SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Synthetic Combinations: A Causal Framework for Combinatorial Interventions
Abhineet Agarwal  
Department of Statistics  
University of California, Berkeley  
aa3797@berkeley.edu
Anish Agarwal*  
Department of IEOR  
Columbia University  
aa5194@columbia.edu
Suhas Vijaykumar $^{\dagger}$   
Amazon, Core AI  
suhasv@mit.edu
We consider a setting where there are  $N$  heterogeneous units and  $p$  interventions. Our goal is to learn unit-specific potential outcomes for any combination of these  $p$  interventions, i.e.,  $N \times 2^p$  causal parameters. Choosing a combination of interventions is a problem that naturally arises in a variety of applications such as factorial design experiments and recommendation engines (e.g., showing a set of movies that maximizes engagement). Running  $N \times 2^p$  experiments to estimate the various parameters is likely expensive and/or infeasible as  $N$  and  $p$  grow. Further, with observational data there is likely confounding, i.e., whether or not a unit is seen under a combination is correlated with its potential outcome. We study this problem under a novel model that imposes latent structure across both units and combinations of interventions. Specifically, we assume latent similarity in potential outcomes across units (i.e., the matrix of potential outcomes is rank  $r$ ) and regularity in how combinations of interventions interact (i.e., the coefficients in the Fourier expansion of the potential outcomes is  $s$  sparse). We establish identification for all  $N \times 2^p$  parameters despite unobserved confounding. We propose an estimation procedure, Synthetic Combinations, and establish finite-sample consistency under precise conditions on the observation pattern. We show that Synthetic Combinations is able to consistently estimate unit-specific potential outcomes given a total of  $\mathrm{poly}(r) \times (N + s^2 p)$  observations. In comparison, previous methods that do not exploit structure across both units and combinations have poorer sample complexity that scales as  $\min(N \times s^2 p, \mathrm{poly}(r) \times (N + 2^p))$ .
Modern-day decision makers—in settings from e-commerce to public policy to medicine—often must select a combination of actions, and would like to do so in a highly personalized manner. Examples include recommending a curated basket of items to customers on a commerce platform, deciding on a combination of therapies for a medical patient, enacting a collection of socio-economic policies for a specific geographic location, conjoint analysis in surveys, selecting important feature sets for machine learning models, etc. Despite the ubiquity of this setting, it comes with significant empirical challenges: with  $p$  interventions and  $N$  units, a decision maker must evaluate  $N \times 2^p$  potential combinations in order to confirm the optimal personalized policy. With large  $N$  and even with relatively small  $p$  (due to exponential dependence), it becomes infeasible to run that many experiments; in observational data there is the additional challenge of potential unobserved confounding. Current methods tackle this problem by following one of two approaches: (i) they impose structure on how combinations of interventions interact, or (ii) they assume latent similarity
*Work done while post-doc at Amazon, Core AI.
†Work done while at MIT.
in potential outcomes across units. However, as we discuss in detail below, these approaches require a large number of observations to estimate all  $N \times 2^p$  potential outcomes because they do not exploit structure across both units and combinations. This motivates the question: how can one effectively share information across both units and combinations of interventions?
Contributions. Our contributions may be summarized as follows. (1) For a unit  $n \in [N]$ , we represent its potential outcomes over the  $2^{p}$  combinations as a Boolean function from  $\{-1,1\}^{p}$  to  $\mathbb{R}$ , expressed in the Fourier basis. To impose structure across combinations, we assume that for a unit  $n$ , the linear coefficients  $\alpha_{n} \in \mathbb{R}^{2^{p}}$  induced by this Fourier basis representation is  $s$ -sparse, i.e., has at most  $s$  nonzero entries. To impose structure across units, we assume that this matrix of Fourier coefficients across units  $A = [\alpha_{n}]_{n \in [N]} \in \mathbb{R}^{N \times 2^{p}}$  has rank  $r$ . This simultaneous sparsity and low-rank assumption allows the researcher to share information across both units and combinations. (2) We establish identification for the  $N \times 2^{p}$  potential outcomes of interest, which requires that any confounding is mediated by the (unobserved) matrix of Fourier coefficients  $A$ . (3) We design a two-step algorithm "Synthetic Combinations" and prove it consistently estimates the various causal parameters, despite potential unobserved confounding. The first step of Synthetic Combinations, termed "horizontal regression", learns the structure across combinations of interventions—assuming sparsity in the Fourier coefficients—via the Lasso. The second step, termed "vertical regression", learns the structure across units—assuming the matrix of Fourier coefficients are low-rank—via principal component regression (PCR). (4) We show that Synthetic Combinations is able to consistently estimate unit-specific potential outcomes given a total of  $\mathrm{poly}(r) \times (N + s^2 p)$  observations (ignoring logarithmic factors). This improves over previous methods that do not exploit structure across both units and combinations, which have sample complexity scaling as  $\min(N \times s^2 p, \mathrm{poly}(r) \times (N + 2^p))$ . A summary of the sample complexities required for different methods can be found in Table 1. A key technical challenge in our proofs is analyzing how the error induced in the first step of Synthetic Combinations percolates through to the second step. To tackle it, we reduce the problem to that of high-dimensional error-in-variables regression with linear model misspecification, and do a novel analysis of this statistical setting.
Learning over Combinations. To place structure on the space of combinations, we use tools from the theory of learning (sparse) Boolean functions, in particular the Fourier transform. Sparsity of the Fourier transform was proposed as a complexity measure to characterize and design learning algorithms for low-depth trees, low-degree polynomials, and small circuits [12, 26]. Learning Boolean functions is now a central topic in learning theory, and is closely related to many important questions in ML more broadly; see e.g. [31] for discussion of the  $k$ -Junta problem and its relation to relevant feature selection. We refer to O'Donnell [34, Chapter 3] for further background on this area. In this paper, we focus on [32] which showed that Lasso can be used to efficiently learn sparse Boolean functions, an essential property in our setting since the dimension of the function class grows exponentially.
Matrix Completion. We build on the observation that imputing counterfactual outcomes in the presence of a latent factor structure can be equivalently expressed as low-rank matrix completion [9, 8, 5]. The observation that low-rank matrices may typically be recovered from a small fraction of the entries by nuclear-norm minimization has had a major impact on modern statistics [15, 35, 13]. In the noisy setting, proposed estimators have generally proceeded by minimizing risk subject to a nuclear-norm penalty, such as in the SoftImpute algorithm of [30], or minimizing risk subject to a rank constraint as in the hard singular-value thresholding (HSVT) algorithms analyzed by [27, 28, 23, 16]. We refer the reader to [19, 33] for a comprehensive overview of this vast literature.
Learning algorithm	Exploits structure across combinations (||αn||0=s)	Exploits structure across units (rank(A)=r)	Sample complexity
Lasso	✓	X	O(N×s2p)
Matrix Completion	X	✓	O(poly(r)×(N+2p))
Synthetic Combinations	✓	✓	O(poly(r)×(N+s2p))
Table 1: Comparison of sample complexity of Synthetic Combinations to other methods.
Econometrics/causal inference. There is a rich literature on how to learn personalized treatment effects for heterogeneous units. This problem is of particular importance in the social sciences and in medicine, where experimental data is limited, and has led to several promising approaches including instrumental variables, difference-in-differences, regression discontinuity, and others; see [7, 25] for an overview. Of particular interest to us here is the "synthetic control" method [1], which exploits an underlying factor structure to effectively "share" counterfactual information between treatment and control units. Building on [1], recent work has shown that the same underlying structure can be used to estimate treatment effects with multiple treatments and heterogeneous units despite unmeasured confounding. The resulting framework, called "synthetic interventions" (SI), uses the factor representation to efficiently share information across similar (yet distinct) units and treatments [5]. We generalize the SI framework to settings where treatments are combinations of interventions and most treatments have no units that receive it, but there is structure across combinations of interventions. In doing so, we enable its use in highly practical cases where multiple interventions are delivered simultaneously, such as recommender systems, medical treatment regimens, and factorial design experiments.
3 Setup and Model
In this section, we first describe requisite notation, background on the Fourier expansion of real-valued functions over booleans, and how it relates to potential outcomes over combinations.
Representation of Combinations as Binary Vectors. Let  $[p] = \{1, \dots, p\}$  denote the set of  $p$  interventions. Denote by  $\Pi$  the power set of  $[p]$ , i.e., the set of all possible combinations of  $p$  interventions, where we note  $|\Pi| = 2^p$ . Then, any given combination  $\pi \in \Pi$  induces the following binary representation  $\mathbf{v}(\pi) \in \{-1, 1\}^p$  defined as follows:  $\mathbf{v}(\pi)_i = 2 \mathbf{1}\{i \in \pi\} - 1$ .
Fourier Expansion of Boolean Functions. Let  $\mathcal{F}_{\mathrm{bool}} = \{f:\{-1,1\}^p\to \mathbb{R}\}$  be the set of all real-valued functions defined on the hypercube  $\{-1,1\} ^p$ . Then  $\mathcal{F}_{\mathrm{bool}}$  forms a Hilbert space defined by the following inner product: for any  $f,g\in \mathcal{F}_{\mathrm{bool}}$ ,  $\langle f,g\rangle_B = \frac{1}{2^p}\sum_{\mathbf{x}\in \{-1,1\}^p}f(\mathbf{x})g(\mathbf{x})$ . This inner product induces the norm  $\langle f,f\rangle_B\coloneqq \| f\| _B^2 = \frac{1}{2^p}\sum_{\mathbf{x}\in \{-1,1\}^p}f^2 (\mathbf{x})$ . We construct an orthonormal basis for  $\mathcal{F}_{\mathrm{bool}}$  as follows: for each subset  $S\subset [p]$ , define a basis function  $\chi_S(\mathbf{x}) = \prod_{i\in S}x_i$  where  $x_{i}$  is the  $i^{\mathrm{th}}$  coefficient of  $\mathbf{x}\in \{-1,1\}^{p}$ . One can verify that for any  $S\subset [p]$  that  $\| \chi_S\| _B = 1$ , and that  $\langle \chi_S,\chi_{S'}\rangle_B = 0$  for any  $S'\neq S$ . Since  $|\{\chi_S:S\subset [p]\} | = 2^p$ , the functions  $\chi_S$  are an orthonormal basis of  $\mathcal{F}_{\mathrm{bool}}$ .
Hence, any  $f \in \mathcal{F}_{\mathrm{bool}}$  can be expressed via the following "Fourier" decomposition:  $f(\mathbf{x}) = \sum_{S \subset [p]} \alpha_S \chi_S(\mathbf{x})$ , where the Fourier coefficient  $\alpha_S$  is given by computing  $\alpha_S = \langle f, \chi_S \rangle_B$ . We will refer to  $\chi_S$  as the Fourier character. Define  $\alpha_f = [\alpha_S]_{S \in [p]} \in \mathbb{R}^{2^p}$  and  $\chi(x) = [\chi_S(\mathbf{x})]_{S \in [p]} \in \{-1, 1\}^{2^p}$  as the vector of Fourier coefficients and characters respectively. Hence any function  $f: \{-1, 1\}^p \to \mathbb{R}$  can be re-expressed as follows:  $f(\mathbf{x}) = \langle \alpha_f, \chi(x) \rangle$ . For  $\pi \in \Pi$ , abbreviate  $\chi_S(\mathbf{v}(\pi))$  and  $\chi(\mathbf{v}(\pi))$  as  $\chi_S^\pi$  and  $\chi^\pi$  respectively.
Observed and potential outcomes. Let  $Y_{n}^{(\pi)} \in \mathbb{R}$  denote the potential outcome for unit  $n$  under combination  $\pi$  and  $Y_{n\pi} \in \{\mathbb{R} \cup \star\}$  as the observed outcome, where  $\star$  indicates a missing value, i.e., the outcome associated with the unit-combination pair  $(n,\pi)$  was not observed. Let  $\mathbf{Y} = [Y_{n\pi}] \in \{\mathbb{R} \cup \star\}^{N \times 2^p}$ . Let  $\mathcal{D} \subset [N] \times [2^p]$ , refer to the subset of unit-combination pairs we do observe, i.e.,
$$
Y _ {n \pi} = \left\{ \begin{array}{l l} Y _ {n} ^ {(\pi)}, & \text {i f} (n, \pi) \in \mathcal {D} \\ \star , & \text {o t h e r w i s e .} \end{array} \right. \tag {1}
$$
Note that (1) implies stable unit treatment value assignment (SUTVA) holds. Let  $\Pi_S \subseteq \Pi$  denote a subset of combinations. For a given unit  $n$ , let  $\mathbf{Y}_{\Pi_S, n} = [Y_{n\pi_i} : \pi_i \in \Pi_S] \in \{\mathbb{R} \cup \star\}^{\left|\Pi_S\right|}$  represent the vector of observed outcomes for all  $\pi \in \Pi_S$ . Similarly, let  $\mathbf{Y}_n^{(\Pi_S)} = [Y_n^{(\pi_i)} : \pi_i \in \Pi_S] \in \mathbb{R}^{\left|\Pi_S\right|}$  represent the vector of potential outcomes. Denote  $\chi(\Pi_S) = [\chi^{\pi_i} : \pi_i \in \Pi_S] \in \{-1, 1\}^{\left|\Pi_S\right| \times 2^p}$ .
3.2 Model & Target Causal Parameter
Define  $Y_{n}^{(\cdot)}: \pi \to \mathbb{R}$  as a real-valued function over the hypercube  $\{-1,1\}^p$  associated with unit  $n$ . It takes as input a combination  $\pi$ , converts it to a  $p$ -dimensional binary vector  $\mathbf{v}(\pi)$ , and outputs a real number  $Y_{n}^{(\pi)}$ . Given the discussion in Section 3.1, it follows that  $Y_{n}^{(\pi)}$  always has the representation  $\langle \alpha_n, \chi^\pi \rangle$  for some  $\alpha_n \in \mathbb{R}^{2^p}$ . Thus, without any loss of generality, the  $\alpha_n$  are unit-specific latent variables (Fourier coefficients) encoding the treatment response function. Below, we state our key assumption on these induced Fourier coefficients.
Assumption 3.1 (Potential Outcome Model). For any unit-combination pair  $(n,\pi)$ , we assume it has the following representation,
$$
Y _ {n} ^ {(\pi)} = \left\langle \boldsymbol {\alpha} _ {n}, \boldsymbol {\chi} ^ {\pi} \right\rangle + \epsilon_ {n} ^ {\pi}, \tag {2}
$$
where  $\alpha_{n}\in \mathbb{R}^{2^{p}}$  and  $\chi^{\pi}\in \{-1,1\}^{2^{p}}$  are the Fourier coefficients and characters, respectively. We assume the following properties: (a) low-rank: the matrix  $\mathcal{A} = [\alpha_n]_{n\in [N]}$  has rank  $r\in [\min \{N,2^p\} ]$  (b) sparsity:  $\alpha_{n}$  is s-sparse (i.e.  $\| \pmb {\alpha}_n\| _0\leq s$  , where  $s\in [2^p ])$  for every unit  $n\in [N]$  ;  $(c)\epsilon_{n}^{\pi}$  is a residual term specific to  $(n,\pi)$  which satisfies  $\mathbb{E}[\epsilon_n^\pi \mid \mathcal{A}] = 0$
The assumption is then that each  $\alpha_{n}$ $s$ -sparse and  $\mathcal{A}$  is rank-  $r$ ;  $\epsilon_n^\pi$  is the residual from this sparse and low-rank approximation, and it serves as the source of uncertainty in our model. Given  $\mathbb{E}[\epsilon_n^{\pi}\mid \mathcal{A}] = 0$ , the matrix  $\mathbb{E}[\mathbf{Y}_N^{(\Pi)}] = [\mathbb{E}[\mathbf{Y}_n^{(\Pi)}]:n\in [N]]\in \mathbb{R}^{2^p\times N}$  also is rank  $r$ , where the expectation is defined with respect to  $\epsilon_n^\pi$ . This is because  $\mathbb{E}[\mathbf{Y}_N^{(\Pi)}]$  can be written as  $\mathbb{E}[\mathbf{Y}_N^{(\Pi)}] = \boldsymbol {\chi}(\Pi)\mathcal{A}^T$ , and since  $\chi (\Pi)$  is an invertible matrix,  $\mathrm{rank}(\mathbb{E}[\mathbf{Y}_N^{(\Pi)}]) = \mathrm{rank}(\mathcal{A})$ . The low-rank property places structure across units; that is, we assume there is sufficient similarity across units so that  $\mathbb{E}[\mathbf{Y}_n^{(\Pi)}]$  for any unit  $n$  can be written as a linear combination of  $r$  other rows of  $\mathbb{E}[\mathbf{Y}_N^{(\Pi)}]$ . This is a standard assumption used to encode latent similarity across units in matrix completion and its related applications (e.g., recommendation engines).
Sparsity establishes unit-specific structure; that is, we assume that the potential outcomes for a given user only depend on a small subset of the functions  $\{\chi_S:S\subset [p]\}$ . We emphasize that this subset of functions can be different across units. As discussed in Section 2, sparsity is commonly employed when studying the learnability of Boolean functions. In the context of recommendation engines, sparsity is implied if the ratings for a set of goods only depend on a small number of combinations of items within that set. Sparsity is also often assumed implicitly in factorial design experiments, where analysts typically only include pairwise interaction effects between interventions and ignore higher-order interactions [24]. We discuss further applications of combinatorial inference in greater detail in Appendix A.
Next, we present an assumption that formalizes the dependence (i.e., confounding) between the missingness pattern induced by the treatment assignments  $\mathcal{D}$  and the potential outcomes  $Y_{n}^{(\pi)}$ , and provide an interpretation of the induced data generating process (DGP) for potential outcomes.
Assumption 3.2 (Selection on Fourier coefficients). For all  $n\in [N]$  and  $\pi \in \Pi$ $Y_{n}^{(\pi)}\perp \mathcal{D}\mid \mathcal{A}$
DGP. Given Assumptions 3.1 and Assumption 3.2, the DGP can be summarized as follows: (i) unit-specific latent Fourier coefficients  $\mathcal{A}$  are either deterministic or sampled from an unknown distribution; we will condition on this quantity throughout. (ii) Given  $\mathcal{A}$ , we sample mean-zero random variables  $\epsilon_{n}^{\pi}$ , and generate potential outcomes according to our model  $Y_{n}^{(\pi)} = \langle \alpha_{n},\chi^{\pi}\rangle +\epsilon_{n}^{\pi}$ . (iii)  $\mathcal{D}$  is allowed to depend on unit-specific latent Fourier coefficients  $\mathcal{A}$  (i.e.  $\mathcal{D} = f(\mathcal{A})$ ). We define all expectations w.r.t. noise,  $\epsilon_{n}^{\pi}$ . This DGP introduces unobserved confounding since  $Y_{n}^{(\pi)} \not\ll \mathcal{D}$ . However, this DGP does imply that Assumption 3.2 holds, i.e., conditional on the Fourier coefficients  $\mathcal{A}$ , the potential outcomes are independent of the treatment assignments  $\mathcal{D}$ . This conditional independence condition can be thought of as "selection on latent Fourier coefficients", which generalizes the widely made assumption of "selection on observables." The latter requires that potential outcomes are independent of treatment assignments conditional on observed covariates—we reemphasize that  $\mathcal{A}$  is unobserved.
Target parameter. For any unit-combination pair  $(n,\pi)$ , we aim to estimate  $\mathbb{E}[Y_n^{(\pi)}|\mathcal{A}]$ , where the expectation is w.r.t.  $\epsilon_{n}^{\pi}$ , and we condition on the set of Fourier coefficients  $\mathcal{A}$ .
4 Identification of Potential Outcomes
We show that  $\mathbb{E}[Y_n^{(\pi)}\mid \mathcal{A}]$  can be written as a function of observed outcomes, i.e., we establish identification of our target causal parameter. As discussed earlier, our model allows for unobserved confounding: whether or not a unit is seen under a combination may be correlated with its potential outcome under that combination due to unobserved factors, as long as certain conditions are met. We introduce necessary notation and assumption required for our result. For a unit  $n\in [N]$ , denote the subset of combinations we observe them under as  $\Pi_{n}\subseteq \Pi$ . For  $\pi \in \Pi$ , let  $\tilde{\chi}_n^\pi \in \mathbb{R}^{2^p}$  denote the vector where we zero out all coordinates of  $\chi^{\pi}\in \mathbb{R}^{2^{p}}$  that correspond to the coefficients of  $\alpha_{n}$  which are zero. For example, if  $\alpha_{n} = (1,1,0,0,\dots 0)$  and  $\chi^{\pi} = (1,1,\dots 1)$ , then  $\tilde{\chi}_n^\pi = (1,1,0,\dots 0)$ . We the make the following assumption.
Assumption 4.1 (Donor Units). We assume there exists a set of "donor units"  $\mathcal{I} \subset [N]$ , such that the following two conditions hold:
(a) Horizontal span inclusion: For any donor unit  $u \in \mathcal{I}$  and combination  $\pi \in \Pi$ , suppose  $\tilde{\chi}_u^\pi \in \text{span}(\tilde{\chi}_u^{\pi_i} : \pi_i \in \Pi_u)$ . That is, there exists exists  $\beta_{\Pi_u}^\pi \in \mathbb{R}^{|\Pi_u|}$  such that  $\tilde{\chi}_u^\pi = \sum_{\pi_i \in \Pi_u} \beta_{\pi_i}^\pi \tilde{\chi}_u^{\pi_i}$ .
(b) Linear span inclusion: For any unit  $n \in [N] \backslash \mathcal{I}$ , suppose  $\alpha_{n} \in \text{span}(\alpha_{u}: u \in \mathcal{I})$ . That is, there exists  $\mathbf{w}^{n}$  such that  $\alpha_{n} = \sum_{u \in \mathcal{I}} w_{u}^{n} \alpha_{u}$
Horizontal span inclusion requires that the set of observed combinations for any donor unit is "diverse" enough that the projection of the Fourier characteristic for a target intervention is in the span of characteristics of observed interventions. Linear span inclusion requires that the donor set is diverse enough such that the Fourier coefficient of any unit is in the span of the Fourier coefficients of the donor set.
4.1 Identification Result
Given these assumptions, we now present our identification theorem.
Theorem 4.2. Let Assumptions 3.1, 3.2, 4.1 hold. Given  $\beta_{\Pi_u}^{\pi}$  and  $\mathbf{w}_u^n$  defined in Assumption 4.1, we have
(a) Donor units: For  $u \in \mathcal{I}$ , and  $\pi \in \Pi$ ,  $\mathbb{E}[Y_u^{(\pi)} \mid \mathcal{A}] = \sum_{\pi_u \in \Pi_u} \beta_{\pi_u}^{\pi} \mathbb{E}[Y_{u,\pi_u} \mid \mathcal{A}, \mathcal{D}]$ .
(b) Non-donor units: For  $n \in [N] \setminus \mathcal{I}$ , and  $\pi \notin \Pi_n$ ,  $\mathbb{E}[Y_n^{(\pi)} \mid \mathcal{A}] = \sum_{u \in \mathcal{I}, \pi_u \in \Pi_u} w_u^n \beta_{\pi_u}^\pi \mathbb{E}[Y_{u,\pi_u} \mid \mathcal{A}, \mathcal{D}]$ .
Theorem 4.2 gives conditions under which the donor set  $\mathcal{I}$  and the observation pattern  $\mathcal{D}$  are sufficient to recover the full set of unit specific potential outcomes  $\mathbb{E}[Y_n^{(\pi)}|\mathcal{A}]$  in the noise-free limit. Part (a) establishes that for every donor unit  $u\in \mathcal{I}$ , the causal estimand can be written as a function of its own observed outcomes  $\mathbb{E}[\mathbf{Y}_{\Pi_u}]$ , given knowledge of  $\beta_{\Pi_u}^{\pi}$ . Part (b) states that the target causal estimand  $\mathbb{E}[Y_n^{(\pi)}]$  for a non-donor unit and combination  $\pi$  can be written as a linear combination of the outcomes of the donor set  $\mathcal{I}$ , given knowledge of  $\mathbf{w}_n^n$ . Previous works that establish identification under a latent factor model require a growing number of donor units to be observed under all treatments [6]. This is infeasible in our setting because the vast majority of combinations have no units that receive it. As a result, we have to first to identify the outcomes of donor units under all combinations (part (a)), before transferring them to non-donor units (part (b)). In order to do so, Theorem 4.2 suggests that the key quantities in estimating  $\mathbb{E}[Y_n^{(\pi)}]$  for any unit-combination pair  $(n,\pi)$  are  $\beta_{\Pi_u}^{\pi}$  and  $\mathbf{w}_u^n$ . In the following section, we provide an algorithm to estimate both  $\beta_{\Pi_u}^{\pi}$  and  $\mathbf{w}_u^n$ , as well as concrete ways of determining the donor set  $\mathcal{I}$ .
5 The Synthetic Combinations Estimator
We now describe the Synthetic Combinations estimator, a simple and flexible two-step procedure for estimating our target causal parameter. See Figure 1 for a pictorial representation.
Step 1: Horizontal Regression. We denote the vector of observed responses  $\mathbf{Y}_{n,\Pi_n} = [Y_{n\pi}:\pi \in \Pi_n]\in \mathbf{R}^{|\Pi_n|}$  for any unit  $n$  as  $\mathbf{Y}_{\Pi_n}$ . Then, for every unit  $u$  in the donor set  $\mathcal{I}$ , we estimate  $\mathbb{E}[Y_u^{(\pi)}]$
via the Lasso, i.e., by solving the following convex program with penalty parameter  $\lambda_{u}$ :