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lecture 2 - linear models Flashcards

1
Q

What is the goal of linear regression?

A

To predict the value of a target variable t for a new input x, given a training dataset comprising N observations {x_n, t_n}.

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2
Q

How does the simplest approach to linear regression work?

A

In the simplest approach, linear regression involves directly constructing an appropriate function y(x) such that for new inputs x, the function predicts corresponding values of t.

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3
Q

How can linear regression be approached from a probabilistic perspective?

A

From a probabilistic perspective, we aim to model the predictive distribution p(t∣x), which expresses the uncertainty about the value of t for each value of x.

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4
Q

Why is the probabilistic approach useful in linear regression?

A

The probabilistic approach allows us to make predictions of t for any new value of x, minimizing the expected value of a suitably chosen loss function.

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5
Q

What is the goal of (polynomial) curve fitting?

A

To exploit the training set to discover the underlying function and make predictions for new inputs, even though individual observations are corrupted by noise.

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6
Q

Why is polynomial curve fitting considered linear in the parameters?

A
  • Although the function y(x,w) is nonlinear in x, it is linear in the parameters w.
  • This means that the model is a linear combination of the coefficients.
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7
Q

How are the coefficients in polynomial curve fitting determined?

A
  • by fitting the polynomial to the training data.
  • this is done by minimizing an error function that measures the misfit between the model predictions and the target values.
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8
Q

What is the error function used in polynomial curve fitting?

A
  • E(w) = (1/2) sum( y(x_n,w) - t_n)^2
  • sums over the number of data points
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9
Q

How is the optimal polynomial determined in polynomial curve fitting?

A
  • finding the value of w that minimizes the error function.
  • Since the error function is quadratic in w, it is convex, ensuring a unique minimum that can be calculated directly.
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10
Q

What is model selection in polynomial curve fitting?

A
  • Model selection is the process of choosing the order M of the polynomial.
  • A higher-order polynomial may fit the training data better, but it can result in overfitting if it poorly represents the underlying function.
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11
Q

What is overfitting in polynomial curve fitting?

A
  • Overfitting occurs when the model fits the training data too closely, including noise, resulting in a poor ability to generalize to new data.
  • This often happens with high-order polynomials.
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12
Q

How can overfitting be detected?

A
  • Overfitting can be detected by comparing the error on a training set and a separate test set.
  • Overfitted models show very low training error but high test error.
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13
Q

What metric is used to assess generalization performance?

A
  • The root-mean-square error (RMSE)
  • for each choice of M, we evaluate the residual value of E(w*) for both the test and training sets (separately)
  • becomes lower on the overfitted training set, but higher on the test set
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14
Q

Why does performance get worse as the polynomial order M increases?

A

As M increases:

  1. The magnitude of the coefficients increases significantly.
  2. Higher-order polynomials fit the training data exactly, including noise, rather than capturing the true trend.
  3. This results in poor generalization to new data.
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15
Q

What happens when M=9 and there are 10 coefficients in polynomial curve fitting?

A
  1. The training error goes to zero because the polynomial exactly fits all data points.
  2. The test error becomes very large due to overfitting, and the function exhibits wild oscillations.
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16
Q

What happens when M=0 and there are 10 coefficients in polynomial curve fitting?

A
  • the model is a straight line
  • w_0 + w_1
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17
Q

What happens when M=1 and there are 10 coefficients in polynomial curve fitting?

A
  • the model is a linear line
  • w_0 + w_1x_1
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18
Q

What general principle can be learned from overfitting?

A

Overfitting is a general property of maximum likelihood estimation. It can also occur in deep learning when training on a small dataset, leading to poor generalization.

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19
Q

What is the effect of adding observations to the training set on training error?

A
  • Adding more data provides more information for the model to learn from.
  • Training error may increase slightly because the model has to fit a more diverse dataset, but this improves generalization and reduces overfitting.
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20
Q

What is the effect of adding observations to the training set on test error?

A

Test error typically decreases because a larger training set helps the model generalize better to unseen data, reducing overfitting.

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21
Q

What is the effect of removing observations from the training set on training error?

A
  • Removing observations reduces the information available for training, making the model less robust.
  • Training error might decrease because the model fits the reduced dataset better, but this often leads to overfitting and worse generalization.
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22
Q

What is the effect of removing observations from the training set on test error?

A

Test error typically increases because the model is more prone to overfitting the smaller training set.

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23
Q

What is the effect of adding observations to the test set?

A
  • Adding observations to the test set does not affect the training error.
  • It provides a more reliable estimate of the model’s generalization performance, as the test set becomes more representative of the underlying data distribution.
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24
Q

What is the effect of removing observations from the test set?

A
  • Removing observations reduces the ability to accurately assess the generalization error.
  • Test error might appear to improve due to fewer diverse samples, but this can give a misleading picture of the model’s true performance.
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25
Q

Summary: How do changes in the training and test sets affect model performance?

A
  1. Adding training data improves generalization and reduces test error.
  2. Removing training data increases overfitting and worsens test error.
  3. Adding test data provides a more reliable estimate of generalization.
  4. Removing test data reduces the ability to assess model performance accurately.
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26
Q

Why can we afford to use more complex models with larger datasets?

A

With larger datasets, the model has more diverse data points to learn from, which reduces the risk of overfitting and improves generalization performance, allowing more flexible models to be used.

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27
Q

What is the goal of ridge regression (L2 regularization)?

A

To discourage large coefficients by adding a penalty term to the error function, ensuring the model is simpler and more generalizable. This reduces overfitting.

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28
Q

How is the error function modified in ridge regression?

A
  • The modified error function \tilde{E}(w) includes
  1. a penalty term proportional to the squared norm of the coefficients
  2. lambda, which controls the strength of the regularization
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29
Q

What happens when lnλ=−∞ ?

A
  • No regularization
  • The polynomial overfits, fitting the noise in the data
  • The coefficients take on very large values
30
Q

What happens as lnλ increases?

A
  1. The polynomial becomes smoother, fitting the general trend without overfitting the noise.
  2. The coefficients shrink significantly, reducing model flexibility and overfitting.
31
Q

What happens when lnλ=0 ?

A
  • Too much regularization
  • The fit becomes very simple, underfitting the data and failing to capture the trend.
  • The coefficients become almost zero, resulting in a straight-line prediction.
32
Q

How does regularization affect the RMSE plot wrt lnλ?

A
  • At small λ, the model overfits, leading to high test error and low training error.
  • As λ increases, the training error increases a bit, and the test error decreases and reaches a minimum, representing a balance between fitting the data and generalizing well (here are the reliable models).
  • At very high λ, the test and training error both increase, showing underfitting of the model
33
Q

What is the trade-off in choosing λ in ridge regression?

A

The trade-off involves balancing bias and variance:

  • Low λ: Low bias but high variance (overfitting).
  • High λ: High bias but low variance (underfitting).
  • The optimal λ minimizes test error by achieving a good bias-variance trade-off.
34
Q

What is the general linear model?

A
  • The general linear model is a linear combination of basis functions
  • ϕ_j(x) are the basis functions.
  • w is the vector of coefficients (parameters).
  • w_0 acts as a bias term when ϕ_0(x)=1
35
Q

What is the purpose of basis functions in the general linear model?

A

Basis functions handle nonlinear relationships between input variables while maintaining the analytical simplicity of a model that is linear in the parameters.

36
Q

What are global basis functions?

A
  • Global basis functions affect the entire input range when the input x changes.
  • Example: Polynomial basis function ϕ_j(x)=x^j
  • Disadvantage: Small approximation errors in specific areas affect the whole function.
37
Q

What are local basis functions?

A
  • Local basis functions affect only a limited region of the input space.
  • Example: Gaussian basis function ϕ_j(x)=exp(-(x-μ_j)^2)/(2s^2))
38
Q

What are sigmoidal basis functions?

A
  • Sigmoidal basis functions transition from 0 to 1 over a certain range of x
  • ϕ_j(x)= σ((x-μ_j)/s)
39
Q

How is the total number of parameters in the general linear model determined?

A

The total number of parameters is M, consisting of w_0 (bias) and M−1 coefficients for the basis functions

40
Q

How is the target variable
t modeled in maximum likelihood estimation?

A
  • t=y(x,w)+ϵ

Where:

  1. y(x,w) is the deterministic model (e.g., linear model).
  2. ϵ is noise, which is normally distributed with mean 0 and variance β^−1
41
Q

How is noise ϵ distributed in maximum likelihood estimation?

A
  • Noise ϵ is normally distributed with mean 0 and variance 𝛽^−1
  • p(ϵ∣β)=N(ϵ∣0,β^−1)
42
Q

What is the conditional probability of t given x,w, and β^−1?

A
  • follows a normal distribution that indicates that t is normally distributed around the model output y(x,w) with variance β^−1
  • p(t∣x,w,β^−1) = N(t∣y(x,w),β^−1)
43
Q

What does the conditional mean E[t∣x] represent?

A
  • the expected value of t given x, which is equal to the model output
  • E[t|x] = y(x,w)
  • the optimal prediction for a new value of x will be given by the conditional mean y(x,w) of the target variable
44
Q

What does it mean when the conditional distribution of
t given x is unimodal?

A
  • A unimodal conditional distribution implies that the probability of observing
    𝑡 is highest around the model output y(x,w)
  • the variance β^−1 determines the spread of the distribution around this prediction.
45
Q

What is the likelihood function in the context of maximum likelihood estimation?

A

The likelihood function p(t∣X,w,β) is the product of the individual Gaussian likelihoods for each input vector

46
Q

Why do we take the logarithm of the likelihood function?

A

Taking the logarithm of the likelihood function:

  1. Turns products into sums, reducing computational cost.
  2. Makes it easier to take derivatives, since differentiation of sums is simpler than products.
47
Q

What is the log-likelihood function?

A
  • ln p(t∣w,β) = 2/N lnβ − 2/N ln(2π) − βE_D(w)
  • where error function E_D(w) = 1/2(sum(t_n-w^Tϕ(x_n))^2)
48
Q

What is the relationship between minimizing the error function and maximizing the log-likelihood?

A
  • Minimizing the error function E_D(w) is equivalent to maximizing the log-likelihood under the Gaussian noise assumption.
  • This provides a motivation for using the error function as a maximum likelihood solution.
49
Q

What is the gradient of the log-likelihood with respect to w?

A

sum(t_n - w^T ϕ(x_n)}ϕ(x_n)^T

50
Q

What is the closed-form solution for the maximum likelihood estimate of w?

A
  • w_ML = ((Φ^T Φ)^−1 Φ^T t
  • this is the moore-penrose pseudo-inverse
  • represents the Ordinary Least Squares (OLS) solution, which minimizes the sum of squared errors (SSE) between the predicted and target values
51
Q

What is the design matrix Φ?

A
  • contains the basis functions
  • each column corresponds to a different basis function
  • each row evaluates all the basis functions on a particular x value
52
Q

What are the key steps in deriving the closed-form solution for w?

A
  1. Start with the likelihood function for the dataset.
  2. Take the log-likelihood to simplify the product into a sum.
  3. Express the log-likelihood as the sum of squared errors (SSE).
  4. Compute the gradient of the SSE with respect to w and set it to zero.
  5. Rewrite the result in matrix form using the design matrix Φ.
  6. Solve for w to get the closed-form solution.
53
Q

What are the differences between a closed-form solution and numerical methods?

A
  • Closed-Form Solution:
  1. Directly computes the exact solution.
  2. Efficient for small problems.
  3. Requires matrix inversion.
  • Numerical Methods:
    1. Approximate the solution iteratively (e.g., gradient descent).
  1. Scales better for large or complex problems.
  2. Examples: Gradient descent, Newton’s method.
54
Q

How can the least-squares solution be represented geometrically?

A

The least-squares solution y=Φw_ML represents the predicted vector y, which is a linear combination of the basis functions weighted by the coefficients w_ML

55
Q

What is the subspace S in the context of the least-squares solution?

A

The subspace S is spanned by the basis functions, meaning it is the space where the predictions y lie. The target vector t represents the actual observed data.

56
Q

What does the predicted vector y represent geometrically?

A
  • The predicted vector y is the orthogonal projection of the target vector t onto the subspace S.
  • This projection minimizes the distance between t and S.
57
Q

What is the role of w_ML in the geometrical interpretation?

A

The weights w_ML are chosen to minimize the distance between the target vector t and its projection onto the subspace S, ensuring the best fit to the data in a least-squares sense.

58
Q

How does the dimensionality M affect the least-squares solution?

A
  • The number of basis functions ϕ_m and corresponding weights w_m depends on the dimensionality M.
  • The weights adjust the model so that the predicted vector y lies as close as possible to the target vector t in the subspace S.
59
Q

What is the objective of regularized least squares?

A

Regularized least squares aims to minimize the sum of the error term and a regularization term to prevent overfitting.

60
Q

How is the error term in regularized least squares formulated?

A

The error term is the sum of the original squared error and a quadratic regularization term.

61
Q

What is the purpose of adding a penalty term in Ridge Regression (L2 regularization)?

A

The penalty term discourages large coefficients, making the model simpler and more generalizable by minimizing both the error on the data and the size of the model’s parameters.

62
Q

How does L1 regularization (Lasso) differ from L2 regularization (Ridge)?

A
  • L1 regularization (Lasso, q=1) penalizes the absolute values of the weights, often resulting in some weights being exactly zero, promoting sparsity.
  • L2 regularization (Ridge, q=2) penalizes the squared values of the weights, shrinking them towards zero without making them exactly zero.
63
Q

What does a high value of the regularization parameter λ imply?

A

High λ implies a stronger penalty, leading to lower coefficients and more shrinkage of the model weights.

64
Q

What happens when λ=0 in regularized least squares?

A

When λ=0, there is no regularization, and the problem reduces to normal least squares regression.

65
Q

What is the penalty applied to a weight w_j in L2 regularization when q=2?

A

The penalty for w_j is proportional to the square of the weight (w_j^2).

66
Q

q = 0.5

A
  • very aggressive regularization for q values lower than 1
  • sharp diamond plot: suggests that even small deviations from zero will be penalized heavily.
67
Q

q = 1

A
  • L1/lasso
  • pushing some weights exactly to zero, making the model “sparse” (some weights will be eliminated)
  • normal diamond plot
68
Q

q = 2

A
  • L2/quadratic
  • penalizes large weights but doesn’t push any of them to zero. Instead, it makes the weights smaller overall. (squared values)
  • round plot
69
Q

q = 4

A
  • as q increases, regularization becomes less aggressive, and large weights are penalized less severely.
  • rounded square plot
70
Q

Why is L1 regularization useful for feature selection?

A

L1 regularization drives some weights to exactly zero, effectively removing the corresponding features from the model.

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