Daily Papers

We introduce our first-generation reasoning models, DeepSeek-R1-Zero and DeepSeek-R1. DeepSeek-R1-Zero, a model trained via large-scale reinforcement learning (RL) without supervised fine-tuning (SFT) as a preliminary step, demonstrates remarkable reasoning capabilities. Through RL, DeepSeek-R1-Zero naturally emerges with numerous powerful and intriguing reasoning behaviors. However, it encounters challenges such as poor readability, and language mixing. To address these issues and further enhance reasoning performance, we introduce DeepSeek-R1, which incorporates multi-stage training and cold-start data before RL. DeepSeek-R1 achieves performance comparable to OpenAI-o1-1217 on reasoning tasks. To support the research community, we open-source DeepSeek-R1-Zero, DeepSeek-R1, and six dense models (1.5B, 7B, 8B, 14B, 32B, 70B) distilled from DeepSeek-R1 based on Qwen and Llama.

Mathematical reasoning poses a significant challenge for language models due to its complex and structured nature. In this paper, we introduce DeepSeekMath 7B, which continues pre-training DeepSeek-Coder-Base-v1.5 7B with 120B math-related tokens sourced from Common Crawl, together with natural language and code data. DeepSeekMath 7B has achieved an impressive score of 51.7% on the competition-level MATH benchmark without relying on external toolkits and voting techniques, approaching the performance level of Gemini-Ultra and GPT-4. Self-consistency over 64 samples from DeepSeekMath 7B achieves 60.9% on MATH. The mathematical reasoning capability of DeepSeekMath is attributed to two key factors: First, we harness the significant potential of publicly available web data through a meticulously engineered data selection pipeline. Second, we introduce Group Relative Policy Optimization (GRPO), a variant of Proximal Policy Optimization (PPO), that enhances mathematical reasoning abilities while concurrently optimizing the memory usage of PPO.

We present DeepSeek-Coder-V2, an open-source Mixture-of-Experts (MoE) code language model that achieves performance comparable to GPT4-Turbo in code-specific tasks. Specifically, DeepSeek-Coder-V2 is further pre-trained from an intermediate checkpoint of DeepSeek-V2 with additional 6 trillion tokens. Through this continued pre-training, DeepSeek-Coder-V2 substantially enhances the coding and mathematical reasoning capabilities of DeepSeek-V2, while maintaining comparable performance in general language tasks. Compared to DeepSeek-Coder-33B, DeepSeek-Coder-V2 demonstrates significant advancements in various aspects of code-related tasks, as well as reasoning and general capabilities. Additionally, DeepSeek-Coder-V2 expands its support for programming languages from 86 to 338, while extending the context length from 16K to 128K. In standard benchmark evaluations, DeepSeek-Coder-V2 achieves superior performance compared to closed-source models such as GPT4-Turbo, Claude 3 Opus, and Gemini 1.5 Pro in coding and math benchmarks.

The rapid development of large language models has revolutionized code intelligence in software development. However, the predominance of closed-source models has restricted extensive research and development. To address this, we introduce the DeepSeek-Coder series, a range of open-source code models with sizes from 1.3B to 33B, trained from scratch on 2 trillion tokens. These models are pre-trained on a high-quality project-level code corpus and employ a fill-in-the-blank task with a 16K window to enhance code generation and infilling. Our extensive evaluations demonstrate that DeepSeek-Coder not only achieves state-of-the-art performance among open-source code models across multiple benchmarks but also surpasses existing closed-source models like Codex and GPT-3.5. Furthermore, DeepSeek-Coder models are under a permissive license that allows for both research and unrestricted commercial use.

We introduce DeepSeek-Prover-V1.5, an open-source language model designed for theorem proving in Lean 4, which enhances DeepSeek-Prover-V1 by optimizing both training and inference processes. Pre-trained on DeepSeekMath-Base with specialization in formal mathematical languages, the model undergoes supervised fine-tuning using an enhanced formal theorem proving dataset derived from DeepSeek-Prover-V1. Further refinement is achieved through reinforcement learning from proof assistant feedback (RLPAF). Beyond the single-pass whole-proof generation approach of DeepSeek-Prover-V1, we propose RMaxTS, a variant of Monte-Carlo tree search that employs an intrinsic-reward-driven exploration strategy to generate diverse proof paths. DeepSeek-Prover-V1.5 demonstrates significant improvements over DeepSeek-Prover-V1, achieving new state-of-the-art results on the test set of the high school level miniF2F benchmark (63.5%) and the undergraduate level ProofNet benchmark (25.3%).

We present DeepSeek-V3, a strong Mixture-of-Experts (MoE) language model with 671B total parameters with 37B activated for each token. To achieve efficient inference and cost-effective training, DeepSeek-V3 adopts Multi-head Latent Attention (MLA) and DeepSeekMoE architectures, which were thoroughly validated in DeepSeek-V2. Furthermore, DeepSeek-V3 pioneers an auxiliary-loss-free strategy for load balancing and sets a multi-token prediction training objective for stronger performance. We pre-train DeepSeek-V3 on 14.8 trillion diverse and high-quality tokens, followed by Supervised Fine-Tuning and Reinforcement Learning stages to fully harness its capabilities. Comprehensive evaluations reveal that DeepSeek-V3 outperforms other open-source models and achieves performance comparable to leading closed-source models. Despite its excellent performance, DeepSeek-V3 requires only 2.788M H800 GPU hours for its full training. In addition, its training process is remarkably stable. Throughout the entire training process, we did not experience any irrecoverable loss spikes or perform any rollbacks. The model checkpoints are available at https://github.com/deepseek-ai/DeepSeek-V3.

In the era of large language models, Mixture-of-Experts (MoE) is a promising architecture for managing computational costs when scaling up model parameters. However, conventional MoE architectures like GShard, which activate the top-K out of N experts, face challenges in ensuring expert specialization, i.e. each expert acquires non-overlapping and focused knowledge. In response, we propose the DeepSeekMoE architecture towards ultimate expert specialization. It involves two principal strategies: (1) finely segmenting the experts into mN ones and activating mK from them, allowing for a more flexible combination of activated experts; (2) isolating K_s experts as shared ones, aiming at capturing common knowledge and mitigating redundancy in routed experts. Starting from a modest scale with 2B parameters, we demonstrate that DeepSeekMoE 2B achieves comparable performance with GShard 2.9B, which has 1.5 times the expert parameters and computation. In addition, DeepSeekMoE 2B nearly approaches the performance of its dense counterpart with the same number of total parameters, which set the upper bound of MoE models. Subsequently, we scale up DeepSeekMoE to 16B parameters and show that it achieves comparable performance with LLaMA2 7B, with only about 40% of computations. Further, our preliminary efforts to scale up DeepSeekMoE to 145B parameters consistently validate its substantial advantages over the GShard architecture, and show its performance comparable with DeepSeek 67B, using only 28.5% (maybe even 18.2%) of computations.

The rapid development of open-source large language models (LLMs) has been truly remarkable. However, the scaling law described in previous literature presents varying conclusions, which casts a dark cloud over scaling LLMs. We delve into the study of scaling laws and present our distinctive findings that facilitate scaling of large scale models in two commonly used open-source configurations, 7B and 67B. Guided by the scaling laws, we introduce DeepSeek LLM, a project dedicated to advancing open-source language models with a long-term perspective. To support the pre-training phase, we have developed a dataset that currently consists of 2 trillion tokens and is continuously expanding. We further conduct supervised fine-tuning (SFT) and Direct Preference Optimization (DPO) on DeepSeek LLM Base models, resulting in the creation of DeepSeek Chat models. Our evaluation results demonstrate that DeepSeek LLM 67B surpasses LLaMA-2 70B on various benchmarks, particularly in the domains of code, mathematics, and reasoning. Furthermore, open-ended evaluations reveal that DeepSeek LLM 67B Chat exhibits superior performance compared to GPT-3.5.

We present DeepSeek-VL, an open-source Vision-Language (VL) Model designed for real-world vision and language understanding applications. Our approach is structured around three key dimensions: We strive to ensure our data is diverse, scalable, and extensively covers real-world scenarios including web screenshots, PDFs, OCR, charts, and knowledge-based content, aiming for a comprehensive representation of practical contexts. Further, we create a use case taxonomy from real user scenarios and construct an instruction tuning dataset accordingly. The fine-tuning with this dataset substantially improves the model's user experience in practical applications. Considering efficiency and the demands of most real-world scenarios, DeepSeek-VL incorporates a hybrid vision encoder that efficiently processes high-resolution images (1024 x 1024), while maintaining a relatively low computational overhead. This design choice ensures the model's ability to capture critical semantic and detailed information across various visual tasks. We posit that a proficient Vision-Language Model should, foremost, possess strong language abilities. To ensure the preservation of LLM capabilities during pretraining, we investigate an effective VL pretraining strategy by integrating LLM training from the beginning and carefully managing the competitive dynamics observed between vision and language modalities. The DeepSeek-VL family (both 1.3B and 7B models) showcases superior user experiences as a vision-language chatbot in real-world applications, achieving state-of-the-art or competitive performance across a wide range of visual-language benchmarks at the same model size while maintaining robust performance on language-centric benchmarks. We have made both 1.3B and 7B models publicly accessible to foster innovations based on this foundation model.

Proof assistants like Lean have revolutionized mathematical proof verification, ensuring high accuracy and reliability. Although large language models (LLMs) show promise in mathematical reasoning, their advancement in formal theorem proving is hindered by a lack of training data. To address this issue, we introduce an approach to generate extensive Lean 4 proof data derived from high-school and undergraduate-level mathematical competition problems. This approach involves translating natural language problems into formal statements, filtering out low-quality statements, and generating proofs to create synthetic data. After fine-tuning the DeepSeekMath 7B model on this synthetic dataset, which comprises 8 million formal statements with proofs, our model achieved whole-proof generation accuracies of 46.3% with 64 samples and 52% cumulatively on the Lean 4 miniF2F test, surpassing the baseline GPT-4 at 23.0% with 64 samples and a tree search reinforcement learning method at 41.0%. Additionally, our model successfully proved 5 out of 148 problems in the Lean 4 Formalized International Mathematical Olympiad (FIMO) benchmark, while GPT-4 failed to prove any. These results demonstrate the potential of leveraging large-scale synthetic data to enhance theorem-proving capabilities in LLMs. Both the synthetic dataset and the model will be made available to facilitate further research in this promising field.

We present DeepSeek-V2, a strong Mixture-of-Experts (MoE) language model characterized by economical training and efficient inference. It comprises 236B total parameters, of which 21B are activated for each token, and supports a context length of 128K tokens. DeepSeek-V2 adopts innovative architectures including Multi-head Latent Attention (MLA) and DeepSeekMoE. MLA guarantees efficient inference through significantly compressing the Key-Value (KV) cache into a latent vector, while DeepSeekMoE enables training strong models at an economical cost through sparse computation. Compared with DeepSeek 67B, DeepSeek-V2 achieves significantly stronger performance, and meanwhile saves 42.5% of training costs, reduces the KV cache by 93.3%, and boosts the maximum generation throughput to 5.76 times. We pretrain DeepSeek-V2 on a high-quality and multi-source corpus consisting of 8.1T tokens, and further perform Supervised Fine-Tuning (SFT) and Reinforcement Learning (RL) to fully unlock its potential. Evaluation results show that, even with only 21B activated parameters, DeepSeek-V2 and its chat versions still achieve top-tier performance among open-source models.

We present DeepSeek-VL2, an advanced series of large Mixture-of-Experts (MoE) Vision-Language Models that significantly improves upon its predecessor, DeepSeek-VL, through two key major upgrades. For the vision component, we incorporate a dynamic tiling vision encoding strategy designed for processing high-resolution images with different aspect ratios. For the language component, we leverage DeepSeekMoE models with the Multi-head Latent Attention mechanism, which compresses Key-Value cache into latent vectors, to enable efficient inference and high throughput. Trained on an improved vision-language dataset, DeepSeek-VL2 demonstrates superior capabilities across various tasks, including but not limited to visual question answering, optical character recognition, document/table/chart understanding, and visual grounding. Our model series is composed of three variants: DeepSeek-VL2-Tiny, DeepSeek-VL2-Small and DeepSeek-VL2, with 1.0B, 2.8B and 4.5B activated parameters respectively. DeepSeek-VL2 achieves competitive or state-of-the-art performance with similar or fewer activated parameters compared to existing open-source dense and MoE-based models. Codes and pre-trained models are publicly accessible at https://github.com/deepseek-ai/DeepSeek-VL2.

In this paper, we propose a simple yet effective method to stabilize extremely deep Transformers. Specifically, we introduce a new normalization function (DeepNorm) to modify the residual connection in Transformer, accompanying with theoretically derived initialization. In-depth theoretical analysis shows that model updates can be bounded in a stable way. The proposed method combines the best of two worlds, i.e., good performance of Post-LN and stable training of Pre-LN, making DeepNorm a preferred alternative. We successfully scale Transformers up to 1,000 layers (i.e., 2,500 attention and feed-forward network sublayers) without difficulty, which is one order of magnitude deeper than previous deep Transformers. Remarkably, on a multilingual benchmark with 7,482 translation directions, our 200-layer model with 3.2B parameters significantly outperforms the 48-layer state-of-the-art model with 12B parameters by 5 BLEU points, which indicates a promising scaling direction.

In this paper, we propose a multimodal search engine that combines visual and textual cues to retrieve items from a multimedia database aesthetically similar to the query. The goal of our engine is to enable intuitive retrieval of fashion merchandise such as clothes or furniture. Existing search engines treat textual input only as an additional source of information about the query image and do not correspond to the real-life scenario where the user looks for 'the same shirt but of denim'. Our novel method, dubbed DeepStyle, mitigates those shortcomings by using a joint neural network architecture to model contextual dependencies between features of different modalities. We prove the robustness of this approach on two different challenging datasets of fashion items and furniture where our DeepStyle engine outperforms baseline methods by 18-21% on the tested datasets. Our search engine is commercially deployed and available through a Web-based application.

Most large multimodal models (LMMs) are implemented by feeding visual tokens as a sequence into the first layer of a large language model (LLM). The resulting architecture is simple but significantly increases computation and memory costs, as it has to handle a large number of additional tokens in its input layer. This paper presents a new architecture DeepStack for LMMs. Considering N layers in the language and vision transformer of LMMs, we stack the visual tokens into N groups and feed each group to its aligned transformer layer from bottom to top. Surprisingly, this simple method greatly enhances the power of LMMs to model interactions among visual tokens across layers but with minimal additional cost. We apply DeepStack to both language and vision transformer in LMMs, and validate the effectiveness of DeepStack LMMs with extensive empirical results. Using the same context length, our DeepStack 7B and 13B parameters surpass their counterparts by 2.7 and 2.9 on average across 9 benchmarks, respectively. Using only one-fifth of the context length, DeepStack rivals closely to the counterparts that use the full context length. These gains are particularly pronounced on high-resolution tasks, e.g., 4.2, 11.0, and 4.0 improvements on TextVQA, DocVQA, and InfoVQA compared to LLaVA-1.5-7B, respectively. We further apply DeepStack to vision transformer layers, which brings us a similar amount of improvements, 3.8 on average compared with LLaVA-1.5-7B.

Background: AI-powered code generation, fueled by Large Language Models (LLMs), is revolutionizing software development. Models like OpenAI's Codex and GPT-4, alongside DeepSeek, leverage vast code and natural language datasets. However, ensuring code quality, correctness, and managing complex tasks remains challenging, necessitating thorough evaluation. Methodology: This research compares ChatGPT (version o1) and DeepSeek (version R1) for Python code generation using online judge coding challenges. It evaluates correctness (online judge verdicts, up to three attempts), code quality (Pylint/Flake8), and efficiency (execution time/memory usage). Results: DeepSeek demonstrated higher correctness, particularly on algorithmic tasks, often achieving 'Accepted' on the first attempt. ChatGPT sometimes requires multiple attempts or failures. ChatGPT encountered fewer issues, used comparable or slightly less memory, consumed less execution times and wrote fewer lines of code. Conclusion: DeepSeek exhibited superior correctness in Python code generation, often requiring fewer attempts, suggesting an advantage in algorithmic problem-solving. Both models showed almost similar efficiency in execution time and memory use. Finally, this research provides insights for developers choosing AI coding assistants and informs future AI-driven software development research.

The irruption of DeepSeek-R1 constitutes a turning point for the AI industry in general and the LLMs in particular. Its capabilities have demonstrated outstanding performance in several tasks, including creative thinking, code generation, maths and automated program repair, at apparently lower execution cost. However, LLMs must adhere to an important qualitative property, i.e., their alignment with safety and human values. A clear competitor of DeepSeek-R1 is its American counterpart, OpenAI's o3-mini model, which is expected to set high standards in terms of performance, safety and cost. In this paper we conduct a systematic assessment of the safety level of both, DeepSeek-R1 (70b version) and OpenAI's o3-mini (beta version). To this end, we make use of our recently released automated safety testing tool, named ASTRAL. By leveraging this tool, we automatically and systematically generate and execute a total of 1260 unsafe test inputs on both models. After conducting a semi-automated assessment of the outcomes provided by both LLMs, the results indicate that DeepSeek-R1 is highly unsafe as compared to OpenAI's o3-mini. Based on our evaluation, DeepSeek-R1 answered unsafely to 11.98% of the executed prompts whereas o3-mini only to 1.19%.

Multi-head Latent Attention (MLA) is an innovative architecture proposed by DeepSeek, designed to ensure efficient and economical inference by significantly compressing the Key-Value (KV) cache into a latent vector. Compared to MLA, standard LLMs employing Multi-Head Attention (MHA) and its variants such as Grouped-Query Attention (GQA) exhibit significant cost disadvantages. Enabling well-trained LLMs (e.g., Llama) to rapidly adapt to MLA without pre-training from scratch is both meaningful and challenging. This paper proposes the first data-efficient fine-tuning method for transitioning from MHA to MLA (MHA2MLA), which includes two key components: for partial-RoPE, we remove RoPE from dimensions of queries and keys that contribute less to the attention scores, for low-rank approximation, we introduce joint SVD approximations based on the pre-trained parameters of keys and values. These carefully designed strategies enable MHA2MLA to recover performance using only a small fraction (0.3% to 0.6%) of the data, significantly reducing inference costs while seamlessly integrating with compression techniques such as KV cache quantization. For example, the KV cache size of Llama2-7B is reduced by 92.19%, with only a 0.5% drop in LongBench performance.

Recently, the DeepSeek series of models, leveraging their exceptional reasoning capabilities and open-source strategy, is reshaping the global AI landscape. Despite these advantages, they exhibit significant safety deficiencies. Research conducted by Robust Intelligence, a subsidiary of Cisco, in collaboration with the University of Pennsylvania, revealed that DeepSeek-R1 has a 100\% attack success rate when processing harmful prompts. Additionally, multiple safety companies and research institutions have confirmed critical safety vulnerabilities in this model. As models demonstrating robust performance in Chinese and English, DeepSeek models require equally crucial safety assessments in both language contexts. However, current research has predominantly focused on safety evaluations in English environments, leaving a gap in comprehensive assessments of their safety performance in Chinese contexts. In response to this gap, this study introduces CHiSafetyBench, a Chinese-specific safety evaluation benchmark. This benchmark systematically evaluates the safety of DeepSeek-R1 and DeepSeek-V3 in Chinese contexts, revealing their performance across safety categories. The experimental results quantify the deficiencies of these two models in Chinese contexts, providing key insights for subsequent improvements. It should be noted that, despite our efforts to establish a comprehensive, objective, and authoritative evaluation benchmark, the selection of test samples, characteristics of data distribution, and the setting of evaluation criteria may inevitably introduce certain biases into the evaluation results. We will continuously optimize the evaluation benchmark and periodically update this report to provide more comprehensive and accurate assessment outcomes. Please refer to the latest version of the paper for the most recent evaluation results and conclusions.

In late January 2025, DeepSeek released their new reasoning model (DeepSeek R1); which was developed at a fraction of the cost yet remains competitive with OpenAI's models, despite the US's GPU export ban. This report discusses the model, and what its release means for the field of Generative AI more widely. We briefly discuss other models released from China in recent weeks, their similarities; innovative use of Mixture of Experts (MoE), Reinforcement Learning (RL) and clever engineering appear to be key factors in the capabilities of these models. This think piece has been written to a tight timescale, providing broad coverage of the topic, and serves as introductory material for those looking to understand the model's technical advancements, as well as its place in the ecosystem. Several further areas of research are identified.

We present the Mixture-of-Tunable-Experts (MoTE), a method that extends the Mixture-of-Experts architecture of Large Language Models (LLMs). Without additional training, MoTE enables meaningful and focused behavior changes in LLMs on-the-fly during inference time. By analyzing the digital LLM brain of DeepSeek-R1 using a technique we dub 'functional Token Resonance Imaging' (fTRI) -- inspired by fMRI and using prompts designed to elicit specific behavior (e.g., 'What happened {time}{place}?') -- we empirically identify distinctive experts associated with behaviors like refusal responses. Using MoTE we are able to intervene and control such specific behavior. We switched off the top 10 most refusal-relevant experts (0.07% of R1's 14,848 routed experts), achieving a 52% refusal reduction on sensitive reference prompts without performance degradation on MT-Bench. Random expert deactivation resulted in smaller behavioral shifts with increased noise, whereas forced expert activation led to significantly higher refusal rates. Our approach shares similarities with sparse autoencoders (SAEs) in terms of explainability and steerability. Unlike SAEs, MoTE does not require large training efforts, as within MoEs with a vast number of experts, specialization already emerged naturally during pretraining. Our findings suggest that significant functional mechanisms in Mixture-of-Experts architectures can at least partially be localized in a small number of specific experts, rather than being distributed throughout the model's weights. Expert subgroups can be tuned to trigger significant behavior variations, providing insights into the inner workings of LLMs.

Large Language Models (LLMs) have achieved remarkable progress in reasoning, alignment, and task-specific performance. However, ensuring harmlessness in these systems remains a critical challenge, particularly in advanced models like DeepSeek-R1. This paper examines the limitations of Reinforcement Learning (RL) as the primary approach for reducing harmful outputs in DeepSeek-R1 and compares it with Supervised Fine-Tuning (SFT). While RL improves reasoning capabilities, it faces challenges such as reward hacking, generalization failures, language mixing, and high computational costs. We propose hybrid training approaches combining RL and SFT to achieve robust harmlessness reduction. Usage recommendations and future directions for deploying DeepSeek-R1 responsibly are also presented.

Traditional evaluations of multimodal large language models (LLMs) have been limited by their focus on single-image reasoning, failing to assess crucial aspects like contextual understanding, reasoning stability, and uncertainty calibration. This study addresses these limitations by introducing a novel benchmark that integrates multi-image reasoning tasks with rejection-based evaluation and positional bias detection. To evaluate these dimensions, we further introduce entropy as a novel metric for quantifying reasoning consistency across reordered answer variants. We applied this benchmark to assess Grok 3, ChatGPT-4o, ChatGPT-o1, Gemini 2.0 Flash Experimental, DeepSeek Janus models, Qwen2.5-VL-72B-Instruct, QVQ-72B-Preview, and Pixtral 12B across eight visual reasoning tasks, including difference spotting and diagram interpretation. Our findings reveal ChatGPT-o1 leading in overall accuracy (82.5\%) and rejection accuracy (70.0\%), closely followed by Gemini 2.0 Flash Experimental (70.8\%). QVQ-72B-Preview demonstrated superior rejection accuracy (85.5\%). Notably, Pixtral 12B (51.7\%) showed promise in specific domains, while Janus models exhibited challenges in bias and uncertainty calibration, reflected in low rejection accuracies and high entropy scores. High entropy scores in Janus models (Janus 7B: 0.8392, Janus 1B: 0.787) underscore their susceptibility to positional bias and unstable reasoning, contrasting with the low entropy and robust reasoning of ChatGPT models. The study further demonstrates that model size is not the sole determinant of performance, as evidenced by Grok 3 underperformance despite its substantial parameter count. By employing multi-image contexts, rejection mechanisms, and entropy-based consistency metrics, this benchmark sets a new standard for evaluating multimodal LLMs, enabling a more robust and reliable assessment of next-generation AI systems.

Set representation has become ubiquitous in deep learning for modeling the inductive bias of neural networks that are insensitive to the input order. DeepSets is the most widely used neural network architecture for set representation. It involves embedding each set element into a latent space with dimension L, followed by a sum pooling to obtain a whole-set embedding, and finally mapping the whole-set embedding to the output. In this work, we investigate the impact of the dimension L on the expressive power of DeepSets. Previous analyses either oversimplified high-dimensional features to be one-dimensional features or were limited to analytic activations, thereby diverging from practical use or resulting in L that grows exponentially with the set size N and feature dimension D. To investigate the minimal value of L that achieves sufficient expressive power, we present two set-element embedding layers: (a) linear + power activation (LP) and (b) linear + exponential activations (LE). We demonstrate that L being poly(N, D) is sufficient for set representation using both embedding layers. We also provide a lower bound of L for the LP embedding layer. Furthermore, we extend our results to permutation-equivariant set functions and the complex field.