Regularisation of Neural Networks by Enforcing Lipschitz Continuity (1804.04368v3)

Abstract: We investigate the effect of explicitly enforcing the Lipschitz continuity of neural networks with respect to their inputs. To this end, we provide a simple technique for computing an upper bound to the Lipschitz constant---for multiple $p$-norms---of a feed forward neural network composed of commonly used layer types. Our technique is then used to formulate training a neural network with a bounded Lipschitz constant as a constrained optimisation problem that can be solved using projected stochastic gradient methods. Our evaluation study shows that the performance of the resulting models exceeds that of models trained with other common regularisers. We also provide evidence that the hyperparameters are intuitive to tune, demonstrate how the choice of norm for computing the Lipschitz constant impacts the resulting model, and show that the performance gains provided by our method are particularly noticeable when only a small amount of training data is available.

• The paper introduces a constrained optimization framework that enforces Lipschitz continuity to mitigate overfitting.
• It systematically computes upper bounds for various p-norms, with ℓ∞ excelling in tabular data and ℓ2 proving effective for image classification.
• Empirical results demonstrate improved sample efficiency and generalization, positioning the method as a robust alternative to heuristic regularization.

Regularisation of Neural Networks by Enforcing Lipschitz Continuity

The paper by Gouk et al. presents an investigation into the regularisation of neural networks by enforcing Lipschitz continuity. The authors introduce a framework to compute an upper bound on the Lipschitz constant for a variety of neural network architectures. This is achieved through a constrained optimization approach that effectively controls the Lipschitz constant during training, thereby mitigating overfitting and improving generalization on unseen data.

Methodology and Contributions

The key contribution of this work lies in formulating the training of neural networks as a constrained optimization problem with Lipschitz continuity. By enforcing an upper bound on the Lipschitz constant, the proposed method provides an inductive bias that aligns with the notion that simpler functions generalize better. The authors offer a systematic approach to calculate these upper bounds for different $p$-norms across layers of feed-forward networks, including fully connected and convolutional layers.

The paper emphasizes practical applicability by ensuring that the constraints can be efficiently incorporated into existing training regimes via projected stochastic gradient methods. This approach is complemented by demonstrating the implementation for various $p$-norms, most notably $\ell_1$, $\ell_2$, and $\ell_\infty$, where each norm's suitability varies across tasks. The pivotal insight that the Lipschitz constant across layers can be bounded individually and multiplied to yield a network-wide bound is leveraged to simplify this process.

Experimental Evaluation

Through comprehensive experimentation across multiple datasets and network architectures, the authors substantiate their claims regarding the efficacy of their regularization technique. Notably, networks with Lipschitz constant constraints exhibit superior performance, especially when trained on limited data. This outcome underscores the method's utility in improving the sample efficiency of neural network training—a critical advantage in scenarios with scarce data resources.

Several experiments reveal that the $\ell_\infty$ Lipschitz constraint performs robustly across various tasks, significantly in tabular data, whereas $\ell_2$ constraints show effectiveness for image classification tasks. This differentiation highlights that distinct regularization norms can be tailored to specific problem domains for optimal results.

Implications and Future Directions

The authors' exploration of Lipschitz continuity extends theoretical notions into practical regularization techniques, offering a well-justified alternative to heuristic methods like dropout. The theoretical underpinning and empirical validation elevate the proposed technique as a reliable tool for neural network regularization.

The findings of this paper implicate broader implications, particularly in settings like GANs, where stability derived from controlled Lipschitz constants can be beneficial. Further research could explore the effectiveness of the technique in recurrent architectures. Moreover, devising mechanisms for per-layer tuning of Lipschitz constants could further harness model capacity, moving beyond the uniform constraints currently applied across layers.

In summary, this paper provides a compelling case for Lipschitz-based regularization, reinforcing the importance of theoretically grounded methods in enhancing neural network performance and underscores promising avenues for further investigation and refinement within the growing field of neural network regularization.