arXiv subjects

Pulkit Grover

Publications and source records attributed to Pulkit Grover.

At least 37 records · Page 2Linked to original sources

arXiv2018

On the Optimal Recovery Threshold of Coded Matrix Multiplication

We provide novel coded computation strategies for distributed matrix-matrix products that outperform the recent "Polynomial code" constructions in recovery threshold, i.e., the required number of successful workers. When $m$-th fraction of each matrix can be stored in each worker node, Polynomial codes require $m^2$ successful workers, while our MatDot codes only require $2m-1$ successful workers, albeit at a higher communication cost from each worker to the fusion node. We also provide a systematic construction of MatDot codes. Further, we propose "PolyDot" coding that interpolates between Polynomial codes and MatDot codes to trade off communication cost and recovery threshold. Finally, we demonstrate a coding technique for multiplying $n$ matrices ($n \geq 3$) by applying MatDot and PolyDot coding ideas.

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arXiv2018

Coded Iterative Computing using Substitute Decoding

In this paper, we propose a new coded computing technique called "substitute decoding" for general iterative distributed computation tasks. In the first part of the paper, we use PageRank as a simple example to show that substitute decoding can make the computation of power iterations solving PageRank on sparse matrices robust to erasures in distributed systems. For these sparse matrices, codes with dense generator matrices can significantly increase storage costs and codes with low-density generator matrices (LDGM) are preferred. Surprisingly, we show through both theoretical analysis and simulations that when substitute decoding is used, coded iterative computing with extremely low-density codes (2 or 3 non-zeros in each row of the generator matrix) can achieve almost the same convergence rate as noiseless techniques, despite the poor error-correction ability of LDGM codes. In the second part of the paper, we discuss applications of substitute decoding beyond solving linear systems and PageRank. These applications include (1) computing eigenvectors, (2) computing the truncated singular value decomposition (SVD), and (3) gradient descent. These examples show that the substitute decoding algorithm is useful in a wide range of applications.

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arXiv2017

Fast Path Localization on Graphs via Multiscale Viterbi Decoding

We consider a problem of localizing a path-signal that evolves over time on a graph. A path-signal can be viewed as the trajectory of a moving agent on a graph in several consecutive time points. Combining dynamic programming and graph partitioning, we propose a path-localization algorithm with significantly reduced computational complexity. We analyze the localization error for the proposed approach both in the Hamming distance and the destination's distance between the path estimate and the true path using numerical bounds. Unlike usual theoretical bounds that only apply to restricted graph models, the obtained numerical bounds apply to all graphs and all non-overlapping graph-partitioning schemes. In random geometric graphs, we are able to derive a closed-form expression for the localization error bound, and a tradeoff between localization error and the computational complexity. Finally, we compare the proposed technique with the maximum likelihood estimate under the path constraint in terms of computational complexity and localization error, and show significant speedup (100 times) with comparable localization error (4 times) on a graph from real data. Variants of the proposed technique can be applied to tracking, road congestion monitoring, and brain signal processing.

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arXiv2017

Coding Method for Parallel Iterative Linear Solver

Computationally intensive distributed and parallel computing is often bottlenecked by a small set of slow workers known as stragglers. In this paper, we utilize the emerging idea of "coded computation" to design a novel error-correcting-code inspired technique for solving linear inverse problems under specific iterative methods in a parallelized implementation affected by stragglers. Example applications include inverse problems in machine learning on graphs, such as personalized PageRank and sampling on graphs. We provably show that our coded-computation technique can reduce the mean-squared error under a computational deadline constraint. In fact, the ratio of mean-squared error of replication-based and coded techniques diverges to infinity as the deadline increases. Our experiments for personalized PageRank performed on real systems and real social networks show that this ratio can be as large as $10^4$. Further, unlike coded-computation techniques proposed thus far, our strategy combines outputs of all workers, including the stragglers, to produce more accurate estimates at the computational deadline. This also ensures that the accuracy degrades "gracefully" in the event that the number of stragglers is large.

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arXiv2017

Computing Linear Transformations with Unreliable Components

We consider the problem of computing a binary linear transformation using unreliable components when all circuit components are unreliable. Two noise models of unreliable components are considered: probabilistic errors and permanent errors. We introduce the "ENCODED" technique that ensures that the error probability of the computation of the linear transformation is kept bounded below a small constant independent of the size of the linear transformation even when all logic gates in the computation are noisy. Further, we show that the scheme requires fewer operations (in order sense) than its "uncoded" counterpart. By deriving a lower bound, we show that in some cases, the scheme is order-optimal. Using these results, we examine the gain in energy-efficiency from use of "voltage-scaling" scheme where gate-energy is reduced by lowering the supply voltage. We use a gate energy-reliability model to show that tuning gate-energy appropriately at different stages of the computation ("dynamic" voltage scaling), in conjunction with ENCODED, can lead to order-sense energy-savings over the classical "uncoded" approach. Finally, we also examine the problem of computing a linear transformation when noiseless decoders can be used, providing upper and lower bounds to the problem.

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arXiv2017

Coded convolution for parallel and distributed computing within a deadline

We consider the problem of computing the convolution of two long vectors using parallel processing units in the presence of "stragglers". Stragglers refer to the small fraction of faulty or slow processors that delays the entire computation in time-critical distributed systems. We first show that splitting the vectors into smaller pieces and using a linear code to encode these pieces provides better resilience against stragglers than replication-based schemes under a simple, worst-case straggler analysis. We then demonstrate that under commonly used models of computation time, coding can dramatically improve the probability of finishing the computation within a target "deadline" time. As opposed to the more commonly used technique of expected computation time analysis, we quantify the exponents of the probability of failure in the limit of large deadlines. Our exponent metric captures the probability of failing to finish before a specified deadline time, i.e. , the behavior of the "tail". Moreover, our technique also allows for simple closed form expressions for more general models of computation time, e.g. shifted Weibull models instead of only shifted exponentials. Thus, through this problem of coded convolution, we establish the utility of a novel asymptotic failure exponent analysis for distributed systems.

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arXiv2017

"Short-Dot": Computing Large Linear Transforms Distributedly Using Coded Short Dot Products

Faced with saturation of Moore's law and increasing dimension of data, system designers have increasingly resorted to parallel and distributed computing. However, distributed computing is often bottle necked by a small fraction of slow processors called "stragglers" that reduce the speed of computation because the fusion node has to wait for all processors to finish. To combat the effect of stragglers, recent literature introduces redundancy in computations across processors, e.g.,~using repetition-based strategies or erasure codes. The fusion node can exploit this redundancy by completing the computation using outputs from only a subset of the processors, ignoring the stragglers. In this paper, we propose a novel technique -- that we call "Short-Dot" -- to introduce redundant computations in a coding theory inspired fashion, for computing linear transforms of long vectors. Instead of computing long dot products as required in the original linear transform, we construct a larger number of redundant and short dot products that can be computed faster and more efficiently at individual processors. In reference to comparable schemes that introduce redundancy to tackle stragglers, Short-Dot reduces the cost of computation, storage and communication since shorter portions are stored and computed at each processor, and also shorter portions of the input is communicated to each processor. We demonstrate through probabilistic analysis as well as experiments that Short-Dot offers significant speed-up compared to existing techniques. We also derive trade-offs between the length of the dot-products and the resilience to stragglers (number of processors to wait for), for any such strategy and compare it to that achieved by our strategy.

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arXiv2017

Graph Codes for Distributed Instant Message Collection in an Arbitrary Noisy Broadcast Network

We consider the problem of minimizing the number of broadcasts for collecting all sensor measurements at a sink node in a noisy broadcast sensor network. Focusing first on arbitrary network topologies, we provide (i) fundamental limits on the required number of broadcasts of data gathering, and (ii) a general in-network computing strategy to achieve an upper bound within factor $\log N$ of the fundamental limits, where $N$ is the number of agents in the network. Next, focusing on two example networks, namely, \textcolor{black}{arbitrary geometric networks and random Erd$\ddot{o}$s-R$\acute{e}$nyi networks}, we provide improved in-network computing schemes that are optimal in that they attain the fundamental limits, i.e., the lower and upper bounds are tight \textcolor{black}{in order sense}. Our main techniques are three distributed encoding techniques, called graph codes, which are designed respectively for the above-mentioned three scenarios. Our work thus extends and unifies previous works such as those of Gallager [1] and Karamchandani~\emph{et. al.} [2] on number of broadcasts for distributed function computation in special network topologies, while bringing in novel techniques, e.g., from error-control coding and noisy circuits, for both upper and lower bounds.

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arXiv2017

Rate Distortion for Lossy In-network Function Computation: Information Dissipation and Sequential Reverse Water-Filling

We consider the problem of distributed lossy linear function computation in a tree network. We examine two cases: (i) data aggregation (only one sink node computes) and (ii) consensus (all nodes compute the same function). By quantifying the accumulation of information loss in distributed computing, we obtain fundamental limits on network computation rate as a function of incremental distortions (and hence incremental loss of information) along the edges of the network. The above characterization, based on quantifying distortion accumulation, offers an improvement over classical cut-set type techniques which are based on overall distortions instead of incremental distortions. This quantification of information loss qualitatively resembles information dissipation in cascaded channels [1]. Surprisingly, this accumulation effect of distortion happens even at infinite blocklength. Combining this observation with an inequality on the dominance of mean-square quantities over relative-entropy quantities, we obtain outer bounds on the rate distortion function that are tighter than classical cut-set bounds by a difference which can be arbitrarily large in both data aggregation and consensus. We also obtain inner bounds on the optimal rate using random Gaussian coding, which differ from the outer bounds by $\mathcal{O}(\sqrt{D})$, where $D$ is the overall distortion. The obtained inner and outer bounds can provide insights on rate (bit) allocations for both the data aggregation problem and the consensus problem. We show that for tree networks, the rate allocation results have a mathematical structure similar to classical reverse water-filling for parallel Gaussian sources.

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arXiv2016

Adaptivity provably helps: information-theoretic limits on $l_0$ cost of non-adaptive sensing

The advantages of adaptivity and feedback are of immense interest in signal processing and communication with many positive and negative results. Although it is established that adaptivity does not offer substantial reductions in minimax mean square error for a fixed number of measurements, existing results have shown several advantages of adaptivity in complexity of reconstruction, accuracy of support detection, and gain in signal-to-noise ratio, under constraints on sensing energy. Sensing energy has often been measured in terms of the Frobenius Norm of the sensing matrix. This paper uses a different metric that we call the $l_0$ cost of a sensing matrix-- to quantify the complexity of sensing. Thus sparse sensing matrices have a lower cost. We derive information-theoretic lower bounds on the $l_0$ cost that hold for any non-adaptive sensing strategy. We establish that any non-adaptive sensing strategy must incur an $l_0$ cost of $Θ\left( N \log_2(N)\right) $ to reconstruct an $N$-dimensional, one--sparse signal when the number of measurements are limited to $Θ\left(\log_2 (N)\right)$. In comparison, bisection-type adaptive strategies only require an $l_0$ cost of at most $\mathcal{O}(N)$ for an equal number of measurements. The problem has an interesting interpretation as a sphere packing problem in a multidimensional space, such that all the sphere centres have minimum non-zero co-ordinates. We also discuss the variation in $l_0$ cost as the number of measurements increase from $Θ\left(\log_2 (N)\right)$ to $Θ\left(N\right)$.

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arXiv2016

Energy Efficient Distributed Coding for Data Collection in a Noisy Sparse Network

We consider the problem of data collection in a two-layer network consisting of (1) links between $N$ distributed agents and a remote sink node; (2) a sparse network formed by these distributed agents. We study the effect of inter-agent communications on the overall energy consumption. Despite the sparse connections between agents, we provide an in-network coding scheme that reduces the overall energy consumption by a factor of $Θ(\log N)$ compared to a naive scheme which neglects inter-agent communications. By providing lower bounds on both the energy consumption and the sparseness (number of links) of the network, we show that are energy-optimal except for a factor of $Θ(\log\log N)$. The proposed scheme extends a previous work of Gallager on noisy broadcasting from a complete graph to a sparse graph, while bringing in new techniques from error control coding and noisy circuits.

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arXiv2015

On the Total-Power Capacity of Regular-LDPC Codes with Iterative Message-Passing Decoders

Motivated by recently derived fundamental limits on total (transmit + decoding) power for coded communication with VLSI decoders, this paper investigates the scaling behavior of the minimum total power needed to communicate over AWGN channels as the target bit-error-probability tends to zero. We focus on regular-LDPC codes and iterative message-passing decoders. We analyze scaling behavior under two VLSI complexity models of decoding. One model abstracts power consumed in processing elements ("node model"), and another abstracts power consumed in wires which connect the processing elements ("wire model"). We prove that a coding strategy using regular-LDPC codes with Gallager-B decoding achieves order-optimal scaling of total power under the node model. However, we also prove that regular-LDPC codes and iterative message-passing decoders cannot meet existing fundamental limits on total power under the wire model. Further, if the transmit energy-per-bit is bounded, total power grows at a rate that is worse than uncoded transmission. Complementing our theoretical results, we develop detailed physical models of decoding implementations using post-layout circuit simulations. Our theoretical and numerical results show that approaching fundamental limits on total power requires increasing the complexity of both the code design and the corresponding decoding algorithm as communication distance is increased or error-probability is lowered.

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arXiv2015

Energy Harvesting Transmitters that Heat Up: Throughput Maximization under Temperature Constraints

Motivated by damage due to heating in sensor operation, we consider the throughput optimal offline data scheduling problem in an energy harvesting transmitter such that the resulting temperature increase remains below a critical level. We model the temperature dynamics of the transmitter as a linear system and determine the optimal transmit power policy under such temperature constraints as well as energy harvesting constraints over an AWGN channel. We first derive the structural properties of the solution for the general case with multiple energy arrivals. We show that the optimal power policy is piecewise monotone decreasing with possible jumps at the energy harvesting instants. We derive analytical expressions for the optimal solution in the single energy arrival case. We show that, in the single energy arrival case, the optimal power is monotone decreasing, the resulting temperature is monotone increasing, and both remain constant after the temperature hits the critical level. We then generalize the solution for the multiple energy arrival case.

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arXiv2015

Energy-efficient Decoders for Compressive Sensing: Fundamental Limits and Implementations

The fundamental problem considered in this paper is "What is the \textit{energy} consumed for the implementation of a \emph{compressive sensing} decoding algorithm on a circuit?". Using the "information-friction" framework, we examine the smallest amount of \textit{bit-meters} as a measure for the energy consumed by a circuit. We derive a fundamental lower bound for the implementation of compressive sensing decoding algorithms on a circuit. In the setting where the number of measurements scales linearly with the sparsity and the sparsity is sub-linear with the length of the signal, we show that the \textit{bit-meters} consumption for these algorithms is order-tight, i.e., it matches the lower bound asymptotically up to a constant factor. Our implementations yield interesting insights into design of energy-efficient circuits that are not captured by the notion of computational efficiency alone.

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arXiv2015

Energy Harvesting Wireless Communications: A Review of Recent Advances

This article summarizes recent contributions in the broad area of energy harvesting wireless communications. In particular, we provide the current state of the art for wireless networks composed of energy harvesting nodes, starting from the information-theoretic performance limits to transmission scheduling policies and resource allocation, medium access and networking issues. The emerging related area of energy transfer for self-sustaining energy harvesting wireless networks is considered in detail covering both energy cooperation aspects and simultaneous energy and information transfer. Various potential models with energy harvesting nodes at different network scales are reviewed as well as models for energy consumption at the nodes.

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arXiv2014

"Information-Friction" and its implications on minimum energy required for communication

Just as there are frictional losses associated with moving masses on a surface, what if there were frictional losses associated with moving information on a substrate? Indeed, many modes of communication suffer from such frictional losses. We propose to model these losses as proportional to "bit-meters," i.e., the product of mass of information (i.e., the number of bits) and the distance of information transport. We use this "information- friction" model to understand fundamental energy requirements on encoding and decoding in communication circuitry. First, for communication across a binary input AWGN channel, we arrive at fundamental limits on bit-meters (and thus energy consumption) for decoding implementations that have a predetermined input-independent length of messages. For encoding, we relax the fixed-length assumption and derive bounds for flexible-message- length implementations. Using these lower bounds we show that the total (transmit + encoding + decoding) energy-per-bit must diverge to infinity as the target error probability is lowered to zero. Further, the closer the communication rate is maintained to the channel capacity (as the target error-probability is lowered to zero), the faster the required decoding energy diverges to infinity.

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arXiv2013

Information embedding and the triple role of control

We consider the problem of information embedding where the encoder modifies a white Gaussian host signal in a power-constrained manner to encode a message, and the decoder recovers both the embedded message and the modified host signal. This partially extends the recent work of Sumszyk and Steinberg to the continuous-alphabet Gaussian setting. Through a control-theoretic lens, we observe that the problem is a minimalist example of what is called the "triple role" of control actions. We show that a dirty-paper-coding strategy achieves the optimal rate for perfect recovery of the modified host and the message for any message rate. For imperfect recovery of the modified host, by deriving bounds on the minimum mean-square error (MMSE) in recovering the modified host signal, we show that DPC-based strategies are guaranteed to attain within a uniform constant factor of 16 of the optimal weighted sum of power required in host signal modification and the MMSE in the modified host signal reconstruction for all weights and all message rates. When specialized to the zero-rate case, our results provide the tightest known lower bounds on the asymptotic costs for the vector version of a famous open problem in decentralized control: the Witsenhausen counterexample. Numerically, this tighter bound helps us characterize the asymptotically optimal costs for the vector Witsenhausen problem to within a factor of 1.3 for all problem parameters, improving on the earlier best known bound of 2.

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arXiv2011

Towards a communication-theoretic understanding of system-level power consumption

Traditional communication theory focuses on minimizing transmit power. However, communication links are increasingly operating at shorter ranges where transmit power can be significantly smaller than the power consumed in decoding. This paper models the required decoding power and investigates the minimization of total system power from two complementary perspectives. First, an isolated point-to-point link is considered. Using new lower bounds on the complexity of message-passing decoding, lower bounds are derived on decoding power. These bounds show that 1) there is a fundamental tradeoff between transmit and decoding power; 2) unlike the implications of the traditional "waterfall" curve which focuses on transmit power, the total power must diverge to infinity as error probability goes to zero; 3) Regular LDPCs, and not their known capacity-achieving irregular counterparts, can be shown to be power order optimal in some cases; and 4) the optimizing transmit power is bounded away from the Shannon limit. Second, we consider a collection of links. When systems both generate and face interference, coding allows a system to support a higher density of transmitter-receiver pairs (assuming interference is treated as noise). However, at low densities, uncoded transmission may be more power-efficient in some cases.

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