ICLR 2024 Oral | To deal with distribution shifts that change over time, the University of Western Ontario and others proposed a learning time series trajectory method

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##The author Zeng Qiulin graduated from Harbin Institute of Technology and received a master's degree from the National University of Singapore. Under the guidance of Professor Wang Bo and Academician Ling Xiaofeng, during his doctoral period, he mainly carried out theoretical demonstration, method and application research on the distribution of random time changes. Currently, he has published many academic papers in ICLR/AAAI/IEEE TNNLS.

Personal homepage: https://hardworkingpearl.github.io/

Distribution shifts over time in real-world machine learning applications is a common question. This situation is framed as time-varying domain generalization (EDG), where the goal is to enable the model to generalize well to unseen target domains in time-varying systems by learning underlying evolving patterns across domains and exploiting these patterns. However, due to the limited number of timestamps in the EDG dataset, existing methods encounter challenges in capturing the evolving dynamics and avoiding overfitting to sparse timestamps, which limits their generalization and adaptability to new tasks.

To solve this problem, we propose a new method SDE-EDG, which collects the infinite subdivision grid evolution trajectory (IFGET) of the data distribution through continuous interpolation samples to overcome the problem of overfitting. . Furthermore, by exploiting the inherent ability of stochastic differential equations (SDEs) to capture continuous trajectories, we propose to align the trajectories modeled by SDEs with those of IFGET through maximum likelihood estimation, thereby achieving the capture of distribution evolution trends.

• Paper title: Latent Trajectory Learning for Limited Timestamps under Distribution Shift over Time

• Paper link: https://openreview.net/pdf?id=bTMMNT7IdW

• Project link: https://github.com/HardworkingPearl/SDE-EDG-iclr2024

Method

Core Idea

To overcome this challenge, SDE-EDG proposes a novel approach by Construct an Infinitely Fined-Grid Evolving Trajectory (IFGET) to create consecutive interpolated samples in the latent representation space to bridge the gap between timestamps. In addition, SDE-EDG utilizes the inherent ability of Stochastic Differential Equations (SDEs) to capture continuous trajectory dynamics, and aligns the SDE modeled trajectories with IFGET through the path alignment regularizer to achieve cross-domain capture of evolving distribution trends. .

Method details

1. Construct IFGET:

First, SDE-EDG builds a sample-to-sample representation for each sample in the latent representation space. Correspondence, collect the evolution trajectory of each individual sample. For any sample

of each category k at time , we search for the closest to it in the feature space at time for its corresponding sample in :

Here is to calculate the distance between two vectors, is the set of samples sampled from the next field .

This correspondence is then used to generate continuous interpolation samples, aiming to connect the time gaps between timestamp intervals and avoid overfitting of sparse timestamps,

Here is sampled from the Beta distribution. By collecting the timing trace of the samples generated in the above way, we get IFGET.

2. Model trajectories using SDE:

SDE-EDG adopts neural SDE to model the continuous time trajectory of data in the latent space. Unlike traditional discrete timestamp-based models, SDE is naturally suitable for simulating continuous time trajectories. SDE-EDG models time series trajectories and can predict samples at any future time through samples at time :

Here the feature space variable is It is predicted by the sample at time , is the drift function, and is the diffusion function.

3. Path alignment and maximum likelihood estimation:

SDE-EDG trains the model by maximizing the likelihood estimation of IFGET,

The final training function is , and the first item is the error loss function for the prediction classification task.

4. Experiment

• 下表展示了 SDE-EDG 与其他基线方法在多个数据集上分类准确率的比较。这些数据集包括 Rotated Gaussian (RG), Circle (Cir), Rotated MNIST (RM), Portraits (Por), Caltran (Cal), PowerSupply (PS), 和 Ocular Disease (OD)。结果显示，SDE-EDG 在所有数据集上的平均准确率均优于其他方法。

• 下图提供了一个直观的比较，展示了 SDE-EDG 算法（左）与传统 DG 方法 IRM（右）在特征表示方面的差异。通过数据特征空间的可视化，我们可以观察到 SDE-EDG 学习到的特征表示具有明显的决策边界，其中不同类别的数据点被清晰地区分开来，以不同形状表示，并且不同域的数据以彩虹条的颜色区分。这表明 SDE-EDG 能够成功捕捉数据随时间演变的动态，并在特征空间中保持类别的可分性。相比之下，IRM 的特征表示则倾向于将数据点坍缩到单一方向，导致决策边界不明显，这反映出 IRM 在捕捉时变分布趋势方面的不足。

• 下图通过一系列子图深入展示了 SDE-EDG 算法在捕捉数据随时间演变的能力方面的优势。子图 (a) 提供了 Sine 数据集的真实标签分布，其中正例和负例用不同颜色的点表示，为后续的比较提供了基准。接着，子图 (b) 和 (c) 分别展示了基于 ERM 的传统方法和 SDE-EDG 算法对同一数据集的预测结果，通过对比可以看出 SDE-EDG 在捕捉数据演变模式上的明显优势。子图 (d) 和 (e) 进一步揭示了 SDE-EDG 学习到的演变路径，其中 (d) 展示了应用了路径对齐损失（最大似然损失函数）后的路径，而 (e) 展示了未应用该损失时的路径。通过这一对比，可以直观地看到路径对齐损失对于确保模型能够正确捕捉和表征数据随时间变化的重要性。

• 下图子图 (a) 展示了在 Portraits 数据集上，使用不同算法进行训练时的准确率收敛轨迹。这个子图提供了一个直观的视角，用以比较 SDE-EDG 算法与其他基线方法（如 ERM、MLDG、GI）在训练过程中性能的变化情况。通过观察训练准确率随时间推移的增长趋势，我们可以评估不同算法的学习能力和收敛速度。SDE-EDG 算法的收敛轨迹尤其值得关注，因为它揭示了该算法在适应不断演变的数据分布时的效率和稳定性。

  下图子图 (b) 和 (c) 分别展示了 RMNIST 和 Circle 数据集上，SDE-EDG 算法在这些数据集上的表现显示出其在处理时变分布时的优越性，即使在面对较大时间跨度的目标域时，也能保持较高的准确率，这表明了 SDE-EDG 算法在捕捉和适应数据演变模式方面的强大能力。

  下图子图 (d) 和 (e) 探讨了最大似然损失（Maximum Likelihood Loss）在 RMNIST 和 PowerSupply 数据集上对 SDE-EDG 性能的影响。通过改变正则化权重 α 的值，这两个子图展示了不同 α 设置对模型性能的具体影响。实验结果表明，适当的 α 值可以显著提高 SDE-EDG 在特定数据集上的性能，这强调了在实际应用中根据数据集特性和任务需求调整超参数的重要性。

结论

论文作者提出了一种新的 SDE-EDG 方法，用于建模时变域泛化（EDG）问题。方法涉及通过识别样本到样本的对应关系并生成连续插值样本来构建 IFGET。随后，作者采用随机微分方程（SDE）并将其与 IFGET 对齐进行训练。文章的贡献在于揭示了通过收集个体的时间轨迹来捕获演变模式的重要性，以及在时间间隔之间进行插值以减轻源时间戳数量有限的问题，这有效地防止了 SDE-EDG 对有限时间戳的过拟合。

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