Understanding Local Minimum, Global Minimum, and Gradient Descent in Optimization

In optimization problems, especially in machine learning and deep learning, concepts like local minima, global minima, and gradient descent are central to how algorithms find optimal solutions. Let’s break down these concepts:

1. Local Minimum vs. Global Minimum

Local Minimum: This is a point in the optimization landscape where the function value is lower than the surrounding points, but it might not be the lowest possible value overall. It’s “locally” the best solution, but there might be a better solution elsewhere in the space.

Global Minimum: This is the point where the function attains the lowest possible value across the entire optimization landscape. It’s the “best” solution globally.

When using gradient-based methods like gradient descent, the goal is to minimize a loss (or cost) function. If the function has multiple minima, we want to find the global minimum, but sometimes the algorithm might get stuck in a local minimum.

2. Why Are Local Minima Considered “Bad”?

Local minima are generally considered problematic because:

• They might not represent the best (i.e., lowest) solution.
• If a gradient-based optimization algorithm, like gradient descent, falls into a local minimum, it may stop improving even though a better solution (the global minimum) exists.

In some cases, local minima may still give acceptable or “good enough” solutions, but the risk is that the algorithm might converge to a point that’s far from optimal.

3. Global Gradient Descent

In practice, what we aim for is global gradient descent, which means we want the algorithm to converge to the global minimum. However, achieving this is not always straightforward due to the following challenges:

• Non-convex functions: Many optimization landscapes (especially in neural networks) are non-convex, meaning they have multiple peaks and valleys. This makes it hard to guarantee that gradient descent will reach the global minimum.
• Saddle points: Besides local minima, the algorithm can also get stuck in saddle points, which are neither minima nor maxima, but flat regions where gradients are small, causing the optimization process to slow down or stop.

4. Why is Gradient Descent Sometimes Inefficient in These Scenarios?

Gradient descent works by following the direction of the steepest decrease in the loss function (i.e., the negative gradient). However, it has some limitations:

• Local minima: In a non-convex landscape, if the algorithm gets trapped in a local minimum, gradient descent might not be able to escape it.
• Saddle points: The gradients near saddle points are close to zero, causing gradient descent to progress very slowly or stagnate.
• Slow convergence: Gradient descent can take a long time to converge to the minimum, especially if the learning rate is not well-tuned.

5. Techniques to Overcome Local Minima and Improve Gradient Descent

To deal with these issues, several techniques are used in practice:

• Momentum: Momentum helps the algorithm build speed in regions with small gradients (like saddle points) and push through local minima by maintaining some velocity.
• Adam Optimizer: This is an adaptive learning rate optimizer that adjusts the learning rate for each parameter, helping the optimization avoid getting stuck in local minima.
• Random restarts: In some cases, running the algorithm multiple times from different starting points (random initializations) can help increase the chances of finding the global minimum.
• Stochastic Gradient Descent (SGD): Instead of using the full dataset at each step, SGD uses a small random subset (mini-batch). The randomness of the data can help it escape local minima and explore other regions of the optimization space.

Terms

Local minimum: A point where the function has a lower value than its neighbors but isn’t the lowest globally.

Global minimum: The absolute lowest point of the function, where the optimization goal is fully achieved.

Local minima can be problematic because they prevent the algorithm from finding the best possible solution.

Gradient descent can sometimes get stuck in local minima or saddle points, but techniques like momentum, adaptive learning rates, and stochasticity (randomness) help mitigate these issues.

Get over local minimum with the Most Popular Optimizer : the Adam Algorithm

Introduction to Adam

Adam (short for Adaptive Moment Estimation) is an optimization algorithm that has gained immense popularity due to its efficiency and effectiveness in training deep neural networks. It combines the benefits of two classical optimization methods: Momentum and RMSProp. Adam is particularly well-suited for non-stationary objectives, problems with noisy gradients, and sparse gradients, which makes it a versatile and powerful optimizer across a wide range of machine learning tasks.

The key intuition behind Adam is that it maintains both the first moment (mean) and second moment (uncentered variance) of the gradients during optimization, which helps in better adapting the learning rates for different parameters. This dual adaptation accelerates the convergence of the optimization process while mitigating the issues of oscillations and vanishing learning rates often encountered in traditional gradient descent methods.

Key Components of Adam

Adam Optimizer: A Combination of Momentum and RMSProp

Adam (Adaptive Moment Estimation) is a combination of two key optimization techniques:

1. Momentum Optimizer
2. RMSProp Optimizer

1. Momentum:

Momentum helps accelerate gradient descent by adding a fraction of the previous update to the current one. This reduces oscillations in gradient directions and speeds up convergence, especially in cases where gradients vary significantly in magnitude.

In Adam, the first moment (mean of the gradients) is computed similarly to how it’s done in the Momentum optimizer. This term accumulates past gradients exponentially, essentially keeping track of a “velocity” to help smooth out the gradient updates.

Adam’s first moment update (momentum-like):

Where:

• is the first moment estimate.
• is the gradient at time step .
• is a hyperparameter that controls the exponential decay rate for the first moment.

2. RMSProp:

RMSProp adjusts the learning rate for each parameter individually based on the magnitude of recent gradients. It prevents the learning rate from becoming too large when gradients are steep and too small when gradients are flat.

In Adam, the second moment (uncentered variance of the gradients) is computed similarly to RMSProp. This term adjusts the learning rate by normalizing the gradients using a running average of their squared values.

Adam’s second moment update (RMSProp-like):

Where:

• is the second moment estimate.
• is the squared gradient at time step .
• is a hyperparameter that controls the exponential decay rate for the second moment.

Final Adam Update:

The final update in Adam combines these two concepts:

Where:

• and are the bias-corrected first and second moment estimates.
• is the learning rate, and is a small constant to prevent division by zero.

Adam combines the momentum aspect from the Momentum optimizer (by keeping track of past gradients to smooth updates) and the adaptive learning rate from RMSProp (by scaling gradients based on the magnitude of past gradients). This makes Adam a powerful optimizer that works well for many types of deep learning tasks. All steps is explained in more details as you continue read this INGOAMPT article

To clarify further, we can say Adam utilizes two crucial mathematical constructs that evolve over time: the first moment (which approximates the gradient’s direction) and the second moment (which helps scale the step size based on gradient magnitudes). Here is an in-depth breakdown of the components and mechanics behind Adam.

1. Gradient Calculation

At each iteration t, Adam begins by computing the gradient of the loss function f(θ_t) with respect to the parameters θ (such as the weights of a neural network):

This represents the steepest direction of change in the objective function at iteration t, just like standard gradient descent.

2. First Moment Estimate (Exponential Moving Average of the Gradient)

Adam updates an exponentially decaying average of the past gradients. This first moment estimate m_t approximates the mean direction of the gradient:

Where:

• m_t is the first moment estimate (running average of gradients),
• β_1 is the decay rate for the moving average (typically set to 0.9).

The inclusion of the first moment helps Adam “smooth out” noisy gradients by looking at a more stable, long-term average of the gradient’s direction. This is akin to Momentum, where we apply a form of inertia to continue moving in the same direction if previous gradients support it.

3. Second Moment Estimate (Exponential Moving Average of Squared Gradient)

In addition to the first moment, Adam calculates an exponentially decaying average of the squared gradients, known as the second moment v_t:

Where:

• v_t is the second moment estimate (running average of squared gradients),
• β_2 is the decay rate for the second moment (typically set to 0.999).

This second moment captures the magnitude of the gradients and ensures that the learning rate for each parameter is adjusted based on how large or small the gradient has been in recent iterations. By accounting for the squared gradients, Adam effectively scales down large gradient updates and prevents exploding gradients from destabilizing the optimization process.

4. Bias Correction

At the beginning of training, the first and second moment estimates are initialized to zero, which leads to biases in these moments, especially during the early stages of training. To correct this, Adam applies bias-corrected estimates:

These corrections ensure that the estimates of the moments are unbiased, particularly during the first few iterations when the raw moments are still small. This bias correction accelerates the early stages of optimization, allowing for more accurate gradient updates right from the start.

5. Parameter Update Rule

The final update step in Adam uses the bias-corrected moments to adjust the parameters. The update rule for each parameter θ is given by:

Where:

• α is the learning rate,
• m_t is the bias-corrected first moment estimate (mean of the gradient),
• v_t is the bias-corrected second moment estimate (squared gradient),
• ε is a small constant (usually set to \(10^{-8}\)) to prevent division by zero.

This update rule combines the first moment (gradient direction) and second moment (gradient magnitude) to compute adaptive learning rates for each parameter. The learning rate is scaled inversely with the second moment estimate, ensuring that large gradient values do not lead to excessively large parameter updates. The presence of ε ensures numerical stability, particularly when the second moment estimate v_t is small.

Advantages of Adam

• Adaptive Learning Rates: Adam adjusts the learning rate for each parameter based on the history of gradients, which is especially useful for problems with large parameter spaces or sparse gradients.
• Efficient Computation: Since Adam only requires first-order gradients and stores a small number of additional parameters (the first and second moments), it is computationally efficient and suitable for large datasets and models.
• Bias Correction: The inclusion of bias correction makes Adam well-behaved during the early stages of optimization when gradients are smaller, allowing it to converge faster and avoid slow starts.
• Robust to Noise: Adam is particularly effective in handling noisy gradients, a common issue in stochastic settings like mini-batch gradient descent.
• Prevents Overshooting: The second moment estimate effectively limits the step size for parameters with large gradients, ensuring stability and preventing oscillations.

Practical Usage and Hyperparameters

• Learning Rate (α): The default value is typically set to 0.001, though it can be tuned depending on the application.
• β₁ and β₂: The default values of β₁ = 0.9 and β₂ = 0.999 generally work well across most problems. β₁ controls the decay rate of the first moment estimate, while β₂ controls the decay rate of the second moment estimate.
• ε: A small constant (e.g., \(10^{-8}\)) ensures numerical stability by preventing division by zero during parameter updates.

Limitations of Adam

• Hyperparameter Sensitivity: Adam can be sensitive to hyperparameters, especially the learning rate.
• Poor Generalization: Some studies have shown that Adam might lead to poorer generalization compared to simpler optimizers like SGD with momentum, as Adam may aggressively adapt learning rates.

Mathematical Proof and Real-World Example Demonstrating Adam’s Efficacy

Introduction

So far, we delved into the mechanics and mathematical foundations of the Adam optimizer. However, Now in this section aims to provide a rigorous mathematical proof illustrating why Adam can effectively navigate complex optimization landscapes, particularly in overcoming local minima. Additionally, we present a real-world example comparing Adam with standard Gradient Descent (GD) to empirically demonstrate Adam’s superiority in escaping local minima.

Mathematical Proof: Adam’s Ability to Overcome Local Minima

Understanding Local Minima in Optimization

In non-convex optimization problems, the objective function f(θ) may contain multiple local minima, saddle points, and flat regions. Traditional optimization algorithms like Gradient Descent (GD) can easily get trapped in these local minima, leading to suboptimal solutions. Adam’s adaptive nature and momentum-based updates provide mechanisms that help traverse these challenging terrains more effectively.

Key Properties of Adam Facilitating Escape from Local Minima

1. Adaptive Learning Rates:

Adam adjusts the learning rate for each parameter based on the first and second moments of the gradients. This dynamic scaling allows Adam to make significant progress even in regions where gradients are minimal, potentially enabling escape from shallow local minima.

2. Momentum (First Moment Estimate):

Adam maintains an exponentially decaying average of past gradients (\( m_t \)), which accumulates the direction of consistent gradient descent. This accumulated momentum helps the optimizer maintain velocity in directions where gradients consistently point, thereby overcoming small oscillations or barriers presented by local minima.

3. Second Moment Estimate (Variance Scaling):

The second moment estimate (\( v_t \)) captures the magnitude of gradients, enabling Adam to adjust the learning rate inversely proportional to the root of this variance. This scaling prevents excessive updates in directions with high variance, ensuring stability and preventing the optimizer from overshooting while still allowing movement in low-gradient regions.

Theoretical Insights

Momentum Helps in Overcoming Shallow Local Minima:

In regions where the gradient is shallow (i.e., near a local minimum), the gradient descent step size becomes small due to small gradient magnitudes. However, with momentum, previous gradients influence the current update, allowing the optimizer to maintain a non-zero velocity even when current gradients are small. This accumulated momentum can help the optimizer traverse out of shallow local minima.

Adaptive Learning Rates Enhance Exploration:

The adaptive learning rates allow Adam to make larger updates for parameters with small gradients and smaller updates for parameters with large gradients. This balance enables the optimizer to explore the parameter space more effectively, potentially bypassing local minima that GD might be stuck in due to uniform small step sizes.

Bias Correction Facilitates Early Acceleration:

The bias correction terms and
ensure that the initial steps of optimization are well-scaled, preventing the optimizer from being overly conservative in the early iterations. This initial acceleration can set the optimizer on a trajectory that avoids being trapped in local minima.

Mathematical Illustration

Consider a simple one-dimensional non-convex function with multiple minima:

This function has a local minimum and a global minimum. Let’s analyze how Adam and GD behave when optimizing this function starting from an initial parameter value near the local minimum.

Function Analysis:

The gradient of the function is:

Solving for the local minima and critical points, we find the optimizer can get stuck in local minima without momentum-based escape mechanisms.

Real-World Example: Comparing Adam and Gradient Descent

Problem Setup:

We aim to fit a linear model:
, where and are the weights and bias, respectively.

Dataset: We use the synthetic dataset:
,

Loss Function:

The goal is to minimize the Mean Squared Error (MSE) loss function:

Gradient Descent:

The gradients with respect to and are:

• 
• 

Adam Optimizer:

Adam uses adaptive learning rates and momentum. The update process involves:

• First moment estimate:
• Second moment estimate:
• Bias correction for first and second moments: ,
• Update rule: ,

First Iteration:

Gradient Descent:

• Error:
• Gradient: ,
• Update: ,

Adam:

• First moment estimates: ,
• Second moment estimates: ,
• Bias-corrected first and second moments: , , ,
• Update: ,

Subsequent Iterations:

Iteration 2:

Gradient Descent:

• Error:
• Gradient: ,
• Update: ,

Adam:

• First moment estimates: ,
• Second moment estimates: ,
• Bias-corrected first and second moments: , , ,
• Update: ,

Iteration 3:

Gradient Descent:

• Error:
• Gradient: ,
• Update: ,

Adam:

• First moment estimates: ,
• Second moment estimates: ,
• Bias-corrected first and second moments: , , ,
• Update: ,

Iteration 4:

Gradient Descent:

• Error:
• Gradient: ,
• Update: ,

Adam:

• First moment estimates: ,
• Second moment estimates: ,
• Bias-corrected first and second moments: , , ,
• Update: ,

Iteration 5:

Gradient Descent:

• Error:
• Gradient: ,
• Update: ,

Adam:

• First moment estimates: ,
• Second moment estimates: ,
• Bias-corrected first and second moments: , , ,
• Update: ,

Final Results after 5 Iterations:

Gradient Descent: ,

Adam: ,

Summary:

Gradient Descent makes larger updates initially, which may cause overshooting and oscillations. Adam adapts the learning rate and uses momentum to make more stable and smaller updates, leading to smoother convergence.

Gradient Descent (GD) and Adam Comparison in Formula

Here are the steps for both GD and Adam:

GD Update Rule:

Adam Update Rules:

Gradient Descent (GD) Weight Update Rule

In vanilla gradient descent, the update rule for the weights w is based purely on the gradient of the loss function L with respect to the weights.

The update rule is:

Where:

• is the weight at time step .
• is the learning rate (a fixed step size).
• is the gradient of the loss function with respect to the weights .

In another words : Adam Optimizer Weight Update Rule

The Adam optimizer combines momentum (via the first moment estimate) and adaptive learning rates (via the second moment estimate) to update the weights more effectively.

Steps to Update the Weight Using Adam:

1. Compute the gradient of the loss function with respect to the weight:
   
   
2. Update biased first moment estimate (this keeps track of the momentum of the gradients):
   
   
3. Update biased second moment estimate (this keeps track of the squared gradients):
   
   
4. Bias correction for the first moment estimate:
   
   
5. Bias correction for the second moment estimate:
   
   
6. Update the weights using both the first and second moment estimates:
   
   

Summary of the Differences

• Gradient Descent (GD) simply uses the gradient to update the weights: .
• Adam uses both the gradient and running averages of past gradients (momentum) and squared gradients (adaptive learning rates) to update the weights, making the process more robust, especially in cases where gradients vary a lot.

Further Example: Rosenbrock Function

For a more complex scenario, we used the Rosenbrock function, a well-known test function for optimization algorithms:

Here, the path to the global minimum requires navigating through steep walls and narrow valleys. Adam outperforms Gradient Descent by converging faster, with fewer oscillations, and more stability in complex optimization landscapes.

Conclusion

The mathematical proof and real-world examples demonstrate Adam’s ability to overcome local minima through its adaptive learning rates and momentum-based updates. These attributes make Adam one of the most effective optimizers for non-convex problems, where local minima and saddle points are common.

How to Use the Adam Optimizer in TensorFlow/Keras and MLX for Apple Silicon

In this article, we’ll explore using the Adam optimizer in deep learning projects with both TensorFlow/Keras and MLX—a machine learning framework optimized for Apple Silicon. We’ll discuss when to use Adam, provide code examples, and highlight the differences between each framework, focusing on the specific capabilities of MLX for Apple Silicon.

What Is MLX on Apple Silicon?

MLX is a machine learning framework built by Apple to optimize machine learning workflows on their M1 and M2 chips. One of the key features of MLX is its use of unified memory architecture, which allows arrays to reside in shared memory. This reduces the need to transfer data between the CPU and GPU, enhancing performance by minimizing data transfer overhead.

MLX provides an API inspired by NumPy and PyTorch, optimized for Apple’s hardware. It supports lazy computation, meaning computations are only performed when necessary, improving performance efficiency. The framework also supports automatic differentiation and vectorization to further optimize deep learning models. This makes it a powerful tool for tasks like transformer models, image generation (e.g., Stable Diffusion), and large-scale text processing on Apple Silicon.

When and Why to Use the Adam Optimizer?

The Adam optimizer is widely used because it dynamically adjusts the learning rate for each parameter, making it well-suited for noisy gradients and sparse datasets. Adam is generally a good choice when:

• Working with complex architectures such as transformers, CNNs, and RNNs.
• Handling sparse data: Adam is effective at adjusting the learning rates of individual parameters, which is helpful when gradients are sparse.
• Exploring new models: If you are unsure which optimizer to choose, Adam is a solid default due to its general efficiency and ease of use.

Using Adam Optimizer in TensorFlow/Keras

In TensorFlow/Keras, integrating the Adam optimizer is straightforward. Here’s an example using Adam in a binary classification model:

import tensorflow as tf
from tensorflow.keras.models import Sequential
from tensorflow.keras.layers import Dense
from tensorflow.keras.optimizers import Adam
from sklearn.datasets import make_classification
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler

# Generate a synthetic dataset
X, y = make_classification(n_samples=1000, n_features=20, n_classes=2, random_state=42)
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)

# Normalize the data
scaler = StandardScaler()
X_train = scaler.fit_transform(X_train)
X_test = scaler.transform(X_test)

# Build a neural network model
model = Sequential([
    Dense(64, input_dim=20, activation='relu'),
    Dense(32, activation='relu'),
    Dense(1, activation='sigmoid')  # Output layer
])

# Add Adam optimizer
adam = Adam(learning_rate=0.001)  # Add the Adam optimizer with a learning rate of 0.001
model.compile(optimizer=adam, loss='binary_crossentropy', metrics=['accuracy'])

# Train the model
history = model.fit(X_train, y_train, epochs=50, batch_size=32, validation_data=(X_test, y_test))

# Evaluate the model
loss, accuracy = model.evaluate(X_test, y_test)
print(f"Test Loss: {loss:.4f}, Test Accuracy: {accuracy:.4f}")

In TensorFlow, you add the Adam optimizer by calling Adam(learning_rate=0.001) when compiling the model. This optimizer dynamically adjusts the learning rate during training, making it highly effective for many tasks.

Using Adam Optimizer in MLX for Apple Silicon

For users working on Apple Silicon, MLX offers an efficient way to use Adam with performance optimizations tailored to Apple’s hardware. Below is the correct method to implement Adam in MLX.

MLX Example Code (Sentiment Analysis with LSTM)

import mlx
from mlx.nn import Sequential, Embedding, LSTM, Linear
from mlx.optimizers import Adam
 
# Define the model architecture
model = Sequential(
    Embedding(vocab_size, embedding_dim),
    LSTM(embedding_dim, hidden_dim, num_layers=2, batch_first=True),
    Linear(hidden_dim, num_classes)
)
 
# Define the loss function and optimizer
criterion = mlx.nn.CrossEntropyLoss()
optimizer = Adam(model.parameters(), lr=0.001)
 
# Training loop
for epoch in range(num_epochs):
    for batch in data_loader:
        inputs, labels = batch
        optimizer.zero_grad()
        outputs = model(inputs)
        loss = criterion(outputs, labels)
        loss.backward()
        optimizer.step()

Key Features of MLX

• Unified Memory: This feature allows arrays to exist in shared memory between CPU and GPU, making the framework faster when switching between devices.
• Lazy Computation: MLX only computes values when necessary, making it more efficient.
• Automatic Differentiation: This feature simplifies the process of calculating gradients for backpropagation during model training.

PyTorch Code Example Using Adam Optimizer

Below is a simple example of how to use the Adam optimizer in PyTorch for training a neural network:

import torch
import torch.nn as nn
import torch.optim as optim
import torchvision
import torchvision.transforms as transforms
from torch.utils.data import DataLoader

# Define a simple feedforward neural network
class SimpleNN(nn.Module):
    def __init__(self, input_size, hidden_size, num_classes):
        super(SimpleNN, self).__init__()
        self.fc1 = nn.Linear(input_size, hidden_size)  # Fully connected layer 1
        self.relu = nn.ReLU()  # ReLU activation
        self.fc2 = nn.Linear(hidden_size, num_classes)  # Fully connected layer 2

    def forward(self, x):
        x = self.fc1(x)
        x = self.relu(x)
        x = self.fc2(x)
        return x

# Hyperparameters
input_size = 784  # MNIST images are 28x28 pixels, flattened into 784 inputs
hidden_size = 128
num_classes = 10  # MNIST has 10 classes (digits 0-9)
num_epochs = 5
batch_size = 64
learning_rate = 0.001

# Load the MNIST dataset
transform = transforms.Compose([transforms.ToTensor(), transforms.Normalize((0.5,), (0.5,))])
train_dataset = torchvision.datasets.MNIST(root='./data', train=True, transform=transform, download=True)
train_loader = DataLoader(dataset=train_dataset, batch_size=batch_size, shuffle=True)

# Instantiate the neural network, loss function, and Adam optimizer
model = SimpleNN(input_size, hidden_size, num_classes)
criterion = nn.CrossEntropyLoss()  # Cross entropy loss for classification
optimizer = optim.Adam(model.parameters(), lr=learning_rate)

# Training loop
for epoch in range(num_epochs):
    for i, (images, labels) in enumerate(train_loader):
        # Reshape images to (batch_size, input_size)
        images = images.view(-1, 28*28)
        
        # Forward pass
        outputs = model(images)
        loss = criterion(outputs, labels)
        
        # Backward pass and optimization
        optimizer.zero_grad()  # Zero the gradient buffers
        loss.backward()  # Compute gradients
        optimizer.step()  # Update weights using Adam
        
        # Print loss every 100 batches
        if (i+1) % 100 == 0:
            print(f'Epoch [{epoch+1}/{num_epochs}], Step [{i+1}/{len(train_loader)}], Loss: {loss.item():.4f}')

print("Training complete.")

This code demonstrates how to use the Adam optimizer to train a neural network on the MNIST dataset using PyTorch.

How to Check if Adam Is Performing Well

To monitor how well Adam is performing:

1. Monitor Loss Over Time: The loss should decrease over training epochs.
   • In TensorFlow, you can track this via history.history['loss'].
   • In MLX, manually log the loss during training.
   • In PyTorch, print the loss during the training loop (e.g., print(f'Epoch [{epoch+1}], Loss: {loss.item()}')).
2. Track Validation Metrics: Use validation loss and accuracy to check if the model is generalizing well to unseen data.
   • In TensorFlow, this can be monitored with history.history['val_accuracy'].
   • In MLX, manually evaluate the model’s performance on validation data.
   • In PyTorch, evaluate on validation data within the training loop by turning off gradients with torch.no_grad() and compute metrics (e.g., loss, accuracy) on the validation set.
3. Adjust Learning Rate: If your model’s performance is not improving or is unstable, try adjusting Adam’s learning rate.
   • Typical values range from 0.001 to 0.0001.
   • In TensorFlow, you can set this when defining the Adam optimizer (e.g., Adam(learning_rate=0.001)).
   • In PyTorch, this is done when instantiating the Adam optimizer (e.g., optimizer = optim.Adam(model.parameters(), lr=0.001)).

Conclusion

The Adam optimizer is a highly efficient and flexible tool for both TensorFlow/Keras, PyTorch, and MLX. With PyTorch, you can leverage dynamic computation graphs and automatic differentiation, offering you flexibility and control over model training. In MLX’s tailored optimizations for Apple Silicon, you can leverage the power of unified memory and lazy computation to maximize performance when training your deep learning models on M1 and M2 chips. Whether you’re working on simple neural networks or advanced transformers, Adam offers dynamic learning rate adaptation, making it an excellent choice for a wide range of machine learning tasks.