On the Convergence Rate of Training Recurrent Neural Networks (1810.12065v4)

Abstract: How can local-search methods such as stochastic gradient descent (SGD) avoid bad local minima in training multi-layer neural networks? Why can they fit random labels even given non-convex and non-smooth architectures? Most existing theory only covers networks with one hidden layer, so can we go deeper? In this paper, we focus on recurrent neural networks (RNNs) which are multi-layer networks widely used in natural language processing. They are harder to analyze than feedforward neural networks, because the $\textit{same}$ recurrent unit is repeatedly applied across the entire time horizon of length $L$, which is analogous to feedforward networks of depth $L$. We show when the number of neurons is sufficiently large, meaning polynomial in the training data size and in $L$, then SGD is capable of minimizing the regression loss in the linear convergence rate. This gives theoretical evidence of how RNNs can memorize data. More importantly, in this paper we build general toolkits to analyze multi-layer networks with ReLU activations. For instance, we prove why ReLU activations can prevent exponential gradient explosion or vanishing, and build a perturbation theory to analyze first-order approximation of multi-layer networks.

• The paper establishes a Polyak-Lojasiewicz condition that quantifies the gradient's role in avoiding bad local minima.
• The investigation demonstrates objective semi-smoothness, ensuring that small perturbations in parameters do not significantly worsen the network's performance.
• The study shows a linear convergence rate for GD and SGD in large-scale RNNs, effectively mitigating vanishing and exploding gradient issues.

Analysis of Gradient Descent and Stochastic Gradient Descent in Recurrent Neural Networks

The paper explores the convergence properties of gradient descent (GD) and stochastic gradient descent (SGD) when applied to training recurrent neural networks (RNNs), specifically focusing on the resilience of these methods against bad local minima in multilayer networks with ReLU activations. This examination is crucial given the complexities and non-convex nature of RNNs that are prominently used in fields like natural language processing, machine translation, and sequential data tasks.

The research emphasizes the scenario where the depth and size of networks introduce several challenges, such as the vanishing or exploding gradient problem. It establishes novel toolkits for analyzing these issues, leading to two main technical results: a Polyak-{\L}ojasiewicz condition and an objective semi-smoothness result. These results contribute insight into why GD and SGD effectively navigate the complex landscape of RNNs, despite their inherent non-convexity.

Key Findings and Contributions

1. Polyak-{\L}ojasiewicz Condition:
   • The paper establishes a condition where the gradient norm is at least proportional to the objective value. This provides a quantitative measure that indicates the absence of bad local minima, supporting the theoretical ground to advocate the robustness of SGD and GD in optimizing RNNs.
   • This is significant because it implies that even though the training objectives are non-convex, the gradient aligns well with decreasing the objective value, offering a theoretical explanation for empirical observations in deep learning.
2. Objective Semi-Smoothness:
   • It demonstrates a form of semi-smoothness which implies that the objective function does not significantly increase despite adversarial perturbations within a certain radius from the initialization. This characteristic ensures that movements in the parameter space during GD and SGD do not lead to notably worse outcomes.
3. Numerical Results:
   • The paper states a linear convergence rate for GD and SGD, with the condition that the number of neurons is sufficiently large (polynomial relative to data size and sequence length). The convergence is achieved in polynomial time concerning the number of layers and data points, mitigating the gradient explosion or vanishing issues over long sequences.
   • It employs a sample complexity that is logarithmic in nature concerning some problem parameters, implying efficiency in the learning processes over complex RNN architectures.

Theoretical Implications and Future Directions

The implications of these findings are profound for both theoretical understanding and practical applications in deep learning and RNN architectures:

• Theoretical Advancements: This work extends the understanding beyond simple network structures, exploring the teachability of more complex multilayer configurations. The methodologies could potentially be adapted to other activation functions and network structures beyond RNNs, providing a broader platform for analyzing non-convex optimization.
• Practical Implementations: Practitioners can leverage these insights to optimize RNN training schedules, mitigate common pitfalls like vanishing/exploding gradients, and employ architectural designs that align well with the proven theoretical foundations.
• Future Developments: The exploration into reducing the dependency on large network sizes for theoretical guarantees presents an interesting venue for ongoing research, potentially leading to an optimization theory that is more resource-efficient.

This paper provides a robust theoretical grounding for the observed effectiveness of GD and SGD in training complex RNNs, illustrating that despite the computational challenges posed by deep networks, well-constructed mathematical frameworks can inform and enhance their real-world application. While the paper primarily explores theoretical aspects, its conclusions broadly inform empirical strategies and the continuous evolution of neural network architectures.