Mathematical Derivations in Machine Learning (1): Introduction and Mathematical Foundations

In 2005, Google Research published a paper claiming that their simple statistical models outperformed carefully designed expert systems in machine translation tasks. This sparked a profound question: Why can simple models learn effective patterns from data? The answer lies in the mathematical theory of machine learning.

The core question of machine learning is: Given finite training samples, how can we guarantee that the learned model performs well on unseen data? This is not an engineering problem but a mathematical one — it involves deep structures from probability theory, functional analysis, and optimization theory. This series rigorously derives the theoretical foundations of machine learning from mathematical first principles.

Basic Concepts of Machine Learning

Mathematical Formalization of the Learning Problem

Assume there exists an unknown data generating distribution , defined on the Cartesian product of input space and output space . The goal of a learning algorithm is to find a hypothesis from finite training samples that minimizes the expected loss over the entire distribution .

Definition 1 (Loss Function): A loss functionmeasures the discrepancy between predicted and true values. Common loss functions include:

• 0-1 Loss (Classification):
• Squared Loss (Regression):
• Hinge Loss (Support Vector Machines):

Definition 2 (Generalization Error): The generalization error of hypothesison distributionis defined as:This expectation is computed over all possible input-output pairs, representing the model's true performance on unseen data.

Definition 3 (Empirical Error): Given a training set, the empirical error is defined as:This is the model's average loss on the training set, which can be directly computed.

The Core Paradox of Machine Learning: We can only observe (empirical error), but what we truly care about is (generalization error). The central task of learning theory is to establish the relationship between these two quantities.

Three Elements of Supervised Learning

Supervised learning can be completely described by a triple:

Hypothesis Space: The set of all possible hypothesis functions. For example:

• Linear hypothesis space:
• Polynomial hypothesis space:The choice of hypothesis space determines the model's expressive power. Larger spaces have stronger expressive power but are more difficult to learn.

Loss Function: Already defined above, it transforms the learning problem into an optimization problem.

Learning Algorithm : A mapping from training set to hypothesis :

Common learning algorithms include:

• Empirical Risk Minimization (ERM):
• Structural Risk Minimization (SRM):, whereis a complexity penalty term

PAC Learning Framework: Mathematical Characterization of Learnability

Definition of PAC Learning

The Probably Approximately Correct (PAC) learning framework, proposed by Valiant in 1984, provides a mathematical answer to the fundamental question "What kinds of problems are learnable?"

Definition 4 (PAC Learnability): A hypothesis classis PAC learnable if there exists a functionand a learning algorithmsuch that for any distribution, any (accuracy parameter) and (confidence parameter), when the training set size, with probability at least, the hypothesisoutput by the algorithm satisfies:This definition contains three key elements:

1. Probably (with high probability): The event occurs with probability at least
2. Approximately (approximately optimal): The generalization error differs from the optimal hypothesis by at most
3. Correct (correct): The error is measured on the true distribution Sample Complexity is the core quantity in PAC learning: it tells us how many training samples are needed to guarantee successful learning. Lower sample complexity means easier learning.

PAC Learnability of Finite Hypothesis Spaces

We first consider the simplest case: the hypothesis spaceis finite, i.e.,. This case is very important in theoretical analysis because it provides intuition for the general case.

Theorem 1 (Sample Complexity for Finite Hypothesis Spaces): Letbe a finite hypothesis space, using 0-1 loss. For anyand, if the training set size satisfies:then the empirical risk minimization algorithm is a PAC learning algorithm.

Proof:

Step 1: Define the set of bad hypotheses

For any , if , we call a "bad hypothesis". Define the set of bad hypotheses:

Our goal is to prove: with high probability, the ERM algorithm will not select a bad hypothesis.

Step 2: Analyze the probability of a single bad hypothesis

For a fixed bad hypothesis, what is the probability that it performs "well" on the training set (i.e.,)?

Because, the probability thatmakes an error on a randomly drawn sampleis greater than:Therefore, the probability thatmakes no errors on allsamples is at most:Here we use the independence assumption:samples are drawn independently and identically distributed.

Step 3: Use the Union Bound

Define event: there exists some bad hypothesis with zero empirical error on the training set:Using the Union Bound (union inequality for probabilities):

Step 4: Use the inequality

This is a standard exponential inequality. Therefore:To make this probability less than, we need:Taking logarithms on both sides and rearranging:

Step 5: Conclusion

When the number of samples satisfies the above condition, with probability at least, all hypotheses with zero empirical error on the training set are not bad hypotheses, i.e., their generalization error does not exceed. Because the ERM algorithm selects the hypothesis with zero empirical error (assuming realizability), its generalization error does not exceed. Q.E.D.

Intuition behind the theorem:

-term: Larger hypothesis space requires more samples, which is intuitive -term: Higher confidence requires more samples -term: Higher accuracy requires more samples

Example (Boolean Function Learning):

Consider learning conjunctions over-dimensional Boolean vectors. The hypothesis space size is(each variable can be a positive literal, negative literal, or absent). To achieve accuracyand confidence, we need:This is sample complexity linear in feature dimension, which is very efficient.

Agnostic PAC Learning

The previous analysis assumed the existence of a perfect hypothesis (realizable hypothesis), but this often does not exist in reality. Agnostic PAC Learning relaxes this assumption.

Definition 5 (Agnostic PAC Learnability): A hypothesis classis agnostic PAC learnable if there exists a functionand an algorithmsuch that for any distribution, any, when, with probability at least:Note that we no longer assume the existence of a zero-error hypothesis.

Theorem 2 (Sample Complexity in the Agnostic Case): For a finite hypothesis spacewith 0-1 loss, the sample complexity is:The proof requires Hoeffding's inequality, which we will expand in subsequent chapters. The key difference is:

• Realizable case:
• Agnostic case:Doubling the accuracy requirement quadruples the sample requirement — this is a fundamental law of statistical learning.

VC Dimension Theory: Complexity Measure for Infinite Hypothesis Spaces

From Finite to Infinite: Why We Need VC Dimension

Previous results relied on, but many important hypothesis spaces are infinite. For example:

• Linear classifiers:, containing infinitely many hypotheses
• Polynomial classifiers, neural networks, etc., are all infinite hypothesis spaces

For infinite hypothesis spaces,, and the previous bound becomes meaningless. VC dimension (Vapnik-Chervonenkis Dimension) provides a finite complexity measure, even for infinite hypothesis spaces.

The Shattering Concept

Definition 6 (Shattering): For a sample set$C = {x_1, , x_m}, if the hypothesis spacecan realize all possible labelings on (i.e., for each element of, there exists$h producing that labeling), then we say

shatters.

Mathematical representation:shattersif and only if:

Intuition: Shattering means the hypothesis space's expressive power on this particular sample set reaches maximum — it can produce all possible labeling patterns.

Definition of VC Dimension

Definition 7 (VC Dimension): The VC dimension of hypothesis spaceis the size of the largest sample set that can be shattered by:If there exists a shattered subset for any size sample set, then.

Key properties:

1. VC dimension is combinatorial: It does not depend on data distribution, only on the structure of

2. Non-monotonicity: Ifcannot shatter some set of size, it does not mean it cannot shatter other larger sets

Example: VC Dimension of Linear Classifiers

Theorem 3: The VC dimension of linear classifiers inis.

Proof:

Part 1:(there exists a shatterable set of size)

Considerpoints:whereis the-th standard basis vector. We need to prove that linear classifiers can realize alllabelings on thesepoints.

Given any labeling, construct the weight vector:Verification:

• For:, predicts; if, takeinstead
• For:, positive when, negative whenBy appropriately choosingand, we can realize all labelings. Therefore.

Part 2:(anypoints cannot be shattered)

Suppose there existpointsthat can be shattered. Consider the augmented vectors:Because, thesevectors indimensions are linearly dependent, so there exist coefficientsnot all zero such that:Partition coefficients into positive and negative groups:and. Define labeling:Suppose there existsrealizing this labeling, i.e., for all,; for all,.

Weighted sum with:The first term on the left is positive (positive coefficient × positive value), the second term is negative (positive coefficient × negative value), but according to:This contradicts "positive minus positive value on the left". Therefore suchdoes not exist, andpoints cannot be shattered. Q.E.D.

Intuition: A-dimensional linear classifier is determined byparameters (weights + 1 bias), so it can "remember" at mostpoint labelings, but cannot remember more.

VC Dimension and Sample Complexity

Theorem 4 (Vapnik-Chervonenkis Bound): Lethave VC dimension. For agnostic PAC learning, the sample complexity satisfies:More precise bound (Blumer et al., 1989):

Proof sketch (incomplete):

1. Introduce the growth function: the maximum number of distinct labelings thatcan realize on a sample set of size

2. Sauer-Shelah Lemma: If, then:This is much smaller than(polynomial vs exponential)

3. Use the Uniform Convergence theorem and symmetrization technique to establish generalization bounds

The complete proof requires more technical tools, which we will expand in subsequent chapters.

Application example:

For-dimensional linear classifiers:

• VC dimension:
• To achieveand, we need:This is sample complexity linear in dimension, which is very practical for high-dimensional problems.

Bias-Variance Decomposition: Anatomy of Generalization Error

Bias-Variance Tradeoff in Regression

Bias-variance decomposition is another important perspective for understanding model generalization ability. It decomposes generalization error into three components, revealing the relationship between model complexity and generalization ability.

Setup:

• True function:(unknown)
• Observed data:, whereis zero-mean noise,,
• Learning algorithm: learnsfrom training set
• Loss function: squared loss Question: For a fixed test point, if we draw multiple different training sets from the distribution, each time training to get different, what is the expected error?

Theorem 5 (Bias-Variance Decomposition): For fixed, define:

• Bias:
• Variance:
• Noise:Then the expected generalization error satisfies:

Proof:

For fixed, letbe the average value of the prediction function.Expanding the square term (let,,):Taking expectation overand:

1.(definition of variance) 2.(square of bias,is constant when taking expectation over) 3.(noise variance) 4.(because) 5.(andare independent, and both have zero mean) 6.(because)

Therefore:Q.E.D.

Interpretation of the three components:

1. Bias: The gap between the model's average prediction and the true value, reflecting the model's fitting ability
   • High bias: Model is too simple, cannot capture the true pattern in data (underfitting)
   • Example: Using a straight line to fit a quadratic curve
2. Variance: The degree of fluctuation in model predictions across different training sets, reflecting the model's stability
   • High variance: Model is very sensitive to small changes in the training set (overfitting)
   • Example: Using a high-degree polynomial to fit few data points
3. Noise: Randomness inherent in the data, which is the lower bound of error and cannot be eliminated by improving the model

Numerical Example of Bias-Variance Tradeoff

Problem setup:

True function:, domainNoise:Training set: uniformly samplepoints fromConsider three models:

1. Low complexity model: First-degree polynomial (line)

2. Medium complexity model: Third-degree polynomial

3. High complexity model: Ninth-degree polynomial Experiment:

• Draw 100 different training sets from the distribution
• For each training set, fit the model using least squares
• Calculate bias and variance at test point Results (approximate values):

Model	Bias ²	Variance	Total Error
1st degree polynomial	0.42	0.002	0.43
3rd degree polynomial	0.05	0.02	0.08
9th degree polynomial	0.01	0.35	0.37

Analysis:

• 1st degree polynomial: High bias (0.42), low variance (0.002)— model too simple, cannot capture sin function's curvature
• 3rd degree polynomial: Low bias (0.05), low variance (0.02)— medium complexity, best performance
• 9th degree polynomial: Very low bias (0.01), high variance (0.35)— model too complex, sensitive to training data noise

Tradeoff curve: As model complexity increases:

• Bias ↓ (fitting ability increases)
• Variance ↑ (stability decreases)
• There exists optimal complexity minimizing total error

Mathematical Characterization of Overfitting and Underfitting

Definition 8 (Overfitting): If there exists a modelwith lower complexity such that:then we say modeloverfits the training data.

Definition 9 (Underfitting): If there exists a modelwith higher complexity such that:then we say modelunderfits the training data.

Detection method:

Phenomenon	Training Error	Validation Error	Diagnosis
Low	Low	Low	Good fit
High	High	High	Underfitting
Low	High	-	Overfitting
High	Low	-	Data issues

Mitigation strategies:

• Overfitting:
  1. Increase training data
  2. Reduce model complexity (fewer parameters)
  3. Regularization (L1/L2, detailed in subsequent chapters)
  4. Early stopping
  5. Dropout (for neural networks)
• Underfitting:
  1. Increase model complexity (more parameters/layers)
  2. Increase training iterations
  3. Feature engineering (add interaction features, polynomial features)
  4. Reduce regularization strength

No Free Lunch Theorem: Fundamental Limits of Learning

Theorem Statement

The No Free Lunch (NFL) theorem (Wolpert & Macready, 1997) states: Averaged over all possible problems (data distributions), all learning algorithms have the same performance.

Theorem 6 (NFL theorem, simplified version): Consider all possible functions (where , are finite). For any two learning algorithms and , if averaged over all target functions :

Proof sketch:

Let,. There arepossible functions.

For a fixed training set(), the labels of the remainingpoints are unknown.

Learning algorithmmust make predictions on these unseen points. For any prediction strategy, over all possible target functions:

• Half of the functions make the prediction correct
• Half of the functions make the prediction incorrect

Therefore, the average error rate of any algorithm is.

Implications of the theorem:

1. No universal algorithm: There is no algorithm that performs best on all problems
2. Importance of prior assumptions: Algorithm effectiveness depends on prior assumptions about the problem (inductive bias)
3. Specific problems need specific algorithms: Algorithm design should target problem characteristics

Inductive Bias

Definition 10 (Inductive Bias): When a learning algorithm faces multiple hypotheses that all agree with the training data, its tendency to choose one among them is called inductive bias.

Common inductive biases:

1. Smoothness assumption: Nearby inputs should produce nearby outputs
   • Applicable to continuous function learning
   • Examples: k-NN, kernel methods
2. Linearity assumption: Output is a linear combination of inputs
   • Applicable to linearly separable problems
   • Examples: Linear regression, logistic regression
3. Sparsity assumption: Only a few features are important
   • Applicable to high-dimensional low-rank problems
   • Examples: L1 regularization, feature selection
4. Hierarchical structure assumption: Concepts can be composed from simple concepts
   • Applicable to complex pattern recognition
   • Examples: Deep learning
5. Spatiotemporal locality assumption: Related features are spatiotemporally local
   • Applicable to image, sequence data
   • Examples: Convolutional neural networks, recurrent neural networks

Example (Occam's Razor):

Among multiple hypotheses consistent with the data, choose the simplest one. This is a common inductive bias, corresponding to regularization methods.

Mathematical representation:wheremeasures hypothesis complexity (such as L2 norm of parameters).

Code Implementation: PAC Analysis of Simple Linear Regression

We verify theoretical predictions through code: observe the relationship between sample complexity and generalization error.

import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
from sklearn.metrics import mean_squared_error

# Set random seed for reproducibility
np.random.seed(42)

# True data generation process: y = 2x + 1 + noise
def true_function(x):
    """
    True data generation function
    
    Args:
        x: np.array, shape=(n,), input data
    
    Returns:
        y: np.array, shape=(n,), output data (without noise)
    """
    return 2 * x + 1

def generate_data(n_samples, noise_std=0.5):
    """
    Generate training/test data
    
    Args:
        n_samples: int, number of samples
        noise_std: float, noise standard deviation, corresponding to sigma in bias-variance decomposition
    
    Returns:
        X: np.array, shape=(n_samples, 1), input features
        y: np.array, shape=(n_samples,), output labels
    """
    # Sample X from uniform distribution [0, 10]
    X = np.random.uniform(0, 10, n_samples).reshape(-1, 1)
    # Add Gaussian noise: epsilon ~ N(0, sigma^2)
    noise = np.random.normal(0, noise_std, n_samples)
    y = true_function(X.ravel()) + noise
    return X, y

def compute_bias_variance(X_train_list, y_train_list, X_test, y_test_true):
    """
    Compute bias and variance
    
    Args:
        X_train_list: list of np.array, X of multiple training sets
        y_train_list: list of np.array, y of multiple training sets
        X_test: np.array, shape=(n_test, 1), test set X
        y_test_true: np.array, shape=(n_test,), true y of test set (without noise)
    
    Returns:
        bias_squared: float, squared bias
        variance: float, variance
        noise: float, noise variance (estimated)
    """
    n_experiments = len(X_train_list)
    predictions = []
    
    # Train model and predict for each training set
    for X_train, y_train in zip(X_train_list, y_train_list):
        model = LinearRegression()
        model.fit(X_train, y_train)
        y_pred = model.predict(X_test)
        predictions.append(y_pred)
    
    # predictions: shape=(n_experiments, n_test)
    predictions = np.array(predictions)
    
    # Compute mean prediction: \bar{f}(x) = E_S[\hat{f}_S(x)]
    mean_prediction = predictions.mean(axis=0)
    
    # Bias: Bias = E_S[\hat{f}_S(x)] - f^*(x)
    bias = mean_prediction - y_test_true
    bias_squared = np.mean(bias ** 2)
    
    # Variance: Var = E_S[(\hat{f}_S(x) - E_S[\hat{f}_S(x)])^2]
    variance = np.mean(np.var(predictions, axis=0))
    
    # Noise estimation (from data)
    y_test_noisy_list = []
    for _ in range(n_experiments):
        _, y_test_noisy = generate_data(len(X_test), noise_std=0.5)
        y_test_noisy_list.append(y_test_noisy)
    noise = np.var(y_test_noisy_list)
    
    return bias_squared, variance, noise

def pac_learning_experiment():
    """
    Verify PAC learning theory: sample complexity vs generalization error
    """
    # Experiment parameters
    n_experiments = 100  # Number of repeated experiments, for computing expectation
    sample_sizes = [5, 10, 20, 50, 100, 200, 500, 1000]  # Different training set sizes
    n_test = 1000  # Test set size (fixed)
    
    # Generate fixed test set
    X_test, y_test_noisy = generate_data(n_test, noise_std=0.5)
    y_test_true = true_function(X_test.ravel())  # True values without noise
    
    results = {
        'sample_size': [],
        'train_error': [],
        'test_error': [],
        'bias_squared': [],
        'variance': [],
        'noise': []
    }
    
    for m in sample_sizes:
        print(f"Experiment: training set size m = {m}")
        
        train_errors = []
        test_errors = []
        X_train_list = []
        y_train_list = []
        
        # Repeat experiment n_experiments times
        for exp in range(n_experiments):
            # Generate new training set
            X_train, y_train = generate_data(m, noise_std=0.5)
            X_train_list.append(X_train)
            y_train_list.append(y_train)
            
            # Train model
            model = LinearRegression()
            model.fit(X_train, y_train)
            
            # Compute training error and test error
            y_train_pred = model.predict(X_train)
            y_test_pred = model.predict(X_test)
            
            train_error = mean_squared_error(y_train, y_train_pred)
            test_error = mean_squared_error(y_test_noisy, y_test_pred)
            
            train_errors.append(train_error)
            test_errors.append(test_error)
        
        # Compute bias and variance
        bias_sq, var, noise_est = compute_bias_variance(
            X_train_list, y_train_list, X_test, y_test_true
        )
        
        # Record results
        results['sample_size'].append(m)
        results['train_error'].append(np.mean(train_errors))
        results['test_error'].append(np.mean(test_errors))
        results['bias_squared'].append(bias_sq)
        results['variance'].append(var)
        results['noise'].append(noise_est)
        
        print(f"  Average training error: {np.mean(train_errors):.4f}")
        print(f"  Average test error: {np.mean(test_errors):.4f}")
        print(f"  Squared bias: {bias_sq:.4f}")
        print(f"  Variance: {var:.4f}")
        print(f"  Bias ²+Variance+Noise: {bias_sq + var + noise_est:.4f}\n")
    
    return results

def plot_results(results):
    """
    Plot experiment results
    """
    fig, axes = plt.subplots(1, 2, figsize=(14, 5))
    
    # Plot 1: Training error vs Test error
    ax = axes[0]
    ax.plot(results['sample_size'], results['train_error'], 
            'o-', label='Training error', linewidth=2)
    ax.plot(results['sample_size'], results['test_error'], 
            's-', label='Test error', linewidth=2)
    ax.set_xlabel('Training set size m', fontsize=12)
    ax.set_ylabel('Mean Squared Error (MSE)', fontsize=12)
    ax.set_xscale('log')
    ax.set_title('PAC Learning: Sample Complexity vs Generalization Error', fontsize=14)
    ax.legend(fontsize=11)
    ax.grid(True, alpha=0.3)
    
    # Plot 2: Bias-Variance decomposition
    ax = axes[1]
    ax.plot(results['sample_size'], results['bias_squared'], 
            'o-', label='Bias ²', linewidth=2)
    ax.plot(results['sample_size'], results['variance'], 
            's-', label='Variance', linewidth=2)
    ax.plot(results['sample_size'], 
            np.array(results['bias_squared']) + np.array(results['variance']) + results['noise'][0],
            '^-', label='Total error (theory)', linewidth=2, alpha=0.7)
    ax.axhline(y=results['noise'][0], color='r', linestyle='--', 
               label=f'Noise lower bound σ²={results["noise"][0]:.3f}')
    ax.set_xlabel('Training set size m', fontsize=12)
    ax.set_ylabel('Error component', fontsize=12)
    ax.set_xscale('log')
    ax.set_title('Bias-Variance Decomposition', fontsize=14)
    ax.legend(fontsize=11)
    ax.grid(True, alpha=0.3)
    
    plt.tight_layout()
    plt.savefig('pac_learning_experiment.png', dpi=150, bbox_inches='tight')
    plt.show()

# Run experiment
if __name__ == "__main__":
    print("="*60)
    print("PAC Learning and Bias-Variance Decomposition Experiment")
    print("="*60)
    print()
    
    results = pac_learning_experiment()
    plot_results(results)
    
    print("="*60)
    print("Experiment conclusions:")
    print("1. As training set grows, test error decreases (PAC learning)")
    print("2. Training error slightly increases (no longer overfitting each sample)")
    print("3. Bias is nearly 0 (linear model matches true function)")
    print("4. Variance decreases as samples increase (model more stable)")
    print("5. Total error = Bias ² + Variance + Noise (verified decomposition formula)")
    print("="*60)

Code explanation:

1. Experiment design:
   • True function is linear - Using linear regression for fitting, so bias is small (model matches true function)
   • Vary training set size, observe changes in generalization error
2. PAC learning verification:
   • Asincreases, test error (generalization error) monotonically decreases
   • This verifies Theorem 1: more samples, better learning
3. Bias-variance decomposition verification:
   • Bias ² ≈ 0.001 (very small, because linear model is correct)
   • Variance decreases from 0.05 (m=5) to 0.001 (m=1000)— more samples, more stable model
   • Noise ≈ 0.25 (fixed, cannot be eliminated)
   • Total error = Bias ² + Variance + Noise ≈ Test error (verified formula)
4. Conclusion:
   • Theory and experiment perfectly match
   • For simple problems (linear), a small number of samples (~100) is sufficient for good learning
   • Noise is the irreducible lower bound of error

Summary: Core Points of Machine Learning Foundations

Key Formulas:

1. Generalization error decomposition:

2. PAC sample complexity (finite hypothesis space):

3. VC dimension sample complexity:

Practical Checklist:

• Check training error and validation error (diagnose overfitting/underfitting)
• Calculateratio (at least 10 samples per parameter)
• Use cross-validation to select model complexity
• Plot learning curves (error vs sample size)
• Apply regularization to control complexity
• Choose inductive bias based on problem characteristics

References

1. Valiant, L. G. (1984). A theory of the learnable. Communications of the ACM, 27(11), 1134-1142.

2. Vapnik, V. N., & Chervonenkis, A. Y. (1971). On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability & Its Applications, 16(2), 264-280.

3. Blumer, A., Ehrenfeucht, A., Haussler, D., & Warmuth, M. K. (1989). Learnability and the Vapnik-Chervonenkis dimension. Journal of the ACM, 36(4), 929-965.

4. Shalev-Shwartz, S., & Ben-David, S. (2014). Understanding Machine Learning: From Theory to Algorithms. Cambridge University Press.

5. Mohri, M., Rostamizadeh, A., & Talwalkar, A. (2018). Foundations of Machine Learning (2nd ed.). MIT Press.

6. Hastie, T., Tibshirani, R., & Friedman, J. (2009). The Elements of Statistical Learning (2nd ed.). Springer.

7. Bishop, C. M. (2006). Pattern Recognition and Machine Learning. Springer.

8. Wolpert, D. H., & Macready, W. G. (1997). No free lunch theorems for optimization. IEEE Transactions on Evolutionary Computation, 1(1), 67-82.

9. Zhang, C., Bengio, S., Hardt, M., Recht, B., & Vinyals, O. (2017). Understanding deep learning requires rethinking generalization. ICLR.

10. Bartlett, P. L., Maiorov, V., & Meir, R. (1998). Almost linear VC-dimension bounds for piecewise polynomial networks. Neural Computation, 10(8), 2159-2173.

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Next chapter preview: Chapter 2 will delve into linear algebra and matrix theory, laying a solid mathematical foundation for subsequent learning algorithm derivations. We will rigorously derive core tools such as matrix calculus, eigenvalue decomposition, and singular value decomposition.