Breaking the Curse of Dimensionality with Convex Neural Networks (1412.8690v2)

Abstract: We consider neural networks with a single hidden layer and non-decreasing homogeneous activa-tion functions like the rectified linear units. By letting the number of hidden units grow unbounded and using classical non-Euclidean regularization tools on the output weights, we provide a detailed theoretical analysis of their generalization performance, with a study of both the approximation and the estimation errors. We show in particular that they are adaptive to unknown underlying linear structures, such as the dependence on the projection of the input variables onto a low-dimensional subspace. Moreover, when using sparsity-inducing norms on the input weights, we show that high-dimensional non-linear variable selection may be achieved, without any strong assumption regarding the data and with a total number of variables potentially exponential in the number of ob-servations. In addition, we provide a simple geometric interpretation to the non-convex problem of addition of a new unit, which is the core potentially hard computational element in the framework of learning from continuously many basis functions. We provide simple conditions for convex relaxations to achieve the same generalization error bounds, even when constant-factor approxi-mations cannot be found (e.g., because it is NP-hard such as for the zero-homogeneous activation function). We were not able to find strong enough convex relaxations and leave open the existence or non-existence of polynomial-time algorithms.

• The paper demonstrates that convex neural networks effectively adapt to unknown low-dimensional structures, reducing approximation and estimation errors.
• It employs sparsity-inducing norms on input weights to perform non-linear variable selection even in exponentially high-dimensional spaces.
• The research offers geometric insights for convex relaxations, outlining theoretical advancements and future directions despite unresolved computational tractability.

Breaking the Curse of Dimensionality with Convex Neural Networks

The paper investigates the potential of single-hidden-layer neural networks with non-decreasing positively homogeneous activation functions, such as ReLU, to counteract the curse of dimensionality. It frames this potential within the context of convex optimization, where the number of hidden units is allowed to grow unbounded, leading to a convex optimization problem through the application of non-Euclidean regularization on the output weights.

Key Contributions

1. Convex Neural Networks and Generalization: The paper demonstrates that this model adapts effectively to unknown linear structures by focusing on projections onto low-dimensional subspaces. This aspect leads to significant reductions in approximation and estimation errors, thus offering a viable approach to managing high-dimensional data more efficiently.
2. Sparsity-Inducing Norms: When applying sparsity-inducing norms on input weights, the model facilitates non-linear variable selection even in exponentially large dimensional spaces relative to the sample size. This feature allows the model to bypass prohibitive assumptions usually needed for standard models in high-dimensional contexts.
3. Infinite Dimensions and Convex Relaxation: Addressing the computational intractability of infinite dimensions, the paper navigates the non-convexity challenge by proposing geometric interpretations and conditions for convex relaxations. While not able to yield polynomial-time algorithms, these relaxations maintain generalization error bounds even without constant-factor approximations.

Numerical Results and Computational Complexity

Analyzing the presented model's computational aspects, the paper asserts that convex relaxations based on semi-definite programming don't fully achieve computational tractability. Despite advances in convex formulations, the existence of polynomial-time solutions remains unresolved, emphasizing the computational difficulty inherent in the non-convex subproblem.

Theoretical Implications and Future Directions

The research underscores critical implications for theoretical advancements in AI, especially in enhancing model adaptivity to intrinsic data structures. It opens directions for developing new optimization approaches or relaxing the convexity conditions further to discover polynomial-time algorithms. Additionally, the challenges pointed out regarding tractable algorithms advocate for further exploration into efficient approximations or heuristics that bear computational feasibility without sacrificing accuracy.

Practical Implications

Practically, this work suggests that leveraging the intrinsic simplicities in data through convex neural networks can advance applicability in tasks traditionally hindered by dimensionality challenges. The ability to efficiently perform non-linear variable selection underscores potential improvements in domains such as high-dimensional data analysis, scientific computing, and machine learning, where interpretations of data structure play a monumental role.

In summary, the paper delivers an insightful analysis developing the relationship between convex neural network estimations and dimensional adaptivity, shaping future considerations in algorithm development and theoretical explorations in neural network architectures.