Hyperparameter optimization (HPO) is a standard procedure in developing effective machine learning models. We tune parameters like learning rates, regularization strengths, or tree depths to maximize performance on a validation set. A significant question arises when using synthetic data: does the optimal set of hyperparameters found using synthetic data align with those found using real data? Understanding this relationship is another facet of assessing the synthetic data's utility.

If the synthetic data perfectly mirrored the real data's underlying structure and complexities, we might expect HPO performed using synthetic data (optimizing on a synthetic or real validation set) to yield hyperparameters ( $H_{synth}$ ) that are very close to, or perform similarly to, the hyperparameters found using real data ( $H_{real}$ ). However, synthetic data generation processes, while aiming for fidelity, might smooth out certain data characteristics, miss complex interactions, or introduce subtle artifacts. These differences can influence the optimization landscape explored during HPO.

Consider a scenario where you perform HPO for a classification model. The optimization process searches for hyperparameters that minimize loss or maximize a metric (like AUC or F1-score) on a validation set. If the synthetic data used for training and validation leads the HPO algorithm (e.g., Bayesian Optimization, Random Search) to identify an optimal region in the hyperparameter space that differs significantly from the region identified using real data, we encounter a potential utility problem. The model tuned using synthetic data might underperform when deployed because its hyperparameters are suboptimal for the true data distribution.

Evaluating Hyperparameter Consistency

To assess the impact of synthetic data on HPO, you can compare the outcomes directly:

Perform HPO using Real Data: Train models using the real training data, optimize hyperparameters based on a real validation set, and record the best hyperparameter set ( $H_{real}$ ) and its corresponding performance on the real test set. This serves as your benchmark.
Perform HPO using Synthetic Data: Train models using only the synthetic training data. You might optimize hyperparameters based on:
- A synthetic validation set (if generated).
- The real validation set (common in practice, simulating a scenario where some real data is available for validation). Record the best hyperparameter set found ( $H_{synth}$ ) and evaluate the model trained on synthetic data with $H_{synth}$ on the real test set.
Compare Results:
- Performance Comparison: How does the real test set performance of the model (trained synth, tuned synth -> $H_{synth}$ ) compare to the benchmark model (trained real, tuned real -> $H_{real}$ )? This directly measures the utility drop resulting from using synthetic data for both training and HPO.
- Hyperparameter Set Comparison: Examine $H_{real}$ and $H_{synth}$ . Are the values for important parameters (e.g., learning rate, regularization strength) vastly different? Large discrepancies suggest the synthetic data provides misleading signals to the HPO process regarding the model's sensitivity to certain parameters.

Isolating the Effect of Hyperparameters

You can further investigate whether the hyperparameters found using synthetic data ( $H_{synth}$ ) are detrimental even when sufficient real training data is available:

Train Real with $H_{synth}$ : Train a model using the real training data but explicitly set the hyperparameters to $H_{synth}$ (found via the synthetic data HPO process). Evaluate this model on the real test set.
Compare: Compare this performance to the benchmark model (trained real, tuned real -> $H_{real}$ ). If the performance using $H_{synth}$ on real data is significantly worse than the benchmark, it confirms that the hyperparameters derived from the synthetic optimization process are suboptimal for the real data distribution, irrespective of the training data source used after HPO.

The following plot illustrates a hypothetical comparison of model performance on the real test set resulting from HPO performed under different conditions.

Comparison of model performance (AUC) on a real test set. Bars show results using optimal hyperparameters derived from HPO on real data ( $H_{real}$ ) versus HPO on synthetic data ( $H_{synth}$ ). Performance is shown for models trained on real data (blue) and synthetic data (orange) using these respective hyperparameter sets. Notice the potential performance drop when using $H_{synth}$ , even when training on real data (blue bar for $H_{synth}$ ).

Interpretation and Practical Considerations

Significant divergence between $H_{real}$ and $H_{synth}$ , or a notable performance drop when using $H_{synth}$ , suggests that the synthetic data may not adequately capture the aspects of the data distribution most relevant for model tuning. This might occur if the synthetic data fails to reproduce complex feature interactions or noise patterns that influence model sensitivity to hyperparameters.

If HPO results differ substantially:

Risk Awareness: Be cautious about deploying models trained and tuned solely on synthetic data without final validation or tuning on real data.
Hybrid Approaches: Consider using synthetic data for initial HPO exploration to narrow down the search space, followed by fine-tuning on a smaller amount of real validation data.
Data Generation Focus: The discrepancy might signal a need to revisit the synthetic data generation process to improve its fidelity concerning aspects that affect hyperparameter sensitivity.

Evaluating the effect on hyperparameter optimization provides a deeper assessment of synthetic data utility. It moves beyond simple performance comparisons using fixed hyperparameters (as in basic TSTR) and examines whether the synthetic data can reliably guide the model tuning process itself, which is often essential for achieving peak performance in real-world applications.