TensorFlow implementation for optimizers

TensorFlow provides a comprehensive suite of optimizers through its `tf.keras.optimizers` module, which includes both pre-built optimizers and the ability to create custom optimizers using the TensorFlow Core low-level APIs.

1. **Pre-built Optimizers**:
- **Adam**: An adaptive learning rate optimizer that computes adaptive learning rates for each parameter based on the first and second moments of the gradients.
- **RMSprop**: An adaptive learning rate optimization algorithm for stochastic optimization that divides the learning rate by an exponentially decaying average of squared gradients.
- **SGD**: Stochastic Gradient Descent, which updates the model parameters using the gradient of the loss function with respect to the weights.
- **Adagrad**: An optimizer that adapts the learning rate for each parameter based on the historical gradients.
- **Adadelta**: An extension of Adagrad that uses a moving window to compute the gradient updates.
- **Adamax**: An extension of Adam that clips the gradients to ensure that the update is not too large.
- **Nadam**: An extension of Adam that includes Nesterov acceleration.
- **Ftrl**: A flexible optimizer that can be used for both online and batch learning, and it supports both L1 and L2 regularization.
- **Lion**: A low-memory optimizer that is suitable for large-scale optimization problems.
- **LossScaleOptimizer**: An optimizer that dynamically scales the loss to prevent underflow.
2. **Custom Optimizers**:
- TensorFlow Core low-level APIs allow for the creation of custom optimizers from scratch, providing full control over the structure, implementation, and behavior of the optimizer.
- This is particularly useful for techniques that require a specific coupling between the model and optimizer, such as Sharpness-Aware Minimization (SAM), which cannot be effectively implemented using the Keras optimizers.

In summary, TensorFlow's `tf.keras.optimizers` module offers a wide range of pre-built optimizers that are suitable for most machine learning tasks. For scenarios that require specialized optimization algorithms or when the developer needs fine-grained control over the optimization process, TensorFlow provides the flexibility to create custom optimizers using the Core APIs.