Newton Losses: Using Curvature Information for Learning with Differentiable Algorithms (2410.19055v1)

Abstract: When training neural networks with custom objectives, such as ranking losses and shortest-path losses, a common problem is that they are, per se, non-differentiable. A popular approach is to continuously relax the objectives to provide gradients, enabling learning. However, such differentiable relaxations are often non-convex and can exhibit vanishing and exploding gradients, making them (already in isolation) hard to optimize. Here, the loss function poses the bottleneck when training a deep neural network. We present Newton Losses, a method for improving the performance of existing hard to optimize losses by exploiting their second-order information via their empirical Fisher and Hessian matrices. Instead of training the neural network with second-order techniques, we only utilize the loss function's second-order information to replace it by a Newton Loss, while training the network with gradient descent. This makes our method computationally efficient. We apply Newton Losses to eight differentiable algorithms for sorting and shortest-paths, achieving significant improvements for less-optimized differentiable algorithms, and consistent improvements, even for well-optimized differentiable algorithms.

• The paper introduces Newton Losses, which use curvature information via a second-order Taylor expansion to reformulate challenging loss functions.
• The paper employs a bifurcated optimization strategy that separates Newton-like loss optimization from standard gradient descent for neural parameters.
• Empirical evaluations on differentiable sorting and shortest-path tasks demonstrate significant improvements in handling non-convex and unstable gradients.

An Overview of "Newton Losses: Using Curvature Information for Learning with Differentiable Algorithms"

In the field of machine learning, effectively training neural networks with complex, non-differentiable objectives presents a significant challenge. The paper "Newton Losses: Using Curvature Information for Learning with Differentiable Algorithms" addresses this by introducing a method termed Newton Losses, aimed at enhancing the performance of these difficult-to-optimize loss functions. The approach leverages second-order information by utilizing the empirical Fisher and Hessian matrices to transform the original loss functions into more tractable ones for gradient descent optimization.

Key Contributions

The central idea of Newton Losses is to incorporate curvature information to locally approximate the loss function through a second-order Taylor expansion. This results in a reformulation whereby the neural network training is conducted using conventional first-order gradient methods, yet the loss function itself is enhanced with second-order characteristics.

1. Integration of Curvature Information: Newton Losses incorporate both Hessian-based and empirical Fisher matrix-based approaches, allowing the method to work even in contexts where computing second derivatives might be infeasible.
2. Bifurcated Optimization Strategy: The paper proposes separating the optimization into two distinct steps: optimizing the loss with a Newton-like approach and optimizing the neural network parameters with gradient descent.
3. Addressing Non-Convexity: By utilizing local convexity introduced through second-order terms, the method mitigates issues related to vanishing and exploding gradients, which often plague non-convex loss landscapes.

Empirical Evaluation

The paper rigorously evaluates Newton Losses across several algorithmic supervision scenarios, notably differentiable sorting and shortest-path tasks:

• Differentiable Sorting: The application of Newton Losses to differentiable sorting algorithms, such as NeuralSort and SoftSort, demonstrated consistent performance enhancements. The experiments highlighted significant improvements in scenarios characterized by challenging gradient issues.
• Shortest-Path Computations: In shortest-path supervision, integrating Newton Losses into stochastic smoothing and analytical relaxation techniques led to notable accuracy gains, underscoring the versatility of the approach across differing algorithmic landscapes.

The results indicate that Newton Losses are most beneficial for loss functions that exhibit optimization difficulty due to issues like non-convexity and gradient instability.

Theoretical and Practical Implications

The Newton Losses methodology has both theoretical significance and practical utility:

• Enhanced Optimization Dynamics: By bridging the gap between first-order and second-order methods, it affords a balanced approach that enhances convergence properties without the computational overhead typically associated with full second-order methods.
• Broader Applicability: The dual variant (Hessian and Fisher) makes it adaptable to a wider array of scenarios, facilitating its integration into existing systems with minimal modification.

Future Directions

The exploration of Newton Losses opens avenues for future research in several domains:

• Generalization to Other Domains: Extending these principles to other non-differentiable and weakly-supervised learning contexts can further illustrate the efficacy of Newton Losses beyond its current application scope.
• Algorithmic Enhancements: Developing more efficient computations for Hessians and Fisher matrices could reduce the latency associated with their incorporation, thereby broadening the method’s applicability in high-dimensional spaces.

In summary, the introduction of Newton Losses offers a promising advancement in optimizing complex algorithmic loss functions. By effectively integrating curvature information, the approach holds potential for improving the training efficacy of neural networks across various domains, advancing both theoretical understanding and empirical performance.