Fast and parallel decoding for transducerWei Kang, Liyong Guo, Fangjun Kuang et al. · nvidia
The transducer architecture is becoming increasingly popular in the field of speech recognition, because it is naturally streaming as well as high in accuracy. One of the drawbacks of transducer is that it is difficult to decode in a fast and parallel way due to an unconstrained number of symbols that can be emitted per time step. In this work, we introduce a constrained version of transducer loss to learn strictly monotonic alignments between the sequences; we also improve the standard greedy search and beam search algorithms by limiting the number of symbols that can be emitted per time step in transducer decoding, making it more efficient to decode in parallel with batches. Furthermore, we propose an finite state automaton-based (FSA) parallel beam search algorithm that can run with graphs on GPU efficiently. The experiment results show that we achieve slight word error rate (WER) improvement as well as significant speedup in decoding. Our work is open-sourced and publicly available\footnote{https://github.com/k2-fsa/icefall}.
Delay-penalized transducer for low-latency streaming ASRWei Kang, Zengwei Yao, Fangjun Kuang et al. · nvidia
In streaming automatic speech recognition (ASR), it is desirable to reduce latency as much as possible while having minimum impact on recognition accuracy. Although a few existing methods are able to achieve this goal, they are difficult to implement due to their dependency on external alignments. In this paper, we propose a simple way to penalize symbol delay in transducer model, so that we can balance the trade-off between symbol delay and accuracy for streaming models without external alignments. Specifically, our method adds a small constant times (T/2 - t), where T is the number of frames and t is the current frame, to all the non-blank log-probabilities (after normalization) that are fed into the two dimensional transducer recursion. For both streaming Conformer models and unidirectional long short-term memory (LSTM) models, experimental results show that it can significantly reduce the symbol delay with an acceptable performance degradation. Our method achieves similar delay-accuracy trade-off to the previously published FastEmit, but we believe our method is preferable because it has a better justification: it is equivalent to penalizing the average symbol delay. Our work is open-sourced and publicly available (https://github.com/k2-fsa/k2).
Predicting Multi-Codebook Vector Quantization Indexes for Knowledge DistillationLiyong Guo, Xiaoyu Yang, Quandong Wang et al. · nvidia
Knowledge distillation(KD) is a common approach to improve model performance in automatic speech recognition (ASR), where a student model is trained to imitate the output behaviour of a teacher model. However, traditional KD methods suffer from teacher label storage issue, especially when the training corpora are large. Although on-the-fly teacher label generation tackles this issue, the training speed is significantly slower as the teacher model has to be evaluated every batch. In this paper, we reformulate the generation of teacher label as a codec problem. We propose a novel Multi-codebook Vector Quantization (MVQ) approach that compresses teacher embeddings to codebook indexes (CI). Based on this, a KD training framework (MVQ-KD) is proposed where a student model predicts the CI generated from the embeddings of a self-supervised pre-trained teacher model. Experiments on the LibriSpeech clean-100 hour show that MVQ-KD framework achieves comparable performance as traditional KD methods (l1, l2), while requiring 256 times less storage. When the full LibriSpeech dataset is used, MVQ-KD framework results in 13.8% and 8.2% relative word error rate reductions (WERRs) for non -streaming transducer on test-clean and test-other and 4.0% and 4.9% for streaming transducer. The implementation of this work is already released as a part of the open-source project icefall.
Make Landscape Flatter in Differentially Private Federated LearningYifan Shi, Yingqi Liu, Kang Wei et al.
To defend the inference attacks and mitigate the sensitive information leakages in Federated Learning (FL), client-level Differentially Private FL (DPFL) is the de-facto standard for privacy protection by clipping local updates and adding random noise. However, existing DPFL methods tend to make a sharper loss landscape and have poorer weight perturbation robustness, resulting in severe performance degradation. To alleviate these issues, we propose a novel DPFL algorithm named DP-FedSAM, which leverages gradient perturbation to mitigate the negative impact of DP. Specifically, DP-FedSAM integrates Sharpness Aware Minimization (SAM) optimizer to generate local flatness models with better stability and weight perturbation robustness, which results in the small norm of local updates and robustness to DP noise, thereby improving the performance. From the theoretical perspective, we analyze in detail how DP-FedSAM mitigates the performance degradation induced by DP. Meanwhile, we give rigorous privacy guarantees with Rényi DP and present the sensitivity analysis of local updates. At last, we empirically confirm that our algorithm achieves state-of-the-art (SOTA) performance compared with existing SOTA baselines in DPFL. Code is available at https://github.com/YMJS-Irfan/DP-FedSAM
Zipformer: A faster and better encoder for automatic speech recognitionZengwei Yao, Liyong Guo, Xiaoyu Yang et al.
The Conformer has become the most popular encoder model for automatic speech recognition (ASR). It adds convolution modules to a transformer to learn both local and global dependencies. In this work we describe a faster, more memory-efficient, and better-performing transformer, called Zipformer. Modeling changes include: 1) a U-Net-like encoder structure where middle stacks operate at lower frame rates; 2) reorganized block structure with more modules, within which we re-use attention weights for efficiency; 3) a modified form of LayerNorm called BiasNorm allows us to retain some length information; 4) new activation functions SwooshR and SwooshL work better than Swish. We also propose a new optimizer, called ScaledAdam, which scales the update by each tensor's current scale to keep the relative change about the same, and also explictly learns the parameter scale. It achieves faster convergence and better performance than Adam. Extensive experiments on LibriSpeech, Aishell-1, and WenetSpeech datasets demonstrate the effectiveness of our proposed Zipformer over other state-of-the-art ASR models. Our code is publicly available at https://github.com/k2-fsa/icefall.
20.0ASJun 23, 2022
Pruned RNN-T for fast, memory-efficient ASR trainingFangjun Kuang, Liyong Guo, Wei Kang et al.
The RNN-Transducer (RNN-T) framework for speech recognition has been growing in popularity, particularly for deployed real-time ASR systems, because it combines high accuracy with naturally streaming recognition. One of the drawbacks of RNN-T is that its loss function is relatively slow to compute, and can use a lot of memory. Excessive GPU memory usage can make it impractical to use RNN-T loss in cases where the vocabulary size is large: for example, for Chinese character-based ASR. We introduce a method for faster and more memory-efficient RNN-T loss computation. We first obtain pruning bounds for the RNN-T recursion using a simple joiner network that is linear in the encoder and decoder embeddings; we can evaluate this without using much memory. We then use those pruning bounds to evaluate the full, non-linear joiner network.
PromptASR for contextualized ASR with controllable styleXiaoyu Yang, Wei Kang, Zengwei Yao et al.
Prompts are crucial to large language models as they provide context information such as topic or logical relationships. Inspired by this, we propose PromptASR, a framework that integrates prompts in end-to-end automatic speech recognition (E2E ASR) systems to achieve contextualized ASR with controllable style of transcriptions. Specifically, a dedicated text encoder encodes the text prompts and the encodings are injected into the speech encoder by cross-attending the features from two modalities. When using the ground truth text from preceding utterances as content prompt, the proposed system achieves 21.9% and 6.8% relative word error rate reductions on a book reading dataset and an in-house dataset compared to a baseline ASR system. The system can also take word-level biasing lists as prompt to improve recognition accuracy on rare words. An additional style prompt can be given to the text encoder and guide the ASR system to output different styles of transcriptions. The code is available at icefall.
16.0LGMar 7, 2023
Amplitude-Varying Perturbation for Balancing Privacy and Utility in Federated LearningXin Yuan, Wei Ni, Ming Ding et al.
While preserving the privacy of federated learning (FL), differential privacy (DP) inevitably degrades the utility (i.e., accuracy) of FL due to model perturbations caused by DP noise added to model updates. Existing studies have considered exclusively noise with persistent root-mean-square amplitude and overlooked an opportunity of adjusting the amplitudes to alleviate the adverse effects of the noise. This paper presents a new DP perturbation mechanism with a time-varying noise amplitude to protect the privacy of FL and retain the capability of adjusting the learning performance. Specifically, we propose a geometric series form for the noise amplitude and reveal analytically the dependence of the series on the number of global aggregations and the $(ε,δ)$-DP requirement. We derive an online refinement of the series to prevent FL from premature convergence resulting from excessive perturbation noise. Another important aspect is an upper bound developed for the loss function of a multi-layer perceptron (MLP) trained by FL running the new DP mechanism. Accordingly, the optimal number of global aggregations is obtained, balancing the learning and privacy. Extensive experiments are conducted using MLP, supporting vector machine, and convolutional neural network models on four public datasets. The contribution of the new DP mechanism to the convergence and accuracy of privacy-preserving FL is corroborated, compared to the state-of-the-art Gaussian noise mechanism with a persistent noise amplitude.
25.5LGFeb 8, 2023
Improving the Model Consistency of Decentralized Federated LearningYifan Shi, Li Shen, Kang Wei et al.
To mitigate the privacy leakages and communication burdens of Federated Learning (FL), decentralized FL (DFL) discards the central server and each client only communicates with its neighbors in a decentralized communication network. However, existing DFL suffers from high inconsistency among local clients, which results in severe distribution shift and inferior performance compared with centralized FL (CFL), especially on heterogeneous data or sparse communication topology. To alleviate this issue, we propose two DFL algorithms named DFedSAM and DFedSAM-MGS to improve the performance of DFL. Specifically, DFedSAM leverages gradient perturbation to generate local flat models via Sharpness Aware Minimization (SAM), which searches for models with uniformly low loss values. DFedSAM-MGS further boosts DFedSAM by adopting Multiple Gossip Steps (MGS) for better model consistency, which accelerates the aggregation of local flat models and better balances communication complexity and generalization. Theoretically, we present improved convergence rates $\small \mathcal{O}\big(\frac{1}{\sqrt{KT}}+\frac{1}{T}+\frac{1}{K^{1/2}T^{3/2}(1-λ)^2}\big)$ and $\small \mathcal{O}\big(\frac{1}{\sqrt{KT}}+\frac{1}{T}+\frac{λ^Q+1}{K^{1/2}T^{3/2}(1-λ^Q)^2}\big)$ in non-convex setting for DFedSAM and DFedSAM-MGS, respectively, where $1-λ$ is the spectral gap of gossip matrix and $Q$ is the number of MGS. Empirically, our methods can achieve competitive performance compared with CFL methods and outperform existing DFL methods.
18.8CRApr 13, 2022
Towards A Critical Evaluation of Robustness for Deep Learning Backdoor CountermeasuresHuming Qiu, Hua Ma, Zhi Zhang et al.
Since Deep Learning (DL) backdoor attacks have been revealed as one of the most insidious adversarial attacks, a number of countermeasures have been developed with certain assumptions defined in their respective threat models. However, the robustness of these countermeasures is inadvertently ignored, which can introduce severe consequences, e.g., a countermeasure can be misused and result in a false implication of backdoor detection. For the first time, we critically examine the robustness of existing backdoor countermeasures with an initial focus on three influential model-inspection ones that are Neural Cleanse (S&P'19), ABS (CCS'19), and MNTD (S&P'21). Although the three countermeasures claim that they work well under their respective threat models, they have inherent unexplored non-robust cases depending on factors such as given tasks, model architectures, datasets, and defense hyper-parameter, which are \textit{not even rooted from delicate adaptive attacks}. We demonstrate how to trivially bypass them aligned with their respective threat models by simply varying aforementioned factors. Particularly, for each defense, formal proofs or empirical studies are used to reveal its two non-robust cases where it is not as robust as it claims or expects, especially the recent MNTD. This work highlights the necessity of thoroughly evaluating the robustness of backdoor countermeasures to avoid their misleading security implications in unknown non-robust cases.
9.7SDSep 1, 2024
LibriheavyMix: A 20,000-Hour Dataset for Single-Channel Reverberant Multi-Talker Speech Separation, ASR and Speaker DiarizationZengrui Jin, Yifan Yang, Mohan Shi et al.
The evolving speech processing landscape is increasingly focused on complex scenarios like meetings or cocktail parties with multiple simultaneous speakers and far-field conditions. Existing methodologies for addressing these challenges fall into two categories: multi-channel and single-channel solutions. Single-channel approaches, notable for their generality and convenience, do not require specific information about microphone arrays. This paper presents a large-scale far-field overlapping speech dataset, crafted to advance research in speech separation, recognition, and speaker diarization. This dataset is a critical resource for decoding ``Who said What and When'' in multi-talker, reverberant environments, a daunting challenge in the field. Additionally, we introduce a pipeline system encompassing speech separation, recognition, and diarization as a foundational benchmark. Evaluations on the WHAMR! dataset validate the broad applicability of the proposed data.
2.3GTApr 9, 2023
Design of Two-Level Incentive Mechanisms for Hierarchical Federated LearningShunfeng Chu, Jun Li, Kang Wei et al.
Hierarchical Federated Learning (HFL) is a distributed machine learning paradigm tailored for multi-tiered computation architectures, which supports massive access of devices' models simultaneously. To enable efficient HFL, it is crucial to design suitable incentive mechanisms to ensure that devices actively participate in local training. However, there are few studies on incentive mechanism design for HFL. In this paper, we design two-level incentive mechanisms for the HFL with a two-tiered computing structure to encourage the participation of entities in each tier in the HFL training. In the lower-level game, we propose a coalition formation game to joint optimize the edge association and bandwidth allocation problem, and obtain efficient coalition partitions by the proposed preference rule, which can be proven to be stable by exact potential game. In the upper-level game, we design the Stackelberg game algorithm, which not only determines the optimal number of edge aggregations for edge servers to maximize their utility, but also optimize the unit reward provided for the edge aggregation performance to ensure the interests of cloud servers. Furthermore, numerical results indicate that the proposed algorithms can achieve better performance than the benchmark schemes.
ZipVoice-Dialog: Non-Autoregressive Spoken Dialogue Generation with Flow MatchingHan Zhu, Wei Kang, Liyong Guo et al.
Generating spoken dialogue is more challenging than monologue text-to-speech (TTS) due to the need for realistic turn-taking and distinct speaker timbres. Existing spoken dialogue generation models, being auto-regressive, suffer from slow and unstable inference. To overcome these limitations, we introduce ZipVoice-Dialog, a non-autoregressive zero-shot spoken dialogue generation model built upon flow matching. Key designs include: 1) speaker-turn embeddings for precise speaker turn-taking; 2) a curriculum learning strategy for stable speech-text alignment; 3) specialized strategies to enable stereo dialogue generation. Additionally, recognizing the lack of open-source large-scale spoken dialogue datasets, we curated OpenDialog, a 6.8k-hour spoken dialogue dataset from in-the-wild speech data. Furthermore, we established a benchmark to comprehensively evaluate various models. Experimental results demonstrate that ZipVoice-Dialog achieves superior performance in intelligibility, speaker turn-taking accuracy, speaker similarity, and inference speed. Our codes, model checkpoints, demo samples, and the OpenDialog dataset are all publicly available at https://github.com/k2-fsa/ZipVoice.
6.4CRJun 11, 2025Code
TooBadRL: Trigger Optimization to Boost Effectiveness of Backdoor Attacks on Deep Reinforcement LearningMingxuan Zhang, Oubo Ma, Kang Wei et al.
Deep reinforcement learning (DRL) has achieved remarkable success in a wide range of sequential decision-making applications, including robotics, healthcare, smart grids, and finance. Recent studies reveal that adversaries can implant backdoors into DRL agents during the training phase. These backdoors can later be activated by specific triggers during deployment, compelling the agent to execute targeted actions and potentially leading to severe consequences, such as drone crashes or vehicle collisions. However, existing backdoor attacks utilize simplistic and heuristic trigger configurations, overlooking the critical impact of trigger design on attack effectiveness. To address this gap, we introduce TooBadRL, the first framework to systematically optimize DRL backdoor triggers across three critical aspects: injection timing, trigger dimension, and manipulation magnitude. Specifically, we first introduce a performance-aware adaptive freezing mechanism to determine the injection timing during training. Then, we formulate trigger selection as an influence attribution problem and apply Shapley value analysis to identify the most influential trigger dimension for injection. Furthermore, we propose an adversarial input synthesis method to optimize the manipulation magnitude under environmental constraints. Extensive evaluations on three DRL algorithms and nine benchmark tasks demonstrate that TooBadRL outperforms five baseline methods in terms of attack success rate while only slightly affecting normal task performance. We further evaluate potential defense strategies from detection and mitigation perspectives. We open-source our code to facilitate reproducibility and further research.
Blank-regularized CTC for Frame Skipping in Neural TransducerYifan Yang, Xiaoyu Yang, Liyong Guo et al.
Neural Transducer and connectionist temporal classification (CTC) are popular end-to-end automatic speech recognition systems. Due to their frame-synchronous design, blank symbols are introduced to address the length mismatch between acoustic frames and output tokens, which might bring redundant computation. Previous studies managed to accelerate the training and inference of neural Transducers by discarding frames based on the blank symbols predicted by a co-trained CTC. However, there is no guarantee that the co-trained CTC can maximize the ratio of blank symbols. This paper proposes two novel regularization methods to explicitly encourage more blanks by constraining the self-loop of non-blank symbols in the CTC. It is interesting to find that the frame reduction ratio of the neural Transducer can approach the theoretical boundary. Experiments on LibriSpeech corpus show that our proposed method accelerates the inference of neural Transducer by 4 times without sacrificing performance. Our work is open-sourced and publicly available https://github.com/k2-fsa/icefall.
2.6LGMay 11, 2024
Robust Model Aggregation for Heterogeneous Federated Learning: Analysis and OptimizationsYumeng Shao, Jun Li, Long Shi et al.
Conventional synchronous federated learning (SFL) frameworks suffer from performance degradation in heterogeneous systems due to imbalanced local data size and diverse computing power on the client side. To address this problem, asynchronous FL (AFL) and semi-asynchronous FL have been proposed to recover the performance loss by allowing asynchronous aggregation. However, asynchronous aggregation incurs a new problem of inconsistency between local updates and global updates. Motivated by the issues of conventional SFL and AFL, we first propose a time-driven SFL (T-SFL) framework for heterogeneous systems. The core idea of T-SFL is that the server aggregates the models from different clients, each with varying numbers of iterations, at regular time intervals. To evaluate the learning performance of T-SFL, we provide an upper bound on the global loss function. Further, we optimize the aggregation weights to minimize the developed upper bound. Then, we develop a discriminative model selection (DMS) algorithm that removes local models from clients whose number of iterations falls below a predetermined threshold. In particular, this algorithm ensures that each client's aggregation weight accurately reflects its true contribution to the global model update, thereby improving the efficiency and robustness of the system. To validate the effectiveness of T-SFL with the DMS algorithm, we conduct extensive experiments using several popular datasets including MNIST, Cifar-10, Fashion-MNIST, and SVHN. The experimental results demonstrate that T-SFL with the DMS algorithm can reduce the latency of conventional SFL by 50\%, while achieving an average 3\% improvement in learning accuracy over state-of-the-art AFL algorithms.
2.7CLSep 18, 2025
Controlling Language Difficulty in Dialogues with Linguistic FeaturesShuyao Xu, Wenguang Wang, Handong Gao et al.
Large language models (LLMs) have emerged as powerful tools for supporting second language acquisition, particularly in simulating interactive dialogues for speaking practice. However, adapting the language difficulty of LLM-generated responses to match learners' proficiency levels remains a challenge. This work addresses this issue by proposing a framework for controlling language proficiency in educational dialogue systems. Our approach leverages three categories of linguistic features, readability features (e.g., Flesch-Kincaid Grade Level), syntactic features (e.g., syntactic tree depth), and lexical features (e.g., simple word ratio), to quantify and regulate text complexity. We demonstrate that training LLMs on linguistically annotated dialogue data enables precise modulation of language proficiency, outperforming prompt-based methods in both flexibility and stability. To evaluate this, we introduce Dilaprix, a novel metric integrating the aforementioned features, which shows strong correlation with expert judgments of language difficulty. Empirical results reveal that our approach achieves superior controllability of language proficiency while maintaining high dialogue quality.
6.6DCMay 2, 2023
Efficient Federated Learning with Enhanced Privacy via Lottery Ticket Pruning in Edge ComputingYifan Shi, Kang Wei, Li Shen et al.
Federated learning (FL) is a collaborative learning paradigm for decentralized private data from mobile terminals (MTs). However, it suffers from issues in terms of communication, resource of MTs, and privacy. Existing privacy-preserving FL methods usually adopt the instance-level differential privacy (DP), which provides a rigorous privacy guarantee but with several bottlenecks: severe performance degradation, transmission overhead, and resource constraints of edge devices such as MTs. To overcome these drawbacks, we propose Fed-LTP, an efficient and privacy-enhanced FL framework with \underline{\textbf{L}}ottery \underline{\textbf{T}}icket \underline{\textbf{H}}ypothesis (LTH) and zero-concentrated D\underline{\textbf{P}} (zCDP). It generates a pruned global model on the server side and conducts sparse-to-sparse training from scratch with zCDP on the client side. On the server side, two pruning schemes are proposed: (i) the weight-based pruning (LTH) determines the pruned global model structure; (ii) the iterative pruning further shrinks the size of the pruned model's parameters. Meanwhile, the performance of Fed-LTP is also boosted via model validation based on the Laplace mechanism. On the client side, we use sparse-to-sparse training to solve the resource-constraints issue and provide tighter privacy analysis to reduce the privacy budget. We evaluate the effectiveness of Fed-LTP on several real-world datasets in both independent and identically distributed (IID) and non-IID settings. The results clearly confirm the superiority of Fed-LTP over state-of-the-art (SOTA) methods in communication, computation, and memory efficiencies while realizing a better utility-privacy trade-off.
Towards the Flatter Landscape and Better Generalization in Federated Learning under Client-level Differential PrivacyYifan Shi, Kang Wei, Li Shen et al.
To defend the inference attacks and mitigate the sensitive information leakages in Federated Learning (FL), client-level Differentially Private FL (DPFL) is the de-facto standard for privacy protection by clipping local updates and adding random noise. However, existing DPFL methods tend to make a sharp loss landscape and have poor weight perturbation robustness, resulting in severe performance degradation. To alleviate these issues, we propose a novel DPFL algorithm named DP-FedSAM, which leverages gradient perturbation to mitigate the negative impact of DP. Specifically, DP-FedSAM integrates Sharpness Aware Minimization (SAM) optimizer to generate local flatness models with improved stability and weight perturbation robustness, which results in the small norm of local updates and robustness to DP noise, thereby improving the performance. To further reduce the magnitude of random noise while achieving better performance, we propose DP-FedSAM-$top_k$ by adopting the local update sparsification technique. From the theoretical perspective, we present the convergence analysis to investigate how our algorithms mitigate the performance degradation induced by DP. Meanwhile, we give rigorous privacy guarantees with Rényi DP, the sensitivity analysis of local updates, and generalization analysis. At last, we empirically confirm that our algorithms achieve state-of-the-art (SOTA) performance compared with existing SOTA baselines in DPFL.
22.4LGFeb 9, 2022
Vertical Federated Learning: Challenges, Methodologies and ExperimentsKang Wei, Jun Li, Chuan Ma et al.
Recently, federated learning (FL) has emerged as a promising distributed machine learning (ML) technology, owing to the advancing computational and sensing capacities of end-user devices, however with the increasing concerns on users' privacy. As a special architecture in FL, vertical FL (VFL) is capable of constructing a hyper ML model by embracing sub-models from different clients. These sub-models are trained locally by vertically partitioned data with distinct attributes. Therefore, the design of VFL is fundamentally different from that of conventional FL, raising new and unique research issues. In this paper, we aim to discuss key challenges in VFL with effective solutions, and conduct experiments on real-life datasets to shed light on these issues. Specifically, we first propose a general framework on VFL, and highlight the key differences between VFL and conventional FL. Then, we discuss research challenges rooted in VFL systems under four aspects, i.e., security and privacy risks, expensive computation and communication costs, possible structural damage caused by model splitting, and system heterogeneity. Afterwards, we develop solutions to addressing the aforementioned challenges, and conduct extensive experiments to showcase the effectiveness of our proposed solutions.
21.0LGJan 18, 2021
Blockchain Assisted Decentralized Federated Learning (BLADE-FL): Performance Analysis and Resource AllocationJun Li, Yumeng Shao, Kang Wei et al.
Federated learning (FL), as a distributed machine learning paradigm, promotes personal privacy by local data processing at each client. However, relying on a centralized server for model aggregation, standard FL is vulnerable to server malfunctions, untrustworthy server, and external attacks. To address this issue, we propose a decentralized FL framework by integrating blockchain into FL, namely, blockchain assisted decentralized federated learning (BLADE-FL). In a round of the proposed BLADE-FL, each client broadcasts the trained model to other clients, aggregates its own model with received ones, and then competes to generate a block before its local training of the next round. We evaluate the learning performance of BLADE-FL, and develop an upper bound on the global loss function. Then we verify that this bound is convex with respect to the number of overall aggregation rounds K, and optimize the computing resource allocation for minimizing the upper bound. We also note that there is a critical problem of training deficiency, caused by lazy clients who plagiarize others' trained models and add artificial noises to disguise their cheating behaviors. Focusing on this problem, we explore the impact of lazy clients on the learning performance of BLADE-FL, and characterize the relationship among the optimal K, the learning parameters, and the proportion of lazy clients. Based on MNIST and Fashion-MNIST datasets, we show that the experimental results are consistent with the analytical ones. To be specific, the gap between the developed upper bound and experimental results is lower than 5%, and the optimized K based on the upper bound can effectively minimize the loss function.