Recovery Guarantees for One-hidden-layer Neural Networks (1706.03175v1)

Abstract: In this paper, we consider regression problems with one-hidden-layer neural networks (1NNs). We distill some properties of activation functions that lead to $\mathit{local~strong~convexity}$ in the neighborhood of the ground-truth parameters for the 1NN squared-loss objective. Most popular nonlinear activation functions satisfy the distilled properties, including rectified linear units (ReLUs), leaky ReLUs, squared ReLUs and sigmoids. For activation functions that are also smooth, we show $\mathit{local~linear~convergence}$ guarantees of gradient descent under a resampling rule. For homogeneous activations, we show tensor methods are able to initialize the parameters to fall into the local strong convexity region. As a result, tensor initialization followed by gradient descent is guaranteed to recover the ground truth with sample complexity $ d \cdot \log(1/\epsilon) \cdot \mathrm{poly}(k,\lambda )$ and computational complexity $n\cdot d \cdot \mathrm{poly}(k,\lambda) $ for smooth homogeneous activations with high probability, where $d$ is the dimension of the input, $k$ ($k\leq d$) is the number of hidden nodes, $\lambda$ is a conditioning property of the ground-truth parameter matrix between the input layer and the hidden layer, $\epsilon$ is the targeted precision and $n$ is the number of samples. To the best of our knowledge, this is the first work that provides recovery guarantees for 1NNs with both sample complexity and computational complexity $\mathit{linear}$ in the input dimension and $\mathit{logarithmic}$ in the precision.

• The paper presents rigorous recovery guarantees for one-hidden-layer neural networks using convexity conditions and advanced tensor methods.
• It demonstrates that popular activation functions satisfy local strong convexity, ensuring linear convergence via gradient descent.
• The study establishes sample complexity linear in the input dimension and validates tensor-based initialization for efficient parameter recovery.

Analysis of "Recovery Guarantees for One-hidden-layer Neural Networks"

The paper "Recovery Guarantees for One-hidden-layer Neural Networks" by Zhong, Song, Jain, Bartlett, and Dhillon presents a comprehensive theoretical analysis of parameter recovery and optimization for one-hidden-layer neural networks (1NNs) under certain conditions. The authors provide novel insights into the conditions required for recovery guarantees, utilizing both convexity properties and advanced tensor methods. This work bridges the gap between empirical success and theoretical understanding of NNs, particularly in regression settings.

Core Contributions

1. Properties of Activation Functions:
   • The paper systematically derives conditions on activation functions that ensure local strong convexity and bounded spectrum of the Hessian near the optimal parameters.
   • Most popular activation functions like ReLU, leaky ReLU, squared ReLU, and sigmoids are demonstrated to satisfy these properties, which are crucial for achieving the desired convexity conditions.
2. Tensor-based Initialization:
   • The authors employ tensor methods to initialize the weights of the neural network within a local strong convexity region. This technique significantly reduces the dependence on the dimension for the required sample complexity.
   • The initialization aims to bring the parameters close enough to the ground truth, facilitating efficient parameter recovery via subsequent optimization.
3. Linear Convergence via Gradient Descent:
   • Given the ensured strong convexity in the local neighborhood, the authors prove that gradient descent can achieve linear convergence under these conditions.
   • They introduce a resampling strategy necessary to preserve convergence guarantees when the iterates depend on the data.
4. Theoretical Guarantees and Sample Complexity:
   • The paper provides theoretical guarantees both for sample complexity and computational efficiency. Notably, the proposed methods yield convergence with sample complexity linear in the input dimension $d$ and logarithmic in precision $\epsilon$.
   • The complexities are expressed as $$d \cdot \log(1/\epsilon) \cdot \text{poly}(k, \lambda)$$ for sample complexity and $n \cdot d \cdot \text{poly}(k, \lambda)$ for computational complexity.

Theoretical and Practical Implications

The results presented in this paper have significant implications for both practical implementations and theoretical explorations in machine learning, especially in understanding non-convex landscape optimizations.

• Practical Implications:
  • The initialization and recovery guarantees could inform the design of more efficient NN training algorithms.
  • By reducing sample and computational complexity, these methods can make NN training more feasible in resource-constrained settings.
• Theoretical Implications:
  • The addressed properties and guarantees offer deeper insights into the dynamics between activation functions and the convergence behavior of gradient-based methods.
  • These results contribute to demystifying why certain architectures and activation functions tend to perform better empirically.

Future Directions

The authors acknowledge several potential extensions to their work. Theoretical expansions to deeper architectures can be particularly intriguing, possibly requiring novel methodological advances or assumptions. Furthermore, addressing non-smooth activations beyond ReLU, such as those involving discontinuities, could be an area for exploration. Lastly, bridging the gap to stochastic optimization (such as SGD), without compromising the convergence guarantees established here, may provide a more holistic understanding of neural network training dynamics.

In summary, this paper methodically breaks down crucial aspects of 1NN training, offering robust guarantees under realistic assumptions that could lead to smarter, more efficient network training paradigms. The blend of convexity analysis and tensor methods presents a powerful toolkit for navigating and optimizing non-convex loss landscapes.