Inductive bias refers to the set of prior assumptions or "biases" that a model relies on during the learning process. These assumptions influence how the model "generalizes from limited training data." In machine learning, we often face challenges such as insufficient training data, noise, and uneven data distribution. A model cannot simply rely on "memorization" to perform well in test sets or real-world scenarios. In such cases, the model needs to "make additional assumptions" to infer information that may not appear in the training data but exists in the real world. These additional assumptions can be understood as the model's "inductive bias."

In simple terms, in machine learning, "inductive bias" represents the prior assumptions a model makes when understanding and processing data. Without these assumptions or biases, it would be difficult for the model to learn generalized rules from limited samples, making it prone to overfitting or failing to acquire useful knowledge.

Why Do We Need Inductive Bias?

• Addressing Challenges from Limited Data: In real-world applications, it is often hard to collect large, perfectly distributed training datasets. For these limited or unevenly distributed datasets, a model cannot rely solely on memorization to achieve good generalization in real-world environments. By incorporating inductive bias (reasonable prior assumptions), we can embed prior knowledge about data distribution or task requirements into the model, enabling it to perform correctly or reasonably even with insufficient data.
• Preventing Overfitting: In deep learning, network architectures often involve millions or even billions of trainable parameters. While this gives models great expressive power, it also introduces severe risks of overfitting. Without reasonable prior constraints, models may use "unexpected" methods to memorize training samples, leading to poor performance on test data. Adding prior constraints (or regularization terms) to the loss function helps reduce overfitting, guiding the model to focus on more "reliable" patterns.
• Enforcing Physical Laws or Prior Knowledge: In certain specific fields, the data often embodies physical laws or domain logic that we already know. For example, in computer vision tasks, prior rules about lighting changes, perspective relationships, and viewpoint changes are well-understood. Similarly, in fluid dynamics or electromagnetism, certain equations or conservation laws have universal significance. Embedding these prior rules into the model reduces redundant search space and allows the model to learn solutions that are more consistent with reality.

Common Forms of Inductive Bias

1. Prior in Network Structure: For instance, in image processing, introducing convolutional layers (CNN) is a structural prior: it assumes local correlations and translational invariance in images, enabling good performance with small-scale training data.
2. Loss Function/Regularization: Adding constraints or penalty terms, such as L2 regularization or sparsity constraints, can be seen as introducing prior assumptions during optimization. In some cases, "Physics-Informed Neural Networks (PINNs)" can be understood as incorporating physical biases (e.g., satisfying specific differential equations) into the loss function, helping neural networks learn solutions consistent with physical laws without requiring large training datasets.
3. Data Augmentation: Techniques like lighting enhancement, rotation, cropping, and flipping are also common biases. Essentially, they tell the model, "Even if an image's lighting or angle changes, the classification or detection results should remain relatively stable." This represents a prior assumption about the distribution of image data.
4. Feature Engineering: Traditional machine learning often relied on expert-designed features (feature engineering). Although deep networks now automate feature extraction, in some scenarios, incorporating domain knowledge (e.g., holiday effects or seasonal variations in time-series data) as prior assumptions can still be important. These are considered inductive biases at the feature engineering level.

Applications and Importance of Inductive Bias

1. Improving Generalization Performance: Deep learning models often face the risk of overfitting due to their large capacity. Incorporating appropriate inductive bias in model architecture and training strategies allows models to focus on "inductive" patterns rather than "memorizing" data randomly. For example, skip connections in ResNet reflect prior assumptions about specific network structures, solving gradient vanishing problems in deep networks. Similarly, Transformers embed sequence dependency into self-attention mechanisms to capture contextual information.
2. Facilitating Multi-Task and Transfer Learning: In multi-task or transfer learning settings, inductive bias helps transfer experience from source tasks to target tasks. For instance, knowing that high or low-frequency components of images are particularly important in certain tasks allows us to use similar network structures or data processing strategies for new tasks, ensuring stable performance.
3. Incorporating Physical Information (PINNs): "Physics-Informed Neural Networks" leverage physical laws as priors, incorporating constraints from physical equations (e.g., partial differential equations) into neural network training. This significantly reduces the need for large-scale labeled data while ensuring the model's solutions adhere to scientific principles. Applications include fluid simulation, heat transfer, and electromagnetism, where PINNs bridge traditional numerical methods and deep learning for a balance between "high fidelity" and "fast computation."

Real-World Example: Inductive Bias in Low-Light Enhancement

In image processing tasks, light enhancement is commonly used to improve image quality and maintain consistent visual effects under varying lighting conditions. An effective method involves applying a multiplicative model, where a mask predicted by the network is element-wise multiplied with the original image to adjust lighting conditions. Mathematically, the model can be expressed as $\hat{y} = f(x) \times x$, where $x$ is the input image and $f(x)$ is a neural network-predicted brightness adjustment gain mask.

Since images are usually quantized to uint8, dequantization is often used together with light enhancement.

This pre-defined multiplicative model denotes the Inductive Bias, $f(x)$ acts as an adjuster, adaptively modifying lighting conditions without altering the core features of the image. This approach inherently reflects an inductive bias: despite changes in lighting, the main content and class of the image should remain unchanged. This method teaches the model to adapt to different lighting conditions while preserving essential image features.

To train such a model, mean squared error (MSE) loss is typically used to optimize the prediction of $f(x) \times x$, ensuring that the mask adjusts lighting accurately without introducing unnecessary distortion or noise. Additionally, smoothness regularization is applied to the prediction of $f(x)$ to enhance generalization and prevent overfitting. This regularization ensures spatial continuity and smoothness in lighting adjustment, avoiding unnatural transitions or excessive local changes.