Inference via Interpolation: Contrastive Representations Provably Enable Planning and Inference (2403.04082v3)

Abstract: Given time series data, how can we answer questions like "what will happen in the future?" and "how did we get here?" These sorts of probabilistic inference questions are challenging when observations are high-dimensional. In this paper, we show how these questions can have compact, closed form solutions in terms of learned representations. The key idea is to apply a variant of contrastive learning to time series data. Prior work already shows that the representations learned by contrastive learning encode a probability ratio. By extending prior work to show that the marginal distribution over representations is Gaussian, we can then prove that joint distribution of representations is also Gaussian. Taken together, these results show that representations learned via temporal contrastive learning follow a Gauss-Markov chain, a graphical model where inference (e.g., prediction, planning) over representations corresponds to inverting a low-dimensional matrix. In one special case, inferring intermediate representations will be equivalent to interpolating between the learned representations. We validate our theory using numerical simulations on tasks up to 46-dimensions.

• The paper introduces contrastive representations that convert inference in time series data into efficient interpolation via linear algebra operations.
• It leverages a Gauss-Markov chain model to theoretically prove that learned representations follow an isotropic Gaussian under regularization.
• Numerical simulations validate the method’s scalability and highlight its potential for applications in robotic control and goal-conditioned planning.

Inference via Interpolation in Time Series Data Using Contrastive Representations

Introduction to the Research

In the contemporary landscape of probabilistic modeling for time series data, a novel approach leveraging contrastive learning has been posited to address the inferential questions of forecasting future states and tracing plausible trajectories between two known states. This method diverges from traditional generative models, offering a framework markedly less computationally demanding and scalable to high-dimensional scenarios. The cornerstone of this methodology rests upon the proposition that contrastive learning, when applied to time series, yields representations that simplify inference tasks by interpolating learned representations, thereby converting the challenge of inference into manageable linear algebra operations.

Theoretical Foundations

The paper elaborates on the fusion of temporal contrastive learning with probabilistic inference, positing that the representations encapsulate the temporal dynamics in a time series through a Gauss-Markov chain model. It's substantiated with a proof demonstrating that, under certain regularizations, the marginal distribution of learned representations adheres to an isotropic Gaussian distribution. This foundation paves the way for inferring future representations to be straightforward, positioning contrastive representations as efficient tools for forecasting and planning. The validity of this theory is extended through numerical simulations, adeptly facilitating inference across dimensions up to 46, showcasing the scalability and efficacy of this approach.

Related Works and Distinction

Addressing the tandem goals of bridging gaps in related work and setting a distinct trajectory, the research navigates through the realms of representation learning for time series data, contrastive learning methodologies, and their potential in goal-oriented decision making and planning. Highlighting the historical reliance on reconstruction-based methods for retention of information critical to predicting future observations, this paper carves a niche by eliminating the need for reconstruction, thereby reducing computational burdens. The divergent approach of applying contrastive learning distinctly for solving prediction and planning over time series data marks a novel path from predecessors, aligning with theoretical and practical implications in goal-conditioned reinforcement learning and planning.

Numerical Simulations and Practical Implications

The practicability of the theorized model is evidenced through numerical simulations, encompassing low-dimensional toy datasets demonstrating spiraling trajectories and extending to higher-dimensional tasks with real-world nuances like robotic control. These simulations not only validate the theoretical framework but also accentuate the potential for application in complex decision-making scenarios like maze navigation and robotic manipulation, where interpreting and planning sequences of actions based on observations are paramount. The comparative analysis with existing methodologies elucidates the superior inference capabilities and computational efficiency of the proposed contrastive representations.

Future Directions and Conclusion

Echoing the paper's insights, there lies a potential future in refining these representations to further accommodate and enhance goal-oriented decision-making processes. The conjecture, supported by empirical evidence, suggests a promising avenue in harnessing contrastive learning for interpreting high-dimensional time series data across diverse applications. While acknowledging limitations pertaining to assumptions underlying the analysis, the research opens a dialogue for extending these methods, illuminating the path towards developing representations with intrinsic geometric properties that could redefine probabilistic inference in time series data.

The ensemble of theoretical advancements and empirical validations presented offers a compelling case for adopting contrastive representations in modeling time series data, fostering a confluence of efficiency, scalability, and practicality in tackling inferential problems in high-dimensional settings.