Distributed Koopman Learning using Partial Trajectories for Control

This paper proposes DDKL-PT, a distributed data-driven framework where multi-agent systems achieve consensus on global dynamics by exchanging local deep Koopman model estimates derived from private partial trajectories, enabling accurate model predictive control without sharing raw training data.

The Gist

Imagine a group of five friends trying to figure out how a complex, mysterious machine works. This machine is a surface vehicle (like a boat or a drone) that moves based on how you push its levers (controls). The problem is, the machine's internal rules are a secret, and no single friend has seen the machine run for a long time.

Here is the situation:

• Friend A only saw the machine move for the first 10 minutes.
• Friend B only saw it for the next 10 minutes.
• Friend C, D, and E saw different, short slices of time.

If they tried to figure out the machine's rules alone, they would fail because their data is too incomplete. If they all sent their private video clips to a central "super-computer" to analyze, they would lose their privacy (maybe they don't want to share where they were or what they were doing).

This paper proposes a clever solution called DDKL-PT (Distributed Deep Koopman Learning using Partial Trajectories). Here is how it works, explained simply:

1. The "Magic Lens" (The Koopman Operator)

Usually, predicting how a machine moves is like trying to predict the path of a chaotic storm; it's non-linear and messy.

The authors use a mathematical trick called the Koopman Operator. Think of this as a "Magic Lens."

• When you look at the machine through the naked eye, it looks chaotic.
• When you look at it through the Magic Lens, the chaos turns into a straight, predictable line.
• Instead of learning the messy rules, the friends just need to learn how to adjust the "lens" and the simple straight-line rules.

2. The "Secret Recipe" Exchange (Distributed Learning)

Instead of sharing their private video clips (the raw data), the friends do something smarter:

1. Local Study: Each friend uses their short video clip to build their own "best guess" of how the Magic Lens works and what the straight-line rules are. They keep their video clips hidden in their pockets.
2. Whispering the Rules: They stand in a circle and whisper their guesses about the rules to their neighbors. They don't say, "I saw the boat turn left at 2 PM." They say, "I think the rule for turning is X."
3. Finding Consensus: They keep whispering and adjusting their guesses based on what their neighbors say. Eventually, they all agree on one single, perfect set of rules that describes the whole machine, even though no one ever saw the whole machine run.

Why is this cool?

• Privacy: No one ever saw anyone else's private data.
• Efficiency: They split the hard work of learning among five people instead of one person doing it all.

3. The "GPS Navigation" (Model Predictive Control)

Once the friends have agreed on the rules, they want to drive the boat to a specific destination (a "goal-tracking" task).

They use a navigation system called Model Predictive Control (MPC).

• Imagine you are driving a car. You don't just look at the road right in front of you; you look 30 seconds ahead. You ask yourself, "If I turn the wheel now, where will I be in 30 seconds? Is that a good spot? If not, I should adjust my turn now."
• The friends use their newly learned "Magic Lens" rules to simulate the future. They calculate the perfect path to the goal, avoiding obstacles and staying on course.

The Results

The paper ran a simulation with a virtual boat:

• The Test: They split a 5,000-second journey into 5 pieces. Each "agent" (friend) only saw one piece.
• The Outcome: Even though they only saw fragments, they successfully agreed on the rules. When they used these rules to drive the boat to a specific spot, they did a great job.
• The Trade-off: Their path was slightly less perfect than if one super-computer had seen the entire 5,000-second journey at once. However, it was "good enough" to get the job done, and they did it while keeping their data private.

The Big Picture

This paper is like teaching a class of students how to solve a giant puzzle. Instead of giving everyone the whole puzzle (which is too big for one person to hold), you give each student a few pieces. They figure out the pattern of their pieces, talk to their neighbors to compare patterns, and eventually, the whole class figures out the picture of the whole puzzle without ever handing their pieces to anyone else.

This allows robots and autonomous systems to learn complex behaviors together, quickly, and securely, without needing a central brain that knows everything.

Technical Summary

Here is a detailed technical summary of the paper "Distributed Koopman Learning using Partial Trajectories for Control" by Hao et al.

1. Problem Statement

The paper addresses the challenge of learning the dynamics of Nonlinear Time-Invariant Systems (NTIS) in a Multi-Agent System (MAS) setting where data is distributed and privacy is a concern.

• System Model: The system is defined by $x(t+1) = f(x(t), u(t))$, where the mapping function $f$ is unknown.
• Data Limitation: Instead of a single centralized dataset containing the full trajectory, the system consists of $N$ agents. Each agent $i$ only observes a partial trajectory $\xi_i$ (a segment of the total state-input history).
• Constraints:
  1. Privacy: Agents cannot share their raw training data (state-input pairs) with neighbors or a central server.
  2. Scalability: Centralized learning methods struggle with large-scale datasets and the computational burden of processing massive state-input pairs.
  3. Insufficiency: A single agent's partial trajectory is insufficient to independently identify the global dynamics.
• Goal: To develop a distributed framework where agents collaboratively learn a global linear representation of the nonlinear dynamics (via the Koopman operator) without sharing raw data, achieving consensus on the learned model parameters.

2. Methodology: DDKL-PT

The authors propose Distributed Deep Koopman Learning using Partial Trajectories (DDKL-PT). This framework combines the Koopman operator theory (lifting nonlinear dynamics to a higher-dimensional linear space) with Deep Neural Networks (DNNs) and Distributed Optimization.

A. Theoretical Foundation

The method seeks to find constant matrices $A^*, B^*, C^*$ and a parameter vector $\theta^*$ for a lifting function $g(\cdot, \theta)$ such that:

1. Lifted Linear Dynamics: $g(x_{t+1}, \theta^*) = A^* g(x_t, \theta^*) + B^* u_t$
2. Reconstruction: $x_{t+1} = C^* g(x_{t+1}, \theta^*)$

B. Algorithm Structure

The algorithm solves a multi-agent optimization problem where each agent minimizes a local loss function while enforcing consensus constraints ($A_1 = \dots = A_N$, etc.). The solution proceeds in two alternating steps:

1. Step 1: Distributed Learning of Dynamics Matrices ($A, B, C$)

   • For a fixed lifting function parameter $\theta$, the problem reduces to a linear regression in the lifted space.
   • The authors adapt a distributed update scheme (based on [21]) that allows agents to update their local estimates of $A_i, B_i, C_i$ using only local data and information exchanged with neighbors.
   • Key Feature: This step uses auxiliary matrices and specific weight matrices to ensure exponential convergence to the optimal solution without requiring a common step size across all agents.

2. Step 2: Distributed Parameter Tuning ($\theta$)

   • With the matrices $A, B, C$ fixed, the non-linear lifting function parameters $\theta$ are updated.
   • Agents use a distributed subgradient method to update $\theta_i$ based on local gradients of the loss function and weighted averages of neighbors' parameters.
   • This ensures that the DNN structure (the lifting function) converges to a consensus across the network.

C. Control Integration (MPC)

Once the dynamics are learned, the paper integrates the learned Koopman model with known kinematic relations to form a Model Predictive Control (MPC) scheme.

• The Koopman model predicts future velocities in local coordinates.
• These predictions are combined with kinematic equations to predict full state evolution (position and orientation).
• The MPC controller uses this combined model to solve an optimal control problem for goal-tracking.

3. Key Contributions

1. Distributed Deep Koopman Framework: The development of DDKL-PT, which enables multi-agent systems to learn global nonlinear dynamics from distributed, partial trajectories without sharing raw data.
2. Privacy Preservation: The algorithm achieves consensus on the global model by exchanging only estimated dynamics matrices ($A, B, C, \theta$), preserving the privacy of local training trajectories.
3. Scalability and Efficiency: By distributing the computational load, the method overcomes the bottlenecks of centralized learning for large datasets.
4. Control Application: The successful integration of the distributedly learned dynamics into an MPC framework for a surface vehicle, demonstrating that the learned model is accurate enough for real-time optimal control.

4. Experimental Results

The method was validated using a 5-agent Multi-Agent System simulating a surface vehicle.

• Setup:
  • A nominal trajectory ($t=0$ to $5000$) was generated.
  • Data was split into non-overlapping (and partially overlapping) segments assigned to 5 agents.
  • The segment $t=4000$ to $5000$ was reserved for testing.
• Consensus Performance:
  • Simulation results (Fig. 3) showed that the local matrices ($A_i, B_i, C_i$) and parameters ($\theta_i$) converged exponentially to the values obtained by a centralized Deep Koopman Operator (DKO) method.
• Prediction Accuracy:
  • Metrics: Mean squared error on the testing dataset.
  • Comparison:
    • Centralized DKO (Ground Truth): $0.0179$
    • Centralized MLP: $0.0205$
    • DDKL-PT (Proposed): $0.0284$
  • Analysis: While DDKL-PT had a slightly higher error than centralized methods (due to data fragmentation), the error was statistically acceptable. ANOVA tests confirmed the ranking of performance but noted the distributed method's error was within a usable range.
• Control Performance:
  • The MPC controller using DDKL-PT dynamics successfully drove all agents from an initial state to a goal state.
  • Convergence was achieved in approximately 300 time steps.
  • While convergence was slightly slower than the centralized DKO-MPC, the tracking errors were small enough to confirm the viability of the approach for model-based optimal control.

5. Significance

This paper bridges the gap between data-driven control, distributed computing, and privacy preservation.

• Practical Impact: It provides a viable solution for scenarios where data is inherently distributed (e.g., fleets of autonomous vehicles, sensor networks) and cannot be aggregated due to bandwidth or privacy constraints.
• Theoretical Advancement: It demonstrates that the Koopman operator framework, traditionally used for centralized learning, can be effectively decentralized using deep learning and consensus optimization.
• Control Relevance: It proves that "good enough" distributed learning is sufficient for high-performance control tasks like MPC, moving beyond pure system identification to practical application.