Learning Long-Term Dependencies in Irregularly-Sampled Time Series (2006.04418v4)

Abstract: Recurrent neural networks (RNNs) with continuous-time hidden states are a natural fit for modeling irregularly-sampled time series. These models, however, face difficulties when the input data possess long-term dependencies. We prove that similar to standard RNNs, the underlying reason for this issue is the vanishing or exploding of the gradient during training. This phenomenon is expressed by the ordinary differential equation (ODE) representation of the hidden state, regardless of the ODE solver's choice. We provide a solution by designing a new algorithm based on the long short-term memory (LSTM) that separates its memory from its time-continuous state. This way, we encode a continuous-time dynamical flow within the RNN, allowing it to respond to inputs arriving at arbitrary time-lags while ensuring a constant error propagation through the memory path. We call these RNN models ODE-LSTMs. We experimentally show that ODE-LSTMs outperform advanced RNN-based counterparts on non-uniformly sampled data with long-term dependencies. All code and data is available at https://github.com/mlech26l/ode-lstms.

Overview of "Learning Long-Term Dependencies in Irregularly-Sampled Time Series"

The paper "Learning Long-Term Dependencies in Irregularly-Sampled Time Series" presents the ODE-LSTM, a recurrent neural network (RNN) model designed for effectively modeling time series data that is sampled at irregular intervals and exhibits long-term dependencies. This work extends upon the existing limitations of continuous-time RNNs, such as ODE-RNNs, which struggle with the vanishing or exploding gradient problem during training—a common obstacle in learning long-term dependencies within sequential data.

Core Contributions

1. Theoretical Analysis: The authors establish that ODE-RNNs, despite their suitability for handling irregular sampling, inherently suffer from vanishing or exploding gradients. This issue arises because the error propagation through time, mathematically expressed by Jacobians in the context of these models, becomes unstable as it exponentially collapses or diverges over time. This analysis is independent of the specific ODE solver employed.
2. ODE-LSTM Architecture: To address this limitation, ODE-LSTMs are introduced, which integrate the strength of long short-term memory (LSTM) networks in managing long-range dependencies with the flexibility of ODE-RNNs. The architecture decouples the memory path from the continuous-time dynamics, allowing for consistent gradient flow, which is crucial for training deep learning models on tasks with long-term dependencies. This design ensures that input data, no matter the irregularity in sampling, can influence the network state robustly over extended temporal gaps.
3. Experimental Validation: Empirical results validate the superiority of ODE-LSTMs over existing RNN-based models (e.g., ODE-RNNs, CT-RNNs, Phased-LSTMs) across diverse tasks involving synthetic and real-world datasets. Notably, ODE-LSTMs excel in tasks requiring the integration of information over large time windows, such as sequential classification problems and physical simulation modeling.

Implications and Future Developments

The immediate practical implication of this research lies in domains frequently encountering irregularly sampled data, such as healthcare and finance, where the temporal patterns are erratic yet critical decisions depend on understanding long-range dependencies. Furthermore, the research highlights a form of neural architecture that can more accurately model systems where input observations are non-uniformly spaced, potentially enabling advancements in fields requiring real-time processing of asynchronous data streams.

Looking forward, this work offers a basis for exploring further extensions of continuous-time RNNs in various applications, including but not limited to robust forecasting in predictive maintenance, biosignal analysis, and event-based data processing in sensor networks. Additionally, future research can focus on enhancing the computational efficiency of such models, exploring hybrid architectures that combine benefits from other neural network types, and leveraging the newfound insights into gradient dynamics for novel training algorithms.

In conclusion, the ODE-LSTM proposed in this paper is a significant methodological advancement for RNN-based modeling of irregularly-sampled sequences, potentially setting a new standard in sequential modeling tasks where robustness to irregular intervals and sensitivity to long-term dependencies is paramount.