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Pursuing causality from data is a fundamental problem in scientific discovery, treatment intervention, and transfer learning. This paper introduces a novel algorithmic method for addressing nonparametric invariance and causality learning in regression models across multiple environments, where the joint distribution of response variables and covariates varies, but the conditional expectations of outcome given an unknown set of quasi-causal variables are invariant. The challenge of finding such an unknown set of quasi-causal or invariant variables is compounded by the presence of endogenous variables that have heterogeneous effects across different environments, including even one of them in the regression would make the estimation inconsistent. The proposed Focused Adversial Invariant Regularization (FAIR) framework utilizes an innovative minimax optimization approach that breaks down the barriers, driving regression models toward prediction-invariant solutions through adversarial testing. Leveraging the representation power of neural networks, FAIR neural networks (FAIR-NN) are introduced for causality pursuit. It is shown that FAIR-NN can find the invariant variables and quasi-causal variables under a minimal identification condition and that the resulting procedure is adaptive to low-dimensional composition structures in a non-asymptotic analysis. Under a structural causal model, variables identified by FAIR-NN represent pragmatic causality and provably align with exact causal mechanisms under conditions of sufficient heterogeneity. Computationally, FAIR-NN employs a novel Gumbel approximation with decreased temperature and stochastic gradient descent ascent algorithm. The procedures are convincingly demonstrated using simulated and real-data examples., Comment: 48 pages, 7 figures with appendix

It is observed empirically that the large language models (LLM), trained with a variant of regression loss using numerous corpus from the Internet, can unveil causal associations to some extent. This is contrary to the traditional wisdom that ``association is not causation'' and the paradigm of traditional causal inference in which prior causal knowledge should be carefully incorporated into the design of methods. It is a mystery why causality, in a higher layer of understanding, can emerge from the regression task that pursues associations. In this paper, we claim the emergence of causality from association-oriented training can be attributed to the coupling effects from the heterogeneity of the source data, stochasticity of training algorithms, and over-parameterization of the learning models. We illustrate such an intuition using a simple but insightful model that learns invariance, a quasi-causality, using regression loss. To be specific, we consider multi-environment low-rank matrix sensing problems where the unknown r-rank ground-truth d*d matrices diverge across the environments but contain a lower-rank invariant, causal part. In this case, running pooled gradient descent will result in biased solutions that only learn associations in general. We show that running large-batch Stochastic Gradient Descent, whose each batch being linear measurement samples randomly selected from a certain environment, can successfully drive the solution towards the invariant, causal solution under certain conditions. This step is related to the relatively strong heterogeneity of the environments, the large step size and noises in the optimization algorithm, and the over-parameterization of the model. In summary, we unveil another implicit bias that is a result of the symbiosis between the heterogeneity of data and modern algorithms, which is, to the best of our knowledge, first in the literature.

This paper considers a multiple environments linear regression model in which data from multiple experimental settings are collected. The joint distribution of the response variable and covariate may vary across different environments, yet the conditional expectation of $y$ given the unknown set of important variables are invariant across environments. Such a statistical model is related to the problem of endogeneity, causal inference, and transfer learning. The motivation behind it is illustrated by how the goals of prediction and attribution are inherent in estimating the true parameter and the important variable set. We construct a novel {\it environment invariant linear least squares (EILLS)} objective function, a multiple-environment version of linear least squares that leverages the above conditional expectation invariance structure and heterogeneity among different environments to determine the true parameter. Our proposed method is applicable without any additional structural knowledge and can identify the true parameter under a near-minimal identification condition. We establish non-asymptotic $\ell_2$ error bounds on the estimation error for the EILLS estimator in the presence of spurious variables. Moreover, we further show that the EILLS estimator is able to eliminate all endogenous variables and the $\ell_0$ penalized EILLS estimator can achieve variable selection consistency in high-dimensional regimes. These non-asymptotic results demonstrate the sample efficiency of the EILLS estimator and its capability to circumvent the curse of endogeneity in an algorithmic manner without any prior structural knowledge., Comment: 65 pages, 4 figures

This paper introduces a Factor Augmented Sparse Throughput (FAST) model that utilizes both latent factors and sparse idiosyncratic components for nonparametric regression. The FAST model bridges factor models on one end and sparse nonparametric models on the other end. It encompasses structured nonparametric models such as factor augmented additive models and sparse low-dimensional nonparametric interaction models and covers the cases where the covariates do not admit factor structures. Via diversified projections as estimation of latent factor space, we employ truncated deep ReLU networks to nonparametric factor regression without regularization and to a more general FAST model using nonconvex regularization, resulting in factor augmented regression using neural network (FAR-NN) and FAST-NN estimators respectively. We show that FAR-NN and FAST-NN estimators adapt to the unknown low-dimensional structure using hierarchical composition models in nonasymptotic minimax rates. We also study statistical learning for the factor augmented sparse additive model using a more specific neural network architecture. Our results are applicable to the weak dependent cases without factor structures. In proving the main technical result for FAST-NN, we establish a new deep ReLU network approximation result that contributes to the foundation of neural network theory. Our theory and methods are further supported by simulation studies and an application to macroeconomic data., Comment: JASA, 81 pages, 6 figures

This paper investigates the stability of deep ReLU neural networks for nonparametric regression under the assumption that the noise has only a finite p-th moment. We unveil how the optimal rate of convergence depends on p, the degree of smoothness and the intrinsic dimension in a class of nonparametric regression functions with hierarchical composition structure when both the adaptive Huber loss and deep ReLU neural networks are used. This optimal rate of convergence cannot be obtained by the ordinary least squares but can be achieved by the Huber loss with a properly chosen parameter that adapts to the sample size, smoothness, and moment parameters. A concentration inequality for the adaptive Huber ReLU neural network estimators with allowable optimization errors is also derived. To establish a matching lower bound within the class of neural network estimators using the Huber loss, we employ a different strategy from the traditional route: constructing a deep ReLU network estimator that has a better empirical loss than the true function and the difference between these two functions furnishes a low bound. This step is related to the Huberization bias, yet more critically to the approximability of deep ReLU networks. As a result, we also contribute some new results on the approximation theory of deep ReLU neural networks., Comment: 79 pages, 5 figures

Analysis of over-parameterized neural networks has drawn significant attention in recentyears. It was shown that such systems behave like convex systems under various restrictedsettings, such as for two-level neural networks, and when learning is only restricted locally inthe so-called neural tangent kernel space around specialized initializations. However, there areno theoretical techniques that can analyze fully trained deep neural networks encountered inpractice. This paper solves this fundamental problem by investigating such overparameterizeddeep neural networks when fully trained. We generalize a new technique called neural feature repopulation, originally introduced in (Fang et al., 2019a) for two-level neural networks, to analyze deep neural networks. It is shown that under suitable representations, overparameterized deep neural networks are inherently convex, and when optimized, the system can learn effective features suitable for the underlying learning task under mild conditions. This new analysis is consistent with empirical observations that deep neural networks are capable of learning efficient feature representations. Therefore, the highly unexpected result of this paper can satisfactorily explain the practical success of deep neural networks. Empirical studies confirm that predictions of our theory are consistent with results observed in practice.

We study the question of how to imitate tasks across domains with discrepancies such as embodiment, viewpoint, and dynamics mismatch. Many prior works require paired, aligned demonstrations and an additional RL step that requires environment interactions. However, paired, aligned demonstrations are seldom obtainable and RL procedures are expensive. We formalize the Domain Adaptive Imitation Learning (DAIL) problem, which is a unified framework for imitation learning in the presence of viewpoint, embodiment, and dynamics mismatch. Informally, DAIL is the process of learning how to perform a task optimally, given demonstrations of the task in a distinct domain. We propose a two step approach to DAIL: alignment followed by adaptation. In the alignment step we execute a novel unsupervised MDP alignment algorithm, Generative Adversarial MDP Alignment (GAMA), to learn state and action correspondences from \emph{unpaired, unaligned} demonstrations. In the adaptation step we leverage the correspondences to zero-shot imitate tasks across domains. To describe when DAIL is feasible via alignment and adaptation, we introduce a theory of MDP alignability. We experimentally evaluate GAMA against baselines in embodiment, viewpoint, and dynamics mismatch scenarios where aligned demonstrations don't exist and show the effectiveness of our approach., Comment: ICML 2020

Most language modeling methods rely on large-scale data to statistically learn the sequential patterns of words. In this paper, we argue that words are atomic language units but not necessarily atomic semantic units. Inspired by HowNet, we use sememes, the minimum semantic units in human languages, to represent the implicit semantics behind words for language modeling, named Sememe-Driven Language Model (SDLM). More specifically, to predict the next word, SDLM first estimates the sememe distribution gave textual context. Afterward, it regards each sememe as a distinct semantic expert, and these experts jointly identify the most probable senses and the corresponding word. In this way, SDLM enables language models to work beyond word-level manipulation to fine-grained sememe-level semantics and offers us more powerful tools to fine-tune language models and improve the interpretability as well as the robustness of language models. Experiments on language modeling and the downstream application of headline gener- ation demonstrate the significant effect of SDLM. Source code and data used in the experiments can be accessed at https:// github.com/thunlp/SDLM-pytorch., Comment: EMNLP 2018. The first three authors contribute equally

In this paper we introduce ZhuSuan, a python probabilistic programming library for Bayesian deep learning, which conjoins the complimentary advantages of Bayesian methods and deep learning. ZhuSuan is built upon Tensorflow. Unlike existing deep learning libraries, which are mainly designed for deterministic neural networks and supervised tasks, ZhuSuan is featured for its deep root into Bayesian inference, thus supporting various kinds of probabilistic models, including both the traditional hierarchical Bayesian models and recent deep generative models. We use running examples to illustrate the probabilistic programming on ZhuSuan, including Bayesian logistic regression, variational auto-encoders, deep sigmoid belief networks and Bayesian recurrent neural networks., Comment: The GitHub page is at https://github.com/thu-ml/zhusuan