# Prediction, Consistency, Curvature: Representation Learning for Locally-Linear Control

arXiv: https://arxiv.org/abs/1909.01506 · HTML: https://mlsanity.com/paper/1909.01506/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Sep 2019.

## Built on by (most cited first, 18 of 18)

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- [Pausing Policy Learning in Non-stationary Reinforcement Learning](https://mlsanity.com/paper/2405.16053/) (2024)
- [Multi-Step Prediction in Linearized Latent State Spaces for Representation Learning](https://mlsanity.com/paper/2209.01127/) (2022)
- [Learn Proportional Derivative Controllable Latent Space from Pixels](https://mlsanity.com/paper/2110.08239/) (2021)
- [Efficient Deep Reinforcement Learning with Predictive Processing Proximal Policy Optimization](https://mlsanity.com/paper/2211.06236/) (2022)
- [Controllability-Constrained Deep Network Models for Enhanced Control of Dynamical Systems](https://mlsanity.com/paper/2311.06438/) (2023)
- [KEEC: Koopman Embedded Equivariant Control](https://mlsanity.com/paper/2312.01544/) (2023)
- [Efficiently Scanning and Resampling Spatio-Temporal Tasks with Irregular Observations](https://mlsanity.com/paper/2410.08681/) (2024)
- [Action Jets: Learning the Local Shape of the Action-conditioned Data Manifold](https://mlsanity.com/paper/2610.19007/) (2026)

## Abstract (arXiv)

Many real-world sequential decision-making problems can be formulated as optimal control with high-dimensional observations and unknown dynamics. A promising approach is to embed the high-dimensional observations into a lower-dimensional latent representation space, estimate the latent dynamics model, then utilize this model for control in the latent space. An important open question is how to learn a representation that is amenable to existing control algorithms? In this paper, we focus on learning representations for locally-linear control algorithms, such as iterative LQR (iLQR). By formulating and analyzing the representation learning problem from an optimal control perspective, we establish three underlying principles that the learned representation should comprise: 1) accurate prediction in the observation space, 2) consistency between latent and observation space dynamics, and 3) low curvature in the latent space transitions. These principles naturally correspond to a loss function that consists of three terms: prediction, consistency, and curvature (PCC). Crucially, to make PCC tractable, we derive an amortized variational bound for the PCC loss function. Extensive experiments on benchmark domains demonstrate that the new variational-PCC learning algorithm benefits from significantly more stable and reproducible training, and leads to superior control performance. Further ablation studies give support to the importance of all three PCC components for learning a good latent space for control.