Songze Li

CR
h-index28
21papers
1,459citations
Novelty63%
AI Score51

21 Papers

15.2ITMar 24, 2022
SwiftAgg+: Achieving Asymptotically Optimal Communication Loads in Secure Aggregation for Federated Learning

Tayyebeh Jahani-Nezhad, Mohammad Ali Maddah-Ali, Songze Li et al.

We propose SwiftAgg+, a novel secure aggregation protocol for federated learning systems, where a central server aggregates local models of $N \in \mathbb{N}$ distributed users, each of size $L \in \mathbb{N}$, trained on their local data, in a privacy-preserving manner. SwiftAgg+ can significantly reduce the communication overheads without any compromise on security, and achieve optimal communication loads within diminishing gaps. Specifically, in presence of at most $D=o(N)$ dropout users, SwiftAgg+ achieves a per-user communication load of $(1+\mathcal{O}(\frac{1}{N}))L$ symbols and a server communication load of $(1+\mathcal{O}(\frac{1}{N}))L$ symbols, with a worst-case information-theoretic security guarantee, against any subset of up to $T=o(N)$ semi-honest users who may also collude with the curious server. Moreover, the proposed SwiftAgg+ allows for a flexible trade-off between communication loads and the number of active communication links. In particular, for $T<N-D$ and for any $K\in\mathbb{N}$, SwiftAgg+ can achieve the server communication load of $(1+\frac{T}{K})L$ symbols, and per-user communication load of up to $(1+\frac{T+D}{K})L$ symbols, where the number of pair-wise active connections in the network is $\frac{N}{2}(K+T+D+1)$.

13.0LGMay 31, 2022
Secure Federated Clustering

Songze Li, Sizai Hou, Baturalp Buyukates et al.

We consider a foundational unsupervised learning task of $k$-means data clustering, in a federated learning (FL) setting consisting of a central server and many distributed clients. We develop SecFC, which is a secure federated clustering algorithm that simultaneously achieves 1) universal performance: no performance loss compared with clustering over centralized data, regardless of data distribution across clients; 2) data privacy: each client's private data and the cluster centers are not leaked to other clients and the server. In SecFC, the clients perform Lagrange encoding on their local data and share the coded data in an information-theoretically private manner; then leveraging the algebraic structure of the coding, the FL network exactly executes the Lloyd's $k$-means heuristic over the coded data to obtain the final clustering. Experiment results on synthetic and real datasets demonstrate the universally superior performance of SecFC for different data distributions across clients, and its computational practicality for various combinations of system parameters. Finally, we propose an extension of SecFC to further provide membership privacy for all data points.

18.4LGApr 26, 2023
FedVS: Straggler-Resilient and Privacy-Preserving Vertical Federated Learning for Split Models

Songze Li, Duanyi Yao, Jin Liu

In a vertical federated learning (VFL) system consisting of a central server and many distributed clients, the training data are vertically partitioned such that different features are privately stored on different clients. The problem of split VFL is to train a model split between the server and the clients. This paper aims to address two major challenges in split VFL: 1) performance degradation due to straggling clients during training; and 2) data and model privacy leakage from clients' uploaded data embeddings. We propose FedVS to simultaneously address these two challenges. The key idea of FedVS is to design secret sharing schemes for the local data and models, such that information-theoretical privacy against colluding clients and curious server is guaranteed, and the aggregation of all clients' embeddings is reconstructed losslessly, via decrypting computation shares from the non-straggling clients. Extensive experiments on various types of VFL datasets (including tabular, CV, and multi-view) demonstrate the universal advantages of FedVS in straggler mitigation and privacy protection over baseline protocols.

1.8LGJun 18, 2022
Secure Embedding Aggregation for Federated Representation Learning

Jiaxiang Tang, Jinbao Zhu, Songze Li et al.

We consider a federated representation learning framework, where with the assistance of a central server, a group of $N$ distributed clients train collaboratively over their private data, for the representations (or embeddings) of a set of entities (e.g., users in a social network). Under this framework, for the key step of aggregating local embeddings trained privately at the clients, we develop a secure embedding aggregation protocol named \scheme, which leverages all potential aggregation opportunities among all the clients, while providing privacy guarantees for the set of local entities and corresponding embeddings \emph{simultaneously} at each client, against a curious server and up to $T < N/2$ colluding clients.

12.0CLSep 25, 2025Code
Enrich-on-Graph: Query-Graph Alignment for Complex Reasoning with LLM Enriching

Songze Li, Zhiqiang Liu, Zhengke Gui et al.

Large Language Models (LLMs) exhibit strong reasoning capabilities in complex tasks. However, they still struggle with hallucinations and factual errors in knowledge-intensive scenarios like knowledge graph question answering (KGQA). We attribute this to the semantic gap between structured knowledge graphs (KGs) and unstructured queries, caused by inherent differences in their focuses and structures. Existing methods usually employ resource-intensive, non-scalable workflows reasoning on vanilla KGs, but overlook this gap. To address this challenge, we propose a flexible framework, Enrich-on-Graph (EoG), which leverages LLMs' prior knowledge to enrich KGs, bridge the semantic gap between graphs and queries. EoG enables efficient evidence extraction from KGs for precise and robust reasoning, while ensuring low computational costs, scalability, and adaptability across different methods. Furthermore, we propose three graph quality evaluation metrics to analyze query-graph alignment in KGQA task, supported by theoretical validation of our optimization objectives. Extensive experiments on two KGQA benchmark datasets indicate that EoG can effectively generate high-quality KGs and achieve the state-of-the-art performance. Our code and data are available at https://github.com/zjukg/Enrich-on-Graph.

6.2CVFeb 10, 2025Code
When Data Manipulation Meets Attack Goals: An In-depth Survey of Attacks for VLMs

Aobotao Dai, Xinyu Ma, Lei Chen et al.

Vision-Language Models (VLMs) have gained considerable prominence in recent years due to their remarkable capability to effectively integrate and process both textual and visual information. This integration has significantly enhanced performance across a diverse spectrum of applications, such as scene perception and robotics. However, the deployment of VLMs has also given rise to critical safety and security concerns, necessitating extensive research to assess the potential vulnerabilities these VLM systems may harbor. In this work, we present an in-depth survey of the attack strategies tailored for VLMs. We categorize these attacks based on their underlying objectives - namely jailbreak, camouflage, and exploitation - while also detailing the various methodologies employed for data manipulation of VLMs. Meanwhile, we outline corresponding defense mechanisms that have been proposed to mitigate these vulnerabilities. By discerning key connections and distinctions among the diverse types of attacks, we propose a compelling taxonomy for VLM attacks. Moreover, we summarize the evaluation metrics that comprehensively describe the characteristics and impact of different attacks on VLMs. Finally, we conclude with a discussion of promising future research directions that could further enhance the robustness and safety of VLMs, emphasizing the importance of ongoing exploration in this critical area of study. To facilitate community engagement, we maintain an up-to-date project page, accessible at: https://github.com/AobtDai/VLM_Attack_Paper_List.

4.9CLNov 11, 2025
Last Layer Logits to Logic: Empowering LLMs with Logic-Consistent Structured Knowledge Reasoning

Songze Li, Zhiqiang Liu, Zhaoyan Gong et al.

Large Language Models (LLMs) achieve excellent performance in natural language reasoning tasks through pre-training on vast unstructured text, enabling them to understand the logic in natural language and generate logic-consistent responses. However, the representational differences between unstructured and structured knowledge make LLMs inherently struggle to maintain logic consistency, leading to \textit{Logic Drift} challenges in structured knowledge reasoning tasks such as Knowledge Graph Question Answering (KGQA). Existing methods address this limitation by designing complex workflows embedded in prompts to guide LLM reasoning. Nevertheless, these approaches only provide input-level guidance and fail to fundamentally address the \textit{Logic Drift} in LLM outputs. Additionally, their inflexible reasoning workflows cannot adapt to different tasks and knowledge graphs. To enhance LLMs' logic consistency in structured knowledge reasoning, we specifically target the logits output from the autoregressive generation process. We propose the \textit{Logits-to-Logic} framework, which incorporates logits strengthening and logits filtering as core modules to correct logical defects in LLM outputs. Extensive experiments show that our approach significantly improves LLMs' logic consistency in structured knowledge reasoning and achieves state-of-the-art performance on multiple KGQA benchmarks.

27.9AINov 14, 2024
Navigating the Risks: A Survey of Security, Privacy, and Ethics Threats in LLM-Based Agents

Yuyou Gan, Yong Yang, Zhe Ma et al.

With the continuous development of large language models (LLMs), transformer-based models have made groundbreaking advances in numerous natural language processing (NLP) tasks, leading to the emergence of a series of agents that use LLMs as their control hub. While LLMs have achieved success in various tasks, they face numerous security and privacy threats, which become even more severe in the agent scenarios. To enhance the reliability of LLM-based applications, a range of research has emerged to assess and mitigate these risks from different perspectives. To help researchers gain a comprehensive understanding of various risks, this survey collects and analyzes the different threats faced by these agents. To address the challenges posed by previous taxonomies in handling cross-module and cross-stage threats, we propose a novel taxonomy framework based on the sources and impacts. Additionally, we identify six key features of LLM-based agents, based on which we summarize the current research progress and analyze their limitations. Subsequently, we select four representative agents as case studies to analyze the risks they may face in practical use. Finally, based on the aforementioned analyses, we propose future research directions from the perspectives of data, methodology, and policy, respectively.

2.3CRApr 17, 2024
OmniLytics+: A Secure, Efficient, and Affordable Blockchain Data Market for Machine Learning through Off-Chain Processing

Songze Li, Mingzhe Liu, Mengqi Chen

The rapid development of large machine learning (ML) models requires a massive amount of training data, resulting in booming demands of data sharing and trading through data markets. Traditional centralized data markets suffer from low level of security, and emerging decentralized platforms are faced with efficiency and privacy challenges. In this paper, we propose OmniLytics+, the first decentralized data market, built upon blockchain and smart contract technologies, to simultaneously achieve 1) data (resp., model) privacy for the data (resp. model) owner; 2) robustness against malicious data owners; 3) efficient data validation and aggregation. Specifically, adopting the zero-knowledge (ZK) rollup paradigm, OmniLytics+ proposes to secret share encrypted local gradients, computed from the encrypted global model, with a set of untrusted off-chain servers, who collaboratively generate a ZK proof on the validity of the gradient. In this way, the storage and processing overheads are securely offloaded from blockchain verifiers, significantly improving the privacy, efficiency, and affordability over existing rollup solutions. We implement the proposed OmniLytics+ data market as an Ethereum smart contract [41]. Extensive experiments demonstrate the effectiveness of OmniLytics+ in training large ML models in presence of malicious data owner, and the substantial advantages of OmniLytics+ in gas cost and execution time over baselines.

6.4CRSep 29, 2025
FuncPoison: Poisoning Function Library to Hijack Multi-agent Autonomous Driving Systems

Yuzhen Long, Songze Li

Autonomous driving systems increasingly rely on multi-agent architectures powered by large language models (LLMs), where specialized agents collaborate to perceive, reason, and plan. A key component of these systems is the shared function library, a collection of software tools that agents use to process sensor data and navigate complex driving environments. Despite its critical role in agent decision-making, the function library remains an under-explored vulnerability. In this paper, we introduce FuncPoison, a novel poisoning-based attack targeting the function library to manipulate the behavior of LLM-driven multi-agent autonomous systems. FuncPoison exploits two key weaknesses in how agents access the function library: (1) agents rely on text-based instructions to select tools; and (2) these tools are activated using standardized command formats that attackers can replicate. By injecting malicious tools with deceptive instructions, FuncPoison manipulates one agent s decisions--such as misinterpreting road conditions--triggering cascading errors that mislead other agents in the system. We experimentally evaluate FuncPoison on two representative multi-agent autonomous driving systems, demonstrating its ability to significantly degrade trajectory accuracy, flexibly target specific agents to induce coordinated misbehavior, and evade diverse defense mechanisms. Our results reveal that the function library, often considered a simple toolset, can serve as a critical attack surface in LLM-based autonomous driving systems, raising elevated concerns on their reliability.

11.7ITFeb 8, 2022
SwiftAgg: Communication-Efficient and Dropout-Resistant Secure Aggregation for Federated Learning with Worst-Case Security Guarantees

Tayyebeh Jahani-Nezhad, Mohammad Ali Maddah-Ali, Songze Li et al.

We propose SwiftAgg, a novel secure aggregation protocol for federated learning systems, where a central server aggregates local models of $N$ distributed users, each of size $L$, trained on their local data, in a privacy-preserving manner. Compared with state-of-the-art secure aggregation protocols, SwiftAgg significantly reduces the communication overheads without any compromise on security. Specifically, in presence of at most $D$ dropout users, SwiftAgg achieves a users-to-server communication load of $(T+1)L$ and a users-to-users communication load of up to $(N-1)(T+D+1)L$, with a worst-case information-theoretic security guarantee, against any subset of up to $T$ semi-honest users who may also collude with the curious server. The key idea of SwiftAgg is to partition the users into groups of size $D+T+1$, then in the first phase, secret sharing and aggregation of the individual models are performed within each group, and then in the second phase, model aggregation is performed on $D+T+1$ sequences of users across the groups. If a user in a sequence drops out in the second phase, the rest of the sequence remain silent. This design allows only a subset of users to communicate with each other, and only the users in a single group to directly communicate with the server, eliminating the requirements of 1) all-to-all communication network across users; and 2) all users communicating with the server, for other secure aggregation protocols. This helps to substantially slash the communication costs of the system.

25.7LGSep 29, 2021
LightSecAgg: a Lightweight and Versatile Design for Secure Aggregation in Federated Learning

Jinhyun So, Chaoyang He, Chien-Sheng Yang et al.

Secure model aggregation is a key component of federated learning (FL) that aims at protecting the privacy of each user's individual model while allowing for their global aggregation. It can be applied to any aggregation-based FL approach for training a global or personalized model. Model aggregation needs to also be resilient against likely user dropouts in FL systems, making its design substantially more complex. State-of-the-art secure aggregation protocols rely on secret sharing of the random-seeds used for mask generations at the users to enable the reconstruction and cancellation of those belonging to the dropped users. The complexity of such approaches, however, grows substantially with the number of dropped users. We propose a new approach, named LightSecAgg, to overcome this bottleneck by changing the design from "random-seed reconstruction of the dropped users" to "one-shot aggregate-mask reconstruction of the active users via mask encoding/decoding". We show that LightSecAgg achieves the same privacy and dropout-resiliency guarantees as the state-of-the-art protocols while significantly reducing the overhead for resiliency against dropped users. We also demonstrate that, unlike existing schemes, LightSecAgg can be applied to secure aggregation in the asynchronous FL setting. Furthermore, we provide a modular system design and optimized on-device parallelization for scalable implementation, by enabling computational overlapping between model training and on-device encoding, as well as improving the speed of concurrent receiving and sending of chunked masks. We evaluate LightSecAgg via extensive experiments for training diverse models on various datasets in a realistic FL system with large number of users and demonstrate that LightSecAgg significantly reduces the total training time.

17.9CRJul 12, 2021
OmniLytics: A Blockchain-based Secure Data Market for Decentralized Machine Learning

Jiacheng Liang, Songze Li, Bochuan Cao et al.

We propose OmniLytics, a blockchain-based secure data trading marketplace for machine learning applications. Utilizing OmniLytics, many distributed data owners can contribute their private data to collectively train an ML model requested by some model owners, and receive compensation for data contribution. OmniLytics enables such model training while simultaneously providing 1) model security against curious data owners; 2) data security against the curious model and data owners; 3) resilience to malicious data owners who provide faulty results to poison model training; and 4) resilience to malicious model owners who intend to evade payment. OmniLytics is implemented as a blockchain smart contract to guarantee the atomicity of payment. In OmniLytics, a model owner splits its model into the private and public parts and publishes the public part on the contract. Through the execution of the contract, the participating data owners securely aggregate their locally trained models to update the model owner's public model and receive reimbursement through the contract. We implement a working prototype of OmniLytics on Ethereum blockchain and perform extensive experiments to measure its gas cost, execution time, and model quality under various parameter combinations. For training a CNN on the MNIST dataset, the MO is able to boost its model accuracy from 62% to 83% within 500ms in blockchain processing time.This demonstrates the effectiveness of OmniLytics for practical deployment.

11.5CRNov 22, 2020
TaiJi: Longest Chain Availability with BFT Fast Confirmation

Songze Li, David Tse

Most state machine replication protocols are either based on the 40-years-old Byzantine Fault Tolerance (BFT) theory or the more recent Nakamoto's longest chain design. Longest chain protocols, designed originally in the Proof-of-Work (PoW) setting, are available under dynamic participation, but has probabilistic confirmation with long latency dependent on the security parameter. BFT protocols, designed for the permissioned setting, has fast deterministic confirmation, but assume a fixed number of nodes always online. We present a new construction which combines a longest chain protocol and a BFT protocol to get the best of both worlds. Using this construction, we design TaiJi, the first dynamically available PoW protocol which has almost deterministic confirmation with latency independent of the security parameter. In contrast to previous hybrid approaches which use a single longest chain to sample participants to run a BFT protocol, our native PoW construction uses many independent longest chains to sample propose actions and vote actions for the BFT protocol. This design enables TaiJi to inherit the full dynamic availability of Bitcoin, as well as its full unpredictability, making it secure against fully-adaptive adversaries with up to 50% of online hash power.

19.4CROct 2, 2019
Coded Merkle Tree: Solving Data Availability Attacks in Blockchains

Mingchao Yu, Saeid Sahraei, Songze Li et al.

In this paper, we propose coded Merkle tree (CMT), a novel hash accumulator that offers a constant-cost protection against data availability attacks in blockchains, even if the majority of the network nodes are malicious. A CMT is constructed using a family of sparse erasure codes on each layer, and is recovered by iteratively applying a peeling-decoding technique that enables a compact proof for data availability attack on any layer. Our algorithm enables any node to verify the full availability of any data block generated by the system by just downloading a $Θ(1)$ byte block hash commitment and randomly sampling $Θ(\log b)$ bytes, where $b$ is the size of the data block. With the help of only one connected honest node in the system, our method also allows any node to verify any tampering of the coded Merkle tree by just downloading $Θ(\log b)$ bytes. We provide a modular library for CMT in Rust and Python and demonstrate its efficacy inside the Parity Bitcoin client.

3.3ITJun 26, 2019
Coded State Machine -- Scaling State Machine Execution under Byzantine Faults

Songze Li, Saeid Sahraei, Mingchao Yu et al.

We introduce an information-theoretic framework, named Coded State Machine (CSM), to securely and efficiently execute multiple state machines on untrusted network nodes, some of which are Byzantine. The standard method of solving this problem is using State Machine Replication, which achieves high security at the cost of low efficiency. We propose CSM, which achieves the optimal linear scaling in storage efficiency, throughput, and security simultaneously with the size of the network. The storage efficiency is scaled via the design of Lagrange coded states and coded input commands that require the same storage size as their origins. The computational efficiency is scaled using a novel delegation algorithm, called INTERMIX, which is an information-theoretically verifiable matrix-vector multiplication algorithm of independent interest. Using INTERMIX, the network nodes securely delegate their coding operations to a single worker node, and a small group of randomly selected auditor nodes verify its correctness, so that computational efficiency can scale almost linearly with the network size, without compromising on security.

16.9LGNov 8, 2018
Pipe-SGD: A Decentralized Pipelined SGD Framework for Distributed Deep Net Training

Youjie Li, Mingchao Yu, Songze Li et al.

Distributed training of deep nets is an important technique to address some of the present day computing challenges like memory consumption and computational demands. Classical distributed approaches, synchronous or asynchronous, are based on the parameter server architecture, i.e., worker nodes compute gradients which are communicated to the parameter server while updated parameters are returned. Recently, distributed training with AllReduce operations gained popularity as well. While many of those operations seem appealing, little is reported about wall-clock training time improvements. In this paper, we carefully analyze the AllReduce based setup, propose timing models which include network latency, bandwidth, cluster size and compute time, and demonstrate that a pipelined training with a width of two combines the best of both synchronous and asynchronous training. Specifically, for a setup consisting of a four-node GPU cluster we show wall-clock time training improvements of up to 5.4x compared to conventional approaches.

16.1LGNov 8, 2018
GradiVeQ: Vector Quantization for Bandwidth-Efficient Gradient Aggregation in Distributed CNN Training

Mingchao Yu, Zhifeng Lin, Krishna Narra et al.

Data parallelism can boost the training speed of convolutional neural networks (CNN), but could suffer from significant communication costs caused by gradient aggregation. To alleviate this problem, several scalar quantization techniques have been developed to compress the gradients. But these techniques could perform poorly when used together with decentralized aggregation protocols like ring all-reduce (RAR), mainly due to their inability to directly aggregate compressed gradients. In this paper, we empirically demonstrate the strong linear correlations between CNN gradients, and propose a gradient vector quantization technique, named GradiVeQ, to exploit these correlations through principal component analysis (PCA) for substantial gradient dimension reduction. GradiVeQ enables direct aggregation of compressed gradients, hence allows us to build a distributed learning system that parallelizes GradiVeQ gradient compression and RAR communications. Extensive experiments on popular CNNs demonstrate that applying GradiVeQ slashes the wall-clock gradient aggregation time of the original RAR by more than 5X without noticeable accuracy loss, and reduces the end-to-end training time by almost 50%. The results also show that GradiVeQ is compatible with scalar quantization techniques such as QSGD (Quantized SGD), and achieves a much higher speed-up gain under the same compression ratio.

21.9CRSep 27, 2018
PolyShard: Coded Sharding Achieves Linearly Scaling Efficiency and Security Simultaneously

Songze Li, Mingchao Yu, Chien-Sheng Yang et al.

Today's blockchain designs suffer from a trilemma claiming that no blockchain system can simultaneously achieve decentralization, security, and performance scalability. For current blockchain systems, as more nodes join the network, the efficiency of the system (computation, communication, and storage) stays constant at best. A leading idea for enabling blockchains to scale efficiency is the notion of sharding: different subsets of nodes handle different portions of the blockchain, thereby reducing the load for each individual node. However, existing sharding proposals achieve efficiency scaling by compromising on trust - corrupting the nodes in a given shard will lead to the permanent loss of the corresponding portion of data. In this paper, we settle the trilemma by demonstrating a new protocol for coded storage and computation in blockchains. In particular, we propose PolyShard: ``polynomially coded sharding'' scheme that achieves information-theoretic upper bounds on the efficiency of the storage, system throughput, as well as on trust, thus enabling a truly scalable system. We provide simulation results that numerically demonstrate the performance improvement over state of the arts, and the scalability of the PolyShard system. Finally, we discuss potential enhancements, and highlight practical considerations in building such a system.

29.3ITJun 4, 2018
Lagrange Coded Computing: Optimal Design for Resiliency, Security and Privacy

Qian Yu, Songze Li, Netanel Raviv et al.

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.

14.9ITMay 24, 2018
Polynomially Coded Regression: Optimal Straggler Mitigation via Data Encoding

Songze Li, Seyed Mohammadreza Mousavi Kalan, Qian Yu et al.

We consider the problem of training a least-squares regression model on a large dataset using gradient descent. The computation is carried out on a distributed system consisting of a master node and multiple worker nodes. Such distributed systems are significantly slowed down due to the presence of slow-running machines (stragglers) as well as various communication bottlenecks. We propose "polynomially coded regression" (PCR) that substantially reduces the effect of stragglers and lessens the communication burden in such systems. The key idea of PCR is to encode the partial data stored at each worker, such that the computations at the workers can be viewed as evaluating a polynomial at distinct points. This allows the master to compute the final gradient by interpolating this polynomial. PCR significantly reduces the recovery threshold, defined as the number of workers the master has to wait for prior to computing the gradient. In particular, PCR requires a recovery threshold that scales inversely proportionally with the amount of computation/storage available at each worker. In comparison, state-of-the-art straggler-mitigation schemes require a much higher recovery threshold that only decreases linearly in the per worker computation/storage load. We prove that PCR's recovery threshold is near minimal and within a factor two of the best possible scheme. Our experiments over Amazon EC2 demonstrate that compared with state-of-the-art schemes, PCR improves the run-time by 1.50x ~ 2.36x with naturally occurring stragglers, and by as much as 2.58x ~ 4.29x with artificial stragglers.