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6.390 Notation and Style Guide

Consistent notation across all course materials: notes, exams, labs, homework, and recitations. If in doubt, the course notes, Shen's lecture slides, and most recent exams are the canonical reference.

Mathematical Notation Conventions

Training Data

  • Data points indexed with superscript in parentheses: x^{(i)}, y^{(i)}
  • Dataset: \mathcal{D} = \{(x^{(i)}, y^{(i)})\}_{i=1}^n
  • n = number of training examples, d = input dimension
  • Feature components use subscripts: x_1, x_2, x_j

Vectors and Matrices

  • Vectors: bold lowercase via \mathbf{} -- e.g., \mathbf{x}
  • Matrices: plain uppercase -- e.g., X, W, A
  • Transpose: X^T or \theta^T
  • Column vectors by default; \theta^T x for dot products
  • Dimensions stated explicitly: X \in \mathbb{R}^{n \times d}, \theta \in \mathbb{R}^d

Parameters and Weights

  • Linear model: \theta (weights), \theta_0 (intercept/bias)
  • Neural network layers: W^{(l)} (weight matrix), W_0^{(l)} or b^{(l)} (bias), with superscript in parens for layer index
  • ML estimate: \theta_{\text{ml}}
  • ERM estimate: \theta_{\text{erm}}
  • Model variant labels: \theta^{\mathrm{multi}}, \theta^{\mathrm{bin}}

Functions

  • Hypothesis: h(x; \theta, \theta_0) -- semicolon separates input from parameters
  • Loss: \mathcal{L}(g, y) for general loss; specific variants \mathcal{L}_{\text{nll}}, \mathcal{L}_{\text{SE}}
  • Objective: J(\theta) or J(\theta, \theta_0)
  • Sigmoid: \sigma(z) = \frac{1}{1 + e^{-z}}
  • Softmax: \operatorname{softmax}
  • ReLU: \text{ReLU}(z) = \max(0, z) -- capitalize "ReLU" in text
  • Feature transform: \phi(x) for transformed feature vector

Neural Networks

  • Pre-activation: Z^{(l)} = (W^{(l)})^T A^{(l-1)} + W_0^{(l)}
  • Post-activation: A^{(l)} = f^{(l)}(Z^{(l)})
  • Input: A^{(0)} = x
  • Activation functions: f^{(l)}(\cdot) per-layer

Gradients

  • Gradient operator: \nabla_\theta J(\theta)
  • Partial derivatives: \frac{\partial J}{\partial \theta}
  • Gradient descent update: \theta^{(t)} = \theta^{(t-1)} - \eta \nabla_\theta J(\theta^{(t-1)})

Attention Mechanism

  • Queries/keys/values: q_i = W_q^T x_i, k_j = W_k^T x_j, v_j = W_v^T x_j
  • Attention scores: s_{ij} = q_i \cdot k_j
  • Softmax'd attention scores: \alpha_{ij} = \frac{e^{s_{ij}}}{\sum_l e^{s_{il}}} (explicit softmax formula) -- do NOT call these "attention weights"
  • Output: z_i = \sum_j \alpha_{ij} v_j

CNNs

  • Filter/kernel: lowercase f or F, shown as small matrices
  • Output size: (\text{input} - \text{filter} + 2\text{padding}) / \text{stride} + 1
  • Receptive field: set notation \{(i,j), \ldots\}

MDPs and Reinforcement Learning

  • States: lowercase s with subscripts (s_0, s_1, s_2) or tuples ((p, m))
  • Actions: lowercase a; named actions in \text{}: \text{Forward}, \text{cash out}
  • State space: \mathcal{S}, action space: \mathcal{A}
  • Reward function: \mathrm{R}(s, a) -- upright R
  • Transition function: \mathrm{T}(s, a, s') -- upright T
  • Discount factor: \gamma \in (0, 1)
  • Horizon: h
  • Optimal value function: \mathrm{V}^*_h(s) (finite horizon), \mathrm{V}^*_\infty(s) or \mathrm{V}^*(s) (infinite horizon) -- upright V
  • Optimal Q-function: \mathrm{Q}^*_h(s, a) (finite horizon), \mathrm{Q}^*_\infty(s, a) (infinite horizon) -- upright Q
  • Policy value function: \mathrm{V}^\pi(s)
  • Policy: \pi or \pi(s); greedy policy: \pi^*(s) = \arg\max_a \mathrm{Q}^*(s, a)
  • All MDP functions (\mathrm{R}, \mathrm{T}, \mathrm{V}, \mathrm{Q}) use upright (\mathrm{}) letters
  • Value iteration (Bellman equation): \mathrm{V}^*_{h+1}(s) = \max_a [\mathrm{R}(s,a) + \gamma \sum_{s'} \mathrm{T}(s,a,s') \mathrm{V}^*_h(s')]
  • Q-value Bellman equation: \mathrm{Q}^*_{h+1}(s,a) = \mathrm{R}(s,a) + \gamma \sum_{s'} \mathrm{T}(s,a,s') \max_{a'} \mathrm{Q}^*_h(s',a')
  • V-Q relationship: \mathrm{V}^*(s) = \max_a \mathrm{Q}^*(s, a)
  • Q-learning update: \mathrm{Q}(s,a) \leftarrow \mathrm{Q}(s,a) + \alpha [r + \gamma \max_{a'} \mathrm{Q}(s',a') - \mathrm{Q}(s,a)]
  • Learning rate (Q-learning): \alpha
  • Epsilon-greedy: \varepsilon-greedy
  • Deep RL / DQN: neural network Q-function Q_\theta(s,a), policy network \pi_\theta
  • DQN target: y = r + \gamma \max_{a'} Q_\theta(s',a')
  • DQN loss (Bellman error): \mathcal{L}(\theta) = \mathbb{E}[(y - Q_\theta(s,a))^2]

Other Conventions

  • Learning rate: \eta
  • Regularization: \lambda
  • Norms: \lVert \cdot \rVert (e.g., \lVert x \rVert); avoid raw ||
  • Absolute value: \left| \cdot \right| (or \lvert \cdot \rvert for proper spacing)
  • Sets: calligraphic \mathcal{} -- \mathcal{D} (dataset), \mathcal{M} (model class), \mathcal{H} (hypothesis class)
  • Real numbers: \mathbb{R}
  • Iteration/time index: superscript in parens \theta^{(t)}