Training Transformers with Enforced Lipschitz Constants (2507.13338v1)

Abstract: Neural networks are often highly sensitive to input and weight perturbations. This sensitivity has been linked to pathologies such as vulnerability to adversarial examples, divergent training, and overfitting. To combat these problems, past research has looked at building neural networks entirely from Lipschitz components. However, these techniques have not matured to the point where researchers have trained a modern architecture such as a transformer with a Lipschitz certificate enforced beyond initialization. To explore this gap, we begin by developing and benchmarking novel, computationally-efficient tools for maintaining norm-constrained weight matrices. Applying these tools, we are able to train transformer models with Lipschitz bounds enforced throughout training. We find that optimizer dynamics matter: switching from AdamW to Muon improves standard methods -- weight decay and spectral normalization -- allowing models to reach equal performance with a lower Lipschitz bound. Inspired by Muon's update having a fixed spectral norm, we co-design a weight constraint method that improves the Lipschitz vs. performance tradeoff on MLPs and 2M parameter transformers. Our 2-Lipschitz transformer on Shakespeare text reaches validation accuracy 60%. Scaling to 145M parameters, our 10-Lipschitz transformer reaches 21% accuracy on internet text. However, to match the NanoGPT baseline validation accuracy of 39.4%, our Lipschitz upper bound increases to 10^264. Nonetheless, our Lipschitz transformers train without stability measures such as layer norm, QK norm, and logit tanh softcapping.

• The paper introduces novel spectral soft cap and spectral hammer techniques to enforce Lipschitz constants in transformers, achieving improved stability and robustness.
• The co-design of the Muon optimizer with architectural modifications enables stable training without conventional normalization while preserving performance.
• Empirical results show that Lipschitz constraints lead to lower activation magnitudes, enhanced adversarial robustness, and a balanced performance tradeoff.

Training Transformers with Enforced Lipschitz Constants: An Expert Overview

This paper addresses the challenge of enforcing Lipschitz continuity in transformer architectures throughout training, a property that has been linked to improved robustness, generalization, and stability in neural networks. While prior work has explored Lipschitz constraints in MLPs, RNNs, and GANs, the application to large-scale transformers—especially with guarantees maintained during training—remained largely unexplored. The authors introduce computationally efficient methods for constraining the spectral norm of weight matrices, enabling the training of transformers with explicit, enforced Lipschitz bounds.

Methodological Contributions

The core technical contributions are as follows:

• Novel Weight Constraint Methods: The paper introduces two new techniques—spectral soft cap and spectral hammer—for constraining the spectral norm of weight matrices. Spectral soft cap is co-designed with the Muon optimizer and uses odd polynomial approximations to efficiently cap singular values, while spectral hammer is tailored for AdamW and directly sets the largest singular value to a threshold.
• Optimizer-Constraint Co-Design: The Muon optimizer, which ensures bounded spectral norm updates, is shown to synergize with spectral normalization and spectral soft cap, yielding a superior tradeoff between model performance and Lipschitz constant compared to AdamW.
• Architectural Adjustments: The authors adapt transformer architectures to facilitate Lipschitz enforcement, including reparameterized residual connections (convex combinations) and $1/d$-scaled attention, following theoretical insights from prior work. Layer normalization and other stability measures are removed to test the sufficiency of Lipschitz constraints for stable training.

Empirical Results

The paper presents extensive empirical evaluation across MLPs and transformers at multiple scales:

• MLPs on CIFAR-10: Muon combined with spectral normalization or spectral soft cap achieves lower Lipschitz bounds and better adversarial robustness than AdamW-based baselines, with comparable or improved accuracy.
• Small Transformers (Shakespeare Dataset): A transformer with a global Lipschitz bound less than 2 achieves 60% validation accuracy, outperforming prior baselines and demonstrating that strong Lipschitz constraints are compatible with nontrivial performance.
• Large Transformers (NanoGPT, 145M parameters): A transformer with a Lipschitz bound less than 10 achieves 21% accuracy on internet text, while relaxing the bound to $10^{264}$ allows matching the NanoGPT baseline accuracy of 39.4%. Notably, these models train stably without layer norm, QK norm, or logit softcapping, and exhibit much lower maximum activation values than unconstrained baselines.

The following table summarizes key results for the NanoGPT-scale experiments:

Model (Constraint)	Lipschitz Bound	Validation Accuracy	Max Activation
Baseline (Speedrun)	∞	39.4%	148,480
Ours ($\sigma_{max}=1$)	10	21.2%	14
Ours ($\sigma_{max}=16$)	$10^{264}$	39.5%	160

Theoretical and Practical Implications

Theoretical Implications:

• The results confirm that it is feasible to train large transformers with explicit, enforced Lipschitz bounds, extending the scope of certifiable robustness and stability guarantees to modern architectures.
• The co-design of optimizers and weight constraints is critical: Muon's fixed-norm updates enable strict norm enforcement, which is not possible with AdamW due to its unconstrained update norm.
• The exponential growth of the global Lipschitz bound with depth (unless all weights are strictly unit-norm) highlights a limitation of current architectural approaches. The inability to maintain depth-independent bounds without sacrificing performance suggests a need for further architectural innovation.

Practical Implications:

• Robustness and Safety: Enforced Lipschitz bounds yield models with improved adversarial robustness and potentially more predictable behavior under input and weight perturbations, which is valuable for safety-critical applications.
• Stability Without Normalization: The ability to train large transformers stably without layer norm or other normalization techniques may simplify deployment and reduce computational overhead.
• Low-Precision Training: The observation of consistently low activation magnitudes in Lipschitz-constrained models suggests an opportunity for aggressive quantization and low-precision inference, with potential efficiency gains.
• Differential Privacy and Generalization: Explicit control over the model's sensitivity to input and weight changes is directly relevant for differentially private training and for theoretical generalization bounds.

Limitations and Open Questions

• Hyperparameter Selection: The process for choosing weight norm, logit scale, and attention scale remains empirical, relying on hyperparameter sweeps rather than principled criteria.
• Loose Global Bounds: The computed global Lipschitz bounds are often loose upper bounds, especially at scale, and may not reflect the true operational sensitivity of the model. Tighter, data-dependent certification methods could improve practical utility.
• Performance Tradeoff: There is a clear tradeoff between the tightness of the Lipschitz bound and model performance, particularly at large scale. Achieving both strong performance and strict global bounds remains challenging.

Future Directions

• Architectural Innovation: Developing architectures that maintain depth-independent Lipschitz bounds without severe performance penalties is a promising direction.
• Tighter Certification: Integrating tighter, possibly data-dependent, Lipschitz certification methods could yield more meaningful guarantees.
• Scalability and Efficiency: Further exploration of low-precision training and inference in Lipschitz-constrained models could unlock significant efficiency gains.
• Broader Applications: The methods developed here are directly applicable to domains requiring robustness, privacy, and safety guarantees, such as autonomous systems, medical AI, and secure federated learning.

Conclusion

This work demonstrates the feasibility and practical benefits of enforcing Lipschitz continuity in transformer models throughout training, enabled by novel weight constraint methods and optimizer co-design. While strict global bounds entail a performance tradeoff at scale, the approach yields models with improved robustness, stability, and potential for efficient deployment. The results motivate further research into architectural and algorithmic advances for certifiable, robust, and efficient deep learning systems.