Why the Adam Optimizer Converges
The Adam optimizer is able to efficiently handle sparse gradients and adaptively adjust learning rates. The convergence of Adam, often observed as a flattening of the cost function, can be attributed to several factors inherent to its design and the characteristics of the dataset being used.
The convergence of the Adam optimizer, resulting in a stable cost value, is a product of its adaptive learning rate, regularization effects, numerical stability mechanisms, and the dataset’s characteristics.
1. Convergence to Local Minimum or Saddle Point
Adaptive Learning Rate: Adam adjusts the learning rate for each parameter individually, allowing it to quickly converge towards a local minimum or saddle point. This adaptability helps in navigating complex cost landscapes but may also cause the optimizer to plateau at a local minimum rather than reaching the global minimum. This plateauing effect can result in a stable final cost value, such as 0.146.
Gradient Behaviour: As Adam converges, the gradients often become small or near zero, slowing down the learning process and leading to a flattened cost curve.
2. Learning Rate
Impact of Learning Rate ((\alpha)): The choice of learning rate significantly influences convergence speed. A larger learning rate might cause overshooting, while a smaller one might lead to slow convergence. Common values like 0.001, 0.005, 0.01, and 0.1 are used to balance these effects.
Stability in Suboptimal Regions: If the learning rate is not optimal, Adam might get stuck in a suboptimal region, such as a local minimum or saddle point, resulting in a stable cost value.
3. Regularization
L2 Regularization: The inclusion of L2 regularization helps prevent overfitting but also affects the optimization process by slightly increasing the final cost value. The observed cost value is a balance between error reduction and the regularization penalty.
4. Numerical Stability
Moment Estimates: Adam uses first and second moment estimates (mean of gradients and squared gradients) to update parameters. As these estimates improve, the magnitude of updates decreases, leading to smaller changes in the cost function and eventual flattening.
Epsilon for Stability: The epsilon parameter ensures numerical stability by preventing division by very small values, which can also lead to reduced update steps when squared gradients are small.
5. Dataset Characteristics
Simplicity of the Dataset: The characteristics of the dataset, such as its simplicity or complexity, can influence convergence. In a simple dataset with few features, the optimizer might reach a plateau quickly due to the limited complexity of the problem.
6. Final Cost Comparison
Reasonable Solution: The stable cost value, such as 0.146, indicates that Adam has found a reasonable solution given the dataset and optimizer settings. It reflects a balance between minimizing error and applying regularization.