Discovering subgroups with the maximum average treatment effect is crucial for targeted decision making in domains such as precision medicine, public policy, and education. While most prior work is formulated in the potential‑outcome framework, the corresponding structural causal model (SCM) for this task has been largely overlooked. In practice, two approaches dominate. The first estimates pointwise conditional treatment effects and then fits a tree on those estimates, effectively turning subgroup estimation into the harder problem of accurate pointwise estimation. The second constructs decision trees or rule sets with ad‑hoc 'causal' heuristics, typically without rigorous justification for why a given heuristic may be used or whether such heuristics are necessary at all. We address these issues by studying the problem directly under the SCM framework. Under the assumption of a partition-based model, we show that optimal subgroup discovery reduces to recovering the data-generating models and hence a standard supervised learning problem (regression or classification). This allows us to adopt any partition-based methods to learn the subgroup from data. We instantiate the approach with CART, arguably one of the most widely used tree-based method, to learn the subgroup with maximum treatment effect. Finally, on a large collection of synthetic and semi‑synthetic datasets, we compare our method against a wide range of baselines and find that our approach, which avoids such causal heuristics, more accurately identifies subgroups with maximum treatment effect.

Hierarchical reinforcement learning (HRL) improves the efficiency of long-horizon reinforcement-learning tasks with sparse rewards by decomposing the task into a hierarchy of subgoals. The main challenge of HRL is efficient discovery of the hierarchical structure among subgoals and utilizing this structure to achieve the final goal. We address this challenge by modeling the subgoal structure as a causal graph and propose a causal discovery algorithm to learn it. Additionally, rather than intervening on the subgoals at random during exploration, we harness the discovered causal model to prioritize subgoal interventions based on their importance in attaining the final goal. These targeted interventions result in a significantly more efficient policy in terms of the training cost. Unlike previous work on causal HRL, which lacked theoretical analysis, we provide a formal analysis of the problem. Specifically, for tree structures and, for a variant of Erdős-Rényi random graphs, our approach results in remarkable improvements. Our experimental results on HRL tasks also illustrate that our proposed framework outperforms existing work in terms of training cost.

We address the challenge of optimizing meta-parameters (hyperparameters) in machine learning, a key factor for efficient training and high model performance. Rather than relying on expensive meta-parameter search methods, we introduce MetaOptimize: a dynamic approach that adjusts meta-parameters, particularly step sizes (also known as learning rates), during training. More specifically, MetaOptimize can wrap around any first-order optimization algorithm, tuning step sizes on the fly to minimize a specific form of regret that considers the long-term impact of step sizes on training, through a discounted sum of future losses. We also introduce lower-complexity variants of MetaOptimize that, in conjunction with its adaptability to various optimization algorithms, achieve performance comparable to those of the best hand-crafted learning rate schedules across diverse machine learning tasks.

This paper investigates causal effect identification in latent variable Linear Non-Gaussian Acyclic Models (lvLiNGAM) using higher-order cumulants, addressing two prominent setups that are challenging in the presence of latent confounding: (1) a single proxy variable that may causally influence the treatment and (2) underspecified instrumental variable cases where fewer instruments exist than treatments. We prove that causal effects are identifiable with a single proxy or instrument and provide corresponding estimation methods. Experimental results demonstrate the accuracy and robustness of our approaches compared to existing methods, advancing the theoretical and practical understanding of causal inference in linear systems with latent confounders.

We study the problem of causal structure learning from a combination of observational and interventional data generated by a linear non-Gaussian structural equation model that might contain cycles. Recent results show that using mere observational data identifies the causal graph only up to a permutation-equivalence class. We obtain a combinatorial characterization of this class by showing that each equivalence class corresponds to a perfect matching in a bipartite graph. This bipartite representation allows us to analyze how interventions modify or constrain the matchings. Specifically, we show that each atomic intervention reveals one edge of the true matching and eliminates all incompatible causal graphs. Consequently, we formalize the optimal experiment design task as an adaptive stochastic optimization problem over the set of equivalence classes with a natural reward function that quantifies how many graphs are eliminated from the equivalence class by an intervention. We show that this reward function is adaptive submodular and provide a greedy policy with a provable near-optimal performance guarantee. A key technical challenge is to efficiently estimate the reward function without having to explicitly enumerate all the graphs in the equivalence class. We propose a sampling-based estimator using random matchings and analyze its bias and concentration behavior. Our simulation results show that performing a small number of interventions guided by our stochastic optimization framework recovers the true underlying causal structure.

Policy gradient (PG) is widely used in reinforcement learning due to its scalability and good performance. In recent years, several variance-reduced PG methods have been proposed with a theoretical guarantee of converging to an approximate first-order stationary point (FOSP) with the sample complexity of $O(\epsilon^{-3})$. However, FOSPs could be bad local optima or saddle points. Moreover, these algorithms often use importance sampling (IS) weights which could impair the statistical effectiveness of variance reduction. In this paper, we propose a variance-reduced second-order method that uses second-order information in the form of Hessian vector products (HVP) and converges to an approximate second-order stationary point (SOSP) with sample complexity of $\tilde{O}(\epsilon^{-3})$. This rate improves the best-known sample complexity for achieving approximate SOSPs by a factor of $O(\epsilon^{-0.5})$. Moreover, the proposed variance reduction technique bypasses IS weights by using HVP terms. Our experimental results show that the proposed algorithm outperforms the state of the art and is more robust to changes in random seeds.

We investigate the estimation of the causal effect of a treatment variable on an outcome in the presence of a latent confounder. We first show that the causal effect is identifiable under certain conditions when data is available from multiple environments, provided that the target causal effect remains invariant across these environments. Secondly, we propose a moment-based algorithm for estimating the causal effect as long as only a single parameter of the data-generating mechanism varies across environments – whether it be the exogenous noise distribution or the causal relationship between two variables. Conversely, we prove that identifiability is lost if both exogenous noise distributions of both the latent and treatment variables vary across environments. Finally, we propose a procedure to identify which parameter of the data-generating mechanism has varied across the environments and evaluate the performance of our proposed methods through experiments on synthetic data.

We study the problem of identifying the unknown intervention targets in structural causal models where we have access to heterogeneous data collected from multiple environments. The unknown intervention targets are the set of endogenous variables whose corresponding exogenous noises change across the environments. We propose a two-phase approach which in the first phase recovers the exogenous noises corresponding to unknown intervention targets whose distributions have changed across environments. In the second phase, the recovered noises are matched with the corresponding endogenous variables. For the recovery phase, we provide sufficient conditions for learning these exogenous noises up to some component-wise invertible transformation. For the matching phase, under the causal sufficiency assumption, we show that the proposed method uniquely identifies the intervention targets. In the presence of latent confounders, the intervention targets among the observed variables cannot be determined uniquely. We provide a candidate intervention target set which is a superset of the true intervention targets. Our approach improves upon the state of the art as the returned candidate set is always a subset of the target set returned by previous work. Moreover, we do not require restrictive assumptions such as linearity of the causal model or performing invariance tests to learn whether a distribution is changing across environments which could be highly sample inefficient. Our experimental results show the effectiveness of our proposed algorithm in practice.

We study the generic identifiability of causal effects in linear non-Gaussian acyclic models (LiNGAM) with latent variables. We consider the problem in two main settings: When the causal graph is known a priori, and when it is unknown. In both settings, we provide a complete graphical characterization of the identifiable direct or total causal effects among observed variables. Moreover, we propose efficient algorithms to certify the graphical conditions. Finally, we propose an adaptation of the reconstruction independent component analysis (RICA) algorithm that estimates the causal effects from the observational data given the causal graph. Experimental results show the effectiveness of the proposed method in estimating the causal effects.

We consider the problem of federated learning in a one-shot setting in which there are$m$machines, each observing$n$sample functions from an unknown distribution on non-convex loss functions. Let$F:[-1,1]^{d}\to {\mathbb {R}} $be the expected loss function with respect to this unknown distribution. The goal is to find an estimate of the minimizer of$F$. Based on its observations, each machine generates a signal of bounded length$B$and sends it to a server. The server collects signals of all machines and outputs an estimate of the minimizer of$F$. We show that the expected loss of any algorithm is lower bounded by$\max \big (1/(\sqrt {n}(mB)^{1/d}), 1/\sqrt {mn}\big)$, up to a logarithmic factor. We then prove that this lower bound is order optimal in$m$and$n$by presenting a distributed learning algorithm, called Multi-Resolution Estimator for Non-Convex loss function (MRE-NC), whose expected loss matches the lower bound for large$mn$up to polylogarithmic factors.

We consider the problem of estimating the causal effect of a treatment on an outcome in linear structural causal models (SCM) with latent confounders when we have access to a single proxy variable. Several methods (such as difference-in-difference (DiD) estimator or negative outcome control) have been proposed in this setting in the literature. However, these approaches require either restrictive assumptions on the data generating model or having access to at least two proxy variables. We propose a method to estimate the causal effect using cross moments between the treatment, the outcome, and the proxy variable. In particular, we show that the causal effect can be identified with simple arithmetic operations on the cross moments if the latent confounder in linear SCM is non-Gaussian. In this setting, DiD estimator provides an unbiased estimate only in the special case where the latent confounder has exactly the same direct causal effects on the outcomes in the pre-treatment and post-treatment phases. This translates to the common trend assumption in DiD, which we effectively relax. Additionally, we provide an impossibility result that shows the causal effect cannot be identified if the observational distribution over the treatment, the outcome, and the proxy is jointly Gaussian. Our experiments on both synthetic and real-world datasets showcase the effectiveness of the proposed approach in estimating the causal effect.

We study experiment design for unique identification of the causal graph of a simple SCM, where the graph may contain cycles. The presence of cycles in the structure introduces major challenges for experiment design as, unlike acyclic graphs, learning the skeleton of causal graphs with cycles may not be possible from merely the observational distribution. Furthermore, intervening on a variable in such graphs does not necessarily lead to orienting all the edges incident to it. In this paper, we propose an experiment design approach that can learn both cyclic and acyclic graphs and hence, unifies the task of experiment design for both types of graphs. We provide a lower bound on the number of experiments required to guarantee the unique identification of the causal graph in the worst case, showing that the proposed approach is order-optimal in terms of the number of experiments up to an additive logarithmic term. Moreover, we extend our result to the setting where the size of each experiment is bounded by a constant. For this case, we show that our approach is optimal in terms of the size of the largest experiment required for uniquely identifying the causal graph in the worst case.

We study the performance of Stochastic Cubic Regularized Newton (SCRN) on a class of functions satisfying gradient dominance property with $1\le\alpha\le2$ which holds in a wide range of applications in machine learning and signal processing. This condition ensures that any first-order stationary point is a global optimum. We prove that the total sample complexity of SCRN in achieving $\epsilon$-global optimum is $\mathcal{O}(\epsilon^{-7/(2\alpha)+1})$ for $1\le\alpha< 3/2$ and $\mathcal{\tilde{O}}(\epsilon^{-2/(\alpha)})$ for $3/2\le\alpha\le 2$. SCRN improves the best-known sample complexity of stochastic gradient descent. Even under a weak version of gradient dominance property, which is applicable to policy-based reinforcement learning (RL), SCRN achieves the same improvement over stochastic policy gradient methods. Additionally, we show that the average sample complexity of SCRN can be reduced to ${\mathcal{O}}(\epsilon^{-2})$ for $\alpha=1$ using a variance reduction method with time-varying batch sizes. Experimental results in various RL settings showcase the remarkable performance of SCRN compared to first-order methods.

We consider distributed statistical optimization in one-shot setting, where there are $m$ machines each observing $n$ i.i.d. samples. Based on its observed samples, each machine sends a $B$-bit-long message to a server. The server then collects messages from all machines, and estimates a parameter that minimizes an expected convex loss function. We investigate the impact of communication constraint, $B$, on the expected error and derive a tight lower bound on the error achievable by any algorithm. We then propose an estimator, which we call Multi-Resolution Estimator (MRE), whose expected error (when $B\ge d\log mn$ where $d$ is the dimension of parameter) meets the aforementioned lower bound up to a poly-logarithmic factor in $mn$. The expected error of MRE, unlike existing algorithms, tends to zero as the number of machines ($m$) goes to infinity, even when the number of samples per machine ($n$) remains upper bounded by a constant. We also address the problem of learning under tiny communication budget, and present lower and upper error bounds for the case that the budget $B$ is a constant.

Given a social network modeled as a weighted graph GG, the influence maximization problem seeks kk vertices to become initially influenced, to maximize the expected number of influenced nodes under a particular diffusion model. The influence maximization problem has been proven to be NP-hard, and most proposed solutions to the problem are approximate greedy algorithms, which can guarantee a tunable approximation ratio for their results with respect to the optimal solution. The state-of-the-art algorithms are based on Reverse Influence Sampling (RIS) technique, which can offer both computational efficiency and non-trivial (1-1/e-ε)(1-1/e-ε)-approximation ratio guarantee for any ε > 0ε>0. RIS-based algorithms, despite their lower computational cost compared to other methods, still require long running times to solve the problem in large-scale graphs with low values of ε. In this article, we present a novel and efficient parallel implementation of a RIS-based algorithm, namely IMM, on GPU. The proposed GPU-accelerated influence maximization algorithm, named gIM, can significantly reduce the running time on large-scale graphs with low values of ε. Furthermore, we show that gIM algorithm can solve other variations of the IM problem, only by applying minor modifications. Experimental results show that the proposed solution reduces the runtime by a factor up to 220 ×. The source code of gIM is publicly available online.

The causal relationships among a set of random variables are commonly represented by a Directed Acyclic Graph (DAG), where there is a directed edge from variable $X$ to variable $Y$ if $X$ is a direct cause of $Y$. From the purely observational data, the true causal graph can be identified up to a Markov Equivalence Class (MEC), which is a set of DAGs with the same conditional independencies between the variables. The size of an MEC is a measure of complexity for recovering the true causal graph by performing interventions. We propose a method for efficient iteration over possible MECs given intervention results. We utilize the proposed method for computing MEC sizes and experiment design in active and passive learning settings. Compared to previous work for computing the size of MEC, our proposed algorithm reduces the time complexity by a factor of $O(n)$ for sparse graphs where $n$ is the number of variables in the system. Additionally, integrating our approach with dynamic programming, we design an optimal algorithm for passive experiment design. Experimental results show that our proposed algorithms for both computing the size of MEC and experiment design outperform the state of the art.

The authors consider the problem of multi‐choice majority voting in a network of n agents where each agent initially selects a choice from a set of K possible choices. The agents try to infer the choice in the majority merely by performing local interactions. Extending the popular ‘Population Protocol’ framework for pairwise interactions between agents, in this study, they propose a new model called ‘Broadcasting Population Protocol’. In the proposed model, each agent broadcasts its messages in such a way that all its neighbours will receive the message simultaneously. They design two distributed algorithms for solving the multi‐choice majority voting problem in this new model. They show that these algorithms return the correct output, i.e. the choice in the majority. They also analyse their performances in terms of time and message complexities. They establish via simulations that the proposed algorithms improve both time and message complexities significantly with respect to previous algorithms proposed in conventional population protocols, and they can be utilised in networks where messages can be transmitted to a subset of agents simultaneously, such as wireless networks.

We consider the problem of learning causal models from observational data generated by linear non-Gaussian acyclic causal models with latent variables. Without considering the effect of latent variables, the inferred causal relationships among the observed variables are often wrong. Under faithfulness assumption, we propose a method to check whether there exists a causal path between any two observed variables. From this information, we can obtain the causal order among the observed variables. The next question is whether the causal effects can be uniquely identified as well. We show that causal effects among observed variables cannot be identified uniquely under mere assumptions of faithfulness and non-Gaussianity of exogenous noises. However, we are able to propose an efficient method that identifies the set of all possible causal effects that are compatible with the observational data. We present additional structural conditions on the causal graph under which causal effects among observed variables can be determined uniquely. Furthermore, we provide necessary and sufficient graphical conditions for unique identification of the number of variables in the system. Experiments on synthetic data and real-world data show the effectiveness of our proposed algorithm for learning causal models.

The main goal in many fields in the empirical sciences is to discover causal relationships among a set of variables from observational data. PC algorithm is one of the promising solutions to learn underlying causal structure by performing a number of conditional independence tests. In this paper, we propose a novel GPU-based parallel algorithm, called cuPC, to execute an order-independent version of PC. The proposed solution has two variants, cuPC-E and cuPC-S, which parallelize PC in two different ways for multivariate normal distribution. Experimental results show the scalability of the proposed algorithms with respect to the number of variables, the number of samples, and different graph densities. For instance, in one of the most challenging datasets, the runtime is reduced from more than 11 hours to about 4 seconds. On average, cuPC-E and cuPC-S achieve 500X and 1300X speedup, respectively, compared to serial implementation on CPU.

A directed acyclic graph (DAG) is the most common graphical model for representing causal relationships among a set of variables. When restricted to using only observational data, the structure of the ground truth DAG is identifiable only up to Markov equivalence, based on conditional independence relations among the variables. Therefore, the number of DAGs equivalent to the ground truth DAG is an indicator of the causal complexity of the underlying structure–roughly speaking, it shows how many interventions or how much additional information is further needed to recover the underlying DAG. In this paper, we propose a new technique for counting the number of DAGs in a Markov equivalence class. Our approach is based on the clique tree representation of chordal graphs. We show that in the case of bounded degree graphs, the proposed algorithm is polynomial time. We further demonstrate that this technique can be utilized for uniform sampling from a Markov equivalence class, which provides a stochastic way to enumerate DAGs in the equivalence class and may be needed for finding the best DAG or for causal inference given the equivalence class as input. We also extend our counting and sampling method to the case where prior knowledge about the underlying DAG is available, and present applications of this extension in causal experiment design and estimating the causal effect of joint interventions.

We consider distributed statistical optimization in one-shot setting, where there are $m$ machines each observing $n$ i.i.d samples. Based on its observed samples, each machine then sends an $O(\log(mn))$-length message to a server, at which a parameter minimizing an expected loss is to be estimated. We propose an algorithm called Multi-Resolution Estimator (MRE) whose expected error is no larger than $\tilde{O}( m^{-1/\max(d,2)} n^{-1/2})$, where $d$ is the dimension of the parameter space. This error bound meets existing lower bounds up to poly-logarithmic factors, and is thereby order optimal. The expected error of MRE, unlike existing algorithms, tends to zero as the number of machines ($m$) goes to infinity, even when the number of samples per machine ($n$) remains upper bounded by a constant. This property of the MRE algorithm makes it applicable in new machine learning paradigms where $m$ is much larger than $n$.

We study the problem of causal structure learning when the experimenter is limited to perform at most $k$ non-adaptive experiments of size $1$. We formulate the problem of finding the best intervention target set as an optimization problem, which aims to maximize the average number of edges whose directions are resolved. We prove that the corresponding objective function is submodular and a greedy algorithm suffices to achieve $(1-\frac{1}{e})$-approximation of the optimal value. We further present an accelerated variant of the greedy algorithm, which can lead to orders of magnitude performance speedup. We validate our proposed approach on synthetic and real graphs. The results show that compared to the purely observational setting, our algorithm orients the majority of the edges through a considerably small number of interventions.

We propose an approach for learning the causal structure in stochastic dynamical systems with a 1-step functional dependency in the presence of latent variables. We propose an information-theoretic approach that allows us to recover the causal relations among the observed variables as long as the latent variables evolve without exogenous noise. We further propose an efficient learning method based on linear regression for the special sub-case when the dynamics are restricted to be linear. We validate the performance of our approach via numerical simulations.

We study causal discovery in a multi-environment setting, in which the functional relations for producing the variables from their direct causes remain the same across environments, while the distribution of exogenous noises may vary. We introduce the idea of using the invariance of the functional relations of the variables to their causes across a set of environments for structure learning. We define a notion of completeness for a causal inference algorithm in this setting and prove the existence of such algorithm by proposing the baseline algorithm. Additionally, we present an alternate algorithm that has significantly improved computational and sample complexity compared to the baseline algorithm. Experiment results show that the proposed algorithm outperforms the other existing algorithms.

In the binary majority voting problem, each node initially chooses between two alternative choices. The goal is to design a distributed algorithm that informs nodes which choice is in majority. In this study, the authors formulate this problem as a hypothesis testing problem and propose fixed‐size and sequential solutions using classical and Bayesian approaches. In the sequential version, the proposed mechanism enables nodes to test which choice is in majority, successively in time. Hence, termination of the algorithm is embedded within it, contrary to the existing approaches which require a monitoring algorithm to indicate the termination. This property makes the algorithm more efficient in terms of message complexity. Furthermore, the authors show that the proposed solution is resilient to Byzantine attacks if network connectivity is F + 1 in the presence of F adversarial nodes. Thus, the proposed algorithm is more robust compared with the previous works which are vulnerable to the existence of adversarial nodes.

In distributed function computation, each node has an initial value and the goal is to compute a function of these values in a distributed manner. In this paper, we propose a novel token-based approach to compute a wide class of target functions to which we refer as “token-based function computation with memory” (TCM) algorithm. In this approach, node values are attached to tokens and travel across the network. Each pair of travelling tokens would coalesce when they meet, forming a token with a new value as a function of the original token values. In contrast to the coalescing random walk (CRW) algorithm, where token movement is governed by random walk, meeting of tokens in our scheme is accelerated by adopting a novel chasing mechanism. We proved that, compared to the CRW algorithm, the TCM algorithm results in a reduction of time complexity by a factor of at least √(n/log(n) in Erdos-Renyi and complete graphs, and by a factor of log (n)/log(log(n)) in torus networks. Simulation results show that there is at least a constant factor improvement in the message complexity of TCM algorithm in all considered topologies. Robustness of the CRW and TCM algorithms in the presence of node failure is analyzed. We show that their robustness can be improved by running multiple instances of the algorithms in parallel.

Averaging consensus on the values of nodes in a network is a principal problem in distributed computation. In the presence of erasure channels, conventional averaging consensus algorithms may not converge to the average value if packets are erased in arbitrary order. In this paper, we propose a “Pseudo-Synchronous Averaging Consensus” (PSAC) algorithm to guarantee averaging consensus over erasure channels by employing tagged packets. We show that the PSAC algorithm has a simple structure and it can work with just two tags “0” and “1”. In asynchronous networks, the PSAC algorithm is a synchronizer in the sense that it keeps the updates of various nodes in step with each other. By exploiting the broadcast nature of wireless links in complete graphs, the PSAC algorithm obtains the exact average value with minimum number of transmissions, in the asynchronous setting.

One of the most challenging problems in Opportunistic Spectrum Access (OSA) is to design channel sensing-based protocol in multi secondary users (SUs) network. Quality of Service (QoS) requirements for SUs have significant implications on this protocol design. In this paper, we propose a new method to find joint policies for SUs which not only tries to guarantee QoS requirements but also maximize network throughput. We use Decentralized Partially Observable Markov Decision Process (Dec-POMDP) to formulate interactions between SUs. Meanwhile, a tractable approach for Dec-POMDP is utilized to extract sub-optimum joint policies for large horizons. Among these policies, the QoS-aware joint policy is selected as the joint sensing strategy for SUs. To show the efficiency of the proposed method, we consider two SUs trying to access two-channel primary users (PUs) network modeled by discrete Markov chains. Simulations demonstrate two interesting findings: 1- Optimum joint policies for large horizons can be obtained using the proposed method. 2- Our method outperforms other related works in terms of network throughput.