Optimization algorithms like gradient descent are typically used in machine learning for which of the following (select all that apply):

Question

BlackTom AI · Accepted Answer

Gradient descent finds model parameters that minimize an objective function, typically expressed as an average loss or negative log-likelihood.

1) Relevant knowledge points and concepts
- Loss function: L(θ) = (1/n) Σ_i ℓ(hθ(x_i), y_i) is the average loss over training examples. Optimization algorithms adjust θ to minimize this.
- Maximum likelihood estimation (MLE): maximize the likelihood L(θ) = Π_i p(y_i | x_i; θ). Equivalently, maximize log-likelihood ℓ(θ) = Σ_i log p(y_i | x_i; θ) or minimize the negative log-likelihood −ℓ(θ) = −Σ_i log p(y_i | x_i; θ).
- Gradient descent: iterative update θ ← θ − α ∇_θ J(θ), where J(θ) is the objective to be minimized (e.g., average loss or negative log-likelihood).

2) Step-by-step reasoning
- If the objective is the average loss J(θ) = (1/n) Σ_i ℓ(...), gradient descent computes ∇J(θ) and updates θ to reduce J. This directly minimizes average loss.
- For MLE, the goal is maximize ℓ(θ). Replace maximization by minimization of negative log-likelihood: J(θ) = −ℓ(θ) = −Σ_i log p(y_i|x_i;θ). Applying gradient descent to J(θ) minimizes −ℓ(θ), which is equivalent to maximizing ℓ(θ) (the likelihood). Gradient ascent could also be used directly on ℓ(θ), but gradient descent on the negative log-likelihood is standard.

3) Calculations / intermediate steps (symbolic)
- Average loss update: θ ← θ − α ∇_θ [(1/n) Σ_i ℓ_i(θ)]
- MLE via negative log-likelihood: θ ← θ − α ∇_θ [−Σ_i log p(y_i|x_i;θ)] ⇒ increases Σ_i log p(y_i|x_i;θ)

4) Why the correct answers are correct
- Minimizing average loss: gradient descent explicitly targets minimizing loss functions.
- Maximizing likelihood: maximizing likelihood = minimizing negative log-likelihood; gradient descent can perform this minimization.

5) Why the other options are incorrect
- Calculating average loss: this is an evaluation step (computing a scalar), not what gradient descent is used for (optimization).
- Calculating probabilities: producing probabilities is model inference or prediction, not the optimization routine’s purpose.

Conclusion: therefore the correct selections are “Minimizing average loss” and “Maximizing likelihood,” which matches the provided answer.

Optimization algorithms like gradient descent are typically used in machine learning for which of the following (select all that apply):多项选择题

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