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Topic 03 Project: Empirical Generalization Bounds on 'Spaceship Titanic'

1. Objective

The goal of this project is to apply statistical learning theory to a real-world dataset. Instead of just aiming for high accuracy, you will calculate Empirical Generalization Bounds to provide a theoretical guarantee on your model's performance. You will explore the trade-offs between optimization, stability, and complexity.

2. Dataset Setup

  • Source: Kaggle Spaceship Titanic
  • Task: Binary classification (Predict if a passenger was transported to another dimension).
  • Features: Numerical (Age, Spending) and Categorical (HomePlanet, Cabin, Destination).

Data Exploration and Preprocessing Tips

  1. Missing Values: Use a sophisticated imputer (like IterativeImputer or KNNImputer) rather than simple mean filling.
  2. Feature Engineering: The Cabin feature can be split into Deck, Num, and Side. These are highly predictive.
  3. Theoretical Constraint: To calculate Rademacher and PAC-Bayes bounds, your inputs must be bounded. Use StandardScaler and then clip values to \([-L, L]\).

3. Step-by-Step Implementation Guide

Step 1: Baseline Training

Train a 3-layer MLP using binary cross-entropy loss.

  • Record the training loss and validation accuracy.
  • Save the model weights \(w^*\).

Step 2: Estimating Rademacher Complexity

How well can your model fit pure noise?

  1. Replace the training labels \(y_i\) with random Rademacher variables \(\sigma_i \in \{1, -1\}\).
  2. Train the model to maximize correlation: \(\max_w \frac{1}{n} \sum \sigma_i \Phi(x_i; w)\).
  3. Repeat for 5 trials. The average correlation is your estimate \(\hat{\mathcal{R}}_S(\mathcal{H})\).
  4. Analysis: Compare this to the training accuracy on real labels. If the model fits noise almost as well as real data, it is likely over-parameterized.

Step 3: PAC-Bayes Certificate

Provide a "guarantee" that the test error will not exceed a certain threshold.

  1. Define Posterior \(Q\): \(\mathcal{N}(w^*, \sigma^2 I)\). Optimize \(\sigma\) to minimize the bound.
  2. Define Prior \(P\): \(\mathcal{N}(0, \lambda I)\), where \(\lambda\) is related to your weight initialization.
  3. Compute McAllester's Bound:
\[ \epsilon = \sqrt{\frac{KL(Q || P) + \ln(2\sqrt{n}/\delta)}{2(n-1)}} \]
  1. Calculate \(R(Q) = \mathbb{E}_{h \sim Q} [\hat{R}_n(h)]\) by sampling model weights and averaging the training loss.
  2. Target: Can you get a bound \(\epsilon < 0.5\) (non-vacuous)?

Step 4: Double Descent Visualization

  1. Vary the number of hidden neurons \(H\) from 1 to 1000.
  2. For each \(H\), train the model until the training loss is near zero.
  3. Plot Train and Test Error vs. \(H\).
  4. Identify the "peak" where the model starts to interpolate and observe if the test error decreases again.

4. Expected Results and Analysis

What to look for?

  1. Complexity vs. Regularization: Show how Weight Decay (\(\lambda\)) affects the estimated Rademacher complexity. Does higher \(\lambda\) consistently lead to lower complexity?
  2. Bound Tightness: PAC-Bayes bounds on neural networks are notoriously loose. Discuss why your bound might be vacuous (\(>1\)) and how "Sharpness-Aware Minimization" might help.
  3. Stability: Perform a "Leave-one-out" experiment. Does removing a single outlier passenger significantly change the model's decision boundary?

Results Table

Model Config Train Error Val Error Estimated Rademacher PAC-Bayes Bound (\(\epsilon\))
MLP (Small)
MLP (Large)
MLP + SAM

5. Deliverables

  • A Jupyter notebook demonstrating the bound calculations.
  • A plot of the "Double Descent" curve for this specific dataset.
  • A short discussion on whether the theoretical bounds (Rademacher/PAC-Bayes) correctly predicted the relative performance of different architectures.