# Write the logistic regression loss function ## Logistic Regression Loss Consider binary logistic regression. - Dataset: \(\{(\mathbf{x}_i, y_i)\}_{i=1}^n\) - Labels: \(y_i \in \{0,1\}\) - Model: \(p_i = P(y_i=1\mid \mathbf{x}_i) = \sigma(\mathbf{w}^\top \mathbf{x}_i + b)\), where \(\sigma(z)=\frac{1}{1+e^{-z}}\). ### Question 1. Write the per-example loss. 2. Write the total loss over \(n\) examples (average or sum). 3. (Optional) Write the L2-regularized objective. ### Constraints & Assumptions - Preserve the scope, facts, inputs, and requested outputs from the prompt above. - If the prompt leaves a detail unspecified, state a reasonable assumption before relying on it. - Keep the answer interview-ready: concise enough to present, but concrete enough to implement or evaluate. ### Clarifying Questions to Ask - Clarify the random variables, distributional assumptions, independence assumptions, and desired output. - Show enough derivation for the interviewer to follow the reasoning. - Explain how you would validate the result with simulation or sensitivity checks. ### What a Strong Answer Covers - A correct setup with definitions, formulas, and boundary conditions. - A step-by-step derivation or estimation plan. - Interpretation of the result, including uncertainty and practical limitations. - Checks for assumptions, edge cases, and numerical stability. ### Follow-up Questions - How would the result change if the assumptions were relaxed? - Can you verify the answer with a simulation? - What is the most likely source of estimation error?