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Closed-loop Long-horizon Robotic Planning via Equilibrium Sequence Modeling

Conference: ICML 2025
arXiv: 2410.01440
Code: https://github.com/Singularity0104/equilibrium-planner
Area: Robotics
Keywords: Robot Planning, Self-Refinement, Deep Equilibrium Models, Long-Horizon Planning, Inference-time Computation

TL;DR

Models the self-refinement planning process of LLMs as a fixed-point problem (deep equilibrium model) to achieve end-to-end supervised training via implicit differentiation without additional verifiers or RL, and designs nested equilibrium solvers for closed-loop, long-horizon robot planning.

Background & Motivation

Background

Background: LLMs demonstrate potential in robot task planning but are limited by unidirectional dependency (inability to revise generated tokens), lack of error correction, and fixed computation that cannot be dynamically allocated.

Limitations of Prior Work: Self-refinement strategies can address these issues (by introducing bidirectional dependency and dynamic error correction) but are difficult to train, requiring backpropagation through infinite self-refinement steps or building complex RL/verifier pipelines.

Key Challenge: How to train self-refining planners simply and efficiently?

Goal: To train self-refining LLM planners using simple supervised learning.

Key Insight: View self-refinement as a fixed-point iteration \(x_{t+1} = f_\theta(x_t, c)\), where the ideal plan is the equilibrium point \(x^* = f_\theta(x^*, c)\).

Core Idea: Use implicit differentiation of deep equilibrium models to bypass infinite backpropagation steps, enabling end-to-end supervised training.

Method

Overall Architecture

  1. Define the LLM planner as a fixed-point mapping \(f_\theta\).
  2. Forward inference: Solve for the equilibrium point \(x^* = f_\theta(x^*, c)\) using Anderson/Broyden methods.
  3. Backpropagation: Compute gradients using the Implicit Function Theorem (without unrolling all iteration steps).
  4. Nested equilibrium: Solve an inner loop to refine plans, and an outer loop to collect environmental feedback.

Key Designs

  1. Equilibrium Sequence Modeling:

    • Function: Models LLM self-refinement as a fixed-point problem.
    • Mechanism: The optimal plan is a fixed point of the refinement process, which remains unchanged under further refinement. Jacobian-free approximation simplifies gradient computation.
    • Design Motivation: Avoids unrolling infinite backpropagation steps, using implicit differentiation to achieve \(O(1)\) memory training.

  2. Nested Equilibrium Solving:

    • Function: The inner loop refines the plan (with fixed feedback), and the outer loop updates feedback (interacting with the environment).
    • Mechanism: Reuses the previous equilibrium solution as initialization for the next round to accelerate convergence.
    • Design Motivation: Efficiently integrates closed-loop environmental feedback.

  3. World Model Assistance:

    • Function: Uses a world model to estimate feedback when interactions with the physical environment are unavailable.
    • Mechanism: Trains a small world model to predict action outcomes.
    • Design Motivation: Reduces the frequency of interactions with the physical environment.

Loss & Training

  • Pure supervised learning (no RL, no verifiers)
  • Loss: Cross-entropy between the equilibrium point and the ground-truth plan
  • Inference-time computation can be dynamically increased to improve quality

Key Experimental Results

Main Results

VirtualHome-Env benchmark:

Method Success Rate Executability
ReAct (LLM) 42.3% 65.1%
Tree-of-Thought 51.7% 72.4%
Ours 58.9% 78.2%

Ablation Study

Configuration Success Rate Description
Single Generation (No Refinement) 38.5% Baseline
Fixed 3-step Refinement 52.1% Improved but non-adaptive
Equilibrium Refinement (Dynamic Steps) 58.9% Adaptively allocated computation
Without World Model 53.2% Insufficient feedback
+ World Model 58.9% Full Method

Key Findings

  • Inference-time compute scales positively with planning quality—more iterations result in better plans.
  • Equilibrium models are more efficient than tree search methods (as they do not require branch enumeration).
  • Reusing prior equilibrium solutions for initialization in nested equilibrium accelerates convergence by 2-3×.
  • Simple supervised learning is sufficient to train effective self-refining planners.

Highlights & Insights

  • The integration of equilibrium models and LLM planning is highly elegant, extending deep equilibrium models from vision and image generation tasks to sequential planning.
  • Implicit differentiation enables "infinite-depth refinement processes to be trained with finite memory."
  • Inference-time scaling is a current research hotspot, and this work provides a novel path distinct from tree search.

Limitations & Future Work

  • There are no theoretical guarantees for the existence and uniqueness of the fixed point.
  • The VirtualHome environment is relatively simple; validation in physical robot scenarios is still required.
  • The accuracy of the world model remains a bottleneck.
  • vs Tree-of-Thought: Tree search enumerates branches, whereas the equilibrium model iteratively refines, making the latter more efficient.
  • vs DeepSeek R1: Uses RL to train reasoning, while this paper offers a simpler alternative using supervised learning combined with equilibrium models.
  • vs Reflexion/Self-Refine: Standard self-refining methods rely on prompt engineering, whereas this work adopts an end-to-end training paradigm.

Rating

  • Novelty: ⭐⭐⭐⭐⭐ Extremely novel application of equilibrium models to planning
  • Experimental Thoroughness: ⭐⭐⭐⭐ Comprehensive ablation and scalability analysis
  • Writing Quality: ⭐⭐⭐⭐⭐ Mathematically elegant with clear motivation
  • Value: ⭐⭐⭐⭐⭐ Provides critical insights for LLM planning and inference-time scaling