Cross-regularization: Adaptive Model Complexity through Validation Gradients

We appreciate your insightful review and constructive technical feedback. Your points have significantly improved the paper's theoretical foundations and connections to existing literature.

Parameter Independence and Bounded Coupling

You correctly identified that our theory requires some degree of parameter independence. Rather than complete independence, our approach requires bounded coupling:

$\|H_{\theta\rho}\| \leq \beta\sqrt{\lambda_{\min}(H_{\theta\theta})\lambda_{\min}(H_{\rho\rho})}$

This condition prevents contradictory optimization objectives from causing instability when training and regularization gradients conflict. Our empirical stability (see extended simulations: https://tinyurl.com/2s45b236) suggests this condition is satisfied in practice, even for neural networks where the noise parameters serve a fundamentally different functional role than weight parameters.

Neural Network Theory

We've addressed your concern about local convergence with two theoretical advances:

1. A refined local structure theorem (https://tinyurl.com/33d2sye4) that establishes precise bounds for the radius of convergence: $r = \min\left(\frac{\mu}{6L_H}, \frac{(1-\gamma)\mu}{2|H|}\right)$.

2. We've developed a new result showing convergence under practical assumptions for neural networks (https://tinyurl.com/4ckfb9ec) that doesn't require global convexity. Our guarantees now only require that:

• The model parameters converge for fixed regularization
• The validation loss has local convexity in regularization parameters
• The gradient coupling satisfies a Lipschitz condition

This theorem establishes that cross-regularization converges for neural networks under assumptions that are consistent with observed empirical behavior, without requiring guarantees of global convergence for the underlying neural network optimization problem.

On the choice of stochastic regularization, we also examined L2 regularization but found minimal empirical effect in our neural network architectures. This is expected because L2 regularization primarily constrains weight magnitudes, which are effectively cancelled by the normalization layers present in our networks.

Connection to Lottery Ticket Hypothesis

Your observation connecting our noise-based regularization to the Lottery Ticket Hypothesis enriches our framework. We've quantified this relationship: our noise levels (σ≈13) create an information bottleneck with capacity C≈0.004 bits/symbol, mathematically equivalent to pruning with sparsity ≈99.4%, aligning with LTH's 98%.

This connection validates our results by demonstrating that two distinct approaches—gradient-based noise adaptation versus iterative weight pruning—converge to similar architecture-specific patterns. In light of this, we will revise our claims about the novelty of the high noise regime, recognizing we're capturing intrinsic properties of neural information capacity rather than method-specific artifacts.

Backpropagation Through Augmentation

For data augmentation, your concern about optimization potentially minimizing all transformations is insightful. Our solution:

1. We parameterize transformations with continuous magnitude parameters (e.g., rotation angle α determining range U[-α,α])
2. During training, we apply single random transformations for regularization effect
3. During validation, we average predictions over multiple transformations: $\mathcal{L}_\text{val} = \mathbb{E}_{\text{transformations}}[\mathcal{L}(f(\text{transform}(x)))]$

This creates an optimization equilibrium where validation performance constrains transformation magnitude: excessive transformations that remove discriminative features increase validation loss, while insufficient transformations lead to overfitting. The validation gradients naturally balance these competing factors.

Our results confirm this behavior: rotation parameters converge to ~3° for SVHN while shear transformations are optimized toward zero. The method identifies which transformations provide regularization benefits for a specific dataset and minimizes those that don't.

For discrete transformations like horizontal flips, we agree this remains a limitation without relaxation techniques, which we're exploring in follow-up work.

Additional Baselines

Based on your feedback, we're implementing:

• Temperature scaling and deep ensembles for calibration comparisons
• EWC for continual learning benchmarks

Our preliminary calibration results (https://tinyurl.com/3mtacsu9) show considerable improvements over post-hoc methods, supporting our claim that uncertainty calibration emerges naturally in our approach.

Thank you for your thoughtful review. The connection to pruning literature and your theoretical insights have improved both the paper and our analysis of cross-regularization's properties.

We thank the reviewer for their thoughtful assessment and constructive feedback. Below, we address the main concerns raised:

Recent Literature

We agree that our literature review requires updating. In the camera-ready version, we will incorporate recent advances in adaptive regularization and complexity control, including:

• Foret et al. (2020) "Sharpness-Aware Minimization for Efficiently Improving Generalization"
• Nado et al. (2020) "Evaluating Prediction-Time Batch Normalization for Robustness under Covariate Shift"
• Chen et al. (2021) "Weighted Training for Cross-Task Learning"

Uncertainty Calibration Comparisons

We have implemented comparisons with established calibration methods:

1. Temperature Scaling (Guo et al., 2017): Our preliminary results (https://tinyurl.com/3mtacsu9) demonstrate that cross-regularization maintains better calibration throughout training compared to temperature scaling with fixed dropout (σ=0.1 in all layers), while achieving higher performance:

2. Deep Ensembles (Lakshminarayanan et al., 2017): This comparison is currently underway and will be included in the camera-ready version with appropriate statistical significance tests and confidence intervals.

These analyses will demonstrate how cross-regularization provides well-calibrated uncertainties without requiring separate post-hoc calibration steps or ensemble overhead, addressing a key limitation in current practice.

Additional Hyperparameter Tuning Comparisons

While we compared with PBT and fixed dropout, we found that Variational Dropout frequently diverged without implementation modifications that would bias the comparison. In the revised version, we will include:

1. Comparisons with Concrete Dropout (Gal et al., 2017)
2. Analysis of noise dynamics for all methods (where convergent)
3. Computational efficiency metrics (wall-clock time and memory usage)

Cross-Validation Equivalence Verification

Regarding the comment on Method 3.3 (Parameter Partition through Gradient Decomposition) verification, we note that Theorem 4.4 establishes the theoretical equivalence. Our empirical validation in Section 5.1 (Figure 1A-C) demonstrates that cross-regularization converges to the same optimal solution as cross-validated regression, confirming the theoretical result. We will clarify this connection in the revised paper.

Large-Scale Dataset Validation

While our primary contribution is methodological—introducing and analyzing a novel regularization framework—we acknowledge the value of validating on larger-scale datasets. Our current experimental suite (CIFAR-10, diabetes dataset) demonstrates the method's applicability across different regularization types and model architectures, though we recognize that larger datasets would provide additional validation. We are exploring experiments on ImageNet with ResNet-50, focusing on:

1. Confirming that the theoretical and empirical advantages scale to larger datasets
2. Identifying any emergent layer-wise regularization patterns specific to deeper architectures
3. Quantifying computational advantages over traditional hyperparameter tuning at scale

These extensions would complement our methodological contribution rather than being essential to validating the core approach.

Neural Network Theory and Stability

We have extended our simulations to 600 epochs (https://tinyurl.com/2s45b236), demonstrating empirical stability of the regularization parameters. For the theoretical foundation, we have developed a new result showing that, under assumptions on regularization parameter behavior, convergence depends only on the standard neural network optimization properties. The full proof is available at: https://tinyurl.com/4ckfb9ec.

Paper Structure

We appreciate the feedback on organization. In the revised version, we will:

1. Merge the Introduction with Background sections
2. Consolidate the application sections (7-9) into a single "Applications and Extensions" section
3. Provide clearer transitions between theoretical development and empirical validation
4. Maintain a consistent narrative flow from problem formulation to practical applications

We thank the reviewer for their detailed feedback, which has highlighted important areas for improvement while acknowledging the novel contributions of our cross-regularization framework.

We appreciate your review and address your concerns below:

1. How We Apply Gradient Descent on Regularization Parameters

Our method works by making regularization parameters explicit and directly optimizable:

L2 Regularization Example:

1. We rewrite weights as w = ρθ where ||θ||₂ = 1 and ρ is the norm
2. Training updates: θₜ₊₁ = θₜ - ηθ∇θL_train(θₜ,ρₜ) - optimizing on training data
3. Regularization updates: ρₜ₊₁ = ρₜ - ηρ∇ρL_val(θₜ₊₁,ρₜ) - optimizing on validation data

This makes regularization strength a parameter receiving direct gradient updates, not a hyperparameter requiring grid search.

2. Neural Network Convergence

While our theoretical guarantees in Section 4 apply to convex settings, our neural network experiments show consistent convergence. The optimization remains stable even with high noise levels, and the validation accuracy improvements are robust across architectures.

We've extended our simulations to 600 epochs (https://tinyurl.com/2s45b236) showing stable noise patterns over time. Our additional theorem (https://tinyurl.com/4ckfb9ec) proves that cross-regularization doesn't introduce instability beyond standard neural network training.

The stability of our method is evidenced by its outperforming fixed regularization, without the divergence issues sometimes seen in other adaptive methods like variational dropout.

3. Application to Other Regularization Techniques

Cross-regularization applies to any regularization with:

• Parameterizable strength
• Differentiable validation performance

For data augmentation (Section 9), we parameterize transformation magnitudes and optimize them through validation gradients. Similar approaches could apply to mixup (parameterizing α) or label smoothing (parameterizing smoothing strength).

We focused on norm-based and noise-based regularization as they provide clear demonstrations of our approach. Extensions to non-differentiable techniques would require additional work beyond the scope of this paper.

4. Distinction from Bilevel Optimization

Our approach differs fundamentally from traditional bilevel optimization:

1. We update parameters directly through validation loss, not through the optimization trajectory
2. We maintain separate parameter spaces for model features versus complexity
3. We don't require approximations of second-order derivatives or parameter history

This distinction makes our method implementable with standard optimizers and applicable to large models where traditional bilevel methods become computationally intractable.

5. Experimental Details

Our experimental implementation follows the algorithm in Section 3:

• L2/L1 experiments: Alternating SGD updates between model parameters and regularization parameters
• Neural networks: Layer-wise noise parameters optimized every 30 training steps using 3-5 MCMC samples
• Augmentation: Parameterized transformations updated through validation-based MC averaging

The code requires only ~20 lines beyond standard training loops, making implementation straightforward for most existing frameworks.

6. Relation to Transfer Learning

We thank you for noting the connection to Chen et al. (2022). Both approaches use validation gradients but for different purposes: their work for weighting source tasks in transfer learning, ours for optimizing regularization parameters. We will include this reference.

We appreciate your feedback and will address these points in our revision.