Deep Layer-wise Networks Have Closed-Form Weights

There is currently a debate within the neuroscience community over the likelihood of the brain performing backpropagation (BP). To better mimic the brain, training a network \textit{one layer at a time} with only a "single forward pass" has been proposed as an alternative to bypass BP; we refer to these networks as "layer-wise" networks. We continue the work on layer-wise networks by answering two outstanding questions. First, $\textit{do they have a closed-form solution?}$ Second, $\textit{how do we know when to stop adding more layers?}$ This work proves that the Kernel Mean Embedding is the closed-form weight that achieves the network global optimum while driving these networks to converge towards a highly desirable kernel for classification; we call it the $\textit{Neural Indicator Kernel}$.

Importance Weighting Approach in Kernel Bayes' Rule

We study a nonparametric approach to Bayesian computation via feature means, where the expectation of prior features is updated to yield expected posterior features, based on regression from kernel or neural net features of the observations. All quantities involved in the Bayesian update are learned from observed data, making the method entirely model-free. The resulting algorithm is a novel instance of a kernel Bayes' rule (KBR). Our approach is based on importance weighting, which results in superior numerical stability to the existing approach to KBR, which requires operator inversion. We show the convergence of the estimator using a novel consistency analysis on the importance weighting estimator in the infinity norm. We evaluate our KBR on challenging synthetic benchmarks, including a filtering problem with a state-space model involving high dimensional image observations. The proposed method yields uniformly better empirical performance than the existing KBR, and competitive performance with other competing methods.

KSD Aggregated Goodness-of-fit Test

We investigate properties of goodness-of-fit tests based on the Kernel Stein Discrepancy (KSD). We introduce a strategy to construct a test, called KSDAgg, which aggregates multiple tests with different kernels. KSDAgg avoids splitting the data to perform kernel selection (which leads to a loss in test power), and rather maximises the test power over a collection of kernels. We provide theoretical guarantees on the power of KSDAgg: we show it achieves the smallest uniform separation rate of the collection, up to a logarithmic term. KSDAgg can be computed exactly in practice as it relies either on a parametric bootstrap or on a wild bootstrap to estimate the quantiles and the level corrections. In particular, for the crucial choice of bandwidth of a fixed kernel, it avoids resorting to arbitrary heuristics (such as median or standard deviation) or to data splitting. We find on both synthetic and real-world data that KSDAgg outperforms other state-of-the-art adaptive KSD-based goodness-of-fit testing procedures.

Composite Goodness-of-fit Tests with Kernels

Model misspecification can create significant challenges for the implementation of probabilistic models, and this has led to development of a range of inference methods which directly account for this issue. However, whether these more involved methods are required will depend on whether the model is really misspecified, and there is a lack of generally applicable methods to answer this question. One set of tools which can help are goodness-of-fit tests, where we test whether a dataset could have been generated by a fixed distribution. Kernel-based tests have been developed to for this problem, and these are popular due to their flexibility, strong theoretical guarantees and ease of implementation in a wide range of scenarios. In this paper, we extend this line of work to the more challenging composite goodness-of-fit problem, where we are instead interested in whether the data comes from any distribution in some parametric family. This is equivalent to testing whether a parametric model is well-specified for the data.

Kernel Methods for Multistage Causal Inference: Mediation Analysis and Dynamic Treatment Effects

We propose kernel ridge regression estimators for mediation analysis and dynamic treatment effects over short horizons. We allow treatments, covariates, and mediators to be discrete or continuous, and low, high, or infinite dimensional. We propose estimators of means, increments, and distributions of counterfactual outcomes with closed form solutions in terms of kernel matrix operations. For the continuous treatment case, we prove uniform consistency with finite sample rates. For the discrete treatment case, we prove root-n consistency, Gaussian approximation, and semiparametric efficiency. We conduct simulations then estimate mediated and dynamic treatment effects of the US Job Corps program for disadvantaged youth.

MMD Aggregated Two-Sample Test

We propose a novel nonparametric two-sample test based on the Maximum Mean Discrepancy (MMD), which is constructed by aggregating tests with different kernel bandwidths. This aggregation procedure, called MMDAgg, ensures that test power is maximised over the collection of kernels used, without requiring held-out data for kernel selection (which results in a loss of test power), or arbitrary kernel choices such as the median heuristic. We work in the non-asymptotic framework, and prove that our aggregated test is minimax adaptive over Sobolev balls. Our guarantees are not restricted to a specific kernel, but hold for any product of one-dimensional translation invariant characteristic kernels which are absolutely and square integrable. Moreover, our results apply for popular numerical procedures to determine the test threshold, namely permutations and the wild bootstrap. Through numerical experiments on both synthetic and real-world datasets, we demonstrate that MMDAgg outperforms alternative state-of-the-art approaches to MMD kernel adaptation for two-sample testing.

KALE Flow: A Relaxed KL Gradient Flow for Probabilities with Disjoint Support

We study the gradient flow for a relaxed approximation to the Kullback-Leibler (KL) divergence between a moving source and a fixed target distribution. This approximation, termed the KALE (KL approximate lower-bound estimator), solves a regularized version of the Fenchel dual problem defining the KL over a restricted class of functions. When using a Reproducing Kernel Hilbert Space (RKHS) to define the function class, we show that the KALE continuously interpolates between the KL and the Maximum Mean Discrepancy (MMD). Like the MMD and other Integral Probability Metrics, the KALE remains well defined for mutually singular distributions. Nonetheless, the KALE inherits from the limiting KL a greater sensitivity to mismatch in the support of the distributions, compared with the MMD. These two properties make the KALE gradient flow particularly well suited when the target distribution is supported on a low-dimensional manifold. Under an assumption of sufficient smoothness of the trajectories, we show the global convergence of the KALE flow. We propose a particle implementation of the flow given initial samples from the source and the target distribution, which we use to empirically confirm the KALE's properties.

Self-Supervised Learning with Kernel Dependence Maximization

We approach self-supervised learning of image representations from a statistical dependence perspective, proposing Self-Supervised Learning with the Hilbert-Schmidt Independence Criterion (SSL-HSIC). SSL-HSIC maximizes dependence between representations of transformed versions of an image and the image identity, while minimizing the kernelized variance of those features. This self-supervised learning framework yields a new understanding of InfoNCE, a variational lower bound on the mutual information (MI) between different transformations. While the MI itself is known to have pathologies which can result in meaningless representations being learned, its bound is much better behaved: we show that it implicitly approximates SSL-HSIC (with a slightly different regularizer). Our approach also gives us insight into BYOL, since SSL-HSIC similarly learns local neighborhoods of samples. SSL-HSIC allows us to directly optimize statistical dependence in time linear in the batch size, without restrictive data assumptions or indirect mutual information estimators. Trained with or without a target network, SSL-HSIC matches the current state-of-the-art for standard linear evaluation on ImageNet, semi-supervised learning and transfer to other classification and vision tasks such as semantic segmentation, depth estimation and object recognition.

Deep Proxy Causal Learning and its Application to Confounded Bandit Policy Evaluation

Proxy causal learning (PCL) is a method for estimating the causal effect of treatments on outcomes in the presence of unobserved confounding, using proxies (structured side information) for the confounder. This is achieved via two-stage regression: in the first stage, we model relations among the treatment and proxies; in the second stage, we use this model to learn the effect of treatment on the outcome, given the context provided by the proxies. PCL guarantees recovery of the true causal effect, subject to identifiability conditions. We propose a novel method for PCL, the deep feature proxy variable method (DFPV), to address the case where the proxies, treatments, and outcomes are high-dimensional and have nonlinear complex relationships, as represented by deep neural network features. We show that DFPV outperforms recent state-of-the-art PCL methods on challenging synthetic benchmarks, including settings involving high dimensional image data. Furthermore, we show that PCL can be applied to off-policy evaluation for the confounded bandit problem, in which DFPV also exhibits competitive performance.

Proximal Causal Learning with Kernels: Two-Stage Estimation and Moment Restriction

We address the problem of causal effect estimation in the presence of unobserved confounding, but where proxies for the latent confounder(s) are observed. We propose two kernel-based methods for nonlinear causal effect estimation in this setting: (a) a two-stage regression approach, and (b) a maximum moment restriction approach. We focus on the proximal causal learning setting, but our methods can be used to solve a wider class of inverse problems characterised by a Fredholm integral equation. In particular, we provide a unifying view of two-stage and moment restriction approaches for solving this problem in a nonlinear setting. We provide consistency guarantees for each algorithm, and we demonstrate these approaches achieve competitive results on synthetic data and data simulating a real-world task. In particular, our approach outperforms earlier methods that are not suited to leveraging proxy variables.