A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks (1810.02281v3)

Abstract: We analyze speed of convergence to global optimum for gradient descent training a deep linear neural network (parameterized as $x \mapsto W_N W_{N-1} \cdots W_1 x$) by minimizing the $\ell_2$ loss over whitened data. Convergence at a linear rate is guaranteed when the following hold: (i) dimensions of hidden layers are at least the minimum of the input and output dimensions; (ii) weight matrices at initialization are approximately balanced; and (iii) the initial loss is smaller than the loss of any rank-deficient solution. The assumptions on initialization (conditions (ii) and (iii)) are necessary, in the sense that violating any one of them may lead to convergence failure. Moreover, in the important case of output dimension 1, i.e. scalar regression, they are met, and thus convergence to global optimum holds, with constant probability under a random initialization scheme. Our results significantly extend previous analyses, e.g., of deep linear residual networks (Bartlett et al., 2018).

• The paper demonstrates that gradient descent achieves linear convergence in deep linear networks when weight matrices are balanced and initial loss is controlled.
• It employs a trajectory-based analysis to overcome traditional non-convex optimization challenges in deep learning models.
• Practical insights highlight improved initialization strategies that enhance training efficiency in deep network configurations.

Convergence Analysis of Gradient Descent for Deep Linear Neural Networks

In the investigated paper, the authors conduct a thorough analysis of the convergence properties of gradient descent when applied to the training of deep linear neural networks over whitened data. The paper is particularly unique in its treatment of non-convexities in deep learning models, overcoming limitations commonly faced by conventional landscape approaches. The analysis focuses on deep linear networks parameterized as a product of weight matrices, exploring conditions under which convergence to a global optimum occurs efficiently.

Key Findings

The authors identify specific conditions that guarantee convergence at a linear rate. These conditions include:

1. The dimensions of the hidden layers must be at least the minimum of the input and output dimensions.
2. The weight matrices must be approximately balanced at initialization.
3. The initial loss must be less than that of any rank-deficient solution.

Notably, violating any of these conditions can lead to convergence failure, highlighting their necessity in the training process.

Theoretical Implications

The paper extends previous analyses by providing rigorous convergence guarantees at a linear rate for general configurations of deep linear networks. These results are significant in that they circumvent typical barriers faced by landscape approaches in proving global convergence for deep models. By focusing on trajectory-based analyses, the authors illuminate aspects of the optimization landscapes that are critical near the paths taken by the optimizer, rather than on the general landscape characteristics.

Practical Implications

The practical implications are rooted in the initialization strategies and network design choices validated by the paper. The trajectory-based approach suggests that initialization schemes ensuring approximate balancedness and maintaining non-degenerate rank in the end-to-end mapping significantly affect convergence. This understanding can guide practitioners in optimizing the training of deep networks, particularly linear neural networks used in applications where rapid training convergence is critical.

Experimental Insights

The empirical demonstrations in the paper reinforce the theoretical claims, displaying how choices in initialization influence convergence behavior under the scrutinized conditions. Balanced initialization strategies were shown to have a stabilizing effect on convergence, outperforming standard independent layer-wise random Gaussian initializations, particularly in deeper network configurations.

Future Directions

This research potentially paves the way for exploring similar convergence guarantees for practical non-linear deep networks, possibly through analogous trajectory analyses. The introduction of balanced initialization strategies in the paper suggests that both theoretical and practical advancements in this area could drive more robust and efficient training of complex models.

By setting a foundational analytical framework for the convergence of linear models, the paper encourages further exploration into the intricate dynamics of network training, with the anticipation that such insights may eventually generalize across broader deep learning paradigms.