arXiv subjects

Saber Salehkaleybar

Publications and source records attributed to Saber Salehkaleybar.

52 records · Page 3Linked to original sources

arXiv2019

Order Optimal One-Shot Distributed Learning

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 $n$.

cs.LG↗

arXiv2019

Interventional Experiment Design for Causal Structure Learning

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.

cs.LG↗

arXiv2019

cuPC: CUDA-based Parallel PC Algorithm for Causal Structure Learning on GPU

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].

cs.DC↗

arXiv2019

Adversarial Orthogonal Regression: Two non-Linear Regressions for Causal Inference

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.

cs.LG↗

arXiv2019

Learning Linear Non-Gaussian Causal Models in the Presence of Latent Variables

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.

cs.LG↗

arXiv2019

Broadcast Distributed Voting Algorithm in Population Protocols

We 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 majority merely by performing local interactions. Population protocols provide a framework for designing pairwise interactions between agents in order to perform tasks in a coordinated manner. In this paper, we propose ``Broadcasting Population Protocol" model as a counterpart model of conventional population protocols for the networks that each agent can send a message to all its neighbors simultaneously. We design two distributed algorithms for solving the multi-choice majority voting problem in the model of broadcasting population protocols. We prove the correctness of these algorithms and analyze their performance in terms of time and message complexities. Experiments show that the proposed algorithm improves both time and message complexities significantly with respect to previous algorithms proposed in conventional population protocols and they can be utilized in networks where messages can be transmitted to a subset of agents simultaneously such as wireless networks.

cs.MA↗

arXiv2018

Counting and Sampling from Markov Equivalent DAGs Using Clique Trees

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.

cs.DS↗

arXiv2018

Budgeted Experiment Design for Causal Structure Learning

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.

cs.LG↗

arXiv2017

Learning Vector Autoregressive Models with Latent Processes

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.

cs.LG↗

arXiv2017

Learning Causal Structures Using Regression Invariance

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.

cs.LG↗

arXiv2017

Token-based Function Computation with Memory

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.

cs.DC↗

arXiv2017

Distributed Voting/Ranking with Optimal Number of States per Node

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.

cs.DC↗

arXiv2017

Optimal Experiment Design for Causal Discovery from Fixed Number of Experiments

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.

cs.LG↗

arXiv2017

Identifying Nonlinear 1-Step Causal Influences in Presence of Latent Variables

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.

cs.IT↗

arXiv2010

A New Framework for Cognitive Medium Access Control: POSG Approach

In this paper, we propose a new analytical framework to solve medium access problem for secondary users (SUs) in cognitive radio networks. Partially Observable Stochastic Games (POSG) and Decentralized Markov Decision Process (Dec-POMDP) are two multi-agent Markovian decision processes which are used to present a solution. A primary network with two SUs is considered as an example to demonstrate our proposed framework. Two different scenarios are assumed. In the first scenario, SUs compete to acquire the licensed channel which is modeled using POSG framework. In the second one, SUs cooperate to access channel for which the solution is based on Dec-POMDP. Besides, the dominant strategy for both of the above mentioned scenarios is presented for a three slot horizon length.

cs.GT↗

arXiv2010

QoS-Aware Joint Policies in Cognitive Radio Networks

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 guarantees QoS requirements but also maximizes 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 joint policy which guarantees QoS requirements 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 three interesting findings: 1- Optimum joint policies for large horizons can be obtained using the proposed method. 2- There exists a joint policy for the assumed QoS constraints. 3- Our method outperforms other related works in terms of network throughput.

cs.GT↗