Learning ReLUs via Gradient Descent (1705.04591v2)

Abstract: In this paper we study the problem of learning Rectified Linear Units (ReLUs) which are functions of the form $max(0,<w,x>)$ with $w$ denoting the weight vector. We study this problem in the high-dimensional regime where the number of observations are fewer than the dimension of the weight vector. We assume that the weight vector belongs to some closed set (convex or nonconvex) which captures known side-information about its structure. We focus on the realizable model where the inputs are chosen i.i.d.~from a Gaussian distribution and the labels are generated according to a planted weight vector. We show that projected gradient descent, when initialization at 0, converges at a linear rate to the planted model with a number of samples that is optimal up to numerical constants. Our results on the dynamics of convergence of these very shallow neural nets may provide some insights towards understanding the dynamics of deeper architectures.

• The paper demonstrates that projected gradient descent, initialized at zero, converges linearly to the planted ReLU model in high-dimensional settings.
• It introduces a minimal sample function using Gaussian width to guarantee near-optimal sample size for effective nonlinear learning.
• The study offers insights for deeper neural architectures, highlighting practical implications for nonconvex optimization in machine learning.

Learning ReLUs via Gradient Descent: An Expert Overview

In the paper titled "Learning ReLUs via Gradient Descent," Mahdi Soltanolkotabi addresses the challenge of learning Rectified Linear Units (ReLUs) through gradient descent in high-dimensional settings. ReLUs are staple nonlinear functions in many neural network architectures, formally expressed as $x \mapsto \max(0, \langle w, x \rangle)$, where $w \in \mathbb{R}^d$ denotes the weight vector. Soltanolkotabi's paper is significant as it deepens our understanding of the convergence dynamics of gradient descent for ReLU-based models, particularly within high-dimensional regimes where the number of observations is less than the dimension of the weight vector $w$.

Core Contributions

1. Projected Gradient Descent Convergence: The paper establishes that a simple projected gradient descent method, initialized at zero, converges linearly to the planted model with a near-optimal sample size that depends on the Gaussian width of the descent cone of a regularizer $\mathcal{R}$. This result holds even when $\mathcal{R}$ is nonconvex, provided the inputs are i.i.d. Gaussian and labeled according to a planted weight vector. The convergence rate is delineated as $O(\log(1/\epsilon))$, making it computationally desirable given the dimensional constraints.
2. Sample Efficiency: Soltanolkotabi introduces the concept of the minimal sample function $n_0 = \mathcal{M}(\mathcal{R}, w^*)$, which implies a Gaussian width characterization of the descent cone. His analysis articulates the necessity for a sample size that is bounded by constants, to guarantee effective learning in high-dimensional nonlinear settings—a noteworthy alignment with the optimal sample requirements for structured signal recovery from linear measurements.
3. Implications for Deeper Architectures: While focusing on very shallow neural networks, the paper provides insights that could potentially inform the behavior of deeper architectures, emphasizing the simplicity of local search heuristics such as gradient descent and its surprising efficacy in practical scenarios.

Theoretical and Practical Implications

From a theoretical perspective, the work enriches the dialogue on nonconvex optimization and structured signal recovery by illustrating convergence in high-dimensional settings—a region traditionally fraught with computational intractability. Practically, these insights pave the way for refined algorithms in machine learning applications involving structured nonlinear models, advocating for broader exploration into nonconvex regularization techniques.

Future Directions

This work calls for further analysis of nonconvex heuristics in neural network training, particularly how similar projected gradient schemes might perform across different activation functions or larger scale networks. The applicability of these convergence guarantees to other model varieties—beyond ReLUs—continues to be an open and inviting prospect for future research.

In conclusion, Soltanolkotabi's paper contributes significantly to our understanding of ReLU learning dynamics, offering a valuable framework for high-dimensional learning using gradient descent. Through robust theoretical foundations, the paper reinforces the efficacy of simple optimization techniques while challenging current perceptions of sample size needs in high dimensionality. This opens avenues for advancements in both theoretical and applied aspects of machine learning and neural network design.