Machine Learning Core Concepts Syllabus

Define regression in machine learning.

Regression is a supervised learning task that predicts a continuous numeric output (e.g., price, temperature) from input features, by modeling the relationship between inputs and a real-valued target.

Write the simple linear regression model equation.

$$\hat{y} = \beta_0 + \beta_1 x$$ where $\beta_0$ is the intercept and $\beta_1$ is the slope (coefficient) of the feature $x$.

What cost function does ordinary linear regression minimize?

Mean Squared Error (the sum of squared residuals): $$MSE = \frac{1}{N}\sum_{i=1}^{N}(y_i - \hat{y}_i)^2$$

What does the coefficient of determination $R^2$ represent?

$$R^2 = 1 - \frac{\sum_i (y_i - \hat{y}_i)^2}{\sum_i (y_i - \bar{y})^2}$$ It is the proportion of variance in the target explained by the model; $R^2 = 1$ is a perfect fit, $R^2 = 0$ means no better than predicting the mean.

Compare Ridge (L2) and Lasso (L1) regularization.

Ridge adds a penalty $\lambda \sum_j \beta_j^2$, shrinking coefficients toward zero but rarely to exactly zero. Lasso adds $\lambda \sum_j |\beta_j|$, which can drive some coefficients to exactly zero, performing automatic feature selection.

What is the gradient descent update rule for a parameter $\theta$?

$$\theta := \theta - \alpha \frac{\partial J(\theta)}{\partial \theta}$$ where $\alpha$ is the learning rate and $J(\theta)$ is the cost function. It iteratively moves parameters in the direction of steepest descent.

What is the difference between batch, stochastic, and mini-batch gradient descent?

Batch GD uses the entire dataset per update (stable, slow). Stochastic GD (SGD) uses one sample per update (fast, noisy). Mini-batch GD uses a small subset per update, balancing speed and stability — the most common in practice.

Define clustering and name two main approaches.

Clustering is unsupervised grouping of data points so that points within a group are more similar to each other than to points in other groups. Two main approaches: partitional (e.g., k-means) and hierarchical (agglomerative/divisive).

Describe the k-means clustering algorithm steps.

(1) Choose $k$ and initialize $k$ centroids; (2) assign each point to its nearest centroid; (3) recompute each centroid as the mean of its assigned points; (4) repeat steps 2-3 until assignments no longer change (convergence). It minimizes within-cluster sum of squares.

What objective (inertia) does k-means minimize?

Within-cluster sum of squares (WCSS): $$J = \sum_{i=1}^{k}\sum_{x \in C_i} \lVert x - \mu_i \rVert^2$$ where $\mu_i$ is the centroid of cluster $C_i$.

What is the elbow method for choosing $k$ in k-means?

Plot the within-cluster sum of squares (inertia) against the number of clusters $k$. The 'elbow' — the point where adding more clusters yields diminishing reduction in inertia — suggests a good value of $k$.

How does DBSCAN differ from k-means?

DBSCAN is density-based: it groups densely packed points (using parameters $\varepsilon$ and minPts) and labels sparse points as noise. Unlike k-means, it does not require specifying the number of clusters, finds arbitrarily shaped clusters, and handles outliers.

What does the silhouette coefficient measure and what is its range?

It measures how well a point fits its own cluster vs. the nearest other cluster: $$s = \frac{b - a}{\max(a, b)}$$ where $a$ is mean intra-cluster distance and $b$ is mean nearest-cluster distance. It ranges from $-1$ (poor) to $+1$ (well-clustered).

What is dimensionality reduction and why is it useful?

It reduces the number of input features while preserving as much meaningful information as possible. Benefits: combats the curse of dimensionality, reduces overfitting and computation, removes redundancy, and enables visualization (2D/3D).

What is the curse of dimensionality?

As the number of features grows, the volume of the feature space increases exponentially, so data becomes sparse. Distances between points become less meaningful, and models need exponentially more data to generalize, degrading performance.

What does Principal Component Analysis (PCA) do?

PCA finds new orthogonal axes (principal components) that are linear combinations of the original features, ordered by the variance they capture. Projecting onto the top components reduces dimensionality while retaining maximal variance.

How are principal components related to the covariance matrix in PCA?

The principal components are the eigenvectors of the data's covariance matrix, and the amount of variance each captures equals its corresponding eigenvalue $\lambda_i$. Components are ranked by descending eigenvalue.

How is the proportion of variance explained by the first $m$ principal components computed?

$$\text{Explained variance ratio} = \frac{\sum_{i=1}^{m} \lambda_i}{\sum_{j=1}^{d} \lambda_j}$$ where $\lambda_i$ are the eigenvalues (component variances) and $d$ is the total number of components.

What is t-SNE primarily used for, and how does it differ from PCA?

t-SNE (t-Distributed Stochastic Neighbor Embedding) is a non-linear technique for visualizing high-dimensional data in 2D/3D by preserving local neighborhood structure. Unlike PCA (linear, variance-preserving), t-SNE focuses on local similarity and is mainly for visualization, not feature transformation.

What is association rule learning?

An unsupervised method for discovering interesting relationships (rules of the form $X \Rightarrow Y$) between items in large transactional datasets — classically used for market basket analysis to find items frequently bought together.

Define support, confidence, and lift for an association rule $X \Rightarrow Y$.

Support $= P(X \cap Y)$ (frequency of the itemset). Confidence $= P(Y \mid X) = \frac{P(X \cap Y)}{P(X)}$ (reliability). Lift $= \frac{P(X \cap Y)}{P(X)P(Y)}$ — lift $> 1$ means positive correlation, $= 1$ independence, $< 1$ negative.

What principle does the Apriori algorithm rely on?

The Apriori (anti-monotone) principle: if an itemset is frequent, all of its subsets are also frequent; equivalently, if an itemset is infrequent, all of its supersets are infrequent. This prunes the search space when mining frequent itemsets.

What are the core components of a reinforcement learning problem?

An agent interacting with an environment via: states $s$, actions $a$, a reward signal $r$, a policy $\pi$ (mapping states to actions), and (often) a value function. The goal is to learn a policy maximizing cumulative reward.

Write the formula for the discounted cumulative return $G_t$ in reinforcement learning.

$$G_t = \sum_{k=0}^{\infty} \gamma^k r_{t+k+1}$$ where $\gamma \in [0,1]$ is the discount factor that weights future rewards less than immediate ones.

Explain the exploration-exploitation trade-off in reinforcement learning.

Exploitation chooses actions known to yield high reward; exploration tries new actions to discover potentially better ones. An agent must balance both — e.g., via an $\varepsilon$-greedy policy — to avoid settling on a suboptimal policy.

State the Q-learning update rule.

$$Q(s,a) \leftarrow Q(s,a) + \alpha \left[ r + \gamma \max_{a'} Q(s',a') - Q(s,a) \right]$$ where $\alpha$ is the learning rate and $\gamma$ the discount factor. It is an off-policy, model-free method.

What is the difference between bias and variance, and what is the bias-variance trade-off?

Bias is error from overly simplistic assumptions (underfitting); variance is error from sensitivity to training-data fluctuations (overfitting). The trade-off: decreasing one tends to increase the other, and total expected error decomposes as $\text{Bias}^2 + \text{Variance} + \text{Irreducible error}$.

What is k-fold cross-validation?

The data is split into $k$ equal folds; the model is trained on $k-1$ folds and validated on the remaining fold, repeating $k$ times so each fold serves once as validation. The performance is averaged, giving a more robust, less variance-prone estimate of generalization.

Distinguish overfitting from underfitting and give one remedy for each.

Overfitting: model fits training noise, low train error but high test error — remedy: regularization, more data, or simpler model. Underfitting: model too simple to capture the pattern, high error everywhere — remedy: a more complex model or better features.

Compare bagging and boosting as ensemble methods.

Bagging trains models in parallel on bootstrap samples and averages them to reduce variance (e.g., Random Forest). Boosting trains models sequentially, each correcting the previous one's errors by reweighting, reducing bias (e.g., AdaBoost, Gradient Boosting / XGBoost).

What is the difference between grid search and random search for hyperparameter optimization?

Grid search exhaustively evaluates every combination in a predefined discrete grid of hyperparameter values. Random search samples combinations randomly from the search space — often more efficient at finding good values when only a few hyperparameters matter, for the same compute budget.