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1890 learning items

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Quiz

MML-14-07: Logistic Regression: Odds, Logit, Sigmoid & Binary Cross-Entropy

40 worked single-select and multiselect questions on Logistic Regression: Odds, Logit, Sigmoid & Binary Cross-Entropy. Cognitive level: Tier 2–3. Prerequisites: Parts 06, 11. Learning objectives: Convert between probabilities, odds and logits; Interpret conditional coefficients and feature transformations; Compute Bernoulli likelihood and binary cross-entropy; Derive and evaluate probability and loss gradients; Diagnose saturation, weighting and misspecification; Separate probability estimates from hard decisions. Each item is self-contained, with calculations, model diagnostics and explanations of every distractor. Study references: [Stanford CS229 notes](https://cs229.stanford.edu/notes2022fall/main_notes.pdf).

mathematics

Nov 1, 2026

40 questions

Quiz

MML-13-05: The Lagrangian Dual Function & Weak Duality

MML-13-05: The Lagrangian Dual Function & Weak Duality. 35 independently reviewed questions, including single-select and multiselect MCQs. Cognitive scope: Tier 2–4. Prerequisites: QS 13.4. Learning goals: Define the dual function and its domain.; Compute quadratic Lagrangian infima.; Recognize unbounded Lagrangian minimizations.; Prove concavity as an infimum of affine functions.; Establish weak duality with correct signs.; Use primal-dual bounds as optimality certificates.; Distinguish dual feasibility, attainment and gap.; Interpret dual gradients and optimization diagnostics. Progress from conceptual checks to worked calculations, ML diagnostics and derivations within the assigned scope. Every question includes content-based feedback for every choice and authoritative references. Select every correct choice when marked multiselect. Photo by Alex Moliski on Pexels. Licensed under the Pexels License: https://www.pexels.com/license/.

mathematics

Nov 1, 2026

35 questions

Quiz

MML-13-04: Complementary Slackness, Active Constraints & Primal/Dual Feasibility

MML-13-04: Complementary Slackness, Active Constraints & Primal/Dual Feasibility. 40 independently reviewed questions, including single-select and multiselect MCQs. Cognitive scope: Tier 2–4. Prerequisites: QS 13.3. Learning goals: Classify active and inactive inequalities.; Interpret complementarity without reversing implications.; Compute active sets and multiplier supports.; Apply LICQ and rank reasoning.; Distinguish strict complementarity and degeneracy.; Analyze feasible tangent cones.; Diagnose tolerance and scaling effects.; Explain local sensitivity and active-set changes. Progress from conceptual checks to worked calculations, ML diagnostics and derivations within the assigned scope. Every question includes content-based feedback for every choice and authoritative references. Select every correct choice when marked multiselect. Photo by Daniel Wells on Pexels. Licensed under the Pexels License: https://www.pexels.com/license/.

mathematics

Nov 1, 2026

40 questions

Quiz

MML-13-03: Inequality Constraints & The Karush-Kuhn-Tucker (KKT) Conditions

MML-13-03: Inequality Constraints & The Karush-Kuhn-Tucker (KKT) Conditions. 45 independently reviewed questions, including single-select and multiselect MCQs. Cognitive scope: Tier 2–4. Prerequisites: QS 13.2. Learning goals: Write nonlinear programs in standard inequality form.; State the four KKT conditions and multiplier conventions.; Compute scalar constrained optima and multipliers.; Solve small multidimensional KKT systems.; Distinguish necessary conditions, qualifications and convex sufficiency.; Diagnose failed KKT residuals.; Derive signs and complementarity from Lagrangian bounds.; Apply KKT reasoning to bounded and normalized ML parameters. Progress from conceptual checks to worked calculations, ML diagnostics and derivations within the assigned scope. Every question includes content-based feedback for every choice and authoritative references. Select every correct choice when marked multiselect. Photo by Soran Ali on Pexels. Licensed under the Pexels License: https://www.pexels.com/license/.

mathematics

Nov 1, 2026

45 questions

Quiz

MML-13-02: Geometric Meaning of Lagrange Multipliers: Gradients & Tangent Hyperplanes

MML-13-02: Geometric Meaning of Lagrange Multipliers: Gradients & Tangent Hyperplanes. 30 independently reviewed questions, including single-select and multiselect MCQs. Cognitive scope: Tier 1–3. Prerequisites: QS 13.1. Learning goals: Identify constraint normals and tangent spaces.; Test feasible first-order directions.; Explain normal-span stationarity geometrically.; Distinguish linearized tangency from exact feasible motion.; Calculate shadow prices with explicit residual conventions.; Use envelope derivatives for several constraints.; Diagnose rank loss and multiplier nonuniqueness.; Connect constrained gradients to ML sensitivity. Progress from conceptual checks to worked calculations, ML diagnostics and derivations within the assigned scope. Every question includes content-based feedback for every choice and authoritative references. Select every correct choice when marked multiselect. Photo by Perry Wunderlich on Pexels. Licensed under the Pexels License: https://www.pexels.com/license/.

mathematics

Nov 1, 2026

30 questions

Quiz

MML-13-01: Equality Constrained Optimization & The Method of Lagrange Multipliers

MML-13-01: Equality Constrained Optimization & The Method of Lagrange Multipliers. 35 independently reviewed questions, including single-select and multiselect MCQs. Cognitive scope: Tier 2–3. Prerequisites: Part 12. Learning goals: Form equality-constrained optimization problems and Lagrangians.; Derive stationarity and feasibility equations.; Solve scalar and two-variable constrained quadratics.; Use matrix KKT systems for affine equalities.; Distinguish necessary conditions from sufficient optimality.; Handle constraint scaling and redundant equations.; Compare elimination and multiplier methods.; Interpret ML normalization and parameter-sharing constraints. Progress from conceptual checks to worked calculations, ML diagnostics and derivations within the assigned scope. Every question includes content-based feedback for every choice and authoritative references. Select every correct choice when marked multiselect. Photo by BI ravencrow on Pexels. Licensed under the Pexels License: https://www.pexels.com/license/.

mathematics

Nov 1, 2026

35 questions

Quiz

MML-12-09: Second-Order Methods: Newton-Raphson, Quasi-Newton (BFGS / L-BFGS) & Natural Gradient

MML-12-09: Second-Order Methods: Newton-Raphson, Quasi-Newton (BFGS / L-BFGS) & Natural Gradient. 40 independently reviewed questions, including single-select and multiselect MCQs. Cognitive scope: Tier 2–4. Prerequisites: QS 12.8, Part 06. Learning goals: Derive Newton directions from quadratic models.; Calculate Newton steps and local convergence orders.; Diagnose indefiniteness, singularity and damping.; Distinguish Hessian from inverse-Hessian secant equations.; Apply and verify BFGS curvature updates.; Explain limited-memory storage and two-loop recursion.; Compute natural-gradient directions using Fisher geometry.; Compare scaling, computation and optimizer assumptions. Progress from conceptual checks to worked calculations, ML diagnostics and derivations within the assigned scope. Every question includes content-based feedback for every choice and authoritative references. Select every correct choice when marked multiselect. Photo by Yoss Traore on Pexels. Licensed under the Pexels License: https://www.pexels.com/license/.

mathematics

Nov 1, 2026

40 questions

Quiz

MML-12-08: Adaptive Learning Rate Algorithms: AdaGrad, RMSProp, Adam & AdamW

MML-12-08: Adaptive Learning Rate Algorithms: AdaGrad, RMSProp, Adam & AdamW. 45 independently reviewed questions, including single-select and multiselect MCQs. Cognitive scope: Tier 2–3. Prerequisites: QS 12.7. Learning goals: Interpret diagonal preconditioning.; Compute AdaGrad accumulators and sparse-coordinate behavior.; Analyze RMSProp exponential memory.; Compute Adam moment recursions.; Apply initialization bias corrections.; Diagnose numerical stability and state handling.; Distinguish coupled regularization from AdamW decay.; Compare optimizer invariance and practical limitations. Progress from conceptual checks to worked calculations, ML diagnostics and derivations within the assigned scope. Every question includes content-based feedback for every choice and authoritative references. Select every correct choice when marked multiselect. Photo by Clickout pixel on Pexels. Licensed under the Pexels License: https://www.pexels.com/license/.

mathematics

Nov 1, 2026

45 questions

Quiz

MML-12-07: Momentum, Nesterov Accelerated Gradient & Heavy-Ball Dynamics

MML-12-07: Momentum, Nesterov Accelerated Gradient & Heavy-Ball Dynamics. 35 independently reviewed questions, including single-select and multiselect MCQs. Cognitive scope: Tier 2–4. Prerequisites: QS 12.6. Learning goals: Translate momentum conventions and memory into updates.; Calculate heavy-ball iterates and gradient histories.; Analyze quadratic characteristic roots and stability.; Compare Nesterov look-ahead with heavy-ball gradients.; State acceleration rates with their assumptions.; Diagnose ravines, noise, overshoot and restarts.; Derive effective dynamics and optimal quadratic tuning.; Connect continuous damping to discrete momentum. Progress from conceptual checks to worked calculations, ML diagnostics and derivations within the assigned scope. Every question includes content-based feedback for every choice and authoritative references. Select every correct choice when marked multiselect. Photo by Barnabas Davoti on Pexels. Licensed under the Pexels License: https://www.pexels.com/license/.

mathematics

Nov 1, 2026

35 questions

Quiz

MML-10-07: Hypothesis Testing Foundations: p-Values, Significance & Errors (Type I/II)

35 worked single-select and multiselect questions on Hypothesis Testing Foundations: $$p$$-Values, Significance & Errors (Type I/II). Intended level: Tier 1–3. Prerequisites: QS 10.6. The sequence develops conceptual interpretation, explicit calculations, ML diagnostics and, where assigned, derivations with carefully stated assumptions. Feedback explains each answer and misconception by content. Learning objectives: State null and alternative hypotheses before seeing results. Interpret tail-area evidence without posterior-probability confusion. Identify Type I and Type II errors and calculate power. Compute simple exact and normal-reference p-values. Separate statistical significance from standardized effect size. Apply Bonferroni and Benjamini–Hochberg adjustments. Diagnose optional stopping, data reuse and selective reporting. Study references: [Shedden, University of Michigan, Introduction to Data Science](https://dept.stat.lsa.umich.edu/~kshedden/introds/topics/hypothesis_tests/); [Stark, Berkeley Statistics 210B](https://www.stat.berkeley.edu/~stark/Teach/S210b/Notes/Lec9/lec9.pdf).

mathematics

Oct 31, 2026

35 questions

Quiz

MML-10-06: Confidence Intervals, Standard Errors & Pivot Quantities

30 worked single-select and multiselect questions on Confidence Intervals, Standard Errors & Pivot Quantities. Intended level: Tier 2–3. Prerequisites: QS 10.5. The sequence develops conceptual interpretation, explicit calculations, ML diagnostics and, where assigned, derivations with carefully stated assumptions. Feedback explains each answer and misconception by content. Learning objectives: Compute and interpret standard errors. Construct normal mean intervals with known population variance. Use exact Student intervals under normal sampling. Interpret pivots and repeated-sampling coverage. Construct and diagnose percentile bootstrap intervals. Explain BCa adjustments and resampling limitations. Connect interval precision with sampling design and ML evaluation. Study references: [NIST/SEMATECH e-Handbook](https://itl.nist.gov/div898/handbook/eda/section3/eda352.htm); [Efron & Narasimhan, The automatic construction of bootstrap confidence intervals](https://pmc.ncbi.nlm.nih.gov/articles/PMC7958418/); [Duke STA 611, Sampling Distributions of Estimators](https://www2.stat.duke.edu/courses/Fall18/sta611.01/Lecture/Lecture13.pdf).

mathematics

Oct 31, 2026

30 questions

Quiz

MML-10-05: Bias, Variance, Mean Squared Error (MSE) & Consistency of Estimators

30 worked single-select and multiselect questions on Bias, Variance, Mean Squared Error (MSE) & Consistency of Estimators. Intended level: Tier 2–4. Prerequisites: QS 10.4. The sequence develops conceptual interpretation, explicit calculations, ML diagnostics and, where assigned, derivations with carefully stated assumptions. Feedback explains each answer and misconception by content. Learning objectives: Calculate bias, variance and mean squared estimation error. Prove and apply the MSE decomposition. Compare biased shrinkage and unbiased alternatives. Distinguish finite-sample unbiasedness from consistency. Use convergence and moment assumptions correctly. Diagnose covariance effects in ensemble estimators. Analyze rates, transformations and irreducible prediction noise. Study references: [Goodfellow, Bengio & Courville, Deep Learning](https://www.deeplearningbook.org/contents/ml.html); [Duke STA 611, Sampling Distributions of Estimators](https://www2.stat.duke.edu/courses/Fall18/sta611.01/Lecture/Lecture13.pdf).

mathematics

Oct 31, 2026

30 questions