Lagrange Coded Computing: Optimal Design for Resiliency, Security and Privacy (1806.00939v4)

Abstract: We consider a scenario involving computations over a massive dataset stored distributedly across multiple workers, which is at the core of distributed learning algorithms. We propose Lagrange Coded Computing (LCC), a new framework to simultaneously provide (1) resiliency against stragglers that may prolong computations; (2) security against Byzantine (or malicious) workers that deliberately modify the computation for their benefit; and (3) (information-theoretic) privacy of the dataset amidst possible collusion of workers. LCC, which leverages the well-known Lagrange polynomial to create computation redundancy in a novel coded form across workers, can be applied to any computation scenario in which the function of interest is an arbitrary multivariate polynomial of the input dataset, hence covering many computations of interest in machine learning. LCC significantly generalizes prior works to go beyond linear computations. It also enables secure and private computing in distributed settings, improving the computation and communication efficiency of the state-of-the-art. Furthermore, we prove the optimality of LCC by showing that it achieves the optimal tradeoff between resiliency, security, and privacy, i.e., in terms of tolerating the maximum number of stragglers and adversaries, and providing data privacy against the maximum number of colluding workers. Finally, we show via experiments on Amazon EC2 that LCC speeds up the conventional uncoded implementation of distributed least-squares linear regression by up to $13.43\times$, and also achieves a $2.36\times$-$12.65\times$ speedup over the state-of-the-art straggler mitigation strategies.

• The paper introduces Lagrange Coded Computing, which optimally balances resiliency, security, and privacy via innovative Lagrange polynomial encoding.
• It establishes a theoretical tradeoff formula for handling stragglers and adversaries in distributed multivariate polynomial computations.
• Practical tests on tasks like linear regression show significant runtime improvements, validating LCC's efficiency in real-world scenarios.

Analysis of Lagrange Coded Computing: Optimal Design for Resiliency, Security, and Privacy

The paper "Lagrange Coded Computing: Optimal Design for Resiliency, Security, and Privacy," introduces Lagrange Coded Computing (LCC), an advanced framework designed to tackle key challenges in distributed computing environments. These challenges include managing stragglers, securing computations against adversarial disruptions, and maintaining data privacy amidst potential collusion among workers. By leveraging the conceptual basis of Lagrange polynomials, this work extends the realms of coded computing far beyond its conventional applications in linear and bilinear operations, adapting it to multivariate polynomial computations prevalent in machine learning and data-heavy tasks.

Technical Overview

The core idea of LCC is to encode input datasets using Lagrange polynomials, creating redundant coded forms spread across multiple workers. This encoding enables the system to be resilient to a predefined number of stragglers, securely protect against a certain number of malicious workers, and guarantee data privacy even when workers collude. The significant contribution of LCC is in establishing an optimal tradeoff among these three critical parameters—resiliency, security, and privacy. The encoding of data is universal for all computations up to a certain degree, which underscores the applicability and utility of LCC in varied computational settings without prior specificity about the task.

Theoretical Contributions

The paper's main theoretical results affirm that LCC achieves the optimal resilience, security, and privacy boundaries for polynomial functions under certain constraints. Specifically, the achieved tradeoff is mathematically characterized by the requirement:

$(K + T - 1)\mathrm{deg}(f) + S + 2A + 1 \leq N$

where $K$ is the number of input batches, $T$ the privacy parameter, $S$ the resiliency parameter, $A$ the security parameter, $\mathrm{deg}(f)$ the degree of polynomial $f$, and $N$ the total number of workers. This relationship underscores the conditions under which LCC can perform reliably, underscoring its optimality and efficiency in balancing these tradeoffs.

Practical Implementation and Results

A practical embodiment of the LCC framework is shown through its application to distributed linear regression tasks. Experiments carried out on the AWS EC2 infrastructure reveal significant performance enhancements. For instance, the LCC implementation reduced computational run-time by up to 13.43 times as compared to uncoded schemes and 12.65 times over conventional straggler mitigation strategies like gradient coding and MVM-based approaches. These results vividly demonstrate LCC's potential as a highly efficient computing paradigm in practical scenarios involving common machine learning tasks.

Implications and Future Directions

Lagrange Coded Computing exemplifies an overview of theoretical elegance and practical utility, paving the way for more robust and efficient distributed systems. The outcomes suggest substantial avenues for further exploration in secure multiparty computations and robust machine learning systems. Given the emphasis on privacy and security, LCC can potentially reshape approaches in scenarios where data confidentiality and integrity are critical.

Future research might expand on optimizing the computational and communication overheads of LCC implementations or adapting the framework to non-polynomial computing tasks. Moreover, the adaptation of LCC in real-world scenarios like blockchain and federated learning could uncover novel advantages, particularly in environments necessitating high resiliency and security standards.

Overall, LCC represents an important step in coded computing research, offering robust solutions to intrinsic challenges in distributed systems while simultaneously maintaining a clear, decipherable mathematical formulation of its capabilities.