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Uncertainty in Multiclass Classification

Dr. Rishabh Pathak
1. What is Uncertainty in Classification? Uncertainty refers to the model’s confidence or doubt in its predictions. Quantifying uncertainty is important to understand how reliable each prediction is. In multiclass classification , uncertainty estimates provide probabilities over multiple classes, reflecting how sure the model is about each possible class. 2. Methods to Estimate Uncertainty in Multiclass Classification Most multiclass classifiers provide methods such as: predict_proba: Returns a probability distribution across all classes. decision_function: Returns scores or margins for each class (sometimes called raw or uncalibrated confidence scores). The probability distribution from predict_proba captures the uncertainty by assigning a probability to each class. 3. Shape and Interpretation of predict_proba in Multiclass Output shape: (n_samples, n_classes) Each row corresponds to the probabilities of ...
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Another algorithm for maximizing `(θ)

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Dr. Rishabh Pathak

Context

·         Recall that in logistic regression, we model the probability of the binary label y∈{0,1} given input x: hθ​(x)=g(θTx)=1+exp(−θTx)1​ where g(⋅) is the sigmoid function.

·         The log-likelihood for n data points is: ℓ(θ)=∑i=1n​logp(y(i)∣x(i);θ) where p(y=1∣x;θ)=hθ​(x),p(y=0∣x;θ)=1−hθ​(x)

·         Maximizing ℓ(θ) corresponds to minimizing the negative log-likelihood, which yields the logistic loss.

Common Approach: Gradient Ascent/Descent

  • Typically, logistic regression parameters are estimated by applying gradient ascent (or descent on negative log-likelihood), updating parameters iteratively based on the gradient of ℓ(θ).

Alternative Algorithm: Newton's Method (Iteratively Reweighted Least Squares - IRLS)

·         Another algorithm for maximizing ℓ(θ) relies on second-order information, specifically the Hessian matrix of second derivatives of ℓ(θ).

·         This method is Newton's method applied to maximize the log-likelihood, sometimes called Iteratively Reweighted Least Squares (IRLS) in logistic regression.

Key Idea of the Algorithm

·         Newton's update iteratively updates parameters by: θ(t+1)=θ(t)−H−1(θ(t))∇ℓ(θ(t)) where

·         ∇ℓ(θ(t)) is the gradient of log-likelihood,

·         H(θ(t)) is the Hessian matrix of second derivatives at iteration t.

·         The Hessian captures curvature of the log-likelihood function, enabling more efficient and often faster convergence than gradient ascent alone.

Logistic Regression (IRLS) Specifics

  • At each iteration:
  • Compute predicted probabilities using current parameters: pi(t)​=hθ(t)​(x(i))
  • Construct a diagonal weight matrix W where: Wii​=pi(t)​(1−pi(t)​)
  • Calculate the adjusted response variable z: z=Xθ(t)+W−1(y−p(t))
  • Update θ by solving the weighted least squares problem: θ(t+1)=(XTWX)−1XTWz

Benefits

  • Faster convergence near the optimum because Newton's method uses curvature information.
  • Quadratic convergence once close to the solution compared to linear convergence of gradient methods.

Tradeoffs

  • Involves computing and inverting the Hessian matrix, which is computationally expensive for large datasets or high-dimensional features.
  • Gradient-based methods (stochastic gradient descent, mini-batch) are more scalable for big data settings.

Summary

  • The alternative algorithm to maximize ℓ(θ) is Newton’s method or IRLS, which uses both first and second derivatives.
  • At each iteration, parameters are updated by solving a weighted least squares problem using the Hessian and gradient of the log-likelihood.
  • This often yields faster and more accurate convergence for logistic regression model fitting.

 

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