Research Agenda and Hints for DA-GP Algorithm

This document outlines the research questions and methodological guidance for a proposed Delayed Acceptance (DA) and Gaussian Process (GP) based algorithm for active learning and surrogate model construction.

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I. Theoretical / Methodological Design

1. What are the theoretical foundations for using Delayed Acceptance in the context of GP-based surrogate modeling?

• Use DA-MCMC literature (e.g., Christen & Fox, 2005) as the basis.
• Frame DA as a two-stage acceptance scheme: use GP to pre-filter proposals before evaluating the true model.

2. How can one formally define and monitor the “fidelity” or “convergence” of the GP surrogate?

• Metrics:
  • Predictive variance (mean or max).
  • Leave-one-out cross-validation error.
  • Stabilization of marginal likelihood.
• Thresholds on these metrics can trigger trust in the GP.

3. What criteria should be used to trigger the switch from evaluating the forward model to relying solely on the GP?

• Example criteria:
  • Predictive variance below a threshold across recent samples.
  • No significant change in GP posterior over several updates.
  • Stagnant improvement in log marginal likelihood.

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II. Algorithm Design

4. How is the Schur complement used to update the Cholesky decomposition in an online fashion?

• When a new point is added: [ K_{n+1} = \begin{bmatrix} K_n & k \ k^T & k_{new} \end{bmatrix} ] Update ( L_{n+1} ) using Schur complement logic to avoid full recomputation.

5. What are the computational benefits of online learning for GP versus full retraining?

• Full retraining: ( \mathcal{O}(n^3) )
• Online update: ( \mathcal{O}(n^2) )
• Enables adaptive, low-latency learning.

6. Can the DA-GP framework maintain posterior accuracy compared to full MCMC or MCMC-GP hybrid approaches?

• Use metrics such as:
  • KL divergence.
  • Wasserstein distance.
  • Posterior moment comparison.
• Empirically validate using controlled test functions.

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III. Evaluation and Benchmarking

7. How does the proposed DA-GP method perform on standard test functions?

• Benchmark on Branin, Hartmann, Rosenbrock, etc.
• Track:
  • Posterior accuracy.
  • Runtime.
  • GP prediction error (RMSE).

8. How does sample efficiency, acceptance rate, and computational cost compare with traditional MCMC-guided GP approaches?

• Evaluate:
  • Number of model evaluations.
  • Acceptance rate pre/post DA.
  • Time per iteration and total convergence time.

9. How does performance scale with dimensionality and noise levels in the model?

• Increase input dimension from 2D to 10D+.
• Add Gaussian noise to the output.
• Measure effect on GP stability, DA rejection rate, and accuracy.

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IV. Broader Impact and Scalability

10. Under what conditions does the DA-GP approach break down or lose effectiveness?

• Scenarios:
  • High-dimensional or sparse data.
  • Poor GP kernel tuning.
  • Non-smooth/multimodal likelihoods.

11. Can this method be generalized to other surrogate models?

• Replace GP with:
  • Bayesian neural networks.
  • Ensemble trees with uncertainty quantification.
• Maintain DA logic: cheap filter before costly evaluation.

12. How can the DA-GP framework be extended for real-time or streaming data scenarios?

• Use online GP updates with:
  • Sliding window or forgetting factors.
  • Real-time adaptation to data drift.

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Next Steps

• Draft pseudocode of the DA-GP algorithm.
• Design experimental setup with benchmarks and metrics.
• Write a literature review comparing MCMC-GP, DA-GP, and pure GP methods.