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

Multi-Stage Predict+Optimize for (Mixed Integer) Linear Programs

Xinyi Hu, Jasper C.H. Lee, Jimmy H.M. Lee, Peter J. Stuckey

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The recently-proposed framework of Predict+Optimize tackles optimization problems with parameters that are unknown at solving time, in a supervised learning setting. Prior frameworks consider only the scenario where all unknown parameters are (eventually) revealed simultaneously. In this work, we propose Multi-Stage Predict+Optimize, a novel extension catering to applications where unknown parameters are revealed in sequential stages, with optimization decisions made in between. We further develop three training algorithms for neural networks (NNs) for our framework as proof of concept, both of which handle all mixed integer linear programs. The first baseline algorithm is a natural extension of prior work, training a single NN which makes a single prediction of unknown parameters. The second and third algorithms instead leverage the possibility of updating parameter predictions between stages, and trains one NN per stage. To handle the interdependency between the neural networks, we adopt sequential and parallelized versions of coordinate descent for training. Experimentation on three benchmarks demonstrates the superior learning performance of our methods over classical approaches.

Poster
Uni-Med: A Unified Medical Generalist Foundation Model For Multi-Task Learning Via Connector-MoE

Xun Zhu, Ying Hu, Fanbin Mo, Miao Li, Ji Wu

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Multi-modal large language models (MLLMs) have shown impressive capabilities as a general-purpose interface for various visual and linguistic tasks. However, building a unified MLLM for multi-task learning in the medical field remains a thorny challenge. To mitigate the tug-of-war problem of multi-modal multi-task optimization in MLLMs, recent advances primarily focus on improving the LLM components, while neglecting the connector that bridges the gap between modalities. In this paper, we introduce Uni-Med, a novel medical generalist foundation model which consists of a universal visual feature extraction module, a connector mixture-of-experts (CMoE) module, and an LLM. Benefiting from the proposed CMoE that leverages a well-designed router with a mixture of projection experts at the connector, Uni-Med achieves efficient solution to the tug-of-war problem and can perform six different medical tasks including question answering, visual question answering, report generation, referring expression comprehension, referring expression generation and image classification. To the best of our knowledge, Uni-Med is the first effort to tackle multi-task interference at the connector in MLLMs. Extensive ablation experiments validate the effectiveness of introducing CMoE under any configuration, with up to an average 8% performance gains. We further provide interpretation analysis of the tug-of-war problem from the perspective of gradient optimization and parameter statistics. Compared to previous state-of-the-art medical MLLMs, Uni-Med achieves competitive or superior evaluation metrics on diverse tasks. Code and resources are available at https://github.com/MSIIP/Uni-Med.

Poster
Direct Preference-Based Evolutionary Multi-Objective Optimization with Dueling Bandits

Tian Huang, Shengbo Wang, Ke Li

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The ultimate goal of multi-objective optimization (MO) is to assist human decision-makers (DMs) in identifying solutions of interest (SOI) that optimally reconcile multiple objectives according to their preferences. Preference-based evolutionary MO (PBEMO) has emerged as a promising framework that progressively approximates SOI by involving human in the optimization-cum-decision-making process. Yet, current PBEMO approaches are prone to be inefficient and misaligned with the DM’s true aspirations, especially when inadvertently exploiting mis-calibrated reward models. This is further exacerbated when considering the stochastic nature of human feedback. This paper proposes a novel framework that navigates MO to SOI by directly leveraging human feedback without being restricted by a predefined reward model nor cumbersome model selection. Specifically, we developed a clustering-based stochastic dueling bandits algorithm that strategically scales well to high-dimensional dueling bandits, and achieves a regret of $\mathcal{O}(K^2\log T)$, where $K$ is the number of clusters and $T$ is the number of rounds. The learned preferences are then transformed into a unified probabilistic format that can be readily adapted to prevalent EMO algorithms. This also leads to a principled termination criterion that strategically manages human cognitive loads and computational budget. Experiments on $48$ benchmark test problems, including synthetic problems, RNA inverse design and protein structure prediction, fully demonstrate the effectiveness of our proposed approach.

Poster
CoMix: A Comprehensive Benchmark for Multi-Task Comic Understanding

Emanuele Vivoli, Marco Bertini, Dimosthenis Karatzas

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The comic domain is rapidly advancing with the development of single-page analysis and synthesis models. However, evaluation metrics and datasets lag behind, often limited to small-scale or single-style test sets. We introduce a novel benchmark, CoMix, designed to evaluate the multi-task capabilities of models in comic analysis. Unlike existing benchmarks that focus on isolated tasks such as object detection or text recognition, CoMix addresses a broader range of tasks including object detection, speaker identification, character re-identification, reading order, and multi-modal reasoning tasks like character naming and dialogue generation. Our benchmark comprises three existing datasets with expanded annotations to support multi-task evaluation. To mitigate the over-representation of manga-style data, we have incorporated a new dataset of carefully selected American comic-style books, thereby enriching the diversity of comic styles. CoMix is designed to assess pre-trained models in zero-shot and limited fine-tuning settings, probing their transfer capabilities across different comic styles and tasks. The validation split of the benchmark is publicly available for research purposes, and an evaluation server for the held-out test split is also provided. Comparative results between human performance and state-of-the-art models reveal a significant performance gap, highlighting substantial opportunities for advancements in comic understanding. The dataset, baseline models, and code are accessible at https://github.com/emanuelevivoli/CoMix-dataset. This initiative sets a new standard for comprehensive comic analysis, providing the community with a common benchmark for evaluation on a large and varied set.

Poster
Towards a "Universal Translator" for Neural Dynamics at Single-Cell, Single-Spike Resolution

Yizi Zhang, Yanchen Wang, Donato M. Jiménez-Benetó, Zixuan Wang, Mehdi Azabou, Blake Aaron Richards, Renee Tung, Olivier Winter, International Brain Laboratory, Eva L Dyer, Liam Paninski, Cole Lincoln Hurwitz

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Neuroscience research has made immense progress over the last decade, but our understanding of the brain remains fragmented and piecemeal: the dream of probing an arbitrary brain region and automatically reading out the information encoded in its neural activity remains out of reach. In this work, we build towards a first foundation model for neural spiking data that can solve a diverse set of tasks across multiple brain areas. We introduce a novel self-supervised modeling approach for population activity in which the model alternates between masking out and reconstructing neural activity across different time steps, neurons, and brain regions. To evaluate our approach, we design unsupervised and supervised prediction tasks using the International Brain Laboratory repeated site dataset, which is comprised of Neuropixels recordings targeting the same brain locations across 48 animals and experimental sessions. The prediction tasks include single-neuron and region-level activity prediction, forward prediction, and behavior decoding. We demonstrate that our multi-task-masking (MtM) approach significantly improves the performance of current state-of-the-art population models and enables multi-task learning. We also show that by training on multiple animals, we can improve the generalization ability of the model to unseen animals, paving the way for a foundation model of the brain at single-cell, single-spike resolution.

Poster
CONTRAST: Continual Multi-source Adaptation to Dynamic Distributions

Sk Miraj Ahmed, Fahim Faisal Niloy, Xiangyu Chang, Dripta S. Raychaudhuri, Samet Oymak, Amit Roy-Chowdhury

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Adapting to dynamic data distributions is a practical yet challenging task. One effective strategy is to use a model ensemble, which leverages the diverse expertise of different models to transfer knowledge to evolving data distributions. However, this approach faces difficulties when the dynamic test distribution is available only in small batches and without access to the original source data. To address the challenge of adapting to dynamic distributions in such practical settings, we propose continual multi-source adaptation to dynamic distributions (CONTRAST), a novel method that optimally combines multiple source models to adapt to the dynamic test data. CONTRAST has two distinguishing features. First, it efficiently computes the optimal combination weights to combine the source models to adapt to the test data distribution continuously as a function of time. Second, it identifies which of the source model parameters to update so that only the model which is most correlated to the target data is adapted, leaving the less correlated ones untouched; this mitigates the issue of ``forgetting" the source model parameters by focusing only on the source model that exhibits the strongest correlation with the test batch distribution. Through theoretical analysis we show that the proposed method is able to optimally combine the source models and prioritize updates to the model least prone to forgetting. Experimental analysis on diverse datasets demonstrates that the combination of multiple source models does at least as well as the best source (with hindsight knowledge), and performance does not degrade as the test data distribution changes over time (robust to forgetting).

Poster
Automated Multi-Task Learning for Joint Disease Prediction on Electronic Health Records

Suhan Cui, Prasenjit Mitra

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In the realm of big data and digital healthcare, Electronic Health Records (EHR) have become a rich source of information with the potential to improve patient care and medical research. In recent years, machine learning models have proliferated for analyzing EHR data to predict patients' future health conditions. Among them, some studies advocate for multi-task learning (MTL) to jointly predict multiple target diseases for improving the prediction performance over single task learning. Nevertheless, current MTL frameworks for EHR data have significant limitations due to their heavy reliance on human experts to identify task groups for joint training and design model architectures. To reduce human intervention and improve the framework design, we propose an automated approach named AutoDP, which can search for the optimal configuration of task grouping and architectures simultaneously. To tackle the vast joint search space encompassing task combinations and architectures, we employ surrogate model-based optimization, enabling us to efficiently discover the optimal solution. Experimental results on real-world EHR data demonstrate the efficacy of the proposed AutoDP framework. It achieves significant performance improvements over both hand-crafted and automated state-of-the-art methods, also maintains a feasible search cost at the same time.

Poster
EMR-Merging: Tuning-Free High-Performance Model Merging

Chenyu Huang, Peng Ye, Tao Chen, Tong He, Xiangyu Yue, Wanli Ouyang

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The success of pretrain-finetune paradigm brings about the release of numerous model weights. In this case, merging models finetuned on different tasks to enable a single model with multi-task capabilities is gaining increasing attention for its practicability. Existing model merging methods usually suffer from (1) significant performance degradation or (2) requiring tuning by additional data or training. In this paper, we rethink and analyze the existing model merging paradigm. We discover that using a single model's weights can hardly simulate all the models' performance. To tackle this issue, we propose Elect, Mask & Rescale-Merging (EMR-Merging). We first (a) elect a unified model from all the model weights and then (b) generate extremely lightweight task-specific modulators, including masks and rescalers, to align the direction and magnitude between the unified model and each specific model, respectively. EMR-Merging is tuning-free, thus requiring no data availability or any additional training while showing impressive performance. We find that EMR-Merging shows outstanding performance compared to existing merging methods under different classical and newly-established settings, including merging different numbers of vision models (up to 30), NLP models, PEFT models, and multi-modal models.

Spotlight
Bandits with Abstention under Expert Advice

Stephen Pasteris, Alberto Rumi, Maximilian Thiessen, Shota Saito, Atsushi Miyauchi, Fabio Vitale, Mark Herbster

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We study the classic problem of prediction with expert advice under bandit feedback. Our model assumes that one action, corresponding to the learner's abstention from play, has no reward or loss on every trial. We propose the CBA (Confidence-rated Bandits with Abstentions) algorithm, which exploits this assumption to obtain reward bounds that can significantly improve those of the classical Exp4 algorithm. Our problem can be construed as the aggregation of confidence-rated predictors, with the learner having the option to abstain from play. We are the first to achieve bounds on the expected cumulative reward for general confidence-rated predictors. In the special case of specialists, we achieve a novel reward bound, significantly improving previous bounds of SpecialistExp (treating abstention as another action). We discuss how CBA can be applied to the problem of adversarial contextual bandits with the option of abstaining from selecting any action. We are able to leverage a wide range of inductive biases, outperforming previous approaches both theoretically and in preliminary experimental analysis. Additionally, we achieve a reduction in runtime from quadratic to almost linear in the number of contexts for the specific case of metric space contexts.

Poster
Improved Bayes Regret Bounds for Multi-Task Hierarchical Bayesian Bandit Algorithms

Jiechao Guan, Hui Xiong

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Hierarchical Bayesian bandit refers to the multi-task bandit problem in which bandit tasks are assumed to be drawn from the same distribution. In this work, we provide improved Bayes regret bounds for hierarchical Bayesian bandit algorithms in the multi-task linear bandit and semi-bandit settings. For the multi-task linear bandit, we first analyze the preexisting hierarchical Thompson sampling (HierTS) algorithm, and improve its gap-independent Bayes regret bound from $O(m\sqrt{n\log{n}\log{(mn)}})$ to $O(m\sqrt{n\log{n}})$ in the case of infinite action set, with $m$ being the number of tasks and $n$ the number of iterations per task. In the case of finite action set, we propose a novel hierarchical Bayesian bandit algorithm, named hierarchical BayesUCB (HierBayesUCB), that achieves the logarithmic but gap-dependent regret bound $O(m\log{(mn)}\log{n})$ under mild assumptions. All of the above regret bounds hold in many variants of hierarchical Bayesian linear bandit problem, including when the tasks are solved sequentially or concurrently. Furthermore, we extend the aforementioned HierTS and HierBayesUCB algorithms to the multi-task combinatorial semi-bandit setting. Concretely, our combinatorial HierTS algorithm attains comparable Bayes regret bound $O(m\sqrt{n}\log{n})$ with respect to the latest one. Moreover, our combinatorial HierBayesUCB yields a sharper Bayes regret bound $O(m\log{(mn)}\log{n})$. Experiments are conducted to validate the soundness of our theoretical results for multi-task bandit algorithms.

Poster
Medformer: A Multi-Granularity Patching Transformer for Medical Time-Series Classification

Yihe Wang, Nan Huang, Taida Li, Yujun Yan, Xiang Zhang

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Medical time series (MedTS) data, such as Electroencephalography (EEG) and Electrocardiography (ECG), play a crucial role in healthcare, such as diagnosing brain and heart diseases. Existing methods for MedTS classification primarily rely on handcrafted biomarkers extraction and CNN-based models, with limited exploration of transformer-based models. In this paper, we introduce Medformer, a multi-granularity patching transformer tailored specifically for MedTS classification. Our method incorporates three novel mechanisms to leverage the unique characteristics of MedTS: cross-channel patching to leverage inter-channel correlations, multi-granularity embedding for capturing features at different scales, and two-stage (intra- and inter-granularity) multi-granularity self-attention for learning features and correlations within and among granularities. We conduct extensive experiments on five public datasets under both subject-dependent and challenging subject-independent setups. Results demonstrate Medformer's superiority over 10 baselines, achieving top averaged ranking across five datasets on all six evaluation metrics. These findings underscore the significant impact of our method on healthcare applications, such as diagnosing Myocardial Infarction, Alzheimer's, and Parkinson's disease. We release the source code at https://github.com/DL4mHealth/Medformer.

Poster
Kraken: Inherently Parallel Transformers For Efficient Multi-Device Inference

Rohan Baskar Prabhakar, Hengrui Zhang, David Wentzlaff

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Large Transformer networks are increasingly used in settings where low inference latency is necessary to enable new applications and improve the end-user experience. However, autoregressive inference is resource intensive and requires parallelism for efficiency. Parallelism introduces collective communication that is both expensive and represents a phase when hardware resources are underutilized. Towards mitigating this, Kraken is an evolution of the standard Transformer architecture that is designed to complement existing tensor parallelism schemes for efficient inference on multi-device systems. By introducing a fixed degree of intra-layer model parallelism, the architecture allows collective operations to be overlapped with compute, decreasing latency and increasing hardware utilization. When trained on OpenWebText, Kraken models reach a similar perplexity as standard Transformers while also preserving their language modeling capabilities as evaluated on the SuperGLUE benchmark. Importantly, when tested on multi-GPU systems using TensorRT-LLM engines, Kraken speeds up Time To First Token by a mean of 35.6% across a range of model sizes, context lengths, and degrees of tensor parallelism.

Poster
Towards Neuron Attributions in Multi-Modal Large Language Models

Junfeng Fang, Zac Bi, Ruipeng Wang, Houcheng Jiang, Yuan Gao, Kun Wang, An Zhang, Jie Shi, Xiang Wang, Tat-Seng Chua

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As Large Language Models (LLMs) demonstrate impressive capabilities, demystifying their internal mechanisms becomes increasingly vital. Neuron attribution, which attributes LLM outputs to specific neurons to reveal the semantic properties they learn, has emerged as a key interpretability approach. However, while neuron attribution has made significant progress in deciphering text-only LLMs, its application to Multimodal LLMs (MLLMs) remains less explored. To address this gap, we propose a novel Neuron Attribution method tailored for MLLMs, termed NAM. Specifically, NAM not only reveals the modality-specific semantic knowledge learned by neurons within MLLMs, but also highlights several intriguing properties of neurons, such as cross-modal invariance and semantic sensitivity. These properties collectively elucidate the inner workings mechanism of MLLMs, providing a deeper understanding of how MLLMs process and generate multi-modal content. Through theoretical analysis and empirical validation, we demonstrate the efficacy of NAM and the valuable insights it offers. Furthermore, leveraging NAM, we introduce a multi-modal knowledge editing paradigm, underscoring the practical significance of our approach for downstream applications of MLLMs.

Poster
MOTE-NAS: Multi-Objective Training-based Estimate for Efficient Neural Architecture Search

Yuming Zhang, Jun Wei Hsieh, Xin Li, Ming-Ching Chang, Chun-Chieh Lee, Kuo-Chin Fan

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Neural Architecture Search (NAS) methods seek effective optimization toward performance metrics regarding model accuracy and generalization while facing challenges regarding search costs and GPU resources. Recent Neural Tangent Kernel (NTK) NAS methods achieve remarkable search efficiency based on a training-free model estimate; however, they overlook the non-convex nature of the DNNs in the search process. In this paper, we develop Multi-Objective Training-based Estimate (MOTE) for efficient NAS, retaining search effectiveness and achieving the new state-of-the-art in the accuracy and cost trade-off. To improve NTK and inspired by the Training Speed Estimation (TSE) method, MOTE is designed to model the actual performance of DNNs from macro to micro perspective by draw loss landscape and convergence speed simultaneously. Using two reduction strategies, the MOTE is generated based on a reduced architecture and a reduced dataset. Inspired by evolutionary search, our iterative ranking-based, coarse-to-fine architecture search is highly effective. Experiments on NASBench-201 show MOTE-NAS achieves 94.32% accuracy on CIFAR-10, 72.81% on CIFAR-100, and 46.38% on ImageNet-16-120, outperforming NTK-based NAS approaches. An evaluation-free (EF) version of MOTE-NAS delivers high efficiency in only 5 minutes, delivering a model more accurate than KNAS.

Poster
Identifiable Shared Component Analysis of Unpaired Multimodal Mixtures

Subash Timilsina, Sagar Shrestha, Xiao Fu

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A core task in multi-modal learning is to integrate information from multiple feature spaces (e.g., text and audio), offering modality-invariant essential representations of data. Recent research showed that, classical tools such as canonical correlation analysis (CCA) provably identify the shared components up to minor ambiguities, when samples in each modality are generated from a linear mixture of shared and private components. Such identifiability results were obtained under the condition that the cross-modality samples are aligned/paired according to their shared information. This work takes a step further, investigating shared component identifiability from multi-modal linear mixtures where cross-modality samples are unaligned. A distribution divergence minimization-based loss is proposed, under which a suite of sufficient conditions ensuring identifiability of the shared components are derived. Our conditions are based on cross-modality distribution discrepancy characterization and density-preserving transform removal, which are much milder than existing studies relying on independent component analysis. More relaxed conditions are also provided via adding reasonable structural constraints, motivated by available side information in various applications. The identifiability claims are thoroughly validated using synthetic and real-world data.

Poster
$C^2M^3$: Cycle-Consistent Multi-Model Merging

Donato Crisostomi, Marco Fumero, Daniele Baieri, Florian Bernard, Emanuele Rodolà

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In this paper, we present a novel data-free method for merging neural networks in weight space. Our method optimizes for the permutations of network neurons while ensuring global coherence across all layers, and it outperforms recent layer-local approaches in a set of challenging scenarios. We then generalize the formulation to the $N$-models scenario to enforce cycle consistency of the permutations with guarantees, allowing circular compositions of permutations to be computed without accumulating error along the path. We qualitatively and quantitatively motivate the need for such a constraint, showing its benefits when merging homogeneous sets of models in scenarios spanning varying architectures and datasets. We finally show that, when coupled with activation renormalization, the approach yields the best results in the task.

Poster
Pre-trained Large Language Models Use Fourier Features to Compute Addition

Tianyi Zhou, Deqing Fu, Vatsal Sharan, Robin Jia

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Pre-trained large language models (LLMs) exhibit impressive mathematical reasoning capabilities, yet how they compute basic arithmetic, such as addition, remains unclear. This paper shows that pre-trained LLMs add numbers using Fourier features---dimensions in the hidden state that represent numbers via a set of features sparse in the frequency domain. Within the model, MLP and attention layers use Fourier features in complementary ways: MLP layers primarily approximate the magnitude of the answer using low-frequency features, while attention layers primarily perform modular addition (e.g., computing whether the answer is even or odd) using high-frequency features. Pre-training is crucial for this mechanism: models trained from scratch to add numbers only exploit low-frequency features, leading to lower accuracy. Introducing pre-trained token embeddings to a randomly initialized model rescues its performance. Overall, our analysis demonstrates that appropriate pre-trained representations (e.g., Fourier features) can unlock the ability of Transformers to learn precise mechanisms for algorithmic tasks.

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
Dual Cone Gradient Descent for Training Physics-Informed Neural Networks

Youngsik Hwang, Dongyoung Lim

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Physics-informed neural networks (PINNs) have emerged as a prominent approach for solving partial differential equations (PDEs) by minimizing a combined loss function that incorporates both boundary loss and PDE residual loss. Despite their remarkable empirical performance in various scientific computing tasks, PINNs often fail to generate reasonable solutions, and such pathological behaviors remain difficult to explain and resolve. In this paper, we identify that PINNs can be adversely trained when gradients of each loss function exhibit a significant imbalance in their magnitudes and present a negative inner product value. To address these issues, we propose a novel optimization framework, *Dual Cone Gradient Descent* (DCGD), which adjusts the direction of the updated gradient to ensure it falls within a dual cone region. This region is defined as a set of vectors where the inner products with both the gradients of the PDE residual loss and the boundary loss are non-negative. Theoretically, we analyze the convergence properties of DCGD algorithms in a non-convex setting. On a variety of benchmark equations, we demonstrate that DCGD outperforms other optimization algorithms in terms of various evaluation metrics. In particular, DCGD achieves superior predictive accuracy and enhances the stability of training for failure modes of PINNs and complex PDEs, compared to existing optimally tuned models. Moreover, DCGD can be further improved by combining it with popular strategies for PINNs, including learning rate annealing and the Neural Tangent Kernel (NTK).

Poster
Optimal Multi-Fidelity Best-Arm Identification

Riccardo Poiani, Rémy Degenne, Emilie Kaufmann, Alberto Maria Metelli, Marcello Restelli

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In bandit best-arm identification, an algorithm is tasked with finding the arm with highest mean reward with a specified accuracy as fast as possible. We study multi-fidelity best-arm identification, in which the algorithm can choose to sample an arm at a lower fidelity (less accurate mean estimate) for a lower cost. Several methods have been proposed for tackling this problem, but their optimality remain elusive, notably due to loose lower bounds on the total cost needed to identify the best arm. Our first contribution is a tight, instance-dependent lower bound on the cost complexity. The study of the optimization problem featured in the lower bound provides new insights to devise computationally efficient algorithms, and leads us to propose a gradient-based approach with asymptotically optimal cost complexity. We demonstrate the benefits of the new algorithm compared to existing methods in experiments. Our theoretical and empirical findings also shed light on an intriguing concept of optimal fidelity for each arm.

Poster