# ============================================================================
# T13 — Kernel choice effects for Gaussian Process regression
# ----------------------------------------------------------------------------
# Unlike paper_replication's P01-P07, the "synthesis" here frames a research
# QUESTION rather than a known paper's method. The model must design the
# experiment (conditions, metrics, datasets) — we only commit to what a
# competent study of this topic would include and what the rubric expects.
# ============================================================================

id: ML13
title: "Evaluating kernel choices for Gaussian Process regression on synthetic 1-D and 5-D functions"
arxiv_id: null
venue: "ARC-Bench 2026"
paper_asset: null

# The "synthesis" plays the role of the upstream briefing: research question,
# background, why the question matters, what "a reasonable experiment" looks
# like. It deliberately does NOT pre-specify a single method to reproduce.
synthesis: |
  Gaussian Process (GP) regression performance depends strongly on the kernel,
  which encodes assumptions about function smoothness and local structure.
  Practitioners often default to the RBF kernel because it is easy to optimize
  and widely available, but this choice can underperform when the target
  function has roughness patterns that are better matched by Matern families or
  when polynomial trends dominate part of the signal. Since kernel
  misspecification can be subtle, a short benchmark should compare predictive
  uncertainty quality as well as point error.

  A CPU-scale study can be done entirely with sklearn GaussianProcessRegressor
  on synthetic datasets where the data-generating process is controlled. In 1-D,
  a mildly nonstationary smooth function with additive Gaussian noise can expose
  differences in extrapolation and uncertainty tails. In 5-D, a mixed
  sinusoidal-plus-quadratic target can stress anisotropy and interactions while
  remaining fast enough for repeated fitting under multiple random seeds.

  To make claims measurable, the experiment should evaluate at least four kernel
  choices (RBF, polynomial, Matern-3/2, Matern-5/2) under the same train/test
  splits, optimizer restarts, and noise model assumptions. Because the topic's
  key metric is negative log-likelihood, uncertainty calibration quality should
  be primary; RMSE and R^2 provide supporting evidence for point prediction.
  Seed averaging is important because synthetic splits and optimizer
  initialization can change marginal likelihood optimization outcomes.

  The core question is whether smoother kernels (RBF, Matern-5/2) consistently
  outperform rougher or mismatched kernels (Matern-3/2, polynomial) on test NLL
  across both low- and moderate-dimensional synthetic tasks, and whether ranking
  by NLL aligns with ranking by RMSE.

  *Which GP kernel among RBF, polynomial, Matern-3/2, and Matern-5/2 yields the best uncertainty-aware generalization (lowest test NLL) on synthetic 1-D and 5-D noisy regression functions?*

hypotheses:
  - id: H1
    statement: "Across the two synthetic datasets, at least one of {RBF, Matern-5/2} achieves the lowest mean test NLL on each dataset (averaged over >=5 seeds)."
    measurable: true
  - id: H2
    statement: "The polynomial kernel has mean test NLL at least 10% worse than the best kernel on at least 1 of the 2 datasets."
    measurable: true
  - id: H3
    statement: "Kernel ranking by mean test NLL and by mean RMSE is not identical on at least 1 dataset, indicating uncertainty quality and point error disagree."
    measurable: true

experiment_design:
  research_question: "How do RBF, polynomial, Matern-3/2, and Matern-5/2 kernels compare for Gaussian Process regression on synthetic 1-D and 5-D noisy functions when judged primarily by test NLL?"
  conditions:
    - name: "gp_rbf"
      description: "GaussianProcessRegressor with ConstantKernel * RBF + WhiteKernel, hyperparameters optimized by log-marginal-likelihood."
    - name: "gp_poly_deg2"
      description: "GaussianProcessRegressor with ConstantKernel * DotProduct^2 (implemented via polynomial feature mapping + DotProduct kernel) + WhiteKernel as a polynomial-trend proxy."
    - name: "gp_matern32"
      description: "GaussianProcessRegressor with ConstantKernel * Matern(nu=1.5) + WhiteKernel."
    - name: "gp_matern52"
      description: "GaussianProcessRegressor with ConstantKernel * Matern(nu=2.5) + WhiteKernel."
  baselines:
    - "gp_rbf as the common default-kernel baseline"
    - "gp_poly_deg2 as a mismatched-trend baseline"
  metrics:
    - name: "test_nll"
      direction: "minimize"
      description: "Mean negative log predictive density on held-out test data using GP predictive mean and variance, averaged over seeds."
    - name: "rmse"
      direction: "minimize"
      description: "Root mean squared error on test targets, averaged over seeds."
    - name: "r2"
      direction: "maximize"
      description: "Coefficient of determination on test targets, averaged over seeds."
  datasets:
    - name: "synthetic_1d_sinmix"
      source: "Generated with numpy: x in [-3,3], y = sin(2x) + 0.3x + epsilon, epsilon~N(0, 0.15^2)."
    - name: "synthetic_5d_mixed"
      source: "Generated with numpy: X in [-2,2]^5, y = sin(x1) + 0.5*x2^2 - 0.7*x3 + 0.3*x4*x5 + epsilon, epsilon~N(0, 0.2^2)."
  compute_requirements:
    gpu_required: false
    estimated_wall_clock_sec: 360

rubric_path: "experiments/arc_bench/config/ml/rubrics/ML13.json"