Using Data from Clustered LQR Systems to Enable…

Using Data from Clustered LQR Systems to Enable Personalized and Collaborative Policy Optimization

This executive plan outlines a method for clustering similar Linear Quadratic Regulator (LQR) processes into K groups, learning a dedicated policy for each cluster, and providing theoretical guarantees with implications for distributed deployment. The core idea is to leverage data from multiple systems to learn generalized control policies that are both personalized and collaborative.

Algorithm Outline

The algorithm follows an Expectation-Maximization (E-step/M-step) approach, utilizing data {x_t^i, u_t^i} from N systems over time T_i. It initializes local system dynamics (A_i, B_i), assigns systems to clusters, and iteratively refines cluster policies.

Stepwise Procedure

1. Initialize cluster centers (A_k, B_k).
2. E-step: Compute responsibilities r_{i,k} indicating system i’s soft assignment to cluster k.
3. M-step: Update cluster models (A_k, B_k) using weighted least squares based on responsibilities.
4. Solve the discrete-time Algebraic Riccati Equation (DARE) for the LQR gain K_k for each cluster.
5. Assign systems to their best-fitting cluster based on responsibilities.
6. Distribute updates via a parameter server or gossip protocol for collaborative learning.
7. Repeat until convergence.

Robustness and Hyperparameters

Robustness is achieved through regularization and robust loss functions to handle model misspecification. A misspecification test aids in dropping outliers and re-clustering. Key hyperparameters include the number of clusters K ∈ {2,3,5,8}, regularization strengths λ_A, λ_B ∈ [1e-4, 1e-2], adequate time steps per system (100–200), and communication rounds per update (up to 5).

Complexity and Relevance

The computational complexity is near-linear in N (number of systems) and moderate in K (number of clusters). The E-step and M-step are O(N K n^2) per iteration, DARE is O(n^3) per cluster, and distributed rounds add O(K n^2) per exchange, where n is the state dimension. This data-driven approach supports scalable mobility/logistics and reflects real-world demand signals in transportation.

Practical Guide: Data Collection and Preprocessing for Clustered LQR

Effective control begins with high-quality data. This section details a practical workflow for collecting, preprocessing, and organizing trajectories to reliably learn local models and cluster them for the clustered LQR approach.

Data Format

For each system i, trajectories {x_t^i, u_t^i} are collected, where x_t^i ∈ R^n is the state and u_t^i ∈ R^m is the control input at time t. The underlying system dynamics are unknown and assumed to be described by local matrices A_i and B_i, which will be estimated.

Preprocessing Steps

• Normalization: Normalize each system’s data to zero mean and unit variance for states (X) and controls (U). This stabilizes least-squares estimates and enhances comparability between systems. Compute per-system statistics (μ_X^i, σ_X^i, μ_U^i, σ_U^i) and apply normalization to obtain X̃^i and Ũ^i. Store both raw and normalized data.
• Local System Identification: Estimate initial local models (A_i, B_i) using regularized least squares: minimize sum_t || x_{t+1}^i − A_i x_t^i − B_i u_t^i ||^2 + λ ( ||A_i||_F^2 + ||B_i||_F^2 ). This fits a linear model while penalizing large parameters to prevent overfitting, especially with limited data.
• Quality Checks: Discard trajectories with missing samples or clear non-stationarity. Interpolate small gaps and align time indices across systems.
• Data Split: Divide data into training (80%), validation (10%), and test (10%) sets for model fitting, hyperparameter tuning, and performance evaluation.
• Storage: Organize per-system artifacts including LS estimates (A_i, B_i), raw trajectories, and derived features (vectorized A_i, B_i, spectral properties) for clustering and reuse.

Practical Guide: Clustered LQR Identification and Policy Learning

When dealing with multiple related but not identical systems, clustering their dynamics and learning a local LQR policy per cluster offers robust performance. This approach combines model identification with control synthesis.

Initialization

Select the number of clusters K. Obtain local dynamics estimates (A_i, B_i) for each system. Vectorize and concatenate these estimates ([vec(A_i); vec(B_i)]) to form feature vectors. Run K-means on these vectors for initial cluster assignment and initialize cluster models (A_k, B_k) using cluster centroids.

E-step: Compute Soft Responsibilities

For each system i and cluster k, calculate residuals e_{i,t,k} = x_{t+1}^i − A_k x_t^i − B_k u_t^i and the cluster-specific cost cost_i(k) = sum_t ||e_{i,t,k}||^2. The soft responsibility r_{i,k} is computed as ∝ exp(−0.5 · cost_i(k) / s^2), normalized such that sum_k r_{i,k} = 1.

M-step: Update Models

For each cluster k, update (A_k, B_k) by minimizing a weighted least squares problem: sum_i r_{i,k} sum_t ||x_{t+1}^i − A_k x_t^i − B_k u_t^i||^2, plus a regularization term. This yields updated A_k and B_k.

Policy Synthesis

For each cluster k, compute the LQR gain K_k by solving the discrete-time Algebraic Riccati Equation (DARE) with A_k, B_k, and chosen cost matrices Q, R. The gain is K_k = −(R + B_k^T P B_k)^{−1} B_k^T P A_k, where P is the DARE solution.

Cluster Assignment and Convergence

Assign each system i to the cluster with the highest responsibility: cluster i → argmax_k r_{i,k}. Repeat the E-step and M-step until convergence criteria (e.g., change in responsibilities < 1e-3) are met.

Validation

Evaluate per-cluster control cost on held-out data, compare against baselines (global LQR, per-system controllers), and visualize cluster assignments and costs to verify meaningful separation.

Distributed Implementation Blueprint

This blueprint facilitates a lean central coordinator (or gossip alternative) with parallel local agents, balancing privacy, robustness, and efficiency by exchanging compact statistics rather than raw data.

Architecture

A central parameter server stores cluster centers (A_k, B_k, K_k) and DARE solutions. Local agents maintain their data and local least-squares (LS) statistics. Agents send aggregated LS terms to the server, which updates A_k, B_k, and broadcasts updated K_k back. A gossip option allows decentralized convergence.

Communication Rounds and Privacy

After each M-step, agents send sufficient statistics (e.g., aggregated sums of x_t x_t^T) to the server. Privacy is enhanced by transmitting aggregated, anonymized statistics, with optional differential privacy noise applied.

Complexity and Failure Handling

Per round complexity is O(n^2 + nm) for agents and O(K n^2) for the server. The system is designed for straggler tolerance, using timeouts and imputation for missing updates, ensuring eventual consistency.

Hyperparameters, Budgets, and Deployment Checklist

Setting hyperparameters effectively tunes model learning and clustering. This guide covers practical aspects for data budgeting, selecting the number of clusters, and ensuring confident deployment.

Hyperparameters to Set

Key parameters include K (number of clusters, 2–8), λ_A and λ_B (regularization on A/B updates, 1e-4 to 1e-2), and cost matrices Q, R. Sigma^2 controls the E-step scale. Start with K in the 2–4 range, use small regularizers, and tune sigma^2 with K.

Data Budget

Aim for at least 100 time steps per system for stable clustering. A total of 1e4 to 1e5 steps across N systems is recommended for robust centers. Prioritize diversity in data if collection is expensive.

Cluster Number Selection

Choose K using methods like the Elbow method, BIC/AIC on residual likelihood, or cross-validated log-likelihood to balance model fit and complexity.

Online Adaptation and Deployment Checklist

For online adaptation, use a sliding window with a reduced learning rate to maintain stability. Before deployment, verify stability via spectral radius of A_k, ensure DARE gains K_k yield closed-loop stability, and test extensively in simulation. Implement safe-fail mechanisms and gradual rollouts.

Comparative Analysis: Baseline vs. Clustered LQR and Deployment Scenarios

Scenario	Summary	Pros	Cons
Baseline global LQR	Single model trained on pooled data.	Simplest, lowest upfront cost.	Ignores heterogeneity; poor performance when dynamics differ.
Clustered LQR (K clusters) with hard assignments	Learns K specialized models with hard cluster assignments.	Personalized policies per cluster.	Requires reliable cluster membership; more data per cluster needed.
Soft-assignment clustered LQR	Allows probabilistic belonging to clusters during learning.	Smoother transitions for fuzzy cluster boundaries.	Higher computational load, potential fragmentation.
Distributed clustered LQR with coordination	Uses parameter server/gossip for updates.	Scalable, privacy-friendly.	Communication overhead, synchronization delays.
Privacy-preserving/decentralized variant	Local learning with periodic secure aggregation.	Strongest data privacy.	Potentially slower convergence, lower accuracy if updates infrequent.

Pros, Cons, and Deployment Considerations

Pros: Personalized control reduces per-cluster regret, improves data efficiency, is robust to heterogeneity, scalable via distributed updates, and has clear hyperparameterization. Cons: Additional complexity, requires careful validation and monitoring, depends on data quality and cluster separability, needs more design/testing, and involves communication/compute overheads. Deployment considerations: Start with simulated data, implement offline evaluation, monitor cluster drift, implement safe-fail mechanisms, and design for gradual rollouts.

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