Using Second-Order Derivatives for Training Models
[E] How can we use second-order derivatives for training models?
Second-order derivatives can be used to train models via optimization techniques that go beyond gradient-based (first-order) methods. One popular second-order method is Newton's method, which uses both the gradient and the Hessian (the matrix of second-order partial derivatives) to update model parameters. The update rule is:
xt+1=xt−H−1∇f(xt)\mathbf{x}_{t+1} = \mathbf{x}_t - H^{-1} \nabla f(\mathbf{x}_t)xt+1=xt−H−1∇f(xt)
where:
- xt\mathbf{x}_txt is the current point (parameter vector),
- HHH is the Hessian matrix (second-order derivatives),
- ∇f(xt)\nabla f(\mathbf{x}_t)∇f(xt) is the gradient (first-order derivative).
The Hessian provides information about the curvature of the loss function, allowing second-order methods to make larger, more accurate updates.
Pros and Cons of Second-Order Optimization
[M] Pros and cons of second-order optimization.
Pros:
- Faster convergence: Second-order methods can achieve faster convergence, especially when the loss function has steep gradients or curvature.
- Better direction: The Hessian gives curvature information, helping to adjust the learning step more accurately than gradient descent.
- Good for ill-conditioned problems: Second-order methods handle optimization problems with ill-conditioned (steep and flat) surfaces better than first-order methods.
Cons:
- Computational cost: Calculating and inverting the Hessian is expensive, particularly for large models.
- Storage complexity: Storing the Hessian requires significant memory, as it is a matrix of size n×nn \times nn×n, where nnn is the number of parameters.
- Risk of non-positive definite Hessians: If the Hessian is not positive definite (i.e., has negative eigenvalues), Newton's method may not converge properly.
Why We Don’t See More Second-Order Optimization in Practice
[M] Why don’t we see more second-order optimization in practice?
Second-order optimization methods are less common because:
- High computational cost: Calculating and storing the Hessian is too expensive for modern deep learning models with millions of parameters.
- Efficient alternatives: First-order methods like stochastic gradient descent (SGD) and its variants (e.g., Adam) are computationally simpler and often sufficient to achieve good results, especially with large datasets.
- Scalability issues: The inversion of the Hessian is particularly impractical for large-scale models, which limits the feasibility of second-order methods in practice.
Hessian and Critical Points
[M] How can we use the Hessian (second derivative matrix) to test for critical points?
To classify a critical point (where the gradient is zero) of a function f(x)f(\mathbf{x})f(x), the Hessian H(x)H(\mathbf{x})H(x) can be used as follows:
- Positive definite Hessian: If the Hessian is positive definite at a critical point (all its eigenvalues are positive), the point is a local minimum.
- Negative definite Hessian: If the Hessian is negative definite (all its eigenvalues are negative), the point is a local maximum.
- Indefinite Hessian: If the Hessian has both positive and negative eigenvalues, the point is a saddle point.
Thus, the eigenvalues of the Hessian provide information about the nature of the critical point.
Jensen's Inequality
[E] What is Jensen’s inequality?
Jensen’s inequality states that for a convex function fff and a random variable XXX, the following inequality holds:
f(E[X])≤E[f(X)]f(\mathbb{E}[X]) \leq \mathbb{E}[f(X)]f(E[X])≤E[f(X)]
In other words, the value of a convex function evaluated at the expected value of a random variable is less than or equal to the expected value of the function applied to the random variable. For concave functions, the inequality is reversed.
Jensen's inequality is a foundational concept in probabilistic inference methods like Expectation-Maximization (EM) and variational inference, where it is used to approximate difficult integrals or optimize likelihoods.
Occam's Razor in Machine Learning
[E] How do we apply Occam's razor in ML?
Occam's Razor, which prefers simpler explanations over complex ones, is applied in machine learning by favoring simpler models (fewer parameters or assumptions) that generalize well to new data. This is often done through:
- Regularization: Techniques like L2 or L1 regularization penalize models with too many parameters to avoid overfitting.
- Model selection: Choosing simpler models when multiple models achieve similar performance to reduce the risk of overfitting and improve generalization.
Wide vs. Deep Neural Networks
[M] If we have a wide NN and a deep NN with the same number of parameters, which one is more expressive and why?
A deep neural network is generally more expressive than a wide neural network with the same number of parameters. Depth allows the network to compose simpler functions into more complex ones, effectively learning hierarchical representations. Each layer can capture different levels of abstraction, which is crucial for tasks like image and speech recognition. In contrast, wider networks tend to model complex relationships in a "shallow" way, which may require many more neurons to achieve the same expressive power.
Universal Approximation Theorem and Neural Networks
[H] The Universal Approximation Theorem states that a neural network with 1 hidden layer can approximate any continuous function for inputs within a specific range. Then why can’t a simple neural network reach an arbitrarily small positive error?
Although the Universal Approximation Theorem states that a neural network with one hidden layer can approximate any continuous function, it does not guarantee efficient approximation. There are several reasons why a simple neural network may fail to achieve arbitrarily small error:
- Number of neurons: The number of neurons required for accurate approximation could be prohibitively large, leading to impractical models.
- Training difficulties: Optimization issues like poor local minima, saddle points, or vanishing gradients can hinder effective learning.
- Generalization: Even if a network can approximate a function well on the training data, it may overfit and fail to generalize to unseen data.
- Non-convexity: The loss function is often non-convex, making it hard to find the global minimum.
Learning Paradigms
[E] Explain supervised, unsupervised, weakly supervised, semi-supervised, and active learning.
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Supervised learning: In this paradigm, the model is trained on labeled data. Each training example consists of an input and a corresponding label (output). The model learns to map inputs to outputs by minimizing the difference between its predictions and the actual labels. Example: image classification where each image has a corresponding label (e.g., "cat", "dog").
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Unsupervised learning: Here, the model is trained on unlabeled data. The goal is to find hidden patterns or structures in the data. Example: clustering (grouping similar data points together) or dimensionality reduction (PCA).
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Weakly supervised learning: In weakly supervised learning, the training data comes with incomplete, noisy, or imprecise labels. The goal is to learn useful patterns from data where the labels might be unreliable or sparse. Example: using noisy crowd-sourced labels or partially labeled datasets.
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Semi-supervised learning: This method uses both labeled and unlabeled data. Typically, a small portion of the data is labeled, and the model leverages the larger unlabeled portion to improve its performance. Example: self-training, where a model trained on labeled data is used to pseudo-label the unlabeled data for further training.
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Active learning: In active learning, the model can query an oracle (often a human annotator) to label new data points that are most informative for the model. The goal is to improve performance with minimal labeling effort by focusing on the most uncertain or impactful examples.
Empirical Risk Minimization (ERM)
[E] What’s the risk in empirical risk minimization?
In empirical risk minimization, "risk" refers to the expected loss or error that the model incurs when making predictions on new, unseen data. Formally, it is the expected value of a loss function L(f(x),y)L(f(x), y)L(f(x),y) over the true data distribution:
R(f)=E(x,y)∼P[L(f(x),y)]R(f) = \mathbb{E}_{(x, y) \sim P} [L(f(x), y)]R(f)=E(x,y)∼P[L(f(x),y)]
However, since the true data distribution PPP is unknown, the empirical risk is the average loss calculated over the training data.
[E] Why is it empirical?
It is called empirical because, in practice, we do not know the true underlying data distribution. Instead, we use the available training data as an approximation to compute the risk. Thus, the risk is based on the empirical distribution of the training set rather than the true population distribution.
[E] How do we minimize that risk?
We minimize empirical risk by finding the model parameters that reduce the average loss over the training data. This is typically done using optimization algorithms like gradient descent, which iteratively adjust the model parameters to minimize the loss function.
Occam's Razor in Machine Learning
[E] How do we apply Occam's razor in ML?
Occam's Razor, which prefers simpler explanations over complex ones, is applied in machine learning by favoring simpler models (fewer parameters or assumptions) that generalize well to new data. This is often done through:
- Regularization: Techniques like L2 or L1 regularization penalize models with too many parameters to avoid overfitting.
- Model selection: Choosing simpler models when multiple models achieve similar performance to reduce the risk of overfitting and improve generalization.
Deep Learning's Rise in Popularity
[E] What are the conditions that allowed deep learning to gain popularity in the last decade?
Several factors contributed to the recent success of deep learning:
- Data availability: The rise of large datasets (e.g., ImageNet) provided enough examples for deep learning models to learn effectively.
- Computational power: Advances in hardware, particularly GPUs and distributed computing, made it feasible to train large neural networks.
- Algorithmic improvements: Techniques like backpropagation, stochastic gradient descent, dropout, and batch normalization enabled more efficient training of deep networks.
- Open-source libraries: The development of tools like TensorFlow, PyTorch, and Keras lowered the barrier to entry for building and training neural networks.
- Better architectures: The invention of architectures like CNNs for image processing and RNNs/LSTMs for sequential data improved performance in specific tasks.
Wide vs. Deep Neural Networks
[M] If we have a wide NN and a deep NN with the same number of parameters, which one is more expressive and why?
A deep neural network is generally more expressive than a wide neural network with the same number of parameters. Depth allows the network to compose simpler functions into more complex ones, effectively learning hierarchical representations. Each layer can capture different levels of abstraction, which is crucial for tasks like image and speech recognition. In contrast, wider networks tend to model complex relationships in a "shallow" way, which may require many more neurons to achieve the same expressive power.
Universal Approximation Theorem and Neural Networks
[H] The Universal Approximation Theorem states that a neural network with 1 hidden layer can approximate any continuous function for inputs within a specific range. Then why can’t a simple neural network reach an arbitrarily small positive error?
Although the Universal Approximation Theorem states that a neural network with one hidden layer can approximate any continuous function, it does not guarantee efficient approximation. There are several reasons why a simple neural network may fail to achieve arbitrarily small error:
- Number of neurons: The number of neurons required for accurate approximation could be prohibitively large, leading to impractical models.
- Training difficulties: Optimization issues like poor local minima, saddle points, or vanishing gradients can hinder effective learning.
- Generalization: Even if a network can approximate a function well on the training data, it may overfit and fail to generalize to unseen data.
- Non-convexity: The loss function is often non-convex, making it hard to find the global minimum.
Saddle Points and Local Minima
[E] What are saddle points and local minima? Which are thought to cause more problems for training large NNs?
- A local minimum is a point where the loss function has a value smaller than that of its neighboring points, but not necessarily the smallest value globally.
- A saddle point is a point where the gradient is zero, but the point is neither a local minimum nor a local maximum. Instead, the function has a minimum in one direction and a maximum in another.
In large neural networks, saddle points are thought to cause more problems than local minima because the loss surface of deep networks is high-dimensional, and there can be many saddle points. At these points, gradient-based methods struggle because the gradient is near zero, causing slow progress in optimization. However, research has shown that local minima in deep networks are often "good enough" and close to the global minimum in terms of performance.
Jensen's Inequality
[E] Jensen’s inequality forms the basis for many algorithms for probabilistic inference, including Expectation-Maximization and variational inference. Explain what Jensen’s inequality is.
Jensen's inequality states that for a convex function fff and a random variable XXX, the following holds:
f(E[X])≤E[f(X)]f(\mathbb{E}[X]) \leq \mathbb{E}[f(X)]f(E[X])≤E[f(X)]
In other words, the value of the convex function applied to the expectation of XXX is less than or equal to the expectation of the function applied to XXX. If fff is concave, the inequality is reversed. This inequality is foundational in probabilistic inference techniques where approximating intractable expectations is crucial (e.g., in variational inference).
L1 vs. L2 Regularization
[M] Why does L1 regularization tend to lead to sparsity while L2 regularization pushes weights closer to 0?
- L1 regularization (Lasso) adds a penalty proportional to the absolute value of the coefficients:
L1 penalty=λ∑∣wi∣L1\text{ penalty} = \lambda \sum |w_i|L1 penalty=λ∑∣wi∣
This tends to shrink some weights exactly to zero, leading to a sparse model where only a few features have non-zero weights. This happens because the gradient of the L1 penalty is constant, which allows some weights to become zero during optimization.
- L2 regularization (Ridge) adds a penalty proportional to the square of the coefficients:
L2 penalty=λ∑wi2L2\text{ penalty} = \lambda \sum w_i^2L2 penalty=λ∑wi2
This tends to push the weights closer to zero but rarely makes them exactly zero. The squared term ensures that large weights are penalized more, but the gradient becomes smaller as weights approach zero, meaning they shrink gradually without becoming zero.
Summary: L1 encourages sparsity by setting weights exactly to zero, while L2 reduces the magnitude of weights, pushing them closer to zero but not to zero.
