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Register a new userFundamental Limits of Approximate Gradient Coding
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Added by
Sinong Wang
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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It has been established that when the gradient coding problem is distributed among $n$ servers, the computation load (number of stored data partitions) of each worker is at least $s+1$ in order to resists $s$ stragglers. This scheme incurs a large overhead when the number of stragglers $s$ is large. In this paper, we focus on a new framework called emph{approximate gradient coding} to mitigate stragglers in distributed learning. We show that, to exactly recover the gradient with high probability, the computation load is lower bounded by $O(log(n)/log(n/s))$. We also propose a code that exactly matches such lower bound. We identify a fundamental three-fold tradeoff for any approximate gradient coding scheme $dgeq O(log(1/epsilon)/log(n/s))$, where $d$ is the computation load, $epsilon$ is the error of gradient. We give an explicit code construction based on random edge removal process that achieves the derived tradeoff. We implement our schemes and demonstrate the advantage of the approaches over the current fastest gradient coding strategies.
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In distributed machine learning (DML), the training data is distributed across multiple worker nodes to perform the underlying training in parallel. One major problem affecting the performance of DML algorithms is presence of stragglers. These are nodes that are terribly slow in performing their task which results in under-utilization of the training data that is stored in them. Towards this, gradient coding mitigates the impact of stragglers by adding sufficient redundancy in the data. Gradient coding and other straggler mitigation schemes assume that the straggler behavior of the worker nodes is identical. Our experiments on the Amazon AWS cluster however suggest otherwise and we see that there is a correlation in the straggler behavior across iterations. To model this, we introduce a heterogeneous straggler model where nodes are categorized into two classes, slow and active. To better utilize training data stored with slow nodes, we modify the existing gradient coding schemes with shuffling of the training data among workers. Our results (both simulation and cloud experiments) suggest remarkable improvement with shuffling over existing schemes. We perform theoretical analysis for the proposed models justifying their utility.
In distributed optimization problems, a technique called gradient coding, which involves replicating data points, has been used to mitigate the effect of straggling machines. Recent work has studied approximate gradient coding, which concerns coding schemes where the replication factor of the data is too low to recover the full gradient exactly. Our work is motivated by the challenge of creating approximate gradient coding schemes that simultaneously work well in both the adversarial and stochastic models. To that end, we introduce novel approximate gradient codes based on expander graphs, in which each machine receives exactly two blocks of data points. We analyze the decoding error both in the random and adversarial straggler setting, when optimal decoding coefficients are used. We show that in the random setting, our schemes achieve an error to the gradient that decays exponentially in the replication factor. In the adversarial setting, the error is nearly a factor of two smaller than any existing code with similar performance in the random setting. We show convergence bounds both in the random and adversarial setting for gradient descent under standard assumptions using our codes. In the random setting, our convergence rate improves upon block-box bounds. In the adversarial setting, we show that gradient descent can converge down to a noise floor that scales linearly with the adversarial error to the gradient. We demonstrate empirically that our schemes achieve near-optimal error in the random setting and converge faster than algorithms which do not use the optimal decoding coefficients.
The maximum possible throughput (or the rate of job completion) of a multi-server system is typically the sum of the service rates of individual servers. Recent work shows that launching multiple replicas of a job and canceling them as soon as one copy finishes can boost the throughput, especially when the service time distribution has high variability. This means that redundancy can, in fact, create synergy among servers such that their overall throughput is greater than the sum of individual servers. This work seeks to find the fundamental limit of the throughput boost achieved by job replication and the optimal replication policy to achieve it. While most previous works consider upfront replication policies, we expand the set of possible policies to delayed launch of replicas. The search for the optimal adaptive replication policy can be formulated as a Markov Decision Process, using which we propose two myopic replication policies, MaxRate and AdaRep, to adaptively replicate jobs. In order to quantify the optimality gap of these and other policies, we derive upper bounds on the service capacity, which provide fundamental limits on the throughput of queueing systems with redundancy.
It has recently been observed that certain extremely simple feature encoding techniques are able to achieve state of the art performance on several standard image classification benchmarks including deep belief networks, convolutional nets, factored RBMs, mcRBMs, convolutional RBMs, sparse autoencoders and several others. Moreover, these triangle or soft threshold encodings are ex- tremely efficient to compute. Several intuitive arguments have been put forward to explain this remarkable performance, yet no mathematical justification has been offered. The main result of this report is to show that these features are realized as an approximate solution to the a non-negative sparse coding problem. Using this connection we describe several variants of the soft threshold features and demonstrate their effectiveness on two image classification benchmark tasks.
This paper focuses on mitigating the impact of stragglers in distributed learning system. Unlike the existing results designed for a fixed number of stragglers, we developed a new scheme called Adaptive Gradient Coding(AGC) with flexible tolerance of various number of stragglers. Our scheme gives an optimal tradeoff between computation load, straggler tolerance and communication cost. In particular, it allows to minimize the communication cost according to the real-time number of stragglers in the practical environments. Implementations on Amazon EC2 clusters using Python with mpi4py package verify the flexibility in several situations.
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