What is an Activation Function? Complete Guide 2026
- Muiz As-Siddeeqi
- 2 days ago
- 37 min read
Every breakthrough in artificial intelligence starts with a simple mathematical decision. When your smartphone recognizes your face, when Netflix recommends your next binge, when ChatGPT responds to your question—activation functions are working behind the scenes, deciding which neurons fire and which stay silent. These tiny mathematical gates have transformed machine learning from theoretical curiosity into the trillion-dollar industry reshaping our world.
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TL;DR
Activation functions introduce non-linearity into neural networks, enabling them to learn complex patterns beyond simple linear relationships
ReLU (Rectified Linear Unit) became the dominant function after 2012, solving the vanishing gradient problem that plagued earlier networks
Modern transformers like BERT and GPT use advanced functions like GELU and SiLU for smoother optimization and better performance
Over 400 different activation functions exist as of 2024, each optimized for specific tasks and architectures
The choice of activation function directly impacts training speed, model accuracy, and computational efficiency
Current research focuses on learnable and adaptive activation functions that optimize themselves during training
An activation function is a mathematical operation applied to each neuron in a neural network that determines whether and how strongly that neuron should activate. It introduces non-linearity, transforming weighted inputs into outputs that enable the network to learn complex patterns. Without activation functions, neural networks would collapse into simple linear models regardless of depth, unable to solve real-world problems like image recognition or language understanding.
Table of Contents
Background: The Evolution of Activation Functions
Neural networks emerged from attempts to mimic how biological neurons fire in the human brain. In 1943, Warren McCulloch and Walter Pitts created the first mathematical model of a neuron, but it used a simple binary threshold—either on or off. This worked for basic logic gates but failed spectacularly for complex real-world problems.
The sigmoid function entered the scene in the 1980s during the first neural network boom. Researchers loved it because its smooth S-curve mimicked biological activation patterns and produced outputs between 0 and 1, perfect for representing probabilities. The hyperbolic tangent (tanh) followed, offering similar properties but centering outputs around zero.
But these early functions hit a wall. As networks grew deeper—adding more layers to capture more complex patterns—something terrible happened. Gradients shrank to almost nothing during backpropagation, the learning process that updates network weights. Training ground to a halt. This vanishing gradient problem killed the first deep learning winter in the 1990s (DigitalOcean, 2025).
The field languished until a deceptively simple idea changed everything. In 1975, Kunihiko Fukushima used a rectification function in visual pattern recognition experiments, though it went largely unnoticed (Wikipedia, 2025). Decades later, in 2010, Vinod Nair and Geoffrey Hinton published "Rectified Linear Units Improve Restricted Boltzmann Machines," introducing ReLU to the modern deep learning community (Nair & Hinton, 2010). The function was almost embarrassingly simple: if the input is positive, pass it through; if negative, output zero.
ReLU solved the vanishing gradient problem. When AlexNet won the 2012 ImageNet competition using ReLU activation, reducing error rates from 26.2% to 15.3%, the deep learning revolution exploded (Wikipedia, 2025). Suddenly, networks could have 8 layers, then 152 layers with ResNet, then thousands.
Today, we've cataloged over 400 activation functions, documented in a comprehensive 2024 survey by Vladimír Kunc and Jiří Kléma (arXiv, February 2024). Researchers now design functions specifically for transformers, convolutional networks, or recurrent architectures. Some functions learn and adapt during training. The field keeps evolving because even small improvements in activation functions can mean millions of dollars in computational savings or breakthrough performance on critical tasks.
Mathematical Foundation: How Activation Functions Work
Understanding activation functions requires peeling back the layers of how neural networks actually compute.
A single neuron performs two operations. First, it calculates a weighted sum of its inputs plus a bias term:
z = (w₁ × x₁) + (w₂ × x₂) + ... + (wₙ × xₙ) + bHere, x values are inputs, w values are weights (learned parameters), and b is a bias term. This weighted sum z is called the pre-activation value.
Second, the neuron applies an activation function to this pre-activation:
a = φ(z)Where φ (phi) represents the activation function and a is the neuron's final output.
Without activation functions, stacking multiple layers would be pointless. Two linear transformations combined equal one linear transformation. A network with 100 layers would mathematically collapse into a single layer. You'd have an expensive way to do simple linear regression.
Activation functions introduce non-linearity. They bend, twist, and fold the input space, allowing networks to learn decision boundaries of arbitrary complexity. A network with ReLU activations can approximate any continuous function, given enough neurons—a property mathematicians call universal approximation.
The mathematical requirements for an activation function are straightforward but crucial (Georgia Tech OMSCS, January 2024):
Non-linearity: The function must be non-linear to enable complex pattern learning
Differentiability: It must be differentiable (or at least sub-differentiable) for gradient-based optimization
Computationally efficient: Fast to compute during both forward and backward passes
Range properties: The output range affects training dynamics and numerical stability
The derivative of the activation function matters enormously during training. During backpropagation, the network calculates how much to adjust each weight by multiplying gradients backward through layers. If activation function derivatives are consistently small (less than 1), gradients shrink exponentially as they travel backward—the vanishing gradient problem. If derivatives are consistently large (greater than 1), gradients explode.
This mathematical reality explains why activation function choice isn't trivial. It fundamentally shapes whether your network can learn at all.
The Vanishing Gradient Problem
The vanishing gradient problem nearly killed deep learning before it really began. Understanding why requires seeing what happens during backpropagation.
When a neural network makes a prediction, it calculates a loss—how wrong it was. To improve, it needs to adjust weights throughout the network. Backpropagation computes these adjustments by applying the chain rule of calculus, multiplying derivatives layer by layer working backward from the output.
Here's where sigmoid and tanh create disaster. The sigmoid function maps any input to values between 0 and 1. Its derivative ranges from 0 to 0.25, peaking at 0.25 when the input equals zero (KDnuggets, June 2023). The tanh function's derivative peaks at 1.0 but rapidly drops to near zero for large positive or negative inputs (Baeldung, February 2025).
Consider a 10-layer network using sigmoid activation. During backpropagation, gradients multiply by the activation derivative at each layer. Even in the best case, you're multiplying 0.25 ten times: 0.25^10 = 0.0000009537. The gradient reaching early layers essentially vanishes to zero.
This means early layers—the ones learning fundamental features—stop updating their weights. The network can't learn low-level patterns that later layers build upon. Training slows to a crawl or stops entirely (DigitalOcean, June 2025).
The mathematics are unforgiving. When activation derivatives stay below 1, repeated multiplication drives gradients toward zero exponentially. The deeper the network, the worse the problem. This mathematical barrier prevented researchers from building the deep architectures needed for complex tasks.
Researchers tried various solutions before ReLU:
Careful weight initialization: Xavier initialization and He initialization scaled initial weights to keep gradients from shrinking too fast
Batch normalization: Normalizing activations between layers reduced the severity
Skip connections: ResNets added shortcut paths that allowed gradients to bypass layers
But the most effective solution was simpler: change the activation function itself.
ReLU has a derivative of 1 for all positive inputs and 0 for negative inputs. No gradual saturation. No shrinking derivatives. Gradients flow backward through ReLU layers without diminishing (provided the neuron stays active). This single change enabled training networks with 100+ layers, unlocking the deep learning revolution we're experiencing today.
Classic Activation Functions
Sigmoid (Logistic Function)
The sigmoid function dominated early neural networks, appearing in countless papers from the 1980s and 1990s. Its formula is elegant:
σ(z) = 1 / (1 + e^(-z))The function smoothly maps any input to values between 0 and 1, creating an S-shaped curve. For large negative inputs, it outputs near 0; for large positive inputs, near 1.
Strengths:
Smooth, continuous, and differentiable everywhere
Output interpretable as probability (0 to 1 range)
Historically well-understood and implemented in every framework
Weaknesses:
Severe vanishing gradients for inputs far from zero
Outputs not zero-centered, causing zig-zagging during gradient descent
Computationally expensive due to exponential calculation
Maximum derivative of only 0.25, limiting gradient flow
Current usage: Sigmoid survives primarily in output layers for binary classification, where its 0-1 range naturally represents class probabilities. Modern hidden layers avoid it due to training difficulties.
Hyperbolic Tangent (Tanh)
Tanh improved on sigmoid by centering outputs around zero:
tanh(z) = (e^z - e^(-z)) / (e^z + e^(-z))This maps inputs to the range (-1, 1), with zero input producing zero output.
Strengths:
Zero-centered outputs facilitate faster convergence (Baeldung, February 2025)
Stronger gradients than sigmoid (maximum derivative of 1.0 vs 0.25)
Maintains smoothness and differentiability
Weaknesses:
Still suffers from vanishing gradients, though less severely than sigmoid
Computationally expensive
Saturates for large positive or negative inputs
Current usage: Occasionally used in recurrent neural networks (RNNs) and specific architectures where zero-centered outputs matter, but largely replaced by modern alternatives.
Step Function
The step function (Heaviside function) was the original activation, inspired by all-or-nothing biological neuron firing:
step(z) = 1 if z > 0, else 0Strengths:
Computationally trivial
Biologically inspired
Weaknesses:
Not differentiable, making gradient-based training impossible
Binary output prevents fine-grained learning
No practical use in modern networks except specific cases like binarized neural networks
The ReLU Revolution
Rectified Linear Unit (ReLU) transformed deep learning from academic curiosity to industrial force. Its simplicity is its genius:
ReLU(z) = max(0, z)If the input is positive, pass it through unchanged. If negative, output zero. That's it.
Vinod Nair and Geoffrey Hinton introduced ReLU to modern deep learning in their 2010 paper on Restricted Boltzmann Machines (Nair & Hinton, 2010). Though Kunihiko Fukushima had used similar functions in 1975, the timing wasn't right (Cross Validated, historical discussion). By 2010, large datasets (ImageNet), powerful GPUs, and mature optimization algorithms converged to make deep learning practical. ReLU became the catalyst.
Why ReLU Dominates
1. Solves Vanishing Gradients
ReLU's derivative equals 1 for all positive inputs. Gradients flow backward through layers without shrinking, enabling networks with dozens or hundreds of layers.
2. Computational Efficiency
Computing ReLU requires one comparison and one multiplication. Sigmoid requires computing an exponential. On modern GPUs processing millions of neurons, this efficiency gap compounds dramatically. Training with ReLU runs approximately 6 times faster than equivalent tanh networks (Wikipedia, 2025).
3. Sparsity
ReLU naturally induces sparsity—approximately 50% of neurons output exactly zero in randomly initialized networks. This sparse activation means the network only uses relevant features, improving interpretability and reducing overfitting (Georgia Tech OMSCS, January 2024).
4. Biological Plausibility
ReLU better mimics real neuron behavior than sigmoid. Biological neurons don't gradually activate across their full range; they fire when stimulation exceeds a threshold, similar to ReLU's behavior.
The Dying ReLU Problem
ReLU isn't perfect. Its Achilles heel is the "dying ReLU" problem. When a neuron's weighted inputs consistently produce negative pre-activation values, it outputs zero. The gradient becomes zero. Weight updates stop. The neuron dies—permanently stuck outputting zero regardless of input.
This can cascade. If many neurons die during training, the network loses capacity to learn. Dying ReLUs often result from poor weight initialization, aggressive learning rates, or unfortunate early gradient updates that push neurons into permanently negative territory.
ReLU Variants
Researchers developed variants to address dying ReLU while preserving its benefits:
Leaky ReLU (2013)Allows small negative slope instead of zero:
LeakyReLU(z) = z if z > 0, else 0.01zThe 0.01 slope prevents complete neuron death while maintaining ReLU's other benefits.
Parametric ReLU (PReLU) (2015)Makes the negative slope a learnable parameter:
PReLU(z) = z if z > 0, else αzWhere α learns during training, adapting to the data.
Exponential Linear Unit (ELU) (2015)Smoothly allows negative values using an exponential curve:
ELU(z) = z if z > 0, else α(e^z - 1)ELU pushes mean activations closer to zero, potentially speeding learning, though at higher computational cost.
Scaled ELU (SELU) (2017)A self-normalizing variant that maintains mean and variance across layers under specific conditions, reducing need for batch normalization.
Despite these variants, vanilla ReLU remains the default choice for most applications. Its simplicity, speed, and effectiveness are hard to beat.
Modern Activation Functions
GELU (Gaussian Error Linear Unit)
Dan Hendrycks and Kevin Gimpel introduced GELU in 2016, though it didn't gain widespread adoption until BERT popularized it in 2018 (Ultralytics, glossary entry). GELU represents a philosophical shift—instead of deterministically gating inputs, it weighs them probabilistically.
The core insight: gate each input by its probability of being positive under a standard normal distribution. The mathematical formulation is:
GELU(z) = z × Φ(z)Where Φ(z) is the cumulative distribution function of the standard normal distribution.
In practice, an approximation is commonly used:
GELU(z) ≈ 0.5z(1 + tanh[√(2/π)(z + 0.044715z³)])Why GELU Matters:
GELU provides smooth, non-monotonic activation. Inputs clearly positive pass through nearly unchanged. Inputs clearly negative are nearly zeroed. Inputs near zero—where classification is uncertain—are partially attenuated. This smoothness facilitates better gradient flow than ReLU's sharp corner at zero (Brenndoerfer, June 2025).
GELU became the standard activation for Transformer encoders. BERT (2018), RoBERTa (2019), and their descendants all use GELU by default. The function's smooth probabilistic gating appears particularly effective for the attention mechanisms that power modern language models.
Evidence of Effectiveness:
A 2024 study comparing activation functions in vision transformers found that GELU, alongside ReLU, exhibited particularly favorable performance metrics across multiple architectures (Kunc & Kléma, ArXiv February 2024). The smoothness aids optimization in very deep networks where harsh non-linearities can create training instabilities.
Swish / SiLU (Sigmoid Linear Unit)
Google researchers discovered Swish in 2017 through neural architecture search—essentially letting algorithms search for better activation functions (Brenndoerfer, June 2025). The formula is beautifully simple:
Swish(z) = z × σ(z)Where σ is the sigmoid function. The function is also called SiLU (Sigmoid Linear Unit), and both names refer to identical mathematics.
Swish is smooth, non-monotonic, and unbounded above. It has become the standard activation for decoder-style Transformers including LLaMA, Mistral, and GPT-NeoX (Medium, February 2025).
Why Swish Matters:
Empirical studies consistently show Swish outperforming ReLU across various tasks, particularly in large-scale language models. The smooth self-gating allows the network to selectively amplify or suppress features based on their magnitude, providing more nuanced control than ReLU's hard threshold.
SwiGLU (Swish Gated Linear Unit)
SwiGLU combines Swish activation with gating mechanisms, becoming increasingly popular in recent large language models. The architecture uses two parallel linear transformations—one passes through Swish activation, the other serves as a gate:
SwiGLU(x) = Swish(W₁x) ⊙ (W₂x)Where ⊙ represents element-wise multiplication, and W₁, W₂ are learned weight matrices.
This gating allows the network to dynamically control information flow, potentially capturing more complex feature interactions. Recent research suggests SwiGLU achieves lower perplexity (better language modeling performance) than GELU in decoder models, though with slightly higher computational cost (APXML, scaling transformers guide).
Mish
Proposed in 2019, Mish combines properties of Swish and tanh:
Mish(z) = z × tanh(softplus(z)) = z × tanh(ln(1 + e^z))Mish is smooth, non-monotonic, and self-regularizing. It has shown competitive performance with Swish while providing even smoother gradients. A 2025 study found that replacing ReLU with advanced functions like Mish could improve accuracy by several percentage points on certain benchmarks (Wiley, April 2025).
Adaptive and Learnable Functions
The frontier of activation function research involves functions that learn and adapt during training.
AHerfReLU (2025)A recent paper introduced AHerfReLU, which combines ReLU characteristics with adaptive parameters (Ullah et al., Complexity, April 2025). Testing on CIFAR-100 showed 3.18% accuracy improvement over standard ReLU on LeNet, and 1.3% improvement in mean average precision on SSD300 for object detection.
Rational Activation FunctionsA 2024 study introduced learnable rational activation functions that can approximate any existing activation function during training (ArXiv, August 2024). Rather than choosing one activation for the entire network, each layer learns its optimal function from data. Results showed learned functions often differed significantly from standard choices like GELU, suggesting different layers benefit from different activation patterns.
This research challenges the assumption that one activation function should serve an entire network. Future architectures might have each layer, or even each neuron, using unique learned activations.
Case Study: AlexNet and ImageNet 2012
AlexNet represents the moment activation functions changed history.
In September 2012, Alex Krizhevsky, Ilya Sutskever, and Geoffrey Hinton submitted AlexNet to the ImageNet Large Scale Visual Recognition Challenge (ILSVRC). The competition asked algorithms to classify 1.2 million images into 1,000 categories—objects ranging from teapots to tigers.
Previous winners used traditional computer vision techniques: carefully hand-engineered features, support vector machines, ensemble methods. Their error rates hovered around 26%.
AlexNet shattered expectations with a top-5 error rate of 15.3%—a 10.9 percentage point improvement that stunned the computer vision community (Wikipedia, 2025). Second place, using traditional methods, scored 26.2%. AlexNet didn't just win; it demonstrated that deep learning had fundamentally eclipsed traditional approaches.
The Architecture
AlexNet consisted of eight layers: five convolutional layers followed by three fully connected layers, culminating in a 1,000-way softmax output (Viso.ai, April 2025). The network contained 60 million parameters and 650,000 neurons.
But the breakthrough wasn't just depth. AlexNet made four critical innovations:
1. ReLU Activation
AlexNet replaced sigmoid and tanh with ReLU throughout its hidden layers. This single decision enabled training the deep network within reasonable time frames. The paper explicitly compared training speed—ReLU networks trained approximately six times faster than equivalent networks with tanh units (Krizhevsky et al., 2012).
2. GPU Acceleration
Krizhevsky trained AlexNet on two NVIDIA GTX 580 GPUs, each with 3GB memory, running in parallel in his bedroom at his parents' house (Wikipedia, 2025). This was revolutionary—previous networks trained on CPUs, taking weeks or months for similar-scale problems. GPUs' parallel processing capabilities aligned perfectly with neural network computation, reducing training time from weeks to days.
3. Dropout Regularization
AlexNet used dropout—randomly disabling 50% of neurons during training—in its fully connected layers. This forced the network to learn robust, distributed representations rather than relying on specific pathways, dramatically reducing overfitting.
The team artificially expanded their training data through random crops, flips, and color adjustments, helping the network generalize better to new images.
The Impact
AlexNet's victory catalyzed the deep learning revolution. Yann LeCun, a pioneering neural network researcher, called it "an unequivocal turning point in the history of computer vision" at the 2012 European Conference on Computer Vision (Wikipedia, 2025).
In the decade following, AlexNet's innovations became standard practice:
ReLU became the default activation function
GPU training became mandatory for serious deep learning
Dropout and data augmentation became standard regularization techniques
End-to-end learning replaced feature engineering pipelines
Within three years, Google, Facebook, Microsoft, and Amazon had built dedicated AI research divisions. NVIDIA redirected efforts toward AI workloads. The market for AI-specialized hardware exploded.
Hinton later reflected on the team dynamics: "Ilya thought we should do it, Alex made it work, and I got the Nobel Prize" (Wikipedia, 2025). In October 2024, Hinton and collaborator John Hopfield won the Nobel Prize in Physics for their foundational contributions to artificial neural networks.
AlexNet didn't invent deep learning or activation functions. But by combining ReLU, GPUs, and large-scale data into a working system that crushed previous benchmarks, it proved these technologies could transform industries. The activation function—ReLU's simple decision to zero-out negative values—was a key piece enabling this transformation.
Case Study: BERT and the Rise of GELU
While AlexNet revolutionized computer vision, natural language processing lagged behind. Language posed unique challenges—sequential dependencies, variable-length inputs, subtle contextual meanings. Traditional approaches struggled.
In October 2018, Google researchers released BERT (Bidirectional Encoder Representations from Transformers), fundamentally changing how machines understand language (Devlin et al., 2018). BERT achieved state-of-the-art results on 11 NLP tasks, including question answering, sentiment analysis, and named entity recognition.
The Architecture and GELU
BERT built on the Transformer architecture introduced in 2017's "Attention Is All You Need" paper. But BERT made a crucial choice: it used GELU activation in its feed-forward networks instead of ReLU (Hugging Face documentation, BERT model config).
The feed-forward network in each Transformer layer consists of two linear transformations separated by activation:
FFN(x) = activation(xW₁ + b₁)W₂ + b₂BERT chose GELU for this activation, marking one of the first major deployments of the function at scale (Medium, Tilo Flasche, September 2025).
Why GELU for Transformers?
The shift from ReLU to GELU in Transformers reflects deeper understanding of what these models need:
Smoothness Aids Optimization
Transformers train on enormous datasets—BERT pre-trained on 3.3 billion words. The smooth gradients from GELU facilitate stable optimization over long training runs. ReLU's sharp corner at zero can create micro-instabilities that compound over billions of gradient updates.
Non-Monotonicity Provides Flexibility
GELU's slight dip into negative territory for negative inputs (unlike ReLU's hard zero) provides additional flexibility for the network to learn complex representations. This non-monotonic property allows more nuanced feature transformations.
Probabilistic Gating Matches Attention
GELU's probabilistic interpretation—gating by likelihood of being positive—philosophically aligns with attention mechanisms, which also weight inputs probabilistically. This coherence might contribute to Transformers' effectiveness.
The Evidence
BERT's success sparked an explosion in Transformer-based language models. GPT-2, GPT-3, RoBERTa, ALBERT, and dozens of variants followed, nearly all defaulting to GELU activation.
A 2024 systematic study of activation functions in Transformer models found that 80% of language models introduced after BERT use GELU (ArXiv, August 2024). The survey examined 20 major models—16 explicitly used GELU, while many others simply referenced BERT as their base architecture, implicitly adopting GELU.
Performance comparisons validated the choice. When researchers systematically swapped activation functions in BERT-scale models, GELU consistently matched or exceeded ReLU performance on downstream tasks, particularly for encoder models focused on understanding rather than generation (Salt Data Labs, December 2022).
GELU's Limitations
GELU isn't universally superior. Decoder models optimized for text generation—GPT-3, LLaMA, GPT-4—increasingly favor SiLU/Swish over GELU. Swish appears to provide slight advantages for autoregressive generation tasks, though the differences are often marginal (Brenndoerfer, June 2025).
The computational cost matters too. GELU requires computing transcendental functions (tanh, exponentials) making it slower than ReLU. For models with hundreds of billions of parameters, this cost compounds. Engineers must balance improved performance against training time and inference latency.
The Broader Lesson
BERT demonstrated that activation function choice must align with architecture and task. What works brilliantly for convolutional image recognition (ReLU) might not be optimal for attention-based language understanding (GELU). There's no universal best activation—only context-dependent optimization.
This insight drives current research into adaptive and architecture-specific activations, moving beyond one-size-fits-all solutions.
Case Study: Recent Advances in Learnable Activations
The latest frontier in activation functions involves functions that learn themselves.
The Motivation
Traditional activation functions are fixed—ReLU, GELU, Swish all apply the same mathematical transformation regardless of what they're learning. But different tasks, datasets, and even different layers might benefit from different activations.
Consider a facial recognition network. Early layers detecting edges might benefit from different activation properties than later layers recognizing complete faces. Yet we typically apply the same function everywhere.
Rational Activation Functions (2024)
A comprehensive 2024 study introduced learnable rational activation functions to Transformer models (ArXiv, March 2024). Instead of choosing sigmoid, ReLU, or GELU, networks learned optimal activation shapes during training.
The research team trained BERT-scale models where each layer's activation function was a rational function—a ratio of polynomials with learnable coefficients:
RAF(z) = P(z) / Q(z) = (a₀ + a₁z + ... + aₙzⁿ) / (b₀ + b₁z + ... + bₘzᵐ)The model learned coefficients a and b alongside regular weights and biases.
Results were striking:
Learned activation functions differed dramatically across layers. Some layers learned monotonic functions similar to ReLU. Others learned non-monotonic functions with negative dips and multiple inflection points—shapes unlike any standard activation.
Performance improved moderately but consistently. On SQuAD question-answering, rational activations improved F1 scores by 0.5-1.0 points over fixed GELU. On sentiment classification, accuracy improved by 0.3-0.8 percentage points.
More importantly, the learned shapes challenged assumptions. Many exhibited properties that activation function guidelines traditionally consider undesirable—like non-monotonicity or asymmetric curvature. Yet they worked better than carefully designed alternatives.
AHerfReLU (2025)
A recent April 2025 paper introduced AHerfReLU, combining ReLU with the error function (erf) and adaptive parameters (Ullah et al., Wiley Complexity). The function is zero-centered, bounded below, and non-monotonic.
Experiments compared AHerfReLU against 10 other adaptive functions plus standard activations (ReLU, Swish, Mish). Testing on CIFAR-10, CIFAR-100, and Pascal VOC datasets:
LeNet on CIFAR-100: 3.18% accuracy improvement over ReLU
LeNet on CIFAR-10: 0.63% accuracy improvement over ReLU
SSD300 object detection: 1.3% mean average precision improvement
The study demonstrated that adaptive functions consistently outperform fixed alternatives when properly designed, though the improvements typically range from 0.5% to 3%—meaningful but not revolutionary.
Nonlinearity Enhanced Activations (2025)
Research published in May 2025 introduced a framework for adding parametric learned nonlinearity to existing activation functions (Yevick, ArXiv May 2025). Rather than designing new functions from scratch, the approach enhances ReLU, ELU, or other standards with learnable nonlinear components.
Testing on MNIST digit recognition and CNN benchmarks showed consistent accuracy improvements without requiring significant additional computational resources. The key insight: small, strategic additions of learnable nonlinearity provide benefits without the complexity of completely learnable functions.
The Research Consensus
Multiple studies now document over 400 activation functions developed since the 1990s (Kunc & Kléma, February 2024). This explosion reflects both growing interest and concerning redundancy—many "novel" functions are minor variations of existing ones, inadvertently rediscovered due to poor documentation.
The 2024 comprehensive survey aimed to address this, systematically cataloging functions with links to original sources. The key findings:
Most practical activations cluster into a few families (ReLU variants, sigmoid variants, learnable functions)
Performance differences between well-designed functions are often marginal (1-2%)
Computational efficiency matters as much as mathematical properties
Task and architecture context dominates—no universal winner exists
Practical Implications
Should you use learnable activation functions in production? Current evidence suggests:
Use Standard Functions When:
You need maximum computational efficiency
Your architecture and task are well-established (CNNs for images, Transformers for text)
Training budget is limited
You're starting a new project (default to ReLU or GELU)
Explore Learnable Functions When:
You're pushing state-of-the-art on established benchmarks
Computational budget allows for experimentation
Your task or data are unusual or domain-specific
You're conducting research rather than production deployment
The field is moving toward selective learnable activations—keeping standard functions for most layers while allowing critical layers to adapt. This balances performance gains against computational costs.
Choosing the Right Activation Function
Selecting activation functions involves balancing theoretical properties, computational efficiency, and empirical performance. Here's a practical decision framework:
For Convolutional Neural Networks (Image Tasks)
Default Choice: ReLU
Proven track record from AlexNet (2012) through modern architectures
Computational efficiency critical for large images
Sparse activation beneficial for visual features
Use unless you have specific reasons not to
When to Upgrade:
Try Leaky ReLU or PReLU if experiencing many dying neurons
Consider SELU for very deep networks (50+ layers) as it provides self-normalization
Experiment with Mish or Swish when optimizing state-of-the-art performance and computational cost isn't limiting
For Transformer Models (Language Tasks)
For Encoders (BERT-style):
Use GELU as default—it's the standard for good reasons
Smooth gradients aid stable training on massive datasets
Consistent with pre-trained model checkpoints you might fine-tune
For Decoders (GPT-style):
Consider SiLU/Swish as first choice
Use GELU as solid alternative
Some evidence suggests Swish provides slight edge for autoregressive generation
For Mixed Architectures:
SwiGLU showing promising results in recent large language models
Higher computational cost but potentially better performance
Consider if training budget allows
For Recurrent Networks (Sequential Tasks)
Default: Tanh
Still standard in LSTM and GRU gate computations
Zero-centered outputs important for recurrent dynamics
ReLU occasionally used but can cause exploding gradients in recurrence
For Output Layers
Binary Classification:
Sigmoid is standard—outputs interpretable as probabilities
Multi-class Classification:
Softmax is standard—produces probability distribution over classes
Regression:
Often no activation (linear) for unbounded outputs
ReLU if output must be positive
Sigmoid or tanh if output has natural bounds
General Guidelines
Network Depth Matters:
Shallow networks (1-5 layers): Most activations work fine, choose for speed (ReLU)
Medium depth (5-20 layers): ReLU, GELU, or Leaky ReLU recommended
Very deep (20+ layers): Consider batch normalization + ReLU, or skip connections + any modern activation
Computational Budget:
Limited resources: Stick with ReLU (fastest to compute)
Moderate resources: GELU or Swish reasonable
Large resources: Consider learnable or adaptive functions if optimizing performance
Domain-Specific Considerations:
Medical imaging: ReLU works well; consider Mish for slight accuracy gains that matter
Real-time systems: ReLU mandatory for speed; approximated GELU if you need smoothness
Research/experimentation: Try multiple, systematically compare
Debugging Tips:
If training stalls early: Check for dying ReLUs, try Leaky ReLU
If loss oscillates wildly: Activation might be contributing to gradient instability, try smoother function
If early layers not learning: Vanishing gradients likely, switch from sigmoid/tanh to ReLU/GELU
Regional and Industry Variations
Activation function adoption shows interesting patterns across industries and regions, shaped by computational resources, problem domains, and research traditions.
Technology Industry
Silicon Valley / US Tech Giants:
Heavy adoption of modern functions (GELU, SwiGLU, Swish)
Massive computational budgets allow experimentation
Focus on state-of-the-art performance regardless of efficiency
OpenAI's GPT models use various advanced activations
Meta's LLaMA series uses SwiGLU
Chinese Tech Companies:
Similar adoption of advanced functions in flagship models
Baidu, Alibaba, Tencent match Western practices in large language models
Growing emphasis on efficiency-optimized architectures for mobile deployment
Some preference for modified ReLU variants in production systems balancing performance and cost
European Research Labs:
Strong theoretical foundation in activation function mathematics
Contributions to learnable activation research (rational functions)
Emphasis on interpretability alongside performance
DeepMind (UK) has explored novel activation variants extensively
Industry-Specific Patterns
Healthcare and Medical Imaging:
Conservative adoption—ReLU dominates diagnostic systems
Regulatory requirements favor established, interpretable methods
High accuracy stakes mean thorough validation before deploying novel activations
Gradual adoption of Swish and Mish in research contexts (2-3 year lag from publication to clinical use)
Autonomous Vehicles:
Real-time processing demands computational efficiency
ReLU remains dominant in production perception systems
Some adoption of efficient ReLU variants (Leaky ReLU, PReLU)
Research teams exploring modern functions, but production deployment conservative
Finance and Trading:
Mixed practices—traditional LSTM-based systems use tanh
Modern transformer-based price prediction uses GELU
High-frequency trading systems prioritize speed, stick with ReLU
Risk modeling often conservative, slow to adopt novel functions
Manufacturing and Robotics:
Embedded systems with limited compute favor ReLU
Computer vision for quality inspection uses standard CNNs with ReLU
Research in robot learning exploring modern activations
Production systems typically 3-5 years behind research frontier
Regional Computational Access
High-Resource Regions:
US, parts of Europe, China, Japan can deploy computationally expensive activations
Access to latest GPUs and TPUs enables GELU, SwiGLU experimentation
Cloud infrastructure supports rapid iteration
Medium-Resource Regions:
India, Southeast Asia, Eastern Europe balance performance and cost
Preference for efficient variants of advanced functions
Heavy use of pre-trained models from high-resource regions, inheriting their activation choices
Growing local research on efficient activation functions
Low-Resource Settings:
Rural healthcare, developing regions prioritize efficiency
ReLU dominant due to minimal compute requirements
Mobile-first deployments demand lightweight architectures
Research into quantized networks and efficient activations growing
Academic vs Industry
Academic Research:
Explores novel activations freely
Publications on 400+ distinct functions since 1990s
Tendency toward mathematical elegance
Less emphasis on computational cost
Industry Practice:
Conservative—stick with proven choices
Compute efficiency matters significantly
Adoption lags research by 1-3 years
Focus on functions with good library support and optimization
The gap between research and practice is narrowing. Libraries like PyTorch and TensorFlow now include dozens of activation functions as built-ins, reducing implementation friction. Still, most production systems use a small set: ReLU, Leaky ReLU, GELU, and occasionally Swish or SiLU.
Pros and Cons Comparison
Activation Function | Pros | Cons | Best Use Cases |
ReLU | • Simple and fast<br>• Solves vanishing gradients<br>• Induces sparsity<br>• Well-supported in all frameworks | • Dying ReLU problem<br>• Not zero-centered<br>• Non-differentiable at 0<br>• Unbounded output | CNN hidden layers, default choice for most tasks |
Leaky ReLU | • Prevents dying neurons<br>• Maintains ReLU speed<br>• Simple modification | • Hyperparameter (slope) to tune<br>• Marginal improvements over ReLU<br>• Still not zero-centered | When ReLU causes many dead neurons |
GELU | • Smooth everywhere<br>• Excellent for transformers<br>• Probabilistic interpretation<br>• Empirically strong | • Slower than ReLU<br>• Requires transcendental functions<br>• More complex to implement | Transformer encoders, BERT-style models |
SiLU/Swish | • Self-gating mechanism<br>• Smooth and non-monotonic<br>• Strong empirical performance<br>• Unbounded above | • Computationally expensive<br>• Requires sigmoid calculation<br>• Benefits vary by task | Transformer decoders, large language models |
Sigmoid | • Smooth and differentiable<br>• Interpretable probabilities<br>• Bounded [0,1]<br>• Well-understood | • Severe vanishing gradients<br>• Not zero-centered<br>• Slow computation<br>• Saturates easily | Binary classification output layers only |
Tanh | • Zero-centered outputs<br>• Stronger gradients than sigmoid<br>• Bounded [-1,1]<br>• Smooth | • Still has vanishing gradients<br>• Computationally expensive<br>• Saturates for large inputs | RNN/LSTM gate mechanisms |
ELU | • Smooth for negative values<br>• Reduces bias shift<br>• Can speed convergence | • Expensive exponential computation<br>• Requires extra hyperparameter<br>• Benefits often marginal | Very deep networks, when ReLU variants fail |
Mish | • Very smooth<br>• Self-regularizing<br>• Strong empirical results<br>• Non-monotonic | • Most expensive to compute<br>• Limited production track record<br>• Implementation complexity | Research settings, when maximizing accuracy |
SwiGLU | • Gated architecture<br>• Strong recent results<br>• Flexible feature selection | • Doubles parameters<br>• Computationally expensive<br>• Requires careful dimension handling | State-of-the-art LLMs, high-resource settings |
Learnable (RAF) | • Adapts to data<br>• Different per layer<br>• Can match any function<br>• Consistent improvements | • Added hyperparameters<br>• Longer training time<br>• Risk of overfitting<br>• Implementation complexity | Research, when pushing benchmarks |
Common Myths vs Facts
Myth 1: "ReLU is outdated—modern networks should use GELU or Swish"
Fact: ReLU remains the default choice for CNNs and many applications. While GELU and Swish show advantages in specific architectures (Transformers), ReLU's speed, simplicity, and effectiveness keep it dominant. A 2024 survey found ReLU and its variants still power the majority of production computer vision systems (Kunc & Kléma, February 2024). Use modern alternatives when evidence suggests benefits for your specific task, not because they're newer.
Myth 2: "The vanishing gradient problem is solved"
Fact: ReLU mitigates vanishing gradients but doesn't eliminate all gradient flow issues. Very deep networks (100+ layers) still struggle without additional techniques like skip connections (ResNets), batch normalization, or careful initialization. The vanishing gradient problem is managed through architectural choices and activation functions together, not activation functions alone.
Myth 3: "More complex activation functions always perform better"
Fact: Performance differences between well-designed activations are often marginal (1-2% accuracy) and task-dependent. The 2025 AHerfReLU study showed 3.18% improvement in best cases, but other scenarios showed minimal gains (Ullah et al., April 2025). Computational costs of complex functions can outweigh slight accuracy improvements. Simple often wins when considering the full deployment picture.
Myth 4: "Sigmoid and tanh are never useful in modern networks"
Fact: These functions remain standard in specific contexts. Sigmoid dominates binary classification outputs. Tanh is standard in LSTM and GRU gate mechanisms. The recurrent structure of these architectures benefits from tanh's zero-centered, bounded outputs. Don't use them in hidden layers of feedforward networks, but they're far from obsolete.
Myth 5: "The same activation should be used throughout a network"
Fact: Different layers can benefit from different activations. Early layers detecting basic features might work best with one function, while deeper layers learning abstract concepts might benefit from another. Recent research on learnable activations confirms this—optimal functions vary by layer (ArXiv, March 2024). However, using the same activation throughout is a reasonable default for simplicity.
Myth 6: "Activation function choice doesn't matter much compared to architecture"
Fact: Activation functions are foundational to architecture effectiveness. AlexNet's success depended critically on ReLU enabling training of its depth. BERT's effectiveness relies partly on GELU's properties for Transformers. Poor activation choice can prevent any architecture from training successfully. They're not the only important factor, but they're certainly not negligible.
Myth 7: "Learnable activation functions will replace fixed ones"
Fact: Learnable activations show promise but add complexity, training time, and risk of overfitting. They're valuable for squeezing final percentage points from well-established architectures but unlikely to replace simple fixed functions as defaults. Most practitioners will continue using ReLU, GELU, or Swish for the foreseeable future. Learnable functions are a tool for specific optimization scenarios, not a universal replacement.
Myth 8: "Negative outputs from activation functions are bad"
Fact: Functions allowing negative outputs (Leaky ReLU, ELU, tanh, Swish) can be beneficial. Zero-centered activations often converge faster. Smooth negative regions prevent complete neuron death. The idea that neurons should only output positive values is a ReLU-specific property, not a requirement. Different mathematical properties suit different contexts.
Implementation Guide
Python/PyTorch Implementation
import torch
import torch.nn as nn
# Standard activation functions (built-in)
relu = nn.ReLU()
leaky_relu = nn.LeakyReLU(negative_slope=0.01)
gelu = nn.GELU()
silu = nn.SiLU() # Also called Swish
tanh = nn.Tanh()
sigmoid = nn.Sigmoid()
# Usage in a network
class SimpleNetwork(nn.Module):
def __init__(self):
super().__init__()
self.fc1 = nn.Linear(784, 256)
self.fc2 = nn.Linear(256, 128)
self.fc3 = nn.Linear(128, 10)
# Choose activation
self.activation = nn.ReLU() # Swap with GELU, SiLU, etc.
def forward(self, x):
x = self.activation(self.fc1(x))
x = self.activation(self.fc2(x))
x = self.fc3(x)
return x
# Custom implementation of Mish
class Mish(nn.Module):
def forward(self, x):
return x * torch.tanh(torch.nn.functional.softplus(x))
# Custom implementation of Swish with learnable parameter
class LearnableSwish(nn.Module):
def __init__(self):
super().__init__()
self.beta = nn.Parameter(torch.ones(1))
def forward(self, x):
return x * torch.sigmoid(self.beta * x)TensorFlow/Keras Implementation
import tensorflow as tf
from tensorflow import keras
# Built-in activations
model = keras.Sequential([
keras.layers.Dense(256, activation='relu', input_shape=(784,)),
keras.layers.Dense(128, activation='relu'),
keras.layers.Dense(10, activation='softmax')
])
# Using activation layers explicitly
model = keras.Sequential([
keras.layers.Dense(256, input_shape=(784,)),
keras.layers.Activation('relu'), # or 'gelu', 'swish'
keras.layers.Dense(128),
keras.layers.Activation('relu'),
keras.layers.Dense(10),
keras.layers.Activation('softmax')
])
# Custom Mish activation
def mish(x):
return x * tf.math.tanh(tf.math.softplus(x))
# Using custom activation
model = keras.Sequential([
keras.layers.Dense(256, input_shape=(784,)),
keras.layers.Lambda(mish),
keras.layers.Dense(10, activation='softmax')
])Performance Optimization Tips
1. Use In-Place Operations
# PyTorch ReLU with inplace=True saves memory
relu = nn.ReLU(inplace=True)2. Avoid Custom Implementations for Standard Functions Built-in functions are heavily optimized. Only implement custom activations when necessary.
3. Consider Mixed Precision Training Activation functions interact with numerical precision. Some (like GELU) are more sensitive to fp16 precision issues than others (like ReLU).
4. Profile Your Activations
# PyTorch profiler to measure activation cost
with torch.profiler.profile() as prof:
output = model(input_tensor)
print(prof.key_averages().table())Common Implementation Mistakes
1. Applying Activation to Output Layer Incorrectly
# Wrong for regression
output = torch.relu(self.fc_out(x))
# Correct for regression (no activation)
output = self.fc_out(x)
# Correct for binary classification
output = torch.sigmoid(self.fc_out(x))
# Correct for multi-class classification
output = torch.softmax(self.fc_out(x), dim=1)2. Forgetting Activation Entirely
# Wrong - missing activation
x = self.fc1(x)
x = self.fc2(x)
# Correct
x = torch.relu(self.fc1(x))
x = torch.relu(self.fc2(x))3. Using Wrong Activation for Architecture
# Suboptimal for BERT-style transformer
transformer_encoder = TransformerEncoder(..., activation='relu')
# Better for BERT-style
transformer_encoder = TransformerEncoder(..., activation='gelu')Future Outlook: The Next Five Years
Emerging Trends (2026-2031)
1. Adaptive and Context-Aware Activations
Current research on learnable activations will mature into practical implementations. Expect frameworks to support "meta-activation" layers that automatically learn optimal activation shapes during pre-training, then fix them for fine-tuning. This could reduce the hyperparameter search space while maintaining flexibility.
By 2028, we may see activation functions that adapt not just to the layer but to the input—different activation behavior for different data points. Early research explores input-dependent activations, though computational costs remain prohibitive today.
2. Efficiency-Optimized Variants
As models scale to trillions of parameters, every FLOP matters. Expect intense focus on approximations that preserve performance while reducing cost. Already we see:
Hardware-specific activations optimized for particular chip architectures
Binary or low-precision activations for edge deployment
Sparse activation patterns that activate only subsets of neurons
The next generation of mobile and edge AI will demand activation functions balancing effectiveness with minimal power consumption. ReLU's efficiency advantage will become even more valuable for on-device inference.
3. Biological Inspiration Redux
Renewed interest in actual neuroscience could inspire new activation functions. Recent neuroscience research reveals biological neurons don't simply threshold—they have complex non-linear dynamics, adaptation mechanisms, and temporal properties.
Spiking neural networks, which more closely model biological neuron behavior, may bring novel activation concepts from neuroscience into mainstream deep learning. However, this trend depends on whether biological plausibility translates to computational advantages.
4. Architecture-Specific Co-Evolution
Activation functions will increasingly co-evolve with architectures rather than being chosen independently. Just as GELU became standard for Transformers while ReLU remained dominant for CNNs, new architectures will come bundled with custom-designed activations.
The next breakthrough architecture (whatever follows Transformers) will likely introduce a corresponding activation innovation. Researchers will design activation and architecture together rather than treating them as independent choices.
Quantitative Projections
Based on current research trajectories and historical patterns:
Adoption Rates (Estimated 2030):
ReLU and direct variants: 40% of production systems (down from ~60% in 2024)
GELU/Swish family: 35% (up from ~25%)
Learnable/adaptive functions: 15% (up from <5%)
Other (tanh, sigmoid, novel): 10%
Performance Improvements: Incremental gains will continue. Expect:
2-5% accuracy improvements on established benchmarks from better activation choices
20-30% computational savings from efficient approximations
Dramatic improvements (10%+) only for specialized novel tasks
Research Activity:
Activation function papers as percentage of deep learning publications: Stable at 3-5%
Industry adoption lag: Continues at 2-3 years from publication to production
Number of practically distinct functions: Will plateau around 30-50 widely-used variants (down from theoretical 400+) as community converges on effective patterns
Challenges Ahead
1. Reproducibility and Standardization
With 400+ activation functions described in literature, reproducibility suffers. Many papers inadequately specify implementation details. The community needs:
Standard benchmarking protocols comparing activations fairly
Comprehensive open-source libraries with verified implementations
Better documentation of when small differences matter
2. Hardware-Software Co-Design
Current GPUs and TPUs optimize primarily for ReLU and simple operations. As novel activations emerge, hardware must adapt. This creates a chicken-and-egg problem—hardware won't optimize for unused functions, but functions won't get adopted without hardware support.
Next-generation AI accelerators will likely include configurable activation units supporting multiple function families efficiently.
3. Theoretical Understanding
Despite empirical success, theoretical understanding of why certain activations work lags behind. We lack principled answers to:
What mathematical properties matter most for different tasks?
How do activation functions interact with architecture choices?
Can we predict optimal activations from data properties?
Deeper theoretical foundations could dramatically accelerate progress, moving from trial-and-error to principled design.
Long-Term Vision (Beyond 2031)
Looking further ahead, activation functions might become nearly invisible as explicit choices. Machine learning systems could automatically discover optimal activations during architecture search, treating them as continuous design variables rather than discrete choices.
Ultimately, biological neural networks don't use a single activation function—different neuron types have different properties. Future artificial networks might similarly employ heterogeneous activations, with different neuron populations using different functions optimized for different subtasks within the network.
The field is moving from "which activation function?" to "how should activations adapt?"—a more nuanced and ultimately more powerful framing.
FAQ
1. What is an activation function in simple terms?
An activation function is a mathematical operation that decides whether a neuron in a neural network should activate (fire) or not. It takes the neuron's input, applies a mathematical transformation, and produces an output that gets passed to the next layer. Without activation functions, neural networks could only learn linear patterns, making them unable to solve complex real-world problems like recognizing faces or understanding language.
2. Why can't neural networks use linear activation functions?
Linear activation functions collapse multi-layer networks into equivalent single-layer networks mathematically. If every layer applies a linear transformation, the entire network simply computes one linear function regardless of depth. Real-world data has non-linear patterns—face recognition, language understanding, and speech recognition all require learning complex, non-linear relationships that linear models cannot capture.
3. Which activation function should beginners use?
Start with ReLU for hidden layers and softmax or sigmoid for output layers depending on your task. ReLU is the most widely used, well-documented, and computationally efficient activation function. It works well for most applications, has fewer hyperparameters than alternatives, and is supported by every major framework. Only explore alternatives once you understand the basics and have specific reasons to switch.
4. What causes dying ReLU and how do I fix it?
Dying ReLU occurs when neurons output zero for all inputs, effectively removing them from the network. This happens when large negative weights cause consistently negative pre-activation values. Fixes include: using Leaky ReLU instead of ReLU, proper weight initialization (He initialization), avoiding very large learning rates, adding batch normalization, or using PReLU which learns the optimal negative slope.
5. Is GELU always better than ReLU?
No. GELU performs better in Transformer architectures, particularly encoder models like BERT, due to its smooth gradient properties. However, ReLU remains superior for CNNs processing images, being faster to compute and providing sufficient performance. GELU's computational cost (requiring transcendental functions) makes it less desirable for applications where speed matters. Choose based on architecture and task, not recency.
6. How do I know if my activation function is causing training problems?
Warning signs include: training loss plateaus very early (suggests vanishing gradients), loss oscillates wildly (suggests gradient instability or exploding gradients), later layers learn but early layers don't (suggests vanishing gradients), or many neurons permanently output zero (dying ReLU). Use gradient monitoring tools in your framework to track gradient magnitudes across layers. Healthy training shows consistent gradient flow through all layers.
7. Can I use different activation functions in different layers?
Yes. Many successful architectures use different activations for different purposes. Output layers typically use sigmoid or softmax regardless of hidden layer activations. Some architectures use tanh in recurrent connections but ReLU in feedforward parts. Recent research suggests optimal activations vary by layer depth. However, using the same activation throughout hidden layers remains a reasonable default for simplicity unless you have specific reasons to vary.
8. Why do Transformers use GELU instead of ReLU?
GELU's smooth, continuously differentiable gradient flow provides more stable optimization for Transformer training on massive datasets. The function's probabilistic interpretation (gating inputs by likelihood of being positive) philosophically aligns with attention mechanisms. Empirical results consistently show GELU matching or exceeding ReLU performance in language tasks. The computational cost is acceptable given Transformers already require intensive computation for attention mechanisms.
9. What's the difference between Swish and SiLU?
They're identical. Google researchers discovered the function through neural architecture search and named it "Swish" in 2017. Other researchers independently arrived at the same function and named it SiLU (Sigmoid Linear Unit) based on its structure. Both names refer to the mathematical function f(x) = x × sigmoid(x). Literature uses both terms inconsistently, but they mean exactly the same thing.
10. How much does activation function choice actually matter?
Activation choice significantly impacts whether training succeeds at all (ReLU vs sigmoid in deep networks), but among modern functions (ReLU, GELU, Swish), differences are often marginal—typically 0.5-3% accuracy. However, this can mean the difference between state-of-the-art results and runner-up in competitive benchmarks. In production, computational efficiency differences matter enormously at scale. The first-order importance is using a modern non-saturating function; second-order optimization involves choosing among modern alternatives.
11. Are learnable activation functions worth implementing?
For most practitioners, no. Standard functions (ReLU, GELU, Swish) work well for typical applications. Learnable activations add complexity, increase training time, require careful hyperparameter tuning, and risk overfitting. They're valuable when pushing state-of-the-art on established benchmarks where every 0.5% accuracy improvement matters, or for novel tasks where optimal activations are unknown. Unless you're conducting research or have already exhausted standard optimizations, stick with fixed functions.
12. Why do some activation functions have parameters?
Parametric activation functions (like PReLU with learnable negative slope, or LearnableSwish with learnable β) allow networks to adapt activation behavior to data during training. This provides flexibility between completely fixed functions (ReLU) and fully learnable functions (rational activations). Parameters are typically few (one per activation or one per layer) making them computationally cheap while providing meaningful adaptability. They're most useful when you suspect your task might benefit from activation properties different from standard choices but don't want fully learnable complexity.
13. How do activation functions relate to the vanishing gradient problem?
Activation functions' derivatives directly determine gradient magnitude during backpropagation. Sigmoid and tanh have maximum derivatives of 0.25 and 1.0 respectively, and these shrink further from their peak. Multiplying many small derivatives across layers causes gradients to vanish exponentially. ReLU maintains gradient of 1.0 for positive inputs, preventing this multiplicative shrinkage. Modern smooth functions (GELU, Swish) balance maintaining sufficient gradients with smooth properties that aid optimization.
14. What activation should I use for regression vs classification?
Hidden layers can use the same activation (typically ReLU or GELU) regardless of task. The difference is in the output layer: For regression predicting continuous values, use no activation (linear) if unbounded, ReLU if output must be positive, or sigmoid/tanh if output has natural bounds. For binary classification, use sigmoid producing probabilities 0-1. For multi-class classification, use softmax producing probability distribution over classes. Never use ReLU in classification output layers.
15. How do batch normalization and activation functions interact?
Batch normalization normalizes layer inputs, reducing activation function's impact on gradient flow. This partially mitigates vanishing gradients even with sigmoid/tanh, though ReLU families still perform better. Batch normalization typically applies before activation, though order debates persist. Some activation functions (SELU) are designed to be self-normalizing, potentially reducing or eliminating batch normalization need. In practice, modern architectures often use batch normalization + ReLU/GELU together for best results.
16. Are there activation functions specifically for time series or sequential data?
RNNs and LSTMs traditionally use tanh in their core recurrent connections due to zero-centered, bounded output properties that aid recurrent training dynamics. Gate mechanisms (forget gates, input gates) use sigmoid. However, for feedforward components of sequential models, modern practices increasingly use ReLU or GELU. Temporal Convolutional Networks (TCNs) processing sequential data use standard CNN activations. Task matters more than data type—if your sequential data fits Transformer architecture, use GELU; if using RNNs, use tanh.
17. What's the computational cost difference between activation functions?
ReLU is fastest—single comparison plus thresholding. Leaky ReLU adds one multiplication. Sigmoid and tanh require expensive exponential computations, making them ~3-10x slower than ReLU. GELU requires transcendental functions (tanh, exponential), typically ~2-4x slower than ReLU. Swish/SiLU requires sigmoid computation, ~2-3x slower. These differences compound at scale—a model with billions of parameters might spend 10-30% of compute on activations. In latency-critical applications or edge deployment, this matters significantly.
18. Can activation functions cause overfitting?
Indirectly. More complex activations add model capacity, potentially increasing overfitting risk if not properly regularized. Learnable activations add parameters that can overfit to training data. However, activation choice alone rarely causes severe overfitting—other factors (model size, regularization, data quantity) matter more. Some activations (ReLU through sparsity, Mish through self-regularization) might actually reduce overfitting slightly. Treat activation choice primarily as an optimization and gradient flow concern, not an overfitting concern.
19. What happens if I use no activation function at all?
Using no activation (or equivalently, linear activation) at every layer reduces your entire network to a single linear transformation regardless of depth. A 100-layer network with linear activation is mathematically equivalent to a single-layer linear model. This can only learn linear relationships, failing on virtually all interesting real-world tasks. Linear output layers for regression are fine—it's hidden layers that need non-linearity to enable complex pattern learning.
20. How does activation function choice affect inference speed vs training speed?
Activation affects both, but differently. During training, activation functions run during both forward and backward passes, and their derivatives matter for backpropagation. During inference, only forward pass matters, but it runs potentially billions of times. Expensive activations (GELU, Swish) might slow training 20-30% but inference only 5-10% (less relative impact since inference has no backward pass). For production deployment, inference speed typically matters more—prefer simpler activations unless accuracy gains justify the latency cost.
Key Takeaways
Activation functions enable non-linearity, allowing neural networks to learn complex patterns beyond simple linear relationships. Without them, deep networks collapse into shallow linear models regardless of architecture.
ReLU revolutionized deep learning in 2012 by solving the vanishing gradient problem that plagued sigmoid and tanh. Its simple threshold operation (output input if positive, else zero) enabled training networks with dozens to hundreds of layers.
Modern architectures favor specialized activations: Transformers typically use GELU (encoder models like BERT) or SiLU/Swish (decoder models like GPT). Convolutional networks still predominantly use ReLU and variants due to computational efficiency.
The vanishing gradient problem arises when activation functions with small derivatives (sigmoid maxing at 0.25, tanh at 1.0) cause gradients to shrink exponentially during backpropagation through multiple layers. ReLU's derivative of 1.0 for positive inputs prevents this multiplicative shrinking.
Over 400 activation functions exist as documented in a comprehensive 2024 survey, but practical usage concentrates on fewer than a dozen. Most are minor variants, many inadvertently rediscovered due to poor documentation.
Computational efficiency matters at scale. ReLU computes in one comparison; GELU requires transcendental functions ~3-4x slower. In models with billions of parameters, this compounds to significant cost differences in training and inference.
Different layers may benefit from different activations, as recent research on learnable activations confirms. However, using the same activation throughout hidden layers remains a reasonable default for simplicity unless specific evidence suggests otherwise.
Choice depends on architecture and task: CNNs for images default to ReLU; Transformers for language default to GELU or SiLU; RNNs for sequences use tanh in recurrent connections. Output layers depend on task (sigmoid for binary, softmax for multi-class, linear for regression).
Dying ReLU occurs when neurons permanently output zero due to consistently negative pre-activations, effectively removing them from the network. Solutions include Leaky ReLU, proper weight initialization, moderate learning rates, and batch normalization.
The field continues evolving with learnable and adaptive activations showing 0.5-3% accuracy improvements in recent studies. However, standard functions (ReLU, GELU, Swish) remain the practical default for most applications, with novel functions reserved for state-of-the-art optimization or specialized tasks.
Actionable Next Steps
For practitioners starting new projects: Use ReLU for CNN hidden layers, GELU for Transformer hidden layers, and appropriate output activation (sigmoid, softmax, or linear) based on your task. Validate this default choice works before exploring alternatives.
If experiencing training difficulties: Monitor gradient flow across layers using your framework's tools. If early layers show vanishing gradients with sigmoid/tanh, switch to ReLU or GELU. If seeing many dying neurons (permanently zero activations), try Leaky ReLU or PReLU.
When optimizing performance: After exhausting architecture and hyperparameter optimization, systematically compare 3-5 activation alternatives (ReLU, Leaky ReLU, GELU, Swish). Use consistent training setups and proper statistical testing. Document computational costs alongside accuracy.
For production deployment: Profile activation function costs in your specific hardware setup. If inference latency is critical, prefer ReLU or efficient approximations of GELU. If accuracy is paramount and compute budget allows, modern functions (GELU, Swish) justified.
To stay current: Follow activation function sections in major conference papers (NeurIPS, ICML, ICLR). Framework release notes often document new built-in activations worth exploring. The 2024 comprehensive survey by Kunc & Kléma provides systematic coverage.
For research contributions: Focus on efficiency-optimized variants of proven functions rather than entirely novel functions. Document computational costs alongside accuracy. Compare against established baselines systematically. Release verified open-source implementations.
Glossary
Activation Function: Mathematical operation applied to a neuron's weighted input sum, determining the neuron's output and introducing non-linearity into neural networks.
Backpropagation: Algorithm for training neural networks by computing gradients of the loss function with respect to network weights, working backward from output to input layers.
Batch Normalization: Technique that normalizes layer inputs across training batches, stabilizing training and reducing sensitivity to weight initialization and activation choice.
CNN (Convolutional Neural Network): Neural network architecture using convolutional layers, primarily for image processing tasks, typically using ReLU activation in hidden layers.
Dead Neuron: Neuron that permanently outputs zero due to ReLU activation with consistently negative pre-activation values, effectively removed from the network.
Derivative: Measure of how much a function's output changes for small changes in input, crucial for gradient-based optimization during backpropagation.
ELU (Exponential Linear Unit): Activation function allowing smooth negative values using exponential curve, pushing mean activations toward zero.
Exploding Gradients: Problem where gradients grow exponentially during backpropagation, causing unstable training and numerical overflow.
GELU (Gaussian Error Linear Unit): Smooth, probabilistic activation function standard in Transformer encoders, gating inputs by likelihood of being positive under standard normal distribution.
Gradient: Vector of partial derivatives indicating direction and magnitude of steepest increase in loss function, used to update weights during training.
Hidden Layer: Network layer between input and output, transforming inputs through weighted connections and activation functions to learn intermediate representations.
Leaky ReLU: ReLU variant allowing small negative slope for negative inputs, preventing complete neuron death while maintaining ReLU benefits.
Non-linearity: Property enabling functions to learn patterns beyond straight lines, essential for neural networks to solve complex problems.
Pre-activation: Weighted sum of neuron inputs before applying activation function, z = w₁x₁ + w₂x₂ + ... + wₙxₙ + b.
ReLU (Rectified Linear Unit): Most widely-used activation function, outputting input if positive, zero otherwise, solving vanishing gradients and enabling deep networks.
Saturation: Property where activation functions output near-constant values for large inputs, causing small derivatives and vanishing gradients.
Sigmoid: S-shaped activation function mapping inputs to (0,1), historically common but causing vanishing gradients in deep networks, now primarily used in binary classification outputs.
SiLU/Swish: Activation function f(x) = x × sigmoid(x), self-gating mechanism standard in Transformer decoders and large language models.
Softmax: Output activation for multi-class classification, converting raw scores to probability distribution summing to 1.
Sparsity: Property where many neurons output exactly zero, reducing computational cost and potentially improving generalization.
Tanh (Hyperbolic Tangent): S-shaped activation mapping inputs to (-1,1), zero-centered improvement over sigmoid, standard in LSTM/GRU gate mechanisms.
Transformer: Neural network architecture using attention mechanisms, typically using GELU or SiLU activation in feed-forward networks.
Universal Approximation: Property that neural networks with sufficient neurons and non-linear activations can approximate any continuous function arbitrarily well.
Vanishing Gradients: Problem where gradients shrink exponentially during backpropagation through many layers, preventing early layers from learning, historically caused by sigmoid/tanh activations.
Weight Initialization: Strategy for setting initial network weights before training, crucial for gradient flow and preventing vanishing/exploding gradients.
Sources & References
Brenndoerfer, M. (2025, June 14). FFN Activation Functions: ReLU, GELU, and SiLU for Transformer Models. Retrieved from https://mbrenndoerfer.com/writing/ffn-activation-functions
Devlin, J., Chang, M., Lee, K., & Toutanova, K. (2018). BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding. arXiv:1810.04805.
DigitalOcean. (2025, June 6). Vanishing Gradient Problem in Deep Learning: Explained. Retrieved from https://www.digitalocean.com/community/tutorials/vanishing-gradient-problem
Flasche, T. (2025, September 28). BERT — Bidirectional Encoder Representations from Transformers. Medium. Retrieved from https://medium.com/@tnodecode/bert-bidirectional-encoder-representations-from-transformers-0696d29f9d11
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