In a linear regression model, what does each learned weight represent?

Question

BlackTom AI · Accepted Answer

In linear regression the model is written y = w0 + w1 x1 + w2 x2 + ... + wn xn, where each wi is a learned coefficient. The key knowledge points: (1) each weight is a coefficient multiplying a feature; (2) the coefficient equals the change in the predicted output when that feature increases by one unit, with all other features held constant; (3) the bias/intercept w0 is the prediction when all features are zero.

Step-by-step reasoning:
1. Start from the model: y(x) = w0 + Σ wi xi.
2. Consider increasing feature xk by exactly 1 while keeping other xi fixed. New prediction: y' = w0 + Σi≠k wi xi + wk(xk + 1) = y + wk.
3. The change in prediction is y' − y = wk. Thus wk directly measures the influence of a one-unit increase in xk on the predicted y.

Intermediate demonstration with numbers:
- Example: y = 2 + 3x1 − 0.5x2. Increasing x1 by 1 (x1 → x1 + 1) gives y' = y + 3, so w1 = 3 means the prediction increases by 3 units per one-unit increase in x1. Increasing x2 by 1 decreases prediction by 0.5 since w2 = −0.5.

Why the correct answer is correct:
- The algebra shows each learned weight equals the marginal effect (partial derivative) ∂y/∂xk = wk, so it quantifies how strongly a feature influences the prediction for a one-unit change.

Why the other options are incorrect:
- “A random value that changes the shape of the dataset”: weights are learned deterministically via optimization, not random, and they describe model behavior, not dataset shape.
- “The number of steps the optimizer will take during training”: optimizer steps are hyperparameters/algorithm behavior, unrelated to coefficient values.
- “The number of samples assigned to each class”: that refers to class counts in classification, not regression coefficients.

Conclusion: therefore each learned weight represents how strongly a feature influences the prediction when it increases by one unit, matching the provided answer.

In a linear regression model, what does each learned weight represent?单项选择题

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