LLGA · Course notes

Notation conventions

One set of symbols across the LLGA course notes

Contents

Why one notation

The same object comes back in many courses. A student who meets PageRank in Graph ML and then APPNP in Advanced GNN should not have to check whether \alpha is the chance of following an edge or of jumping away. When one symbol carries two meanings, the reader pays twice: once to translate, once to doubt the result.

These notes therefore give each symbol one meaning, family by family. Each table lists the object, its symbol and convention, and the course pages that use it; the defining equations follow on their own lines. This is the notes’ own editorial convention. Lecturers’ slides and the papers cited may use other letters, and a page that quotes them translates once.

Graphs and spectra

Graph pages share one set of matrices. The letter P is kept for the random walk alone, so a permutation is \Pi and the GCN propagator is \hat A.

Object Symbol and convention Courses that use it (links)
Adjacency and degrees A adjacency, D = \operatorname{diag}(d_v); with self-loops \tilde A = A + I and its degree matrix \tilde D Graph ML ch. 0 · Advanced GNN ch. 0 · Shapes ch. 5
Laplacians L = D - A (combinatorial, positive semi-definite); L_{\mathrm{sym}} (normalized); L_{\mathrm{rw}} (random-walk) Graph ML ch. 0, ch. 4 · Shapes ch. 5 · Emerging ML ch. 2 · Computer Vision ch. 8 · Machine Learning ch. 9 · Signal Processing ch. 8
Random-walk matrix P = D^{-1}A, row-stochastic; P means nothing else on graph pages Graph ML ch. 0 · Advanced GNN ch. 3
GCN propagator \hat A, the renormalized adjacency with self-loops Graph ML ch. 5 · Advanced GNN ch. 0, ch. 3
Permutation \Pi, acting on a graph as (\Pi X, \Pi A \Pi^\top) Graph ML ch. 2 · Geometric Data ch. 7
Laplacian spectrum ascending and 1-based, 0 = \lambda_1 \le \lambda_2 \le \dots; Fiedler pair (\lambda_2, u_2) Graph ML ch. 0, ch. 4 · Shapes ch. 8 · Geometric Data ch. 2
Adjacency spectrum descending, \mu_1 \ge \mu_2 \ge \dots (eigenvector centrality, over-smoothing) Graph ML ch. 0, ch. 5

L = D - A, \qquad L_{\mathrm{sym}} = I - D^{-1/2} A D^{-1/2}, \qquad L_{\mathrm{rw}} = I - D^{-1} A

P = D^{-1} A, \qquad \hat A = \tilde D^{-1/2}\, \tilde A\, \tilde D^{-1/2}, \qquad \tilde A = A + I

0 = \lambda_1 \le \lambda_2 \le \dots \le \lambda_n \ \ (\text{eigenvalues of } L), \qquad \mu_1 \ge \mu_2 \ge \dots \ge \mu_n \ \ (\text{eigenvalues of } A)

“Normalized Laplacian” without a qualifier always means L_{\mathrm{sym}}. The matrix I - D^{-1}A is the random-walk Laplacian.

PageRank and personalized PageRank

One parameter, one meaning: \alpha is the teleport (restart) probability, typically 0.15. The probability of following an edge is 1 - \alpha.

Object Symbol and convention Courses that use it (links)
PageRank stationary row vector \pi; teleport probability \alpha Graph ML ch. 0 · Text Mining and NLP ch. 8 (TextRank)
Personalized PageRank from u \pi_u, restarting at the indicator vector e_u Graph ML ch. 2, ch. 7 · Advanced GNN ch. 3
APPNP propagation the same \alpha, propagator \hat A, base predictions H Advanced GNN ch. 3

\pi^\top = (1 - \alpha)\, \pi^\top P + \frac{\alpha}{n}\, \mathbf 1^\top

\pi_u = \alpha \big(I - (1 - \alpha) P^\top\big)^{-1} e_u

Z^{(k+1)} = (1 - \alpha)\, \hat A\, Z^{(k)} + \alpha H

The damping factor of the original PageRank paper and of TextRank is 1 - \alpha. The letter d is not used for it, since d already names a document and a dimension.

Gossip and consensus

Decentralized training mixes models through a matrix on the communication graph. “Spectral gap” refers to that mixing matrix only.

Object Symbol and convention Courses that use it (links)
Mixing matrix W, symmetric and doubly stochastic Emerging ML ch. 2 · Deep Learning P2 ch. 8
Spectral gap \delta_W, one minus the second-largest eigenvalue modulus of W Emerging ML ch. 2 · Deep Learning P2 ch. 8
Algebraic connectivity \lambda_2(L) of the communication graph Emerging ML ch. 2 · Graph ML ch. 0
Gossip step \eta_g, with W = I - \eta_g L Emerging ML ch. 2

\delta_W = 1 - \max_{i \ge 2} \lvert \lambda_i(W) \rvert, \qquad W = I - \eta_g L

Manifolds and curvature

The continuous Laplace–Beltrami operator is negative semi-definite. Its discrete counterpart is a pair of positive semi-definite matrices, and every sign below follows from that choice.

Object Symbol and convention Courses that use it (links)
Laplace–Beltrami operator \Delta_{\mathcal M} = \operatorname{div}_{\mathcal M} \nabla_{\mathcal M}, negative semi-definite Geometric Data ch. 1, ch. 2 · Shapes ch. 5
Discrete operator cotangent stiffness L_c (symmetric, positive semi-definite) and lumped mass matrix M Geometric Data ch. 2 · Shapes ch. 5
Spectrum generalized problem L_c \varphi = \lambda M \varphi, 1-based: 0 = \lambda_1 < \lambda_2 \le \dots Geometric Data ch. 2, ch. 3 · Shapes ch. 8
Heat kernel signature h_t, built on the eigenpairs (\lambda_k, \varphi_k) Geometric Data ch. 3 · Shapes ch. 8
Curvatures mean curvature H, Gaussian curvature K (K means nothing else); outward unit normal \mathbf n Geometric Data ch. 1 · Shapes ch. 7
Mesh and mass the mesh is \mathcal M, the mass matrix is M Geometric Data ch. 1, ch. 2 · Shapes ch. 1

(L_c)_{ij} = -\tfrac12 \big(\cot \alpha_{ij} + \cot \beta_{ij}\big) \ \ (j \sim i), \qquad (L_c)_{ii} = -\sum_{j \sim i} (L_c)_{ij}

L_c\, \varphi_k = \lambda_k\, M\, \varphi_k, \qquad -\Delta_{\mathcal M} \approx M^{-1} L_c

h_t = \sum_{k} e^{-\lambda_k t}\, \varphi_k \varphi_k^\top

\Delta_{\mathcal M}\, \mathbf x = -2H\, \mathbf n

With the outward normal, H > 0 on a sphere. On the unit sphere, \Delta_{\mathcal M} \mathbf x = -2\mathbf x, which checks the sign.

Symmetry and equivariance

A group acts on inputs and outputs through representations. Invariance is the special case where the output representation is the identity.

Object Symbol and convention Courses that use it (links)
Group action group G, representations \rho_{\mathrm{in}}(g) and \rho_{\mathrm{out}}(g) Advanced DL ch. 4 · Geometric Data ch. 7 · Graph ML ch. 2
Invariance \rho_{\mathrm{out}}(g) = I for every g Graph ML ch. 2 · Geometric Data ch. 7
Euclidean group E(n) element (Q, t) acting as x \mapsto Qx + t, with Q \in O(n) Graph ML ch. 8 · Geometric Data ch. 7 · Advanced GNN ch. 8
Permutations \Pi, acting on a graph as (\Pi X, \Pi A \Pi^\top) Graph ML ch. 2 · Advanced GNN ch. 2

f\big(\rho_{\mathrm{in}}(g)\, x\big) = \rho_{\mathrm{out}}(g)\, f(x) \qquad \text{for all } g \in G

f(\Pi X,\, \Pi A \Pi^\top) = \Pi\, f(X, A) \qquad \text{for every permutation matrix } \Pi

Data, loss and risk

Every learning page uses the same sample, loss and risk. Calligraphic \mathcal L always denotes a quantity that is minimized.

Object Symbol and convention Courses that use it (links)
Sample \mathcal D_n = \{(x_i, y_i)\}_{i=1}^n, i.i.d. from P Machine Learning ch. 1 · Statistical Learning Theory ch. 1 · Refresher Statistics ch. 9 · Deep Learning P1 ch. 1
Sizes n samples, d features, design matrix X \in \mathbb R^{n \times d} Machine Learning ch. 2 · Boosting to Foundation Models ch. 0 · Optimization for AI ch. 5
Loss \ell(\hat y, y), prediction first, as in PyTorch loss(input, target) Machine Learning ch. 1 · Boosting to Foundation Models ch. 2 · Deep Learning P1 ch. 2
Risk R(f) and empirical risk \hat R_n(f) Machine Learning ch. 1 · Statistical Learning Theory ch. 1 · Refresher Statistics ch. 9
Training objective F(\theta) = \hat R_n(f_\theta) + \Omega(\theta), when the page declares it Optimization for AI ch. 2 · Machine Learning ch. 2
Minimized quantity \mathcal L, with a subscript naming it (\mathcal L_{\mathrm{VAE}}, \mathcal L_{\mathrm{KD}}, …) Deep Learning P1 ch. 2 · Multimodal GenAI ch. 1 · LLMs ch. 5
Smoothness constant L (on graph pages, L is the Laplacian) Optimization for AI ch. 2 · Machine Learning ch. 0
Mini-batch B with \lvert B \rvert = b Deep Learning P1 ch. 3 · Optimization for AI ch. 5

R(f) = \mathbb E\, \ell\big(f(X), Y\big), \qquad \hat R_n(f) = \frac1n \sum_{i=1}^n \ell\big(f(x_i), y_i\big)

Two orders coexist and are both fixed. A loss takes the prediction first, as in \ell(\hat y, y). A divergence takes the reference distribution first, as in \mathrm{KL}(p \,\|\, q) (see the ELBO section).

Optimizers

Optimization pages and training recipes share the same letters for steps, momenta and constants.

Object Symbol and convention Courses that use it (links)
Learning rate \eta, also for deterministic methods and Robbins–Monro steps Optimization for AI ch. 2 · Deep Learning P1 ch. 3 · Reinforcement Learning ch. 3
Momentum \beta; Nesterov’s coefficient is \beta_k Refresher Neural Nets ch. 5 · Optimization for AI ch. 3 · Deep Learning P1 ch. 3
Adam (\beta_1, \beta_2, \epsilon) with moments m_t, v_t Deep Learning P1 ch. 3 · Optimization for AI ch. 10 · Refresher Neural Nets ch. 5
Weight decay \lambda Deep Learning P1 ch. 3 · Optimization for AI ch. 10
Constants smoothness L, strong convexity \mu, condition number \kappa = L/\mu, gradient-noise variance \sigma^2 Optimization for AI ch. 2, ch. 5 · Emerging ML ch. 0
Mini-batch size b; Emerging ML writes m (declared deviation: b is its Laplace scale) Optimization for AI ch. 5 · Deep Learning P1 ch. 3

v_{t+1} = \beta\, v_t + g_t, \qquad \theta_{t+1} = \theta_t - \eta\, v_{t+1}

m_t = \beta_1 m_{t-1} + (1 - \beta_1)\, g_t, \qquad v_t = \beta_2 v_{t-1} + (1 - \beta_2)\, g_t^2

\theta_t = \theta_{t-1} - \eta\, \frac{\hat m_t}{\sqrt{\hat v_t} + \epsilon}, \qquad \hat m_t = \frac{m_t}{1 - \beta_1^t}, \quad \hat v_t = \frac{v_t}{1 - \beta_2^t}

Linear layers and attention

A single example is a column; a batch stacks examples as rows. The weight matrix is stored as in nn.Linear.weight, so code and formulas match.

Object Symbol and convention Courses that use it (links)
Linear layer, one example column form h = Wx + b, W \in \mathbb R^{d_{\mathrm{out}} \times d_{\mathrm{in}}} Refresher Neural Nets ch. 2, ch. 3 · Deep Learning P1 ch. 1, ch. 2 · Deep Learning P2 ch. 0
Linear layer, a batch rows: H_{\mathrm{out}} = H_{\mathrm{in}} W^\top + \mathbf 1 b^\top Refresher Neural Nets ch. 3 · Deep Learning P1 ch. 2
GNN layers row-major HW with W \in \mathbb R^{d_{\mathrm{in}} \times d_{\mathrm{out}}}, if the page says so Graph ML ch. 5 · Advanced GNN ch. 0
LoRA update BA, B \in \mathbb R^{d_{\mathrm{out}} \times r}, A \in \mathbb R^{r \times d_{\mathrm{in}}} Deep Learning P1 ch. 7 · MLOps ch. 7
Attention inputs tokens as rows, X \in \mathbb R^{n \times d}; Q = X W_Q with W_Q \in \mathbb R^{d \times d_k}, same for K and V Deep Learning P1 ch. 6 · Text Mining and NLP ch. 7 · LLMs ch. 2 · Advanced DL ch. 0
Heads and scale h heads, d_k = d/h; the scale is \sqrt{d_k} Deep Learning P1 ch. 6 · Deep Learning P2 ch. 0, ch. 1 · Multimodal GenAI ch. 0 · Speech ch. 0

h = W x + b, \quad W \in \mathbb R^{d_{\mathrm{out}} \times d_{\mathrm{in}}} \qquad \Longleftrightarrow \qquad H_{\mathrm{out}} = H_{\mathrm{in}} W^\top + \mathbf 1 b^\top

\mathrm{Att}(Q, K, V) = \mathrm{softmax}\Big(\frac{Q K^\top}{\sqrt{d_k}}\Big) V \qquad (\text{softmax taken row by row})

Vocabulary and word embeddings

The vocabulary is calligraphic, so that V stays free for the value matrix and, on graph pages, for the vertex set.

Object Symbol and convention Courses that use it (links)
Vocabulary \mathcal V, of size \lvert \mathcal V \rvert Text Mining and NLP ch. 1 · LLMs ch. 1 · Deep Learning P1 ch. 6
Embedding matrix E \in \mathbb R^{\lvert \mathcal V \rvert \times d} Text Mining and NLP ch. 4 · LLMs ch. 0 · Deep Learning P1 ch. 6
Skip-gram with negative sampling two tables: center vectors v_w, context vectors u_c Text Mining and NLP ch. 4 · Graph ML ch. 3 · Deep Learning P1 ch. 6 · LLMs ch. 0
Negative samples k negatives n_1, \dots, n_k \sim p_n, with p_n(w) \propto \mathrm{freq}(w)^{3/4} Text Mining and NLP ch. 4 · Graph ML ch. 3

\mathcal L_{\mathrm{neg}} = -\log \sigma\big(u_c^\top v_w\big) - \sum_{i=1}^{k} \log \sigma\big(-u_{n_i}^\top v_w\big)

On graph pages, the “word” w is the center node of a random-walk window and c a node in its context.

Scaling laws, experts and the KV cache

Large-model pages count parameters, tokens, experts and bytes with the letters below. E is the irreducible loss, so the number of experts gets its own symbol.

Object Symbol and convention Courses that use it (links)
Model and data size N parameters, D training tokens, compute C \approx 6ND FLOPs Advanced DL ch. 9 · LLMs ch. 3 · Deep Learning P2 ch. 1 · Emerging ML ch. 7
Loss scaling law \mathcal L(N, D) with irreducible loss E (Hoffmann et al. 2022 write L) LLMs ch. 3 · Advanced DL ch. 9
Mixture of experts n_E experts, top-k routing Advanced DL ch. 6 · LLMs ch. 2 · Advanced Topics in LLMs ch. 5
Load-balancing loss \mathcal L_{\mathrm{aux}} with weight \alpha_{\mathrm{aux}}; f_i routed fraction, P_i mean router probability LLMs ch. 2 · Advanced Topics in LLMs ch. 5
KV cache M_{\mathrm{KV}} in bytes; n_{\mathrm{layers}} layers, n_{\mathrm{kv}} key/value heads of size d_{\mathrm{head}} LLMs ch. 2, ch. 6 · Deep Learning P2 ch. 5 · MLOps ch. 3

C \approx 6ND, \qquad \mathcal L(N, D) = E + \frac{A}{N^{\alpha}} + \frac{B}{D^{\beta}}

\mathcal L_{\mathrm{aux}} = \alpha_{\mathrm{aux}}\, n_E \sum_{i=1}^{n_E} f_i P_i, \qquad \sum_{i=1}^{n_E} f_i = 1

The fraction f_i is normalized by k times the number of tokens, so the fractions sum to one even with top-k routing.

M_{\mathrm{KV}} = 2\, n_{\mathrm{layers}}\, n_{\mathrm{kv}}\, d_{\mathrm{head}}\; b\, T\, s

Here b is the batch size, T the number of cached tokens per sequence and s the bytes per element (s = 2 in fp16). For Llama-3-8B in fp16 this gives 128 KiB per token.

Temperatures and distillation

Four letters share the “temperature” role, each in one place: T, \tau, \eta and \beta.

Object Symbol and convention Courses that use it (links)
Softmax with temperature \mathrm{softmax}_T, with T = 1 by default LLMs ch. 6 · Privacy and UQ ch. 7 · Advanced DL ch. 3
Decoding, calibration, distillation temperature T LLMs ch. 6 · Privacy and UQ ch. 7 · Advanced DL ch. 3 · CV Applications ch. 8 · Deep Learning P2 ch. 5
Contrastive similarity (CLIP, NT-Xent) \tau, and only there Multimodal GenAI ch. 5 · Advanced DL ch. 7
Gibbs and exponential weights inverse temperature \eta Machine Learning ch. 9 · Statistical Learning Theory ch. 7 · Probability and Monte Carlo ch. 8
KL-regularized RL and RLHF coefficient \beta LLMs ch. 5 · Control in GenAI ch. 5
Soft actor-critic entropy coefficient \alpha, flagged as a temperature Reinforcement Learning ch. 6
Distillation weight \lambda, on the teacher term Advanced DL ch. 3 · CV Applications ch. 8 · Responsible AI ch. 3

\mathrm{softmax}_T(z)_i = \frac{e^{z_i / T}}{\sum_j e^{z_j / T}}

\pi_j \propto \pi_{0,j}\, e^{-\eta\, n \hat R_n(f_j)}, \qquad \pi^\star \propto \pi_{\mathrm{ref}}\, e^{r / \beta}

\mathcal L_{\mathrm{KD}} = (1 - \lambda)\, \ell_{\mathrm{CE}}(z_s, y) + \lambda\, T^2\, \mathrm{KL}\big(\mathrm{softmax}(z_t / T) \,\big\|\, \mathrm{softmax}(z_s / T)\big)

Here z_t and z_s are the teacher and student logits. The factor T^2 keeps the gradient of the soft term on the scale of the hard term. Where \eta or \beta is read as a temperature, the temperature is its inverse.

Probabilistic models and the ELBO

The KL divergence always puts the reference distribution first. The ELBO is a bound that is maximized, so it never borrows the letter of a loss.

Object Symbol and convention Courses that use it (links)
KL divergence \mathrm{KL}(p \,\Vert\, q) = \mathbb E_p[\log p/q] every row below
Maximum likelihood minimizes \mathrm{KL}(p_{\mathrm{data}} \,\Vert\, p_\theta) Machine Learning ch. 3, ch. 7 · Refresher Statistics ch. 5
VAE regularizer \mathrm{KL}\big(q_\phi(z \mid x) \,\Vert\, p(z)\big) Deep Learning P1 ch. 9 · Multimodal GenAI ch. 0, ch. 1
RLHF and DPO \beta\, \mathrm{KL}(\pi_\theta \,\Vert\, \pi_{\mathrm{ref}}) LLMs ch. 5 · Control in GenAI ch. 5
Trust region (TRPO, PPO) \mathrm{KL}(\pi_{\mathrm{old}} \,\Vert\, \pi_\theta) Reinforcement Learning ch. 5
Distillation \mathrm{KL}(p_{\mathrm{teacher}} \,\Vert\, p_{\mathrm{student}}) Advanced DL ch. 3 · CV Applications ch. 8
Entropic optimal transport \varepsilon\, \mathrm{KL}(\Gamma \,\Vert\, a b^\top) Optimal Transport ch. 3
Bayesian deep learning each KL bound states its direction Privacy and UQ ch. 8
ELBO \mathrm{ELBO}(\theta, \phi; x), maximized; the EM bound is \mathrm{ELBO}(q, \theta) Deep Learning P1 ch. 9 · Multimodal GenAI ch. 0, ch. 1 · Machine Learning ch. 7
VAE training loss \mathcal L_{\mathrm{VAE}} = -\mathrm{ELBO} Deep Learning P1 ch. 9 · Multimodal GenAI ch. 1

\mathrm{KL}(p \,\|\, q) = \mathbb E_{x \sim p}\Big[\log \frac{p(x)}{q(x)}\Big]

\mathrm{ELBO}(\theta, \phi; x) = \mathbb E_{q_\phi(z \mid x)}\big[\log p_\theta(x \mid z)\big] - \mathrm{KL}\big(q_\phi(z \mid x) \,\|\, p(z)\big) \;\le\; \log p_\theta(x)

\mathcal L_{\mathrm{VAE}}(\theta, \phi; x) = -\,\mathrm{ELBO}(\theta, \phi; x)

Hidden Markov models

Sequence pages share one HMM notation. The forward variable includes the emission at time t; the backward variable does not.

Object Symbol and convention Courses that use it (links)
States and observations hidden states z_t \in \{1, \dots, K\}, observations o_t Text Mining and NLP ch. 6 · Speech ch. 1
Parameters initial law \pi, transitions A = (a_{ij}), emissions b_j(o) Text Mining and NLP ch. 6 · Speech ch. 1
Forward and backward \alpha_t(j) and \beta_t(j) Text Mining and NLP ch. 6 · Speech ch. 1, ch. 2
Viterbi best-path score \delta_t(j) Text Mining and NLP ch. 6 · Speech ch. 1
Topic models LDA hyperparameters \alpha_{\mathrm{LDA}}, \beta_{\mathrm{LDA}}, so that \alpha and \beta stay free Text Mining and NLP ch. 5

\alpha_t(j) = p\big(o_{1:t},\, z_t = j\big), \qquad \beta_t(j) = p\big(o_{t+1:T} \mid z_t = j\big)

\alpha_{t+1}(j) = b_j(o_{t+1}) \sum_{i=1}^{K} \alpha_t(i)\, a_{ij}, \qquad \delta_{t+1}(j) = b_j(o_{t+1}) \max_{i}\, \delta_t(i)\, a_{ij}

The CTC forward–backward follows the same rule: \alpha_t(s) contains the frame-t emission and \beta_t(s) does not. The original CTC paper puts it in both, so its product \alpha_t \beta_t is divided by the emission.

Diffusion

Time runs from data at t = 0 to noise at t = T. The guidance weight w equals 1 for plain conditional sampling.

Object Symbol and convention Courses that use it (links)
Noise schedule \beta_t, \alpha_t = 1 - \beta_t, \bar\alpha_t = \prod_{s \le t} \alpha_s Multimodal GenAI ch. 3 · Deep Learning P2 ch. 7 · Advanced DL ch. 10 · Probability and Monte Carlo ch. 9
Noise predictor \varepsilon_\theta(x_t, t, c); condition c, null condition \varnothing Multimodal GenAI ch. 3, ch. 4 · Control in GenAI ch. 7
Classifier-free guidance weight w: w = 0 unconditional, w = 1 conditional, w > 1 over-guided Multimodal GenAI ch. 4 · Control in GenAI ch. 7 · Graph Generative Models ch. 7 · Advanced DL ch. 10
Discrete diffusion one-hot row vectors; Q_t row-stochastic, \bar Q_t = Q_1 \cdots Q_t Graph ML ch. 9 · Graph Generative Models ch. 5
Other weights named β \beta_{\mathrm{KL}} for the β-VAE weight, \beta_c for the VQ commitment weight Multimodal GenAI ch. 1

x_t = \sqrt{\bar\alpha_t}\, x_0 + \sqrt{1 - \bar\alpha_t}\, \varepsilon, \qquad \varepsilon \sim \mathcal N(0, I)

\nabla_{x_t} \log p_t(x_t) \approx -\frac{\varepsilon_\theta(x_t, t)}{\sqrt{1 - \bar\alpha_t}}

\tilde\varepsilon_w = \varepsilon_\theta(x_t, t, \varnothing) + w \big(\varepsilon_\theta(x_t, t, c) - \varepsilon_\theta(x_t, t, \varnothing)\big)

Some papers write guidance in the form below. Their weight w' is ours minus one, w' = w - 1, so their w' = 0 is plain conditional sampling.

\tilde\varepsilon = (1 + w')\, \varepsilon_\theta(x_t, t, c) - w'\, \varepsilon_\theta(x_t, t, \varnothing)

q(x_t \mid x_{t-1}) = \mathrm{Cat}\big(x_t;\ x_{t-1} Q_t\big), \qquad q(x_t \mid x_0) = \mathrm{Cat}\big(x_t;\ x_0 \bar Q_t\big)

Q_t = \alpha_t I + \beta_t\, \mathbf 1 m^\top \quad \Longrightarrow \quad \bar Q_t = \bar\alpha_t I + (1 - \bar\alpha_t)\, \mathbf 1 m^\top

With marginal transitions toward a law m, as in DiGress, \bar\alpha_t is the probability of keeping the original category. The training loss splits into minimized terms:

-\mathrm{ELBO} = \mathcal L_T + \sum_{t \ge 2} \mathcal L_{t-1} + \mathcal L_0

Reinforcement learning and RLHF

Value functions always carry their policy. A bare G is the GRPO group size; the return always has an index or an argument.

Object Symbol and convention Courses that use it (links)
Markov decision process (\mathcal S, \mathcal A, P, r, \gamma), policy \pi(a \mid s) Reinforcement Learning ch. 1 · Deep Learning P1 ch. 1 · Deep Learning P2 ch. 7
Return G_t, and G(\tau) for a trajectory Reinforcement Learning ch. 1 · Deep Learning P2 ch. 7
Value functions V^\pi, Q^\pi, A^\pi, always with the superscript; optimal V^\star, Q^\star Reinforcement Learning ch. 1, ch. 3 · Deep Learning P1 ch. 1 · Real-time AI ch. 7
Occupancy and objective d^\pi, defined where it is used; J(\pi) Reinforcement Learning ch. 5
Policy optimization ratio \rho_t(\theta), Fisher matrix F(\theta), GAE estimate \hat A_t Reinforcement Learning ch. 5 · LLMs ch. 5
RLHF \pi_\theta, \pi_{\mathrm{ref}}, reward r_\phi(x, y), preference y_w \succ y_l, coefficient \beta, GRPO group size G LLMs ch. 4, ch. 5 · Control in GenAI ch. 5
Bandits and tree search counts N_a(t), means \hat\mu_a(t), index \mathrm{UCB}_a(t); UCT uses the same form Reinforcement Learning ch. 4 · Real-time AI ch. 6

G_t = \sum_{k \ge 0} \gamma^k R_{t+k+1}, \qquad V^\pi(s) = \mathbb E_\pi\big[G_t \mid S_t = s\big], \qquad A^\pi = Q^\pi - V^\pi

\rho_t(\theta) = \frac{\pi_\theta(a_t \mid s_t)}{\pi_{\theta_{\mathrm{old}}}(a_t \mid s_t)}

\max_{\pi}\ \mathbb E_{\pi}[r] - \beta\, \mathrm{KL}(\pi \,\|\, \pi_{\mathrm{ref}}) \quad \Longrightarrow \quad \pi^\star(y \mid x) \propto \pi_{\mathrm{ref}}(y \mid x)\, e^{r(x, y)/\beta}

\mathrm{UCB}_a(t) = \hat\mu_a(t) + c \sqrt{\frac{\ln t}{N_a(t)}}, \qquad c = \sqrt 2 \ \text{ for UCB1}

The soft actor-critic keeps \alpha for its entropy coefficient (see temperatures). Step sizes are \eta, as on the optimizer pages.

Differential privacy

Privacy pages share one budget, one neighbor relation and one randomized-response parameter. A bare \varepsilon is always a privacy budget.

Object Symbol and convention Courses that use it (links)
Privacy guarantee (\varepsilon, \delta)-DP with add/remove neighbors D \simeq D', unless stated Emerging ML ch. 5 · Privacy and UQ ch. 1 · Explainability, Security, Privacy of LLMs ch. 8
Sensitivities \Delta_1 and \Delta_2 Privacy and UQ ch. 2 · Emerging ML ch. 5
Mechanisms Laplace noise of scale \Delta_1 / \varepsilon; Gaussian noise of standard deviation \sigma_G Privacy and UQ ch. 2 · Explainability, Security, Privacy of LLMs ch. 8
DP-SGD clip C, noise multiplier \sigma, sampling rate q = b/n, steps T, dataset size n Privacy and UQ ch. 4 · Explainability, Security, Privacy of LLMs ch. 8 · Emerging ML ch. 5
Randomized response p, the probability of a truthful answer Privacy and UQ ch. 4 · Advanced GNN ch. 7
Attack budget \varepsilon_{\mathrm{atk}} (adversarial radius); smoothing noise \sigma Security and Robustness ch. 2, ch. 3, ch. 5

\Pr\big[\mathcal M(D) \in S\big] \le e^{\varepsilon}\, \Pr\big[\mathcal M(D') \in S\big] + \delta \qquad \text{for all } D \simeq D' \text{ and all } S

\sigma_G \ge \frac{\Delta_2 \sqrt{2 \ln(1.25/\delta)}}{\varepsilon} \qquad (0 < \varepsilon < 1)

\tilde g = \frac1b \Big(\sum_{i \in B} \mathrm{clip}_C(g_i) + \mathcal N\big(0, \sigma^2 C^2 I\big)\Big)

\varepsilon = \ln \frac{p}{1 - p} \qquad (\text{randomized response, } p \ge \tfrac12)

When a page deviates

Sometimes a page must keep another symbol: it follows a paper line by line, or a letter is already taken in its own chapter. The rule is simple. A page that deviates says so at the start of its notation block, the definition or paragraph where it first fixes its symbols, usually in chapter 0.

That opening line points here and lists only the deviations, each with its translation to the site-wide symbol. The deviation then holds on the whole page: no silent switch halfway through. A course page could open its notation block like this.

**Notation.** Site-wide conventions: see [Notation conventions](../../notation/index.html).
Deviations on this page: $d$ is the TextRank damping factor, that is $1 - \alpha$ in the site-wide convention.

Letters are scoped to their family. On graph pages \lambda is an eigenvalue, in an optimizer it is the weight decay, in distillation it weights the teacher. When two families meet on one page and a letter clashes, the page uses the subscripted forms of the tables: \eta_g, \alpha_{\mathrm{aux}}, \beta_{\mathrm{KL}}, \alpha_{\mathrm{LDA}}, \varepsilon_{\mathrm{atk}}.

The pages are being aligned with this table one by one. Until a page carries its notation line, read its own definitions first. A new convention is added here before a page relies on it.