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Inference via Interpolation: Contrastive Representations Provably Enable Planning and Inference
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Shows that representations learned by a regularized temporal contrastive (infoNCE) objective follow a Gauss-Markov chain, so prediction and multi-step planning reduce to closed-form Gaussian inference (often linear interpolation), validated on 2D synthetic, maze, and 39/46-D robotic tasks.
Problem
Answering prediction ("what happens next") and planning/in-filling ("how did we get here") questions over high-dimensional time series typically requires generative models or iterative inference; the paper asks whether discriminative contrastive representations can support these inferences in closed
Approach
Combine a norm-regularized symmetrized infoNCE objective (whose optimum encodes a probability ratio) with an isotropic-Gaussian marginal assumption to prove that learned representations are jointly Gaussian along a Markov chain; with a rotation-matrix dynamics and large c, the posterior waypoint means reduce to linear interpolation.
Key results
- On the 2D maze, planning waypoints with the learned representations and a proportional controller increases success rate by up to 4.5x, with the hardest goals going from 18% to 84%.
- On 39-dimensional door-opening, the method yields lower waypoint MSE than PCA, VIP, and naive interpolation baselines, averaged over 500 validation trajectories.
- Preliminary experiments on the 46-dimensional hammer task reproduce the same qualitative ordering (results in appendix).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use inferred paths over contrastive representations to control | Control on the synthetic dataset · Synthetic dataset (hardest goals) | success rate: 18% -> 84% | Helped |
| Use contrastive representations to plan intermediate waypoints | Proportional controller · Maze | success rate: up to 4.5× | Helped |
| Interpolate learned nonlinear contrastive representations instead of PCA representations | Maze planning · Maze | performance | Helped |
| Interpolate contrastive representations instead of VIP representations | Maze planning · Maze | performance | Helped |
| Use contrastive representations for waypoint inference instead of PCA or VIP | Robotic door opening · 39-dimensional robotic door opening | mean squared error | Helped |
Surprising findings
- Naive interpolation between observations and PCA interpolation perform poorly, indicating the time series is highly nonlinear and that learned representations are doing more than denoising.
Limitations
- Theory relies on two assumptions (isotropic Gaussian marginal; infoNCE optimum encodes a probability ratio) that may not hold in practice and whose error propagation is not analyzed.
- Experiments are preliminary and didactic, on small tasks up to 46 dimensions.
- Planning is defined as inferring intermediate states rather than optimal control.
Experimental setup
Datasets: D4RL door-human-v0 D4RL hammer-human-v0 maze dataset from prior work synthetic 2D spiral trajectories stock opening price data
Benchmarks: D4RL door-human-v0 D4RL hammer-human-v0 2D maze
Metrics: mean squared error of inferred waypoints (via nearest neighb success rate success rate on hardest goals
Baselines: PCA VIP no planning (naive interpolation)
Builds on: infoNCE temporal contrastive learning (goal-conditioned RL) VIP word embedding analogy analysis
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