Saber Salehkaleybar

Saeed Masiha, Saber Salehkaleybar, Niao He et al.

We study the performance of Stochastic Cubic Regularized Newton (SCRN) on a class of functions satisfying gradient dominance property with $1\leα\le2$ 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 $ε$-global optimum is $\mathcal{O}(ε^{-7/(2α)+1})$ for $1\leα< 3/2$ and $\mathcal{\tilde{O}}(ε^{-2/(α)})$ for $3/2\leα\le 2$. 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 ${\mathcal{O}}(ε^{-2})$ for $α=1$ 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.

Saber Salehkaleybar, Sadegh Khorasani, Negar Kiyavash et al.

Variance-reduced gradient estimators for policy gradient methods have been one of the main focus of research in the reinforcement learning in recent years as they allow acceleration of the estimation process. We propose a variance-reduced policy-gradient method, called SHARP, which incorporates second-order information into stochastic gradient descent (SGD) using momentum with a time-varying learning rate. SHARP algorithm is parameter-free, achieving $ε$-approximate first-order stationary point with $O(ε^{-3})$ number of trajectories, while using a batch size of $O(1)$ at each iteration. Unlike most previous work, our proposed algorithm does not require importance sampling which can compromise the advantage of variance reduction process. Moreover, the variance of estimation error decays with the fast rate of $O(1/t^{2/3})$ where $t$ is the number of iterations. Our extensive experimental evaluations show the effectiveness of the proposed algorithm on various control tasks and its advantage over the state of the art in practice.

Saeed Masiha, Saber Salehkaleybar, Niao He et al.

We study the local convergence rate of stochastic first-order methods under a local $α$-Polyak-Lojasiewicz ($α$-PL) condition in a neighborhood of a target connected component $\mathcal{M}$ of the local minimizer set. The parameter $α\in [1,2]$ is the exponent of the gradient norm in the $α$-PL inequality: $α=2$ recovers the classical PL case, $α=1$ corresponds to Holder-type error bounds, and intermediate values interpolate between these regimes. Our performance criterion is the number of oracle queries required to output $\hat{x}$ with $F(\hat{x})-l \le \varepsilon$, where $l := F(y)$ for any $y \in \mathcal{M}$. We work in a local regime where the algorithm is initialized near $\mathcal{M}$ and, with high probability, its iterates remain in that neighborhood. We establish a lower bound $Ω(\varepsilon^{-2/α})$ for all stochastic first-order methods in this regime, and we obtain a matching upper bound $\mathcal{O}(\varepsilon^{-2/α})$ for $1 \le α< 2$ via a SARAH-type variance-reduced method with time-varying batch sizes and step sizes. In the convex setting, assuming a local $α$-PL condition on the $\varepsilon$-sublevel set, we further show a complexity lower bound $\widetildeΩ(\varepsilon^{-2/α})$ for reaching an $\varepsilon$-global optimum, matching the $\varepsilon$-dependence of known accelerated stochastic subgradient methods.

Ehsan Mokhtarian, Saber Salehkaleybar, AmirEmad Ghassami et al.

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.

Sadegh Khorasani, Saber Salehkaleybar, Negar Kiyavash et al.

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 $O(ε^{-3})$. 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 $\tilde{O}(ε^{-3})$. This rate improves the best-known sample complexity for achieving approximate SOSPs by a factor of $O(ε^{-0.5})$. 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.

Ramin Safaeian, Saber Salehkaleybar, Mahmoud Tabandeh

Causal relationships among a set of variables are commonly represented by a directed acyclic graph. The orientations of some edges in the causal DAG can be discovered from observational/interventional data. Further edges can be oriented by iteratively applying so-called Meek rules. Inferring edges' orientations from some previously oriented edges, which we call Causal Orientation Learning (COL), is a common problem in various causal discovery tasks. In these tasks, it is often required to solve multiple COL problems and therefore applying Meek rules could be time-consuming. Motivated by Meek rules, we introduce Meek functions that can be utilized in solving COL problems. In particular, we show that these functions have some desirable properties, enabling us to speed up the process of applying Meek rules. In particular, we propose a dynamic programming (DP) based method to apply Meek functions. Moreover, based on the proposed DP method, we present a lower bound on the number of edges that can be oriented as a result of intervention. We also propose a method to check whether some oriented edges belong to a causal DAG. Experimental results show that the proposed methods can outperform previous work in several causal discovery tasks in terms of running-time.

Lincen Yang, Zhong Li, Matthijs van Leeuwen et al.

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 methods, 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. Our source code is available at https://github.com/ylincen/causal-subgroup.

Mohammad Reza Samsami, Mohammadhossein Bahari, Saber Salehkaleybar et al.

Imitation learning is a powerful approach for learning autonomous driving policy by leveraging data from expert driver demonstrations. However, driving policies trained via imitation learning that neglect the causal structure of expert demonstrations yield two undesirable behaviors: inertia and collision. In this paper, we propose Causal Imitative Model (CIM) to address inertia and collision problems. CIM explicitly discovers the causal model and utilizes it to train the policy. Specifically, CIM disentangles the input to a set of latent variables, selects the causal variables, and determines the next position by leveraging the selected variables. Our experiments show that our method outperforms previous work in terms of inertia and collision rates. Moreover, thanks to exploiting the causal structure, CIM shrinks the input dimension to only two, hence, can adapt to new environments in a few-shot setting. Code is available at https://github.com/vita-epfl/CIM.

Sadegh Khorasani, Saber Salehkaleybar, Negar Kiyavash et al.

Causal probing methods aim to test and control how internal representations influence the behavior of generative models. In causal probing, an intervention modifies hidden states so that a property takes on a different value. Most existing approaches define such interventions by training an auxiliary probe classifier, which ties the method to a specific task or model and risks misalignment with the model's predictive geometry. We propose Hidden-state Driven Margin Intervention (HDMI), a probe-free, gradient-based technique that directly steers hidden states using the model's native output. HDMI applies a margin objective that increases the probability of a target continuation while decreasing that of the source, without relying on probe classifiers. We further introduce a lookahead variant (LA-HDMI) for text editing that backpropagates through the softmax embeddings, modifying the current hidden state so that the likelihood of user-specified tokens increases in next token generations while preserving fluency. To evaluate interventions, we measure completeness (whether the targeted property changes as intended) and selectivity (whether unrelated properties are preserved), and report their harmonic mean as an overall measure of reliability. HDMI consistently achieves higher reliability than prior methods on the LGD agreement corpus and the CausalGym benchmark, across Meta-Llama-3-8B-Instruct, and Pythia-70M.

Ana Leticia Garcez Vicente, Gijs van Seeventer, Saber Salehkaleybar

Estimating causal effects from observational data requires identifying valid adjustment sets. This task is especially challenging in realistic settings where latent confounding and feedback loops are present. Existing approaches typically assume acyclicity or rely on global causal structure learning, limiting applicability and computational efficiency. In this work, we study a local, data-driven method for covariate selection based on conditional independence information. While this method is known to be sound and complete in acyclic causal models, its validity in the presence of cycles has remained unclear. Our main contribution is to show that these guarantees extend to cyclic causal models. In particular, our result relies on the invariance of conditional independence assertions under $σ$-acyclification. These findings establish a unified, cycle-agnostic perspective on covariate selection and causal effect estimation, showing that the method applies across cyclic and acyclic settings without modification. Empirically, we validate this on extensive synthetic data, showing reliable performance in cyclic causal models.

Arsalan Sharifnassab, Saber Salehkaleybar, Sina Ghiassian et al.

We propose Soft Preference Optimization (SPO), a method for aligning generative models, such as Large Language Models (LLMs), with human preferences, without the need for a reward model. SPO optimizes model outputs directly over a preference dataset through a natural loss function that integrates preference loss with a regularization term across the model's entire output distribution rather than limiting it to the preference dataset. Although SPO does not require the assumption of an existing underlying reward model, we demonstrate that, under the Bradley-Terry (BT) model assumption, it converges to a softmax of scaled rewards, with the distribution's "softness" adjustable via the softmax exponent, an algorithm parameter. We showcase SPO's methodology, its theoretical foundation, and its comparative advantages in simplicity, computational efficiency, and alignment precision.

Arsalan Sharifnassab, Saber Salehkaleybar, Richard Sutton

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.

Yuqin Yang, Saber Salehkaleybar, Negar Kiyavash

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.

Andreas Sauter, Saber Salehkaleybar, Aske Plaat et al.

Predicting the distribution of outcomes under hypothetical interventions is crucial across healthcare, economics, and policy-making. However, existing methods often require restrictive assumptions, and are typically limited by the lack of amortization across problem instances. We propose ACTIVA, a transformer-based conditional variational autoencoder (VAE) architecture for amortized causal inference, which estimates interventional distributions directly from observational data without. ACTIVA learns a latent representation conditioned on observational inputs and intervention queries, enabling zero-shot inference by amortizing causal knowledge from diverse training scenarios. We provide theoretical insights showing that ACTIVA predicts interventional distributions as mixtures over observationally equivalent causal models. Empirical evaluations on synthetic and semi-synthetic datasets confirm the effectiveness of our amortized approach and highlight promising directions for future real-world applications.

Kasra Jalaldoust, Saber Salehkaleybar, Negar Kiyavash

We consider the problem of causal discovery (a.k.a., causal structure learning) in a multi-domain setting. We assume that the causal functions are invariant across the domains, while the distribution of the exogenous noise may vary. Under causal sufficiency (i.e., no confounders exist), we show that the causal diagram can be discovered under less restrictive functional assumptions compared to previous work. What enables causal discovery in this setting is bijective generation mechanisms (BGM), which ensures that the functional relation between the exogenous noise $E$ and the endogenous variable $Y$ is bijective and differentiable in both directions at every level of the cause variable $X = x$. BGM generalizes a variety of models including additive noise model, LiNGAM, post-nonlinear model, and location-scale noise model. Further, we derive a statistical test to find the parents set of the target variable. Experiments on various synthetic and real-world datasets validate our theoretical findings.

Sadegh Khorasani, Saber Salehkaleybar, Negar Kiyavash et al.

We present Dual-Feedback Actor (DFA), a reinforcement learning algorithm that fuses both individual rewards and pairwise preferences (if available) into a single update rule. DFA uses the policy's log-probabilities directly to model the preference probability, avoiding a separate reward-modeling step. Preferences can be provided by human-annotators (at state-level or trajectory-level) or be synthesized online from Q-values stored in an off-policy replay buffer. Under a Bradley-Terry model, we prove that minimizing DFA's preference loss recovers the entropy-regularized Soft Actor-Critic (SAC) policy. Our simulation results show that DFA trained on generated preferences matches or exceeds SAC on six control environments and demonstrates a more stable training process. With only a semi-synthetic preference dataset under Bradley-Terry model, our algorithm outperforms reward-modeling reinforcement learning from human feedback (RLHF) baselines in a stochastic GridWorld and approaches the performance of an oracle with true rewards.

Yaroslav Kivva, Sina Akbari, Saber Salehkaleybar et al.

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.

Ehsan Sharifian, Saber Salehkaleybar, Negar Kiyavash

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 graph in an 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.

Kacper Kadziolka, Saber Salehkaleybar

Causal inference remains a fundamental challenge for large language models. Recent advances in internal reasoning with large language models have sparked interest in whether state-of-the-art reasoning models can robustly perform causal discovery-a task where conventional models often suffer from severe overfitting and near-random performance under data perturbations. We study causal discovery on the Corr2Cause benchmark using the emergent OpenAI's o-series and DeepSeek-R model families and find that these reasoning-first architectures achieve significantly greater native gains than prior approaches. To capitalize on these strengths, we introduce a modular in-context pipeline inspired by the Tree-of-Thoughts and Chain-of-Thoughts methodologies, yielding nearly three-fold improvements over conventional baselines. We further probe the pipeline's impact by analyzing reasoning chain length, complexity, and conducting qualitative and quantitative comparisons between conventional and reasoning models. Our findings suggest that while advanced reasoning models represent a substantial leap forward, carefully structured in-context frameworks are essential to maximize their capabilities and offer a generalizable blueprint for causal discovery across diverse domains.

Sadegh Khorasani, Saber Salehkaleybar, Negar Kiyavash et al.

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.

Daniele Tramontano, Yaroslav Kivva, Saber Salehkaleybar et al.

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.

Daniele Tramontano, Yaroslav Kivva, Saber Salehkaleybar et al.

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.

Arsalan Sharifnassab, Saber Salehkaleybar, S. Jamaloddin Golestani

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.

Sepehr Dehdashtian, Matin Hashemi, Saber Salehkaleybar

We consider the problem of recovering channel code parameters over a candidate set by merely analyzing the received encoded signals. We propose a deep learning-based solution that I) is capable of identifying the channel code parameters for any coding scheme (such as LDPC, Convolutional, Turbo, and Polar codes), II) is robust against channel impairments like multi-path fading, III) does not require any previous knowledge or estimation of channel state or signal-to-noise ratio (SNR), and IV) outperforms related works in terms of probability of detecting the correct code parameters.

Amir Amirinezhad, Saber Salehkaleybar, Matin Hashemi

We study the problem of experiment design to learn causal structures from interventional data. We consider an active learning setting in which the experimenter decides to intervene on one of the variables in the system in each step and uses the results of the intervention to recover further causal relationships among the variables. The goal is to fully identify the causal structures with minimum number of interventions. We present the first deep reinforcement learning based solution for the problem of experiment design. In the proposed method, we embed input graphs to vectors using a graph neural network and feed them to another neural network which outputs a variable for performing intervention in each step. Both networks are trained jointly via a Q-iteration algorithm. Experimental results show that the proposed method achieves competitive performance in recovering causal structures with respect to previous works, while significantly reducing execution time in dense graphs.

Ali AhmadiTeshnizi, Saber Salehkaleybar, Negar Kiyavash

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.

Arsalan Sharifnassab, Saber Salehkaleybar, S. Jamaloddin Golestani

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 $\tilde{O}\big(m^{-{1}/{\max(d,2)}} n^{-1/2}\big)$, 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 $n$.

AmirEmad Ghassami, Saber Salehkaleybar, Negar Kiyavash

It is known that from purely observational data, a causal DAG is identifiable only up to its Markov equivalence class, and for many ground truth DAGs, the direction of a large portion of the edges will be remained unidentified. The golden standard for learning the causal DAG beyond Markov equivalence is to perform a sequence of interventions in the system and use the data gathered from the interventional distributions. We consider a setup in which given a budget $k$, we design $k$ interventions non-adaptively. We cast the problem of finding the best intervention target set as an optimization problem which aims to maximize the number of edges whose directions are identified due to the performed interventions. First, we consider the case that the underlying causal structure is a tree. For this case, we propose an efficient exact algorithm for the worst-case gain setup, as well as an approximate algorithm for the average gain setup. We then show that the proposed approach for the average gain setup can be extended to the case of general causal structures. In this case, besides the design of interventions, calculating the objective function is also challenging. We propose an efficient exact calculator as well as two estimators for this task. We evaluate the proposed methods using synthetic as well as real data.

M. Reza Heydari, Saber Salehkaleybar, Kun Zhang

We propose two nonlinear regression methods, named Adversarial Orthogonal Regression (AdOR) for additive noise models and Adversarial Orthogonal Structural Equation Model (AdOSE) for the general case of structural equation models. Both methods try to make the residual of regression independent from regressors while putting no assumption on noise distribution. In both methods, two adversarial networks are trained simultaneously where a regression network outputs predictions and a loss network that estimates mutual information (in AdOR) and KL-divergence (in AdOSE). These methods can be formulated as a minimax two-player game; at equilibrium, AdOR finds a deterministic map between inputs and output and estimates mutual information between residual and inputs, while AdOSE estimates a conditional probability distribution of output given inputs. The proposed methods can be used as subroutines to address several learning problems in causality, such as causal direction determination (or more generally, causal structure learning) and causal model estimation. Synthetic and real-world experiments demonstrate that the proposed methods have a remarkable performance with respect to previous solutions.

Saber Salehkaleybar, AmirEmad Ghassami, Negar Kiyavash et al.

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, one usually infers wrong causal relationships among the observed variables. 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 them. The next question is then whether or not the causal effects can be uniquely identified as well. It can be shown that causal effects among observed variables cannot be identified uniquely even under the assumptions of faithfulness and non-Gaussianity of exogenous noises. However, we will propose an efficient method to identify the set of all possible causal effects that are compatible with the observational data. Furthermore, we present some structural conditions on the causal graph under which we can learn causal effects among observed variables uniquely. We also 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 on learning causal models.

Saber Salehkaleybar, Arsalan Sharifnassab, S. Jamaloddin Golestani

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 $B\ge\log mn$) meets the aforementioned lower bound up to poly-logarithmic factors, and is thereby order optimal. We also address the problem of learning under tiny communication budget, and present lower and upper error bounds when $B$ is a constant. 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 $n$.

Behrooz Zarebavani, Foad Jafarinejad, Matin Hashemi et al.

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 500 X and 1300 X speedup, respectively, compared to serial implementation on CPU. The source code of cuPC is available online [1].

AmirEmad Ghassami, Saber Salehkaleybar, Negar Kiyavash et al.

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.

AmirEmad Ghassami, Saber Salehkaleybar, Negar Kiyavash et al.

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.

AmirEmad Ghassami, Saber Salehkaleybar, Negar Kiyavash et al.

We study causal inference 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. 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. The experiment results show that the proposed algorithm outperforms the other existing algorithms.

Saber Salehkaleybar, Arsalan Sharif-Nassab, S. Jamaloddin Golestani

Considering a network with $n$ nodes, where each node initially votes for one (or more) choices out of $K$ possible choices, we present a Distributed Multi-choice Voting/Ranking (DMVR) algorithm to determine either the choice with maximum vote (the voting problem) or to rank all the choices in terms of their acquired votes (the ranking problem). The algorithm consolidates node votes across the network by updating the states of interacting nodes using two key operations, the union and the intersection. The proposed algorithm is simple, independent from network size, and easily scalable in terms of the number of choices $K$, using only $K\times 2^{K-1}$ nodal states for voting, and $K\times K!$ nodal states for ranking. We prove the number of states to be optimal in the ranking case, this optimality is conjectured to also apply to the voting case. The time complexity of the algorithm is analyzed in complete graphs. We show that the time complexity for both ranking and voting is $O(\log(n))$ for given vote percentages, and is inversely proportional to the minimum of the vote percentage differences among various choices.

Saber Salehkaleybar, S. Jamaloddin Golestani

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 $\sqrt{n/\log(n)}$ in Erdös-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.

Saber Salehkaleybar, Jalal Etesami, Negar Kiyavash et al.

We study the problem of learning the support of transition matrix between random processes in a Vector Autoregressive (VAR) model from samples when a subset of the processes are latent. It is well known that ignoring the effect of the latent processes may lead to very different estimates of the influences among observed processes, and we are concerned with identifying the influences among the observed processes, those between the latent ones, and those from the latent to the observed ones. We show that the support of transition matrix among the observed processes and lengths of all latent paths between any two observed processes can be identified successfully under some conditions on the VAR model. From the lengths of latent paths, we reconstruct the latent subgraph (representing the influences among the latent processes) with a minimum number of variables uniquely if its topology is a directed tree. Furthermore, we propose an algorithm that finds all possible minimal latent graphs under some conditions on the lengths of latent paths. Our results apply to both non-Gaussian and Gaussian cases, and experimental results on various synthetic and real-world datasets validate our theoretical results.

AmirEmad Ghassami, Saber Salehkaleybar, Negar Kiyavash

We study the problem of causal structure learning over a set of random variables when the experimenter is allowed to perform at most $M$ experiments in a non-adaptive manner. We consider the optimal learning strategy in terms of minimizing the portions of the structure that remains unknown given the limited number of experiments in both Bayesian and minimax setting. We characterize the theoretical optimal solution and propose an algorithm, which designs the experiments efficiently in terms of time complexity. We show that for bounded degree graphs, in the minimax case and in the Bayesian case with uniform priors, our proposed algorithm is a $ρ$-approximation algorithm, where $ρ$ is independent of the order of the underlying graph. Simulations on both synthetic and real data show that the performance of our algorithm is very close to the optimal solution.

Saber Salehkaleybar, Jalal Etesami, Negar Kiyavash

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.