show that our approach outperforms other methods, notably those which explicitly recover the latent confounder.

Practical Kernel Tests of Conditional Independence

We describe a data-efficient, kernel-based approach to statistical testing of conditional independence. A major challenge of conditional independence testing, absent in tests of unconditional independence, is to obtain the correct test level (the specified upper bound on the rate of false positives), while still attaining competitive test power. Excess false positives arise due to bias in the test statistic, which is obtained using nonparametric kernel ridge regression. We propose three methods for bias control to correct the test level, based on data splitting, auxiliary data, and (where possible) simpler function classes. We show these combined strategies are effective both for synthetic and real-world data.

A Distributional Analogue to the Successor Representation

This paper contributes a new approach for distributional reinforcement learning which elucidates a clean separation of transition structure and reward in the learning process. Analogous to how the successor representation (SR) describes the expected consequences of behaving according to a given policy, our distributional successor measure (SM) describes the distributional consequences of this behaviour. We formulate the distributional SM as a distribution over distributions and provide theory connecting it with distributional and model-based reinforcement learning. Moreover, we propose an algorithm that learns the distributional SM from data by minimizing a two-level maximum mean discrepancy. Key to our method are a number of algorithmic techniques that are independently valuable for learning generative models of state. As an illustration of the usefulness of the distributional SM, we show that it enables zero-shot risk-sensitive policy evaluation in a way that was not previously possible.

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.

Distributional Bellman Operators over Mean Embeddings

We propose a novel algorithmic framework for distributional reinforcement learning, based on learning finite-dimensional mean embeddings of return distributions. We derive several new algorithms for dynamic programming and temporal-difference learning based on this framework, provide asymptotic convergence theory, and examine the empirical performance of the algorithms on a suite of tabular tasks. Further, we show that this approach can be straightforwardly combined with deep reinforcement learning, and obtain a new deep RL agent that improves over baseline distributional approaches on the Arcade Learning Environment.