Input Convex Neural Networks (1609.07152v3)

Abstract: This paper presents the input convex neural network architecture. These are scalar-valued (potentially deep) neural networks with constraints on the network parameters such that the output of the network is a convex function of (some of) the inputs. The networks allow for efficient inference via optimization over some inputs to the network given others, and can be applied to settings including structured prediction, data imputation, reinforcement learning, and others. In this paper we lay the basic groundwork for these models, proposing methods for inference, optimization and learning, and analyze their representational power. We show that many existing neural network architectures can be made input-convex with a minor modification, and develop specialized optimization algorithms tailored to this setting. Finally, we highlight the performance of the methods on multi-label prediction, image completion, and reinforcement learning problems, where we show improvement over the existing state of the art in many cases.

• The paper introduces Input Convex Neural Networks (ICNNs) that enforce convexity through non-negative weights and convex activations.
• ICNNs replace standard feedforward prediction with optimization-based inference using methods like the bundle entropy technique.
• Empirical results on datasets like BibTeX, Olivetti, and OpenAI Gym demonstrate improved performance in multi-label classification, image completion, and reinforcement learning.

Analysis of Input Convex Neural Networks

The paper "Input Convex Neural Networks" introduces a novel architecture for scalar-valued neural networks, termed as Input Convex Neural Networks (ICNNs). This architecture is designed to enforce convexity with respect to a subset of the inputs, providing a significant advantage in certain machine learning applications. This discourse will explore the architecture, applications, empirical results, and potential implications.

Architectural Framework

ICNNs impose structural constraints on neural network parameters to ensure that network outputs are convex functions with respect to certain inputs. This is achieved by using ReLU or similar non-linearities that are both convex and non-decreasing, coupled with non-negative weights in specific layers. Two variants of ICNNs are explored: Fully Input Convex Neural Networks (FICNNs), which ensure joint convexity over all inputs, and Partially Input Convex Neural Networks (PICNNs), which maintain convexity over a subset of inputs while allowing non-convexity over others.

Methodology and Inference

ICNNs enable inference via optimization, replacing the traditional feedforward prediction model with an optimization task over the convex input dimensions. Efficient optimization is achieved through novel algorithms such as the bundle entropy method, which offers a structured approach to handling ICNN-specific inference tasks.

Empirical Evaluation

The paper reports empirical results across various domains demonstrating the ICNN's capacity to improve performance over existing methods:

1. Multi-Label Classification: On the BibTeX dataset, ICNNs showed competitive performance with a macro-F1 score of 0.415, surpassing feedforward baseline models.
2. Image Completion: On the Olivetti dataset, ICNNs demonstrated effective image inpainting, with lower mean squared error than traditional techniques.
3. Reinforcement Learning: For continuous control tasks in the OpenAI Gym, ICNNs outperformed several strong baselines such as DDPG and NAF in several environments, showcasing their robust applicability.

Implications and Future Work

The introduction of ICNNs provides a new paradigm for scenarios where convexity is beneficial, such as structured prediction and data imputation. The enforced convexity during inference can lead to more reliable and robust convergence in optimization-based tasks, adding theoretical strengths to practical implementations.

Future research could explore the integration and scalability of ICNNs with more complex neural architectures and real-world applications requiring high-dimensional structured outputs. Moreover, addressing any limitations due to the non-negative weight constraints could expand the applicability of ICNNs to an even broader range of problems.

Conclusion

ICNNs represent a significant advancement in neural network design, opening avenues for integrating convex optimization with deep learning. This architecture not only enhances the representational power in important application domains but also offers promising directions for further theoretical and empirical exploration.