Activation functions are essential components in neural networks. Without them, neural networks would be restricted to learning only linear relationships. This limitation would severely affect their performance on most real-world tasks, which involve nonlinear patterns. One of the most influential and widely adopted activation functions in deep learning is the Rectified Linear Unit, or ReLU. This part of the guide explores the fundamental ideas behind activation functions, how they work, and why ReLU plays such a pivotal role in deep neural networks.
Neural Networks and the Need for Activation Functions
Artificial neural networks are computational models inspired by the human brain. They consist of layers of interconnected nodes, also known as neurons. Each node receives input, processes it, and passes the output to the next layer in the network. A deep neural network includes multiple hidden layers between the input and output, enabling it to learn and represent complex patterns in data. These layers alone, however, are not sufficient to allow the network to make sophisticated decisions. What adds the critical layer of intelligence and decision-making ability to these networks are activation functions. Activation functions introduce non-linearities into the model, which allows neural networks to learn patterns that cannot be represented through simple linear transformations. Without them, a network would essentially behave like a single-layer perceptron, regardless of how many layers it had, limiting its capacity to model real-world phenomena.
What Is an Activation Function
An activation function is a mathematical function that determines the output of a neural network node. It takes the input signal produced by a neuron (often a linear combination of its inputs and weights), applies a transformation to that signal, and passes the result to the next layer in the network. This process is critical because it decides which information is carried forward in the network and which is discarded. There are various types of activation functions. Some, like the sigmoid and hyperbolic tangent (tanh), produce smooth outputs and are differentiable across their entire domain. Others, like ReLU, are simpler and more efficient. What all activation functions have in common is that they help the network model complex, nonlinear relationships between variables. For example, in a banking application, the relationship between the number of children and monthly spending may not be linear. Activation functions allow the network to model such irregularities accurately.
The Importance of Non-Linearity in Neural Networks
Non-linearity is crucial because it enables neural networks to go beyond the limitations of linear models. A model based solely on linear transformations can only capture straight-line relationships between variables. This constraint is insufficient for most tasks like image recognition, natural language processing, or financial forecasting, where the data patterns are inherently complex and multidimensional. By introducing non-linear activation functions at various layers, neural networks can approximate any function, a property known as the universal approximation theorem. This means that, given enough layers and the right activation functions, a neural network can theoretically model any distribution of data, no matter how complex. The choice of activation function directly influences how well the model learns these patterns. ReLU is especially effective because it adds non-linearity in a simple yet powerful way, without the computational complexity of functions like sigmoid or tanh.
Introducing the Rectified Linear Unit (ReLU)
The Rectified Linear Unit, commonly known as ReLU, is an activation function that has become the default choice for many deep learning models. The mathematical expression for ReLU is: f(x) = max(0, x). This function outputs the input directly if it is positive; otherwise, it returns zero. The simplicity of this formula is one of its greatest strengths. It is easy to compute, differentiable in most of its domain, and avoids some of the major pitfalls associated with earlier activation functions. For instance, functions like sigmoid and tanh suffer from the vanishing gradient problem, which can stall learning in deep networks. ReLU, by contrast, maintains gradients for positive inputs, enabling faster and more stable training. Its sparse activation, which means that only some neurons are activated at any given time, also improves efficiency and reduces the risk of overfitting.
How ReLU Works in Practice
To understand how ReLU operates, consider a single neuron in a neural network. This neuron receives an input x, which is typically a weighted sum of the outputs from the previous layer. The ReLU activation function is then applied to this input. If x is positive, the neuron outputs x. If x is negative or zero, the output is zero. This behavior can be visualized as a graph that is flat for all negative inputs and diagonal for positive inputs. This creates a piecewise linear function that is not only simple but also highly effective in practice. During training, the ReLU function allows the model to propagate gradients effectively, speeding up convergence. Since the derivative of ReLU is 1 for positive inputs and 0 for negative inputs, it maintains gradient flow where needed and stops it where it is not, which can also help in regularizing the model.
ReLU’s Efficiency Compared to Other Functions
One of the main reasons ReLU has become the default activation function is its computational efficiency. Unlike the sigmoid or tanh functions, which require expensive exponential computations, ReLU only requires a comparison operation. This makes it much faster to evaluate, particularly in large networks where millions of neurons are involved. In practice, this speed can translate into significantly reduced training times and lower computational costs. Furthermore, because ReLU outputs zero for all negative inputs, it naturally results in sparse representations. In other words, a large number of neurons are inactive at any given moment, which reduces the number of operations and helps prevent overfitting by encouraging the network to develop more focused and efficient representations of data.
Why ReLU Became the Default Choice
Before the widespread adoption of ReLU, functions like sigmoid and tanh were commonly used. However, these functions have significant limitations, especially when applied to deep networks. One of the biggest challenges with these earlier functions is the vanishing gradient problem. As gradients are propagated back through the network during training, they can become extremely small, especially for layers close to the input. This slows learning to a crawl and may prevent the network from converging altogether. ReLU largely solves this issue by providing constant gradients for positive inputs, which allows the network to learn much faster. In addition, because of its sparse activations, ReLU also helps in reducing the likelihood of overfitting, making it a more robust choice for many types of problems. These practical advantages have led to ReLU becoming the standard activation function in many deep learning architectures.
Limitations of ReLU and the Dying Neuron Problem
Despite its advantages, ReLU is not without flaws. One well-known issue is the dying neuron problem. This occurs when a neuron’s input becomes negative for all examples in the dataset, causing the neuron to output zero every time. Since the gradient of ReLU is also zero for negative inputs, the neuron stops learning and effectively dies. Once this happens, the weights associated with this neuron are no longer updated during backpropagation, and the neuron becomes inactive permanently. In some models, especially those with poor initialization or high learning rates, a significant number of neurons can die early in training, reducing the model’s capacity and efficiency. To address this problem, several variations of ReLU have been developed, including Leaky ReLU, Parametric ReLU, and Exponential Linear Unit, each of which modifies the function in a way that ensures some gradient is preserved even for negative inputs.
When to Use ReLU in Neural Networks
ReLU is a solid default choice for many deep learning models, especially those involving image processing, natural language processing, and general pattern recognition tasks. Its simplicity, speed, and effectiveness in deep networks make it ideal for architectures like convolutional neural networks (CNNs) and feedforward neural networks. However, there are situations where ReLU might not be the best option. In recurrent neural networks (RNNs), for instance, activation functions like tanh or gated units are often preferred due to their ability to handle sequences and maintain long-term dependencies. Also, in tasks where the dying neuron problem is prevalent or when using very deep networks without batch normalization, variants like Leaky ReLU or PReLU may offer improved performance. Ultimately, the choice of activation function should be guided by experimentation and a clear understanding of the task at hand.
Implementing ReLU in PyTorch: A Practical Guide
In this section, we explore how the Rectified Linear Unit (ReLU) is implemented and used in practice using PyTorch, one of the most popular deep learning frameworks. By the end, you’ll understand how ReLU behaves during forward and backward passes and how to incorporate it into real neural network models.
Using ReLU in PyTorch
PyTorch provides a built-in ReLU class under torch.nn, as well as a functional version in torch.nn.functional. Both are easy to use. Here’s how to apply ReLU in its two forms:
Option 1: Using torch.nn.ReLU
python
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import torch
import torch.nn as nn
relu = nn.ReLU()
# Example input tensor
x = torch.tensor([[-1.0, 0.0, 2.0, -3.0]], requires_grad=True)
# Apply ReLU
output = relu(x)
print(output)
Output:
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tensor([[0., 0., 2., 0.]], grad_fn=<ReluBackward0>)
As expected, all negative values are set to zero, and positive values remain unchanged.
Option 2: Using torch.nn.functional.relu
python
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import torch.nn.functional as F
output = F.relu(x)
print(output)
Both approaches are equivalent in terms of behavior. The choice depends on whether you are using functional layers or building a model class with modules.
ReLU in Forward and Backward Passes
Let’s examine how ReLU behaves during the forward and backward passes in training. We’ll compute a simple loss and perform backpropagation:
python
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# Simple model with ReLU
output = F.relu(x)
loss = output.sum() # Simple sum as a dummy loss
loss.backward()
print(x.grad)
Output:
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tensor([[0., 0., 1., 0.]])
This output shows the gradient of the loss with respect to the input x. The gradient is propagated only through the positive input (2.0 in this case), while negative or zero values receive a gradient of 0. This behavior illustrates why neurons with negative inputs can “die” (i.e., stop learning) during training.
Building a Neural Network with ReLU
Here’s how to use ReLU in a complete neural network using PyTorch’s nn.Module:
python
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class SimpleNN(nn.Module):
def __init__(self):
super(SimpleNN, self).__init__()
self.fc1 = nn.Linear(4, 8)
self.relu = nn.ReLU()
self.fc2 = nn.Linear(8, 1)
def forward(self, x):
x = self.fc1(x)
x = self.relu(x)
x = self.fc2(x)
return x
# Example usage
model = SimpleNN()
sample_input = torch.randn(1, 4)
output = model(sample_input)
print(output)
In this example, ReLU is applied after the first linear layer to introduce non-linearity before the output layer.
Visualizing ReLU’s Behavior
Understanding the shape of the ReLU function helps reinforce how it transforms data:
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import matplotlib.pyplot as plt
import numpy as np
x = np.linspace(-10, 10, 100)
y = np.maximum(0, x)
plt.plot(x, y)
plt.title(“ReLU Activation Function”)
plt.xlabel(“Input”)
plt.ylabel(“Output”)
plt.grid(True)
plt.show()
This plot clearly shows ReLU’s piecewise nature: it’s flat (zero) for all negative values and linear for positive ones.
Handling ReLU’s Limitations
To overcome the dying ReLU problem in practice, you might try:
- Better weight initialization (e.g., He initialization)
- Lower learning rates to prevent neurons from dying
- Using Leaky ReLU, which allows a small, non-zero gradient for negative inputs:
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leaky_relu = nn.LeakyReLU(negative_slope=0.01)
output = leaky_relu(x)
This allows gradients to flow even when the input is negative, which can keep neurons active.
Exploring ReLU Variants: Overcoming the Limitations of ReLU
While the Rectified Linear Unit (ReLU) has become the standard activation function in deep learning, it is not perfect. One of the most significant limitations is the “dying ReLU” problem, where neurons become inactive and stop learning. To address this issue, researchers have proposed several variants of ReLU, including Leaky ReLU, Parametric ReLU (PReLU), Exponential Linear Unit (ELU), and others. These modified functions aim to preserve the benefits of ReLU while mitigating its drawbacks.
Leaky ReLU: Allowing a Small Gradient for Negative Inputs
Leaky ReLU addresses the dying ReLU problem by allowing a small, non-zero slope for negative inputs instead of outputting zero.
Formula:
f(x)={x,if x≥0αx,if x<0f(x) = \begin{cases} x, & \text{if } x \geq 0 \\ \alpha x, & \text{if } x < 0 \end{cases}f(x)={x,αx,if x≥0if x<0
Where α\alphaα is a small constant, typically 0.01.
PyTorch Implementation:
python
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import torch.nn as nn
leaky_relu = nn.LeakyReLU(negative_slope=0.01)
x = torch.tensor([[-3.0, -1.0, 0.0, 2.0]])
output = leaky_relu(x)
print(output)
Advantages:
- Prevents neurons from dying.
- Allows learning even when inputs are negative.
Disadvantages:
- The slope α\alphaα is fixed and may not be optimal for all tasks.
Parametric ReLU (PReLU): Learnable Slope for Negative Inputs
PReLU extends Leaky ReLU by making the negative slope a learnable parameter, allowing the network to adapt during training.
Formula:
f(x)={x,if x≥0ax,if x<0f(x) = \begin{cases} x, & \text{if } x \geq 0 \\ a x, & \text{if } x < 0 \end{cases}f(x)={x,ax,if x≥0if x<0
Where aaa is a parameter learned during training.
PyTorch Implementation:
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prelu = nn.PReLU()
output = prelu(x)
print(output)
Advantages:
- Adapts to data.
- Avoids manual tuning of negative slope.
Disadvantages:
- Adds extra parameters.
- Slightly more computationally expensive.
Exponential Linear Unit (ELU): Smooth and Differentiable Everywhere
ELU goes beyond piecewise linear functions and introduces a smooth exponential curve for negative values.
Formula:
f(x)={x,if x≥0α(ex−1),if x<0f(x) = \begin{cases} x, & \text{if } x \geq 0 \\ \alpha (e^x – 1), & \text{if } x < 0 \end{cases}f(x)={x,α(ex−1),if x≥0if x<0
Where α\alphaα is typically set to 1.0.
PyTorch Implementation:
python
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elu = nn.ELU(alpha=1.0)
output = elu(x)
print(output)
Advantages:
- Smooth output helps with gradient flow.
- Outputs can be negative, helping the mean activation stay closer to zero, which can speed up learning.
Disadvantages:
- More computationally expensive than ReLU.
- Introduces non-linearity that may not be necessary in some tasks.
ReLU6: Capped ReLU for Mobile and Embedded Devices
ReLU6 is a variant of ReLU used primarily in mobile networks like MobileNet. It caps the output at 6 to improve quantization robustness.
Formula:
f(x)=min(max(0,x),6)f(x) = \min(\max(0, x), 6)f(x)=min(max(0,x),6)
PyTorch Implementation:
python
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relu6 = nn.ReLU6()
output = relu6(x)
print(output)
Use Case:
- Efficient and stable for low-precision hardware (e.g., mobile devices, edge computing).
- Use ReLU when training deep CNNs or fully connected networks—default, fast, and effective.
- Use Leaky ReLU or PReLU if you’re seeing many dead neurons during training.
- Use ELU if you want smooth activation with centered outputs and can afford extra computation.
- Use ReLU6 for mobile or embedded applications where quantization is critical.
Choosing the Right Activation Function
While ReLU is a powerful default, understanding its variants helps fine-tune your model’s behavior. Each variant offers specific benefits tailored to different tasks and architectures. When training neural networks, it’s always a good idea to start with ReLU, monitor your model’s behavior, and experiment with alternatives like Leaky ReLU or ELU if performance plateaus or training becomes unstable.
Advanced Activation Functions in Deep Learning: Beyond ReLU
As deep learning evolves, so do its building blocks. Activation functions, which have long played a fundamental role in neural networks, have also advanced significantly to meet the demands of modern architectures. While ReLU and its variants still dominate many applications due to their simplicity and effectiveness, cutting-edge models—particularly those in natural language processing (NLP) and computer vision—have begun to rely on more sophisticated activation functions that offer smoother gradients, better generalization, or improved performance on large-scale datasets.
This part of the guide explores the landscape of modern activation functions beyond ReLU, including Swish and GELU, and their applications in architectures like Transformers, LSTMs, and large-scale foundation models. It also examines the unique activation needs of these models, their empirical performance, and the mathematical intuition that explains why some functions work better than others in specific contexts.
The Evolution from Simplicity to Sophistication
The journey of activation functions began with step functions and moved to sigmoid and tanh. These early choices introduced non-linearity and allowed neural networks to model complex relationships. However, they came with significant drawbacks such as the vanishing gradient problem, which severely limited the depth and effectiveness of networks. The introduction of ReLU marked a major breakthrough by solving many of these issues while offering a simple, efficient, and sparsely active alternative. ReLU became the default choice for convolutional networks, fully connected layers, and many basic models.
Yet as models grew deeper and more complex, especially with the rise of sequence-based architectures and transformer models, new challenges emerged. These architectures required activation functions that were not just non-linear but also smooth, differentiable across their entire domain, and more expressive in subtle ways. Researchers began to experiment with novel formulations that could offer performance gains in specific contexts. Two of the most influential developments in this area are the Swish and GELU activation functions.
Swish: A Self-Gated Smooth Function
Swish is a smooth, non-monotonic activation function introduced by researchers at Google. Unlike ReLU, which is piecewise linear, Swish is defined by the function f(x) = x * sigmoid(x). This formulation gives it a smooth curve with a slight negative slope for negative inputs and a gentle upward curve for positive values. The key feature of Swish is that it retains small negative values instead of outright discarding them, as ReLU does. This allows gradient information to flow more easily during backpropagation, especially in deep networks.
The smoothness of Swish makes it particularly effective in deep architectures where sharp transitions in the activation function could introduce optimization difficulties. The non-monotonic nature of Swish also means it can respond differently to different input magnitudes, leading to richer representations and a potential improvement in generalization. Swish has been shown to outperform ReLU in several image classification tasks, especially in deeper convolutional networks like EfficientNet. It is also often preferred in architectures where regularization and gradient flow are critical for training success.
One of the appealing aspects of Swish is its self-gating property. By multiplying the input by its own sigmoid, Swish behaves like a learned gate that modulates the input dynamically based on its magnitude. This is conceptually similar to attention mechanisms, which adaptively control how much each component contributes to the output. As a result, Swish can be seen as a lightweight, neuron-wise gating mechanism that enhances the representational power of the network.
Swish is differentiable everywhere, which also makes it favorable for gradient-based optimization. In contrast, ReLU is not differentiable at zero, and its flat negative region can halt gradient flow. These factors contribute to Swish’s popularity in modern deep learning research and its use in production-grade models.
GELU: Gaussian Error Linear Unit
The Gaussian Error Linear Unit, or GELU, was popularized by the Transformer architecture, particularly in models like BERT and GPT. GELU is defined mathematically as f(x) = x * Φ(x), where Φ(x) is the cumulative distribution function (CDF) of the standard normal distribution. In practice, this is often approximated for computational efficiency using a combination of tanh and polynomial terms. What distinguishes GELU is that it probabilistically weights inputs based on how likely they are to be positive under a Gaussian distribution. Rather than applying a hard threshold, like ReLU, or a smooth squashing function, like sigmoid, GELU softly gates inputs based on their position on the normal curve.
The result is a smooth, differentiable function that is more nuanced than ReLU or Swish. GELU has become the activation function of choice in many large language models because it performs particularly well on tasks involving long-range dependencies and complex, hierarchical structures. Its probabilistic gating nature may contribute to more stable gradient propagation across long sequences, which is a common challenge in transformer-based models. GELU introduces an elegant blend of mathematical rigor and empirical performance, offering a refined control over neuron activation without the sharp cutoffs of ReLU or the exponential behavior of ELU.
In transformer models, GELU is used immediately after the first linear layer in each feedforward block. This placement is strategic: the first linear layer expands the representation space, and GELU ensures that information flows smoothly and meaningfully through this higher-dimensional space before being projected back down. The function’s shape helps preserve important features while dampening noise, contributing to the model’s capacity to capture subtle patterns in language or images.
Moreover, GELU’s soft behavior for negative inputs—rather than outright suppression—supports a richer and more diverse activation pattern across neurons. This is important for models that are trained on vast and diverse datasets, such as web-scale corpora or high-resolution images. A network that suppresses too much input, as ReLU can do, might lose important nuances. GELU maintains a balance between sensitivity and stability, allowing neurons to activate just enough without overfitting or causing instability.
Activation Functions in LSTMs and Recurrent Networks
While GELU and Swish have gained traction in transformer models, recurrent architectures such as Long Short-Term Memory networks (LSTMs) and Gated Recurrent Units (GRUs) traditionally use different activation functions. These architectures rely heavily on sigmoid and tanh functions, not because they are the most efficient, but because their mathematical properties align well with the gating mechanisms inherent to RNNs.
In LSTMs, the sigmoid function is used to control the input, forget, and output gates. The sigmoid’s bounded output between 0 and 1 makes it ideal for gating behavior, acting like a soft switch that determines how much information should be passed through each gate. The tanh function, with outputs ranging from -1 to 1, is used to squash the cell state values and outputs, ensuring that the model remains numerically stable across time steps.
These functions are not without their drawbacks. The vanishing gradient problem is especially acute in RNNs because of repeated multiplications over long sequences. That’s why LSTMs and GRUs were developed in the first place—to mitigate these issues with carefully designed gating mechanisms. Although ReLU has been experimented with in some RNN variants, it is rarely used in standard LSTMs because of its unbounded nature, which can lead to exploding gradients and instability over time.
Nevertheless, research continues into using alternative activations in recurrent networks. Leaky ReLU, ELU, and even Swish have been tested in gated architectures, but the benefits remain task-dependent and often come with trade-offs in stability or interpretability. As of now, sigmoid and tanh remain the standard choices in recurrent models, but as training techniques and optimization methods improve, there may be greater room to experiment with more modern alternatives.
Activation Functions in Transformers
The transformer architecture introduced in the paper “Attention Is All You Need” revolutionized deep learning by replacing recurrence with self-attention mechanisms. However, activation functions remain an essential part of transformers. Each transformer block includes a feedforward neural network that typically consists of two linear layers with a non-linear activation function between them. In many early transformer models, ReLU was used. However, GELU quickly became the preferred choice due to its smoother gradient behavior and better empirical performance.
GELU helps transformers deal with large inputs and long sequences more effectively. In natural language processing, where input sequences can vary in length and complexity, GELU’s smooth activation function helps reduce abrupt changes in gradients, contributing to more stable and reliable learning. Moreover, transformers often operate with extremely large batch sizes and learning rates during pretraining. Under such conditions, the robustness of the activation function becomes critical. GELU’s probabilistic nature offers a form of regularization that may help the model generalize better across diverse samples.
In vision transformers, which apply the transformer architecture to image patches rather than word tokens, GELU continues to be the dominant activation function. The benefits observed in NLP translate well to vision, where spatial relationships and texture patterns also require stable, nuanced feature extraction. Activation functions like Swish and GELU enable these models to learn subtle correlations that would be lost with harsher functions like ReLU.
Practical Considerations and Performance Trends
While newer activation functions offer compelling advantages, they also come with trade-offs. GELU and Swish are more computationally expensive than ReLU. This cost may be negligible in small to medium models but becomes significant in massive architectures with billions of parameters. In some production environments, practitioners still prefer ReLU or Leaky ReLU for their simplicity and speed, especially when inference latency is a concern.
Nevertheless, the trend in research and large-scale applications clearly favors more sophisticated activation functions. Nearly all state-of-the-art models in NLP and vision use GELU or its variations. BERT, GPT-2, GPT-3, T5, ViT (Vision Transformer), and others rely heavily on these modern activations. The performance boost they provide in terms of training speed, stability, and generalization is often worth the extra cost in computation.
The choice of activation function also depends on the optimizer used, the batch size, learning rate schedule, and initialization strategy. There is no universally optimal activation function; rather, the best choice is context-dependent. As networks continue to grow in size and complexity, the interplay between activation functions and other components of the architecture becomes more important. Fine-tuning this balance is part of the art and science of building high-performing neural networks.
The Future of Activation Functions
As deep learning models continue to evolve, activation functions will likely continue to diversify. There is ongoing research into learnable activations, where the shape of the function itself is learned from data. These include dynamic functions that adapt over time or across layers, such as Adaptive Piecewise Linear Units or gated activation functions that resemble attention mechanisms in their responsiveness.
Another area of exploration is the use of normalization-aware activations, where the activation function interacts closely with normalization layers like BatchNorm or LayerNorm to improve stability and convergence. The boundaries between architectural elements are becoming increasingly blurred, and activation functions may become more integrated with other components, such as routing, attention, or memory mechanisms.
What remains constant, however, is the essential role of activation functions in shaping how a neural network transforms and interprets data. From the simplicity of ReLU to the mathematical elegance of GELU, activation functions are silent yet powerful contributors to the success of deep learning. They modulate information, control gradient flow, and ultimately define the expressive capacity of neural networks.
Conclusion
The rise of Swish, GELU, and other advanced activation functions reflects the deepening understanding of how non-linearity affects training dynamics and model performance. As the complexity and scale of deep learning models continue to increase, so does the importance of choosing the right activation function. While ReLU remains a reliable choice for many applications, modern models benefit significantly from smoother and more expressive alternatives like Swish and GELU. These functions improve gradient flow, enable richer feature extraction, and lead to more stable training, especially in transformer-based architectures and large-scale pretraining scenarios.
In summary, activation functions are no longer just mathematical conveniences; they are critical design decisions that influence the entire learning process. Whether you are building a lightweight CNN for mobile applications or training a billion-parameter transformer for language modeling, understanding the strengths and weaknesses of different activation functions can make the difference between a good model and a great one.