Learning One-hidden-layer Neural Networks with Landscape Design (1711.00501v2)

Abstract: We consider the problem of learning a one-hidden-layer neural network: we assume the input $x\in \mathbb{R}^d$ is from Gaussian distribution and the label $y = a^\top \sigma(Bx) + \xi$, where $a$ is a nonnegative vector in $\mathbb{R}^m$ with $m\le d$, $B\in \mathbb{R}^{m\times d}$ is a full-rank weight matrix, and $\xi$ is a noise vector. We first give an analytic formula for the population risk of the standard squared loss and demonstrate that it implicitly attempts to decompose a sequence of low-rank tensors simultaneously. Inspired by the formula, we design a non-convex objective function $G(\cdot)$ whose landscape is guaranteed to have the following properties: 1. All local minima of $G$ are also global minima. 2. All global minima of $G$ correspond to the ground truth parameters. 3. The value and gradient of $G$ can be estimated using samples. With these properties, stochastic gradient descent on $G$ provably converges to the global minimum and learn the ground-truth parameters. We also prove finite sample complexity result and validate the results by simulations.

• The paper’s main contribution is that a carefully designed objective function ensures all local minima are global, aligning with true parameters.
• It employs landscape design using Hermite polynomials and tensor decompositions to address non-convex optimization challenges.
• Empirical results show finite sample efficiency and reliable convergence, even with simple initialization strategies.

Summary of "Learning One-hidden-layer Neural Networks with Landscape Design"

The paper "Learning One-hidden-layer Neural Networks with Landscape Design" provides a rigorous approach to solving the problem of learning one-hidden-layer neural networks. Specifically, it tackles the challenge of non-convex optimization encountered when training such networks and proposes a strategy that yields global convergence guarantees.

Main Contributions and Theoretical Insights

The core contribution of this work lies in designing an objective function for training one-hidden-layer neural networks that does not suffer from spurious local minima. The authors assume the input data follows a Gaussian distribution and consider a model where the label $y$ is generated from a linear transformation of a non-linear activation of the input matrix $x$, plus some noise $\xi$. They propose a novel objective function $G(\cdot)$, with the following essential properties:

1. Global Optimality of Local Minima: All local minima of the designed function are global minima, ensuring convergence to the correct parameters.
2. Correspondence to True Parameters: The global minima correspond exactly to the ground truth parameters (up to designated symmetries).
3. Sample Accessibility: The value and gradient of this objective can be estimated using data samples.

Moreover, the authors prove finite sample complexity, indicating that their method can achieve results with a polynomially bounded number of samples. This ensures practical applicability in scenarios with limited data availability.

Landscape Design Approach

The research develops a theoretical framework that reformulates the optimization landscape to systematically eliminate undesired local minima. The proposed method hinges on the properties of Hermite polynomials and tensor decompositions, facilitating an analysis of the optimization landscape that ensures favorable properties for gradient-based methods.

Numerical and Theoretical Guarantees

Empirical evaluations verify the predicted convergence properties of the proposed method. The authors demonstrate via simulations that standard approaches like SGD for the $\ell_2$ loss on one-hidden-layer neural networks can fail to converge to global minima, particularly under realistic non-convex conditions. Conversely, the newly defined objective $G(\cdot)$ ensures that gradient descent converges to the global minimum, even with simple initialization strategies.

Implications and Future Directions

From a theoretical standpoint, this work underscores the importance of objective function design in non-convex optimization scenarios, particularly in neural network training. Practically, it provides a pathway to more reliable and theoretically grounded neural network training methods, especially in configurations where classic convergence guarantees do not hold.

The insights from the paper could be extended to other architectures and learning scenarios, potentially providing a broader framework for designing training objectives with desired properties. Future research directions include extending the approach to non-Gaussian inputs and other forms of neural network architectures beyond the one-hidden-layer case.

Conclusion

The paper provides a solid theoretical and empirical foundation for the training of one-hidden-layer neural networks under non-convex settings. By ensuring that all local minima are global and aligning them with the true model parameters, this work represents a significant step towards more robust and reliable neural network training. The framework introduced could inspire further reforms in the optimization landscape of different learning models, ultimately enhancing the efficacy of neural network applications across various domains.