A Survey on Causal Inference <s> Balanced representation learning. <s> This paper addresses the general problem of domain adaptation which arises in a variety of applications where the distribution of the labeled sample available somewhat differs from that of the test data. Building on previous work by Ben-David et al. (2007), we introduce a novel distance between distributions, discrepancy distance, that is tailored to adaptation problems with arbitrary loss functions. We give Rademacher complexity bounds for estimating the discrepancy distance from finite samples for different loss functions. Using this distance, we derive new generalization bounds for domain adaptation for a wide family of loss functions. We also present a series of novel adaptation bounds for large classes of regularization-based algorithms, including support vector machines and kernel ridge regression based on the empirical discrepancy. This motivates our analysis of the problem of minimizing the empirical discrepancy for various loss functions for which we also give several algorithms. We report the results of preliminary experiments that demonstrate the benefits of our discrepancy minimization algorithms for domain adaptation. <s> BIB001 </s> A Survey on Causal Inference <s> Balanced representation learning. <s> Observational studies are rising in importance due to the widespread accumulation of data in fields such as healthcare, education, employment and ecology. We consider the task of answering counterfactual questions such as, "Would this patient have lower blood sugar had she received a different medication?". We propose a new algorithmic framework for counterfactual inference which brings together ideas from domain adaptation and representation learning. In addition to a theoretical justification, we perform an empirical comparison with previous approaches to causal inference from observational data. Our deep learning algorithm significantly outperforms the previous state-of-the-art. <s> BIB002 </s> A Survey on Causal Inference <s> Balanced representation learning. <s> There is intense interest in applying machine learning to problems of causal inference in fields such as healthcare, economics and education. In particular, individual-level causal inference has important applications such as precision medicine. We give a new theoretical analysis and family of algorithms for predicting individual treatment effect (ITE) from observational data, under the assumption known as strong ignorability. The algorithms learn a "balanced" representation such that the induced treated and control distributions look similar. We give a novel, simple and intuitive generalization-error bound showing that the expected ITE estimation error of a representation is bounded by a sum of the standard generalization-error of that representation and the distance between the treated and control distributions induced by the representation. We use Integral Probability Metrics to measure distances between distributions, deriving explicit bounds for the Wasserstein and Maximum Mean Discrepancy (MMD) distances. Experiments on real and simulated data show the new algorithms match or outperform the state-of-the-art. <s> BIB003 </s> A Survey on Causal Inference <s> Balanced representation learning. <s> Learning representations for counterfactual inference from observational data is of high practical relevance for many domains, such as healthcare, public policy and economics. Counterfactual inference enables one to answer "What if...?" questions, such as "What would be the outcome if we gave this patient treatment $t_1$?". However, current methods for training neural networks for counterfactual inference on observational data are either overly complex, limited to settings with only two available treatment options, or both. Here, we present Perfect Match (PM), a method for training neural networks for counterfactual inference that is easy to implement, compatible with any architecture, does not add computational complexity or hyperparameters, and extends to any number of treatments. PM is based on the idea of augmenting samples within a minibatch with their propensity-matched nearest neighbours. Our experiments demonstrate that PM outperforms a number of more complex state-of-the-art methods in inferring counterfactual outcomes across several real-world and semi-synthetic datasets. <s> BIB004 </s> A Survey on Causal Inference <s> Balanced representation learning. <s> Predictive models that generalize well under distributional shift are often desirable and sometimes crucial to building robust and reliable machine learning applications. We focus on distributional shift that arises in causal inference from observational data and in unsupervised domain adaptation. We pose both of these problems as prediction under a shift in design. Popular methods for overcoming distributional shift make unrealistic assumptions such as having a well-specified model or knowing the policy that gave rise to the observed data. Other methods are hindered by their need for a pre-specified metric for comparing observations, or by poor asymptotic properties. We devise a bound on the generalization error under design shift, incorporating both representation learning and sample re-weighting. Based on the bound, we propose an algorithmic framework that does not require any of the above assumptions and which is asymptotically consistent. We empirically study the new framework using two synthetic datasets, and demonstrate its effectiveness compared to previous methods. <s> BIB005 </s> A Survey on Causal Inference <s> Balanced representation learning. <s> Estimating individual treatment effect (ITE) is a challenging problem in causal inference, due to missing counterfactuals and the selection bias. Existing ITE estimation methods mainly focus on balancing the distributions of control and treated groups, but ignore the local similarity information that is helpful. In this paper, we propose a local similarity preserved individual treatment effect (SITE) estimation method based on deep representation learning. SITE preserves local similarity and balances data distributions simultaneously, by focusing on several hard samples in each mini-batch. Experimental results on synthetic and three real-world datasets demonstrate the advantages of the proposed SITE method, compared with the state-of-the-art ITE estimation methods. <s> BIB006 </s> A Survey on Causal Inference <s> Balanced representation learning. <s> Treatment effect estimation refers to the estimation of causal effects, which benefits decision-making process across various domains, but it is a challenging problem in real practice. The estimation of causal effects from observational data at the individual level faces two major challenges, i.e., treatment selection bias and missing counterfactuals. Existing methods tackle the selection bias problem by learning a balanced representation and infer the missing counterfactuals based on the learned representation. However, most existing methods learn the representation in a global manner and ignore the local similarity information, which is essential for an accurate estimation of causal effects. Motivated by the above observations, we propose a novel representation learning method, which adaptively extracts fine-grained similarity information from the original feature space and minimizes the distance between different treatment groups as well as the similarity loss during the representation learning procedure. Experiments on three public datasets demonstrate that the proposed method achieves the best performance in causal effect estimation among all the compared methods and is robust to the treatment selection bias. <s> BIB007
The most basic assumption used in statistical learning theory is that training data and test data are drawn from the same distribution. However, in most practical cases, the test data are drawn from a distribution that is only related, but not identical, to the distribution of the training data. In causal inference, this is also a big challenge. Unlike the randomized control trials, the mechanism of treatment assignment is not explicit in observational data. Therefore, interventions of interest are not independent of the property of the subjects. For example, in an observational study of the treatment effect of a medicine, the medicine is assigned to individuals based on several factors, including the known confounders and some unknown confounders. As a result, the counterfactual distribution will generally be different from the factual distribution. Thus, it is necessary to predict counterfactual outcomes by learning from the factual data, which converts the causal inference problem to a domain adaptation problem. Extracting effective feature representations is critical for domain adaptation. A model  with a generalization bound is proposed to formalize this intuition theoretically, which can not only explicitly minimize the difference between the source and target domains, but also maximize the margin of the training set. Building on this work  , the discrepancy distance between distributions is tailored to adaptation problems with arbitrary loss functions BIB001 . In the following discussions, the discrepancy distance plays an important role in addressing the domain adaptation problem in causal inference. So far, we can see a clear connection between counterfactual inference and domain adaptation. An intuitive idea is to enforce the similarity between the distributions of different treatment groups in the representation space. The learned representations trade-off three objectives: (1) low-error prediction over the factual representation, (2) low-error prediction over counterfactual outcomes by taking into account relevant factual outcomes, and (3) the distance between the distribution of treatment population and that of control population BIB002 . Following this motivation, BIB003 gives a simple and intuitive generalization-error bound. It shows that the expected ITE estimation error of representation is bounded by a sum of the standard generalization-error of that representation and the distance between the treated and control distributions based on representation. Integral probability metric (IPM) is used to measure the distances between distributions, and explicit bounds are derived for the Wasserstein distance and Maximum Mean Discrepancy (MMD) distance. The goal is to find a representation Φ : X → R and hypothesis h : X × {0, 1} → Y that minimizes the following objective function: W i , and the weights r i compensate for the difference in treatment group size. R is a model complexity term. Given two probability density functions p, q defined over S ⊆ R d , and a function family G of functions д : S → R, the IPM is defined as: This model allows for learning complex nonlinear representations and hypotheses with large flexibility. When the dimension of Φ is high, it risks losing the influence of t on h if the concatenation of Φ and W is treated as input. To address this problem, one approach is to parameterize h 1 (Φ) and h 0 (Φ) as two separate "heads" of the joint network. h 1 (Φ) is used to estimate the outcome under treatment and h 0 (Φ) is for the control group. Each sample is used to update only the head corresponding to the observed treatment. The advantage is that statistical power is shared in the common representation layers and the influence of treatment is retained in the separate heads BIB003 . This model can also be extended to any number of treatments, as described in the perfect match (PM) approach BIB004 . Following this idea, a few improved models have been proposed and discussed. For example, BIB005 brings together shift-invariant representation learning and re-weighting methods.  presents a new context-aware weighting scheme based on the importance sampling technique, on top of representation learning, to alleviate the selection bias problem in ITE estimation. Existing ITE estimation methods mainly focus on balancing the distributions of control and treated groups, but ignore the local similarity information that provides meaningful constraints on the ITE estimation. In BIB006 BIB007 , a local similarity preserved individual treatment effect (SITE) estimation method is proposed based on deep representation learning. SITE preserves local similarity and balances data distributions simultaneously. The framework of SITE contains five major components: representation network, triplet pairs selection, positiondependent deep metric (PDDM), middle point distance minimization (MPDM), and the outcome prediction network. To improve the model efficiency, SITE takes input units in a mini-batch fashion, and triplet pairs could be selected from every mini-batch. The representation network learns latent embeddings for the input units. With the selected triplet pairs, PDDM and MPDM can preserve the local similarity information and meanwhile achieve the balanced distributions in the latent space. Finally, the embeddings of mini-batch are fed forward to a dichotomous outcome prediction network to get the potential outcomes. The loss function of SITE is as follows: where L F L is the factual loss between the estimated and observed factual outcomes. L P DDM and L M P DM are the loss functions for PDDM and MPDM, respectively. The last term is L 2 regularization on model parameters M (except the bias term). Most models focus on covariates with numerical values, while how to handle covariates with textual information for treatment effect estimation is still an open question. One major challenge is how to filter out the nearly instrumental variables which are the variables more predictive to the treatment than the outcome. Conditioning on those variables to estimate the treatment effect would amplify the estimation bias. To address this challenge, a conditional treatment-adversarial learning based matching (CTAM) method is proposed in  . CTAM incorporates the treatment-adversarial learning to filter out the information related to nearly instrumental variables when learning the representations, and then it performs matching among the learned representations to estimate the treatment effects. The CTAM contains three major components: text processing, representation learning, and conditional treatment discriminator. Through the text processing component, the original text is transformed into vectorized representation S. After that, S is concatenated with the non-textual covariates X to construct a unified feature vector, which is then fed into the representation neural network to get the latent representation Z . After learning the representation, Z , together with potential outcomes Y , are fed into the conditional treatment discriminator. During the training procedures, the representation learner plays a minimax game with the conditional treatment discriminator: By preventing the discriminator from assigning correct treatment, the representation learner can filter out the information related to nearly instrumental variables. The final matching procedure is performed in the representation space Z . The conditional treatment-adversarial learning helps reduce the bias of treatment effect estimation. Compared to the above regression-based methods after representation learning, matching method is more interpretable, because any sample's counterfactual outcome is directly set to be the factual outcome of its nearest neighbor in the group receiving the opposite treatment. Nearest neighbor matching (NNM) sets the counterfactual outcome of any treatment (control) sample to be equal to the factual outcome of its nearest neighbor in the control (treatment) group. Although being simple, flexible and interpretable, most NNM approaches could be easily misled by variables that do not affect the outcome. To address this challenge, matching can be performed on subspaces that are predictive of the outcome variable for both the treatment group and the control group. Applying NNM in the learned subspaces leads to a more accurate estimation of the counterfactual outcomes and therefore the accurate estimation of treatment effects.  estimates the counterfactual outcomes of treatment samples by learning a projection matrix that maximizes the nonlinear dependence between the subspace and outcome variable for control samples. Then it directly applies the learned projection matrix to all the samples and finds every treatment sample's matched control sample in the subspace.