NeurIPS 2024 Papers

Conference on Neural Information Processing Systems

Decision breakdown

4,562papers

Counted from the official paper list; may differ slightly from the number the conference announced.

  • Oral
    721.6%
  • Spotlight
    3828.4%
  • Poster
    4,06489.1%
  • Other / unlabeled
    441.0%

Browse all papers

ViLCo-Bench: VIdeo Language COntinual learning Benchmark

Tianqi Tang, Shohreh Deldari, Hao Xue, Celso M de Melo, Flora D. Salim

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Video language continual learning involves continuously adapting to information from video and text inputs, enhancing a model’s ability to handle new tasks while retaining prior knowledge. This field is a relatively under-explored area, and establishing appropriate datasets is crucial for facilitating communication and research in this field. In this study, we present the first dedicated benchmark, ViLCo-Bench, designed to evaluate continual learning models across a range of video-text tasks. The dataset comprises ten-minute-long videos and corresponding language queries collected from publicly available datasets. Additionally, we introduce a novel memory-efficient framework that incorporates self-supervised learning and mimics long-term and short-term memory effects. This framework addresses challenges including memory complexity from long video clips, natural language complexity from open queries, and text-video misalignment. We posit that ViLCo-Bench, with greater complexity compared to existing continual learning benchmarks, would serve as a critical tool for exploring the video-language domain, extending beyond conventional class-incremental tasks, and addressing complex and limited annotation issues. The curated data, evaluations, and our novel method are available at https://github.com/cruiseresearchgroup/ViLCo.

Poster
LESS: Label-Efficient and Single-Stage Referring 3D Segmentation

Xuexun Liu, Xiaoxu Xu, Jinlong Li, Qiudan Zhang, Xu Wang, Nicu Sebe, Lin Ma

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Referring 3D Segmentation is a visual-language task that segments all points of the specified object from a 3D point cloud described by a sentence of query. Previous works perform a two-stage paradigm, first conducting language-agnostic instance segmentation then matching with given text query. However, the semantic concepts from text query and visual cues are separately interacted during the training, and both instance and semantic labels for each object are required, which is time consuming and human-labor intensive. To mitigate these issues, we propose a novel Referring 3D Segmentation pipeline, Label-Efficient and Single-Stage, dubbed LESS, which is only under the supervision of efficient binary mask. Specifically, we design a Point-Word Cross-Modal Alignment module for aligning the fine-grained features of points and textual embedding. Query Mask Predictor module and Query-Sentence Alignment module are introduced for coarse-grained alignment between masks and query. Furthermore, we propose an area regularization loss, which coarsely reduces irrelevant background predictions on a large scale. Besides, a point-to-point contrastive loss is proposed concentrating on distinguishing points with subtly similar features. Through extensive experiments, we achieve state-of-the-art performance on ScanRefer dataset by surpassing the previous methods about 3.7% mIoU using only binary labels. Code is available at https://github.com/mellody11/LESS.

Poster
Autoregressive Policy Optimization for Constrained Allocation Tasks

David Winkel, Niklas Alexander Strauß, Maximilian Bernhard, Zongyue Li, Thomas Seidl, Matthias Schubert

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Allocation tasks represent a class of problems where a limited amount of resources must be allocated to a set of entities at each time step. Prominent examples of this task include portfolio optimization or distributing computational workloads across servers. Allocation tasks are typically bound by linear constraints describing practical requirements that have to be strictly fulfilled at all times. In portfolio optimization, for example, investors may be obligated to allocate less than 30\% of the funds into a certain industrial sector in any investment period. Such constraints restrict the action space of allowed allocations in intricate ways, which makes learning a policy that avoids constraint violations difficult. In this paper, we propose a new method for constrained allocation tasks based on an autoregressive process to sequentially sample allocations for each entity. In addition, we introduce a novel de-biasing mechanism to counter the initial bias caused by sequential sampling. We demonstrate the superior performance of our approach compared to a variety of Constrained Reinforcement Learning (CRL) methods on three distinct constrained allocation tasks: portfolio optimization, computational workload distribution, and a synthetic allocation benchmark. Our code is available at: https://github.com/niklasdbs/paspo

Poster
An engine not a camera: Measuring performative power of online search

Celestine Mendler-Dünner, Gabriele Carovano, Moritz Hardt

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The power of digital platforms is at the center of major ongoing policy and regulatory efforts. To advance existing debates, we designed and executed an experiment to measure the performative power of online search providers. Instantiated in our setting, performative power quantifies the ability of a search engine to steer web traffic by rearranging results. To operationalize this definition we developed a browser extension that performs unassuming randomized experiments in the background. These randomized experiments emulate updates to the search algorithm and identify the causal effect of different content arrangements on clicks. Analyzing tens of thousands of clicks, we discuss what our robust quantitative findings say about the power of online search engines, using the Google Shopping antitrust investigation as a case study. More broadly, we envision our work to serve as a blueprint for how the recent definition of performative power can help integrate quantitative insights from online experiments with future investigations into the economic power of digital platforms.

Poster
A Systematic Review of NeurIPS Dataset Management Practices

Yiwei Wu, Leah Hope Ajmani, Shayne Longpre, Hanlin Li

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As new machine learning methods demand larger training datasets, researchers and developers face significant challenges in dataset management. Although ethics reviews, documentation, and checklists have been established, it remains uncertain whether consistent dataset management practices exist across the community. This lack of a comprehensive overview hinders our ability to diagnose and address fundamental tensions and ethical issues related to managing large datasets. We present a systematic review of datasets published at the NeurIPS Datasets and Benchmarks track, focusing on four key aspects: provenance, distribution, ethical disclosure, and licensing. Our findings reveal that dataset provenance is often unclear due to ambiguous filtering and curation processes. Additionally, a variety of sites are used for dataset hosting, but only a few offer structured metadata and version control. These inconsistencies underscore the urgent need for standardized data infrastructures for the publication and management of datasets.

Poster
Autobidder's Dilemma: Why More Sophisticated Autobidders Lead to Worse Auction Efficiency

Yuan Deng, Jieming Mao, Vahab Mirrokni, Hanrui Zhang, Song Zuo

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The recent increasing adoption of autobidding has inspired the growing interest in analyzing the performance of classic mechanism with value-maximizing autobidders both theoretically and empirically. It is known that optimal welfare can be obtained in first-price auctions if autobidders are restricted to uniform bid-scaling and the price of anarchy is $2$ when non-uniform bid-scaling strategies are allowed. In this paper, we provide a fine-grained price of anarchy analysis for non-uniform bid-scaling strategies in first-price auctions, demonstrating the reason why more powerful (individual) non-uniform bid-scaling strategies may lead to worse (aggregated) performance in social welfare. Our theoretical results match recent empirical findings that a higher level of non-uniform bid-scaling leads to lower welfare performance in first-price auctions.

Poster
Attack-Aware Noise Calibration for Differential Privacy

Bogdan Kulynych, Juan Felipe Gomez, Georgios Kaissis, Flavio Calmon, Carmela Troncoso

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Differential privacy (DP) is a widely used approach for mitigating privacy risks when training machine learning models on sensitive data. DP mechanisms add noise during training to limit the risk of information leakage. The scale of the added noise is critical, as it determines the trade-off between privacy and utility. The standard practice is to select the noise scale to satisfy a given privacy budget ε. This privacy budget is in turn interpreted in terms of operational attack risks, such as accuracy, sensitivity, and specificity of inference attacks aimed to recover information about the training data records. We show that first calibrating the noise scale to a privacy budget ε, and then translating ε to attack risk leads to overly conservative risk assessments and unnecessarily low utility. Instead, we propose methods to directly calibrate the noise scale to a desired attack risk level, bypassing the step of choosing ε. For a given notion of attack risk, our approach significantly decreases noise scale, leading to increased utility at the same level of privacy. We empirically demonstrate that calibrating noise to attack sensitivity/specificity, rather than ε, when training privacy-preserving ML models substantially improves model accuracy for the same risk level. Our work provides a principled and practical way to improve the utility of privacy-preserving ML without compromising on privacy.

Poster
RoleAgent: Building, Interacting, and Benchmarking High-quality Role-Playing Agents from Scripts

Jiaheng Liu, Zehao Ni, Haoran Que, Tao Sun, Noah Wang, Jian Yang, JiakaiWang, Hongcheng Guo, Z.Y. Peng, Ge Zhang, Jiayi Tian, Xingyuan Bu, Ke Xu, Wenge Rong, Junran Peng, Zhaoxiang Zhang

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Believable agents can empower interactive applications ranging from immersive environments to rehearsal spaces for interpersonal communication. Recently, generative agents have been proposed to simulate believable human behavior by using Large Language Models. However, the existing method heavily relies on human-annotated agent profiles (e.g., name, age, personality, relationships with others, and so on) for the initialization of each agent, which cannot be scaled up easily. In this paper, we propose a scalable RoleAgent framework to generate high-quality role-playing agents from raw scripts, which includes building and interacting stages. Specifically, in the building stage, we use a hierarchical memory system to extract and summarize the structure and high-level information of each agent for the raw script. In the interacting stage, we propose a novel innovative mechanism with four steps to achieve a high-quality interaction between agents. Finally, we introduce a systematic and comprehensive evaluation benchmark called RoleAgentBench to evaluate the effectiveness of our RoleAgent, which includes 100 and 28 roles for 20 English and 5 Chinese scripts, respectively. Extensive experimental results on RoleAgentBench demonstrate the effectiveness of RoleAgent.

Poster
Faster Algorithms for User-Level Private Stochastic Convex Optimization

Andrew Lowy, Daogao Liu, Hilal Asi

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We study private stochastic convex optimization (SCO) under user-level differential privacy (DP) constraints. In this setting, there are $n$ users (e.g., cell phones), each possessing $m$ data items (e.g., text messages), and we need to protect the privacy of each user's entire collection of data items. Existing algorithms for user-level DP SCO are impractical in many large-scale machine learning scenarios because: (i) they make restrictive assumptions on the smoothness parameter of the loss function and require the number of users to grow polynomially with the dimension of the parameter space; or (ii) they are prohibitively slow, requiring at least $(mn)^{3/2}$ gradient computations for smooth losses and $(mn)^3$ computations for non-smooth losses. To address these limitations, we provide novel user-level DP algorithms with state-of-the-art excess risk and runtime guarantees, without stringent assumptions. First, we develop a linear-time algorithm with state-of-the-art excess risk (for a non-trivial linear-time algorithm) under a mild smoothness assumption. Our second algorithm applies to arbitrary smooth losses and achieves optimal excess risk in $\approx (mn)^{9/8}$ gradient computations. Third, for non-smooth loss functions, we obtain optimal excess risk in $n^{11/8} m^{5/4}$ gradient computations. Moreover, our algorithms do not require the number of users to grow polynomially with the dimension.

Poster
Empowering and Assessing the Utility of Large Language Models in Crop Science

Hang Zhang, Jiawei Sun, Renqi Chen, Wei Liu, Zhonghang Yuan, Xinzhe Zheng, Zhefan Wang, Zhiyuan Yang, Hang Yan, Han-Sen Zhong, Xiqing Wang, Wanli Ouyang, Fan Yang, Nanqing Dong

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Large language models (LLMs) have demonstrated remarkable efficacy across knowledge-intensive tasks. Nevertheless, their untapped potential in crop science presents an opportunity for advancement. To narrow this gap, we introduce CROP, which includes a novel instruction tuning dataset specifically designed to enhance LLMs’ professional capabilities in the crop science sector, along with a benchmark that serves as a comprehensive evaluation of LLMs’ understanding of the domain knowledge. The CROP dataset is curated through a task-oriented and LLM-human integrated pipeline, comprising 210,038 single-turn and 1,871 multi-turn dialogues related to crop science scenarios. The CROP benchmark includes 5,045 multiple-choice questions covering three difficulty levels. Our experiments based on the CROP benchmark demonstrate notable enhancements in crop science-related tasks when LLMs are fine-tuned with the CROP dataset. To the best of our knowledge, CROP dataset is the first-ever instruction tuning dataset in the crop science domain. We anticipate that CROP will accelerate the adoption of LLMs in the domain of crop science, ultimately contributing to global food production.

Poster
Contrastive losses as generalized models of global epistasis

David H Brookes, Jakub Otwinowski, Sam Sinai

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Fitness functions map large combinatorial spaces of biological sequences to properties of interest. Inferring these multimodal functions from experimental data is a central task in modern protein engineering. Global epistasis models are an effective and physically-grounded class of models for estimating fitness functions from observed data. These models assume that a sparse latent function is transformed by a monotonic nonlinearity to emit measurable fitness. Here we demonstrate that minimizing supervised contrastive loss functions, such as the Bradley-Terry loss, is a simple and flexible technique for extracting the sparse latent function implied by global epistasis. We argue by way of a fitness-epistasis uncertainty principle that the nonlinearities in global epistasis models can produce observed fitness functions that do not admit sparse representations, and thus may be inefficient to learn from observations when using a Mean Squared Error (MSE) loss (a common practice). We show that contrastive losses are able to accurately estimate a ranking function from limited data even in regimes where MSE is ineffective and validate the practical utility of this insight by demonstrating that contrastive loss functions result in consistently improved performance on empirical benchmark tasks.

Poster
Rethinking Optimal Transport in Offline Reinforcement Learning

Arip Asadulaev, Rostislav Korst, Alexander Korotin, Vage Egiazarian, Andrey Filchenkov, Evgeny Burnaev

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We propose a novel algorithm for offline reinforcement learning using optimal transport. Typically, in offline reinforcement learning, the data is provided by various experts and some of them can be sub-optimal. To extract an efficient policy, it is necessary to \emph{stitch} the best behaviors from the dataset. To address this problem, we rethink offline reinforcement learning as an optimal transportation problem. And based on this, we present an algorithm that aims to find a policy that maps states to a \emph{partial} distribution of the best expert actions for each given state. We evaluate the performance of our algorithm on continuous control problems from the D4RL suite and demonstrate improvements over existing methods.

Poster
On the Target-kernel Alignment: a Unified Analysis with Kernel Complexity

Chao Wang, Xin He, Yuwen Wang, Junhui Wang

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This paper investigates the impact of alignment between the target function of interest and the kernel matrix on a variety of kernel-based methods based on a general loss belonging to a rich loss function family, which covers many commonly used methods in regression and classification problems. We consider the truncated kernel-based method (TKM) which is estimated within a reduced function space constructed by using the spectral truncation of the kernel matrix and compare its theoretical behavior to that of the standard kernel-based method (KM) under various settings. By using the kernel complexity function that quantifies the complexity of the induced function space, we derive the upper bounds for both TKM and KM, and further reveal their dependencies on the degree of target-kernel alignment. Specifically, for the alignment with polynomial decay, the established results indicate that under the just-aligned and weakly-aligned regimes, TKM and KM share the same learning rate. Yet, under the strongly-aligned regime, KM suffers the saturation effect, while TKM can be continuously improved as the alignment becomes stronger. This further implies that TKM has a strong ability to capture the strong alignment and provide a theoretically guaranteed solution to eliminate the phenomena of saturation effect. The minimax lower bound is also established for the squared loss to confirm the optimality of TKM. Extensive numerical experiments further support our theoretical findings. The Python code for reproducing the numerical experiments is available at https://github.com/wywangen.

Poster
DFA-GNN: Forward Learning of Graph Neural Networks by Direct Feedback Alignment

Gongpei Zhao, Tao Wang, Congyan Lang, Yi Jin, Yidong Li, Haibin Ling

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Graph neural networks (GNNs) are recognized for their strong performance across various applications, with the backpropagation (BP) algorithm playing a central role in the development of most GNN models. However, despite its effectiveness, BP has limitations that challenge its biological plausibility and affect the efficiency, scalability and parallelism of training neural networks for graph-based tasks. While several non-backpropagation (non-BP) training algorithms, such as the direct feedback alignment (DFA), have been successfully applied to fully-connected and convolutional network components for handling Euclidean data, directly adapting these non-BP frameworks to manage non-Euclidean graph data in GNN models presents significant challenges. These challenges primarily arise from the violation of the independent and identically distributed (i.i.d.) assumption in graph data and the difficulty in accessing prediction errors for all samples (nodes) within the graph. To overcome these obstacles, in this paper we propose DFA-GNN, a novel forward learning framework tailored for GNNs with a case study of semi-supervised learning. The proposed method breaks the limitations of BP by using a dedicated forward training mechanism. Specifically, DFA-GNN extends the principles of DFA to adapt to graph data and unique architecture of GNNs, which incorporates the information of graph topology into the feedback links to accommodate the non-Euclidean characteristics of graph data. Additionally, for semi-supervised graph learning tasks, we developed a pseudo error generator that spreads residual errors from training data to create a pseudo error for each unlabeled node. These pseudo errors are then utilized to train GNNs using DFA. Extensive experiments on 10 public benchmarks reveal that our learning framework outperforms not only previous non-BP methods but also the standard BP methods, and it exhibits excellent robustness against various types of noise and attacks.

Poster
Fast Last-Iterate Convergence of Learning in Games Requires Forgetful Algorithms

Yang Cai, Gabriele Farina, Julien Grand-Clément, Christian Kroer, Chung-Wei Lee, Haipeng Luo, Weiqiang Zheng

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Self play via online learning is one of the premier ways to solve large-scale zero-sum games, both in theory and practice. Particularly popular algorithms include optimistic multiplicative weights update (OMWU) and optimistic gradient-descent-ascent (OGDA). While both algorithms enjoy $O(1/T)$ ergodic convergence to Nash equilibrium in two-player zero-sum games, OMWU offers several advantages, including logarithmic dependence on the size of the payoff matrix and $\tilde{O}(1/T)$ convergence to coarse correlated equilibria even in general-sum games. However, in terms of last-iterate convergence in two-player zero-sum games, an increasingly popular topic in this area, OGDA guarantees that the duality gap shrinks at a rate of $(1/\sqrt{T})$, while the best existing last-iterate convergence for OMWU depends on some game-dependent constant that could be arbitrarily large. This begs the question: is this potentially slow last-iterate convergence an inherent disadvantage of OMWU, or is the current analysis too loose? Somewhat surprisingly, we show that the former is true. More generally, we prove that a broad class of algorithms that do not forget the past quickly all suffer the same issue: for any arbitrarily small $\delta>0$, there exists a $2\times 2$ matrix game such that the algorithm admits a constant duality gap even after $1/\delta$ rounds. This class of algorithms includes OMWU and other standard optimistic follow-the-regularized-leader algorithms.

Poster
HARMONIC: Harnessing LLMs for Tabular Data Synthesis and Privacy Protection

Yuxin Wang, Duanyu Feng, Yongfu Dai, Zhengyu Chen, Jimin Huang, Sophia Ananiadou, Qianqian Xie, Hao Wang

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Data serves as the fundamental basis for advancing deep learning. The tabular data presented in a structured format is highly valuable for modeling and training. However, even in the era of LLM, obtaining tabular data from sensitive domains remains a challenge due to privacy or copyright concerns. Therefore, exploring the methods for effectively using models like LLMs to generate synthetic tabular data, which is privacy-preserving but similar to original one, is urgent. In this paper, we introduce a new framework HARMONIC for tabular data generation and evaluation by LLMs. In the data generation part of our framework, we employ fine-tuning to generate tabular data and enhance privacy rather than continued pre-training which is often used by previous small-scale LLM-based methods. In particular, we construct an instruction fine-tuning dataset based on the idea of the k-nearest neighbors algorithm to inspire LLMs to discover inter-row relationships. By such fine-tuning, LLMs are trained to remember the format and connections of the data rather than the data itself, which reduces the risk of privacy leakage. The experiments find that our tabular data generation achieves equivalent performance as existing methods but with better privacy by the metric of MLE, DCR, etc. In the evaluation part of our framework, we develop a specific privacy risk metric DLT for LLM synthetic data generation, which quantifies the extent to which the generator itself leaks data. We also developed LLE, a performance evaluation metric for downstream LLM tasks, which is more practical and credible than previous metrics. The experiments show that our data generation method outperform the previous methods in the metrics DLT and LLE.

Poster
Identifying Selections for Unsupervised Subtask Discovery

Yiwen Qiu, Yujia Zheng, Kun Zhang

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When solving long-horizon tasks, it is intriguing to decompose the high-level task into subtasks. Decomposing experiences into reusable subtasks can improve data efficiency, accelerate policy generalization, and in general provide promising solutions to multi-task reinforcement learning and imitation learning problems. However, the concept of subtasks is not sufficiently understood and modeled yet, and existing works often overlook the true structure of the data generation process: subtasks are the results of a *selection* mechanism on actions, rather than possible underlying confounders or intermediates. Specifically, we provide a theory to identify, and experiments to verify the existence of selection variables in such data. These selections serve as subgoals that indicate subtasks and guide policy. In light of this idea, we develop a sequential non-negative matrix factorization (seq- NMF) method to learn these subgoals and extract meaningful behavior patterns as subtasks. Our empirical results on a challenging Kitchen environment demonstrate that the learned subtasks effectively enhance the generalization to new tasks in multi-task imitation learning scenarios. The codes are provided at this [*link*](https://anonymous.4open.science/r/Identifying\_Selections\_for\_Unsupervised\_Subtask\_Discovery/README.md).

Poster
Low-Rank Optimal Transport through Factor Relaxation with Latent Coupling

Peter Halmos, XinHao Liu, Julian Gold, Benjamin Raphael

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Optimal transport (OT) is a general framework for finding a minimum-cost transport plan, or coupling, between probability distributions, and has many applications in machine learning. A key challenge in applying OT to massive datasets is the quadratic scaling of the coupling matrix with the size of the dataset. [Forrow et al. 2019] introduced a factored coupling for the k-Wasserstein barycenter problem, which [Scetbon et al. 2021] adapted to solve the primal low-rank OT problem. We derive an alternative parameterization of the low-rank problem based on the _latent coupling_ (LC) factorization previously introduced by [Lin et al. 2021] generalizing [Forrow et al. 2019]. The LC factorization has multiple advantages for low-rank OT including decoupling the problem into three OT problems and greater flexibility and interpretability. We leverage these advantages to derive a new algorithm _Factor Relaxation with Latent Coupling_ (FRLC), which uses _coordinate_ mirror descent to compute the LC factorization. FRLC handles multiple OT objectives (Wasserstein, Gromov-Wasserstein, Fused Gromov-Wasserstein), and marginal constraints (balanced, unbalanced, and semi-relaxed) with linear space complexity. We provide theoretical results on FRLC, and demonstrate superior performance on diverse applications -- including graph clustering and spatial transcriptomics -- while demonstrating its interpretability.

Poster
Web2Code: A Large-scale Webpage-to-Code Dataset and Evaluation Framework for Multimodal LLMs

Sukmin Yun, Haokun Lin, Rusiru Thushara, Mohammad Qazim Bhat, Yongxin Wang, Zutao Jiang, Mingkai Deng, Jinhong Wang, Tianhua Tao, Junbo Li, Haonan Li, Preslav Nakov, Timothy Baldwin, Zhengzhong Liu, Eric P. Xing, Xiaodan Liang, Zhiqiang Shen

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Multimodal large language models (MLLMs) have shown impressive success across modalities such as image, video, and audio in a variety of understanding and generation tasks. However, current MLLMs are surprisingly poor at understanding webpage screenshots and generating their corresponding HTML code. To address this problem, we propose Web2Code, a benchmark consisting of a new large-scale webpage-to-code dataset for instruction tuning and an evaluation framework for the webpage understanding and HTML code translation abilities of MLLMs. For dataset construction, we leverage pretrained LLMs to enhance existing webpage-to-code datasets as well as generate a diverse pool of new webpages rendered into images. Specifically, the inputs are webpage images and instructions, while the responses are the webpage's HTML code. We further include diverse natural language QA pairs about the webpage content in the responses to enable a more comprehensive understanding of the web content. To evaluate model performance in these tasks, we develop an evaluation framework for testing MLLMs' abilities in webpage understanding and web-to-code generation. Extensive experiments show that our proposed dataset is beneficial not only to our proposed tasks but also in the general visual domain. We hope our work will contribute to the development of general MLLMs suitable for web-based content generation and task automation. Our data and code are available at https://github.com/MBZUAI-LLM/web2code.

Poster
Fixed Confidence Best Arm Identification in the Bayesian Setting

Kyoungseok Jang, Junpei Komiyama, Kazutoshi Yamazaki

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We consider the fixed-confidence best arm identification (FC-BAI) problem in the Bayesian setting. This problem aims to find the arm of the largest mean with a fixed confidence level when the bandit model has been sampled from the known prior. Most studies on the FC-BAI problem have been conducted in the frequentist setting, where the bandit model is predetermined before the game starts. We show that the traditional FC-BAI algorithms studied in the frequentist setting, such as track-and-stop and top-two algorithms, result in arbitrarily suboptimal performances in the Bayesian setting. We also obtain a lower bound of the expected number of samples in the Bayesian setting and introduce a variant of successive elimination that has a matching performance with the lower bound up to a logarithmic factor. Simulations verify the theoretical results.

Poster