Machine Learning & Deep Learning,
in Brief.

Every method below reduces, ultimately, to four pillars: linear algebra (how we represent data and transformations), probability (how we quantify uncertainty), optimization (how we learn from data), and information theory (how we measure signal). A researcher who has internalised these pillars can read any new architecture or paper and reconstruct its logic from first principles.

Linear Algebra

Inner product

$⟨\mathbf{x},\mathbf{y}⟩={\mathbf{x}}^{\top }\mathbf{y}={\sum }_i{x}_i{y}_i$. Orthogonality: $⟨\mathbf{x},\mathbf{y}⟩=0$.

Norms

$\Vert \mathbf{x}{\Vert }_2=\sqrt{{\mathbf{x}}^{\top }\mathbf{x}}$, $\Vert \mathbf{x}{\Vert }_1={\sum }_i|x_i|$, $\Vert \mathbf{x}{\Vert }_{\infty }={\max }_i|x_i|$. All ${\ell }_p$ for $p\ge 1$ are norms.

Rank

$\mathrm{rank}(A)$ is the dimension of the column space; $A\in {\mathbb{R}}^{m\times n}$ has $\mathrm{rank}(A)\le \min (m,n)$.

Positive definite

$A\succ 0⟺{\mathbf{x}}^{\top }A\mathbf{x}>0$ for all $\mathbf{x}\ne 0$; equivalently all eigenvalues are positive.

Eigendecomposition

For symmetric $A\in {\mathbb{R}}^{n\times n}$ there exists an orthogonal $Q$ and diagonal $\Lambda$ such that $A=Q\Lambda Q^{\top }$, with eigenvectors as columns of $Q$ and eigenvalues on the diagonal of $\Lambda$. Symmetric matrices have real eigenvalues and orthogonal eigenvectors.

Singular Value Decomposition (SVD)

For any $A\in {\mathbb{R}}^{m\times n}$, $A=U\Sigma V^{\top }$ with $U\in {\mathbb{R}}^{m\times m}$, $V\in {\mathbb{R}}^{n\times n}$ orthogonal and $\Sigma$ diagonal with singular values ${\sigma }_1\ge {\sigma }_2\ge \cdots \ge 0$. SVD gives the best rank-$k$ approximation in Frobenius and spectral norms (Eckart–Young):

$$A_k=\sum _{i=1}^k{\sigma }_i{u}_i{v}_i^{\top },\Vert A-A_k{\Vert }_F^2=\sum _{i>k}{\sigma }_i^2.$$

Matrix calculus — identities to memorise

Probability & Statistics

Bayes' rule is the only rational procedure for updating beliefs given evidence; all probabilistic ML is a specialization of it.

Bayes' rule. $p(\theta \mid x)=\frac{p(x\mid \theta )p(\theta )}{p(x)}$. The denominator is the marginal likelihood, $p(x)=\int p(x\mid \theta )p(\theta )d\theta$.

Maximum likelihood (MLE). ${\hat{\theta }}_{\mathrm{MLE}}=\arg {\max }_{\theta }{\sum }_i\log p(x_i\mid \theta )$. Asymptotically normal with variance given by the inverse Fisher information $\mathcal{I}(\theta {)}^{-1}$.

Maximum a posteriori (MAP). ${\hat{\theta }}_{\mathrm{MAP}}=\arg {\max }_{\theta }{\sum }_i\log p(x_i\mid \theta )+\log p(\theta )$. Gaussian prior $\Rightarrow$ ${\ell }_2$ regularization; Laplace prior $\Rightarrow$ ${\ell }_1$.

Useful distributions & conjugacies

Expectation, variance, covariance. $\mathrm{Var}(X)=\mathbb{E}[X^2]-\mathbb{E}[X{]}^2$. For independent $X,Y$: $\mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)$. Covariance matrix ${\Sigma }_{ij}=\mathrm{Cov}(X_i,X_j)$ is symmetric PSD.

Law of Large Numbers & CLT. For i.i.d. $X_i$ with mean $\mu$, variance ${\sigma }^2$: ${\bar{X}}_n\to \mu$ a.s. and $\sqrt{n}({\bar{X}}_n-\mu )\Rightarrow \mathcal{N}(0,{\sigma }^2)$.

Delta method. If $\sqrt{n}(\hat{\theta }-\theta )\Rightarrow \mathcal{N}(0,{\sigma }^2)$ and $g$ is differentiable at $\theta$, then $\sqrt{n}(g(\hat{\theta })-g(\theta ))\Rightarrow \mathcal{N}(0,(g^{\prime }(\theta ){)}^2{\sigma }^2)$.

Optimization

Convexity. $f$ is convex iff $f(\lambda x+(1-\lambda )y)\le \lambda f(x)+(1-\lambda )f(y)$. A twice-differentiable $f$ is convex iff ${\nabla }^{2}f⪰0$. For convex problems, every local min is global.

Gradient descent. ${\theta }_{t+1}={\theta }_t-\eta \nabla f({\theta }_t)$. For $L$-smooth convex $f$ (i.e. $\nabla f$ is $L$-Lipschitz), choosing $\eta \le 1/L$ guarantees $f({\theta }_t)-f^{\star }\le \mathcal{O}(1/t)$. For $\mu$-strongly convex, linear convergence $\mathcal{O}(e^{-t\mu /L})$.

Stochastic gradient descent. Replaces $\nabla f$ with an unbiased mini-batch estimator ${\hat{g}}_t$, $\mathbb{E}[{\hat{g}}_t]=\nabla f$. Requires decreasing step sizes $\sum {\eta }_t=\infty$, $\sum {\eta }_t^2<\infty$ (Robbins–Monro) for convergence of the iterates under convexity and bounded variance.

Constrained optimization — KKT. For $\min f(x)$ s.t. $g_i(x)\le 0$, $h_j(x)=0$, the Lagrangian is $\mathcal{L}(x,\lambda ,\nu )=f(x)+\sum {\lambda }_i{g}_i(x)+\sum {\nu }_j{h}_j(x)$. At an optimum (under constraint qualifications): stationarity, primal/dual feasibility, and complementary slackness ${\lambda }_i{g}_i(x^{\star })=0$.

Information Theory

Entropy

$H(p)=-{\sum }_{x}p(x)\log p(x)$. Concave; maximised by uniform.

Cross-entropy

$H(p,q)=-{\sum }_{x}p(x)\log q(x)=H(p)+D_{\mathrm{KL}}(p\Vert q)$.

KL divergence

$D_{\mathrm{KL}}(p\Vert q)={\sum }_{x}p(x)\log \frac{p(x)}{q(x)}\ge 0$, non-symmetric.

Mutual information

$I(X;Y)=D_{\mathrm{KL}}(p(x,y)\Vert p(x)p(y))=H(X)-H(X\mid Y)$.

Why cross-entropy is the classification loss. Minimising $H(p_{\text{data}},q_{\theta })={\mathbb{E}}_{x\sim p_{\text{data}}}[-\log q_{\theta }(x)]$ is equivalent to minimising $D_{\mathrm{KL}}(p_{\text{data}}\Vert q_{\theta })$, since $H(p_{\text{data}})$ is constant in $\theta$. This is maximum likelihood.

Jensen's inequality. For convex $\phi$: $\phi (\mathbb{E}[X])\le \mathbb{E}[\phi (X)]$. Underlies the ELBO, importance sampling bounds, and most variational arguments.

We describe the invariants that apply to every model — the decomposition of error, the shape of regularization, and the vocabulary of evaluation.

The Bias–Variance Decomposition

For a point $x_0$ and a learning algorithm producing $\hat{f}$ on random training sets:

$$\underset{\text{expected squared error}}{\underbrace{\mathbb{E}\left[(y-\hat{f}(x_0){)}^2\right]}}=\underset{{\text{Bias}}^2}{\underbrace{(\mathbb{E}[\hat{f}(x_0)]-f(x_0){)}^2}}+\underset{\text{Variance}}{\underbrace{\mathrm{Var}[\hat{f}(x_0)]}}+\underset{\text{irreducible}}{\underbrace{{\sigma }_{\epsilon }^2}}.$$

Regularization

Ridge (${\ell }_2$). Minimise $\Vert \mathbf{y}-X\mathbf{\beta }{\Vert }_2^2+\lambda \Vert \mathbf{\beta }{\Vert }_2^2$. Closed form: $\hat{\mathbf{\beta }}=(X^{\top }X+\lambda I{)}^{-1}{X}^{\top }\mathbf{y}$. Always well-conditioned; shrinks coefficients uniformly.

Lasso (${\ell }_1$). Minimise $\Vert \mathbf{y}-X\mathbf{\beta }{\Vert }_2^2+\lambda \Vert \mathbf{\beta }{\Vert }_1$. No closed form; solved via coordinate descent or proximal gradient. Induces sparsity — exact zeros — because the ${\ell }_1$ ball has corners along the coordinate axes.

Elastic net. ${\lambda }_1\Vert \mathbf{\beta }{\Vert }_1+{\lambda }_2\Vert \mathbf{\beta }{\Vert }_2^2$. Retains sparsity of Lasso while handling correlated features (groups them rather than arbitrarily picking one).

Equivalences. ${\ell }_2$ penalty ≡ Gaussian prior on weights; ${\ell }_1$ ≡ Laplace prior; data augmentation ≡ marginalising over an implicit prior.

Evaluation & Model Selection

Cross-validation

• k-Fold — default for i.i.d. tabular data; pick $k\in \{5,10\}$.
• Stratified k-Fold — preserves class balance in classification.
• Leave-one-out (LOO) — low bias, high variance; expensive.
• Nested CV — outer loop estimates generalisation, inner loop tunes hyperparameters; avoids optimistic bias from reusing the same data for both.
• Time-series CV — use forward chaining (expanding/rolling window). Never shuffle.

Classification metrics

Regression metrics

MSE (penalises outliers quadratically), MAE (robust), RMSE (same units as $y$), $R^2=1-\mathrm{RSS}/\mathrm{TSS}$, MAPE (scale-free but undefined at $y=0$), Pearson / Spearman / Kendall correlation — indispensable in quant research where rank of predictions matters more than absolute values.

Feature engineering checklist

• Scaling: Standardise or min-max for distance-based models (kNN, SVM, k-means) and any gradient-based optimization (faster, more stable convergence). Trees are scale-invariant.
• Encoding: One-hot for low-cardinality categoricals; target/mean encoding with out-of-fold statistics for high-cardinality (prevents target leakage).
• Missing data: Explicit missing indicator + imputation beats silent imputation, since "missing" is often informative.
• Leakage audit: Any feature that would not be available at prediction time is leakage. The most expensive bugs in ML are silent leakages.

Before neural networks, these are the tools that set the baselines. In quantitative research, linear methods and gradient-boosted trees still dominate tabular tasks — they are compact, interpretable, and hard to beat with modest data.

Linear Models

Ordinary Least Squares

$\hat{\mathbf{\beta }}=(X^{\top }X{)}^{-1}{X}^{\top }\mathbf{y}$. Assumes linearity, no multicollinearity, homoscedastic and uncorrelated errors. The Gauss–Markov theorem: OLS is BLUE (Best Linear Unbiased Estimator) under these assumptions.

Logistic Regression

Models $p(y=1\mid \mathbf{x})=\sigma ({\mathbf{w}}^{\top }\mathbf{x}+b)$ with $\sigma (z)=1/(1+e^{-z})$. Negative log-likelihood:

$$\mathcal{L}(\mathbf{w})=-\sum _i[y_i\log \sigma ({\mathbf{w}}^{\top }{\mathbf{x}}_i)+(1-y_i)\log (1-\sigma ({\mathbf{w}}^{\top }{\mathbf{x}}_i))].$$

Gradient: ${\nabla }_{\mathbf{w}}\mathcal{L}={\sum }_i(\sigma ({\mathbf{w}}^{\top }{\mathbf{x}}_i)-y_i){\mathbf{x}}_i=X^{\top }(\hat{\mathbf{p}}-\mathbf{y})$. Hessian is PSD $\Rightarrow$ convex; optimizable by IRLS / L-BFGS / SGD.

Multinomial / softmax. $p(y=k\mid \mathbf{x})=e^{{\mathbf{w}}_k^{\top }\mathbf{x}}/{\sum }_j{e}^{{\mathbf{w}}_j^{\top }\mathbf{x}}$. One coefficient vector is redundant (shift invariance); often fix ${\mathbf{w}}_K=0$.

Generalised linear models (GLMs)

Exponential-family likelihood with link function $g$: $g({\mu }_i)={\mathbf{x}}_i^{\top }\mathbf{\beta }$. Linear regression ($g=\mathrm{id}$, Gaussian), logistic ($g=\mathrm{logit}$, Bernoulli), Poisson regression ($g=\log$, Poisson). Inference via IRLS.

Trees & Ensembles

Decision trees recursively split on the feature and threshold that maximise impurity reduction. Classification uses Gini ${\sum }_k{p}_k(1-p_k)$ or entropy $-{\sum }_k{p}_k\log p_k$; regression uses variance reduction. Single trees overfit; depth and leaf-count are the main regularisers.

Trees are scale-invariant, handle mixed types and missing values natively, and are the only mainstream family that can represent non-smooth, interaction-heavy functions without feature engineering.

Random Forests. Bagging: train $B$ trees on bootstrap samples, and at each split consider only a random $m$ of $p$ features. Variance-reduction ensemble: $\mathrm{Var}(\bar{f})=\rho {\sigma }^2+(1-\rho ){\sigma }^2/B$, where $\rho$ is the between-tree correlation. Feature subsampling lowers $\rho$.

Gradient Boosting. Fit tree $h_t$ to the negative gradient of the loss w.r.t. the current prediction $F_{t-1}$: $F_t=F_{t-1}+\eta h_t$. With squared loss, the gradient is the residual; with logistic loss, it is $y-\hat{p}$. Implementations (XGBoost, LightGBM, CatBoost) add second-order terms, regularized leaves, histogram splits, and native categorical handling.

Kernels, k-NN, Naive Bayes

Support Vector Machines

Maximise the margin $2/\Vert \mathbf{w}\Vert$ subject to $y_i({\mathbf{w}}^{\top }{\mathbf{x}}_i+b)\ge 1-{\xi }_i$, ${\xi }_i\ge 0$. Primal:

$$\underset{\mathbf{w},b,\mathbf{\xi }}{\min }\frac{1}{2}\Vert \mathbf{w}{\Vert }^2+C\sum _i{\xi }_i\text{s.t.}{y}_i({\mathbf{w}}^{\top }{\mathbf{x}}_i+b)\ge 1-{\xi }_i.$$

Dual only depends on inner products $⟨{\mathbf{x}}_i,{\mathbf{x}}_j⟩$, enabling the kernel trick: replace with $K({\mathbf{x}}_i,{\mathbf{x}}_j)$ corresponding to an inner product in some RKHS.

Common kernels: linear ${\mathbf{x}}^{\top }\mathbf{y}$; polynomial $({\mathbf{x}}^{\top }\mathbf{y}+c{)}^d$; RBF $\exp (-\Vert \mathbf{x}-\mathbf{y}{\Vert }^2/2{\sigma }^2)$; Matérn (GP-friendly).

k-Nearest Neighbours

No training; classify by majority among $k$ nearest points. Curse of dimensionality: in high $d$, distances concentrate and all points become roughly equidistant. Useful for small, low-$d$ problems and as a local baseline.

Naive Bayes

Assumes features are conditionally independent given class: $p(\mathbf{x}\mid y)={\prod }_{j}p(x_j\mid y)$. Decision: $\arg {\max }_{y}p(y){\prod }_{j}p(x_j\mid y)$. Despite the crude independence assumption, it is an extraordinarily strong text-classification baseline.

Unsupervised Learning

k-Means

Minimises ${\sum }_i\Vert {\mathbf{x}}_i-{\mathbf{\mu }}_{c(i)}{\Vert }^2$ by alternating (i) assign each point to nearest centroid, (ii) update centroids to cluster means. Non-convex; use k-means++ seeding. Assumes spherical, equal-variance clusters.

Gaussian Mixture Models (EM)

$p(\mathbf{x})={\sum }_{k=1}^K{\pi }_k\mathcal{N}(\mathbf{x}\mid {\mathbf{\mu }}_k,{\Sigma }_k)$. Fit via EM: E-step compute responsibilities ${\gamma }_{ik}={\pi }_k\mathcal{N}({\mathbf{x}}_i\mid {\mathbf{\mu }}_k,{\Sigma }_k)/{\sum }_j{\pi }_j\mathcal{N}({\mathbf{x}}_i\mid {\mathbf{\mu }}_j,{\Sigma }_j)$; M-step update ${\pi }_k,{\mathbf{\mu }}_k,{\Sigma }_k$ by weighted MLE. EM monotonically increases the log-likelihood.

Principal Component Analysis

Find orthonormal directions of maximum variance. Center $X$ and compute top-$k$ eigenvectors of the covariance $\frac{1}{n}{X}^{\top }X$ — equivalently, top-$k$ right singular vectors of $X$. The first PC maximises $\mathrm{Var}(X\mathbf{v})$ subject to $\Vert \mathbf{v}\Vert =1$, giving ${\mathbf{v}}_1=\arg \max {\mathbf{v}}^{\top }\Sigma \mathbf{v}$ — the leading eigenvector.

t-SNE / UMAP. Non-linear embeddings for visualisation. Preserve local neighbourhoods but not global distances; never interpret cluster sizes or between-cluster distances literally.

A neural network is a parameterised function $f_{\theta }:\mathcal{X}\to \mathcal{Y}$ built by composing affine maps and non-linearities. Everything below — activations, initializations, normalizations, optimizers — exists to make the composition of very many such layers trainable at scale.

Multi-layer perceptron

${\mathbf{h}}^{(\ell )}=\varphi (W^{(\ell )}{\mathbf{h}}^{(\ell -1)}+{\mathbf{b}}^{(\ell )})$, ${\mathbf{h}}^{(0)}=\mathbf{x}$. Universal approximation: a single hidden layer with enough units can approximate any continuous function on a compact set. In practice, depth buys far more expressivity per parameter than width.

Activation functions

Backpropagation

Backprop is reverse-mode automatic differentiation applied to a DAG of differentiable primitives. For each node, given $\partial \mathcal{L}/\partial {\mathbf{h}}^{(\ell )}$, compute $\partial \mathcal{L}/\partial {\mathbf{h}}^{(\ell -1)}$ and $\partial \mathcal{L}/\partial W^{(\ell )}$ by the chain rule. For the MLP above, with ${\mathbf{z}}^{(\ell )}=W^{(\ell )}{\mathbf{h}}^{(\ell -1)}+{\mathbf{b}}^{(\ell )}$:

$${\mathbf{\delta }}^{(\ell )}=\frac{\partial \mathcal{L}}{\partial {\mathbf{z}}^{(\ell )}}=\left(W^{(\ell +1)\top }{\mathbf{\delta }}^{(\ell +1)}\right)\odot {\varphi }^{\prime }({\mathbf{z}}^{(\ell )}),\frac{\partial \mathcal{L}}{\partial W^{(\ell )}}={\mathbf{\delta }}^{(\ell )}{\mathbf{h}}^{(\ell -1)\top }.$$

Vanishing & exploding gradients. Products of Jacobian norms ${\prod }_{\ell }\Vert J^{(\ell )}\Vert$ either collapse to zero or diverge as depth grows. Mitigations: careful initialization, normalization layers, residual connections, gradient clipping.

Initialization

Xavier / Glorot

$W_{ij}\sim \mathcal{N}(0,2/(n_{\mathrm{in}}+n_{\mathrm{out}}))$. Suited to tanh / sigmoid.

He / Kaiming

$W_{ij}\sim \mathcal{N}(0,2/n_{\mathrm{in}})$. Suited to ReLU.

Orthogonal

$W$ a random orthogonal matrix; preserves activation norms through depth. Good for RNNs.

Optimizers

All practical optimizers for deep learning fit the pattern: maintain a state of first- and second-moment estimates of the gradient, apply scaled updates, optionally decouple weight decay.

Learning-rate schedules

• Step / exponential decay — classical, robust.
• Cosine annealing ${\eta }_t={\eta }_{\min }+\frac{1}{2}({\eta }_{\max }-{\eta }_{\min })(1+\cos (\pi t/T))$ — smooth, widely used.
• Warmup — linearly ramp $\eta$ for the first few thousand steps; essential for Transformers, where early updates are otherwise catastrophic.
• One-cycle — warm up, then cosine down; enables aggressive max LRs.

Regularization & Normalization

Weight decay

Adds $\frac{\lambda }{2}\Vert \theta {\Vert }^2$ to the loss; equivalent to Gaussian prior. Implement as decoupled (AdamW) for adaptive optimizers.

Dropout

Randomly zero each activation with probability $p$ at training time, scale by $1/(1-p)$. Approximates averaging over an exponential ensemble of sub-networks.

Early stopping

Halt when validation loss stops improving. Implicit regularization equivalent to an ${\ell }_2$ ball in some linear settings.

Data augmentation

Learn invariances (flips, crops, MixUp, CutMix, noise). Essentially the most effective regularizer in computer vision.

Label smoothing

Replace one-hot target $y$ with $(1-\epsilon )y+\epsilon /K$. Prevents overconfident logits; improves calibration.

Normalization

Loss functions

MSE

$\frac{1}{n}\sum (y_i-{\hat{y}}_i{)}^2$. MLE under Gaussian noise.

MAE / Huber

Robust to outliers; Huber is quadratic near zero, linear in the tail.

Binary cross-entropy

$-[y\log \hat{p}+(1-y)\log (1-\hat{p})]$. MLE for Bernoulli.

Categorical CE

$-{\sum }_k{y}_k\log {\hat{p}}_k$. MLE for Categorical / softmax.

Contrastive / InfoNCE

$-\log \frac{e^{s(\mathbf{x},{\mathbf{x}}^{+})/\tau }}{{\sum }_j{e}^{s(\mathbf{x},{\mathbf{x}}_j)/\tau }}$. Lower bound on mutual information; core of self-supervised learning.

Convolutional Networks (CNN)

A convolution layer applies a small shared kernel $K$ across spatial positions: $(X\ast K{)}_{i,j}={\sum }_{u,v}{X}_{i+u,j+v}{K}_{u,v}$. Parameter-sharing + translation equivariance are the inductive biases that make CNNs so efficient on images.

Output spatial size after a conv with kernel $k$, padding $p$, stride $s$: $\lfloor (n+2p-k)/s\rfloor +1$. Receptive field grows linearly with depth for fixed $k$; dilations expand it exponentially.

Pooling (max, average) downsamples and builds local invariance. Modern architectures often replace pooling with strided convolutions.

Key designs

• ResNet. ${\mathbf{h}}^{(\ell )}={\mathbf{h}}^{(\ell -1)}+\mathcal{F}({\mathbf{h}}^{(\ell -1)})$. Residual connections let gradients flow through identity paths — the reason networks can go to hundreds of layers without collapsing.
• Inception / MobileNet. Factorised / depthwise-separable convolutions: $k\times k\times c$ → ($k\times k$ depthwise) + ($1\times 1$ pointwise). Same expressivity, a fraction of the FLOPs.
• U-Net. Encoder–decoder with skip connections at each resolution. Dominant for segmentation; the backbone of most diffusion models.

Recurrent Networks (RNN / LSTM / GRU)

Vanilla RNN: ${\mathbf{h}}_t=\tanh (W_h{\mathbf{h}}_{t-1}+W_x{\mathbf{x}}_t+\mathbf{b})$. Backprop through time unrolls the recurrence; gradients flow through $T$ products of Jacobians, producing vanishing/exploding gradients on long sequences.

LSTM

Gated cell with explicit memory ${\mathbf{c}}_t$:

$$\begin{array}{rl}{\mathbf{f}}_t& =\sigma (W_f[{\mathbf{h}}_{t-1},{\mathbf{x}}_t])\text{(forget)}\\ {\mathbf{i}}_t& =\sigma (W_i[{\mathbf{h}}_{t-1},{\mathbf{x}}_t])\text{(input)}\\ {\mathbf{o}}_t& =\sigma (W_o[{\mathbf{h}}_{t-1},{\mathbf{x}}_t])\text{(output)}\\ {\tilde{\mathbf{c}}}_t& =\tanh (W_c[{\mathbf{h}}_{t-1},{\mathbf{x}}_t])\\ {\mathbf{c}}_t& ={\mathbf{f}}_t\odot {\mathbf{c}}_{t-1}+{\mathbf{i}}_t\odot {\tilde{\mathbf{c}}}_t\\ {\mathbf{h}}_t& ={\mathbf{o}}_t\odot \tanh ({\mathbf{c}}_t)\end{array}$$

The additive cell update is the key: gradients propagate through $\mathbf{c}$ without repeated multiplication by Jacobians, taming vanishing gradients. GRU is a streamlined variant with 2 gates and no separate cell state.

Attention & Transformers

Scaled dot-product attention. Given queries $Q\in {\mathbb{R}}^{n\times d_k}$, keys $K\in {\mathbb{R}}^{n\times d_k}$, values $V\in {\mathbb{R}}^{n\times d_v}$:

$$\mathrm{Attention}(Q,K,V)=\mathrm{softmax}\left(\frac{Q{K}^{\top }}{\sqrt{d_k}}\right)V.$$

The $1/\sqrt{d_k}$ keeps the dot products at a scale where softmax is not saturated. Complexity is $\mathcal{O}(n^{2}d)$ in sequence length.

Multi-head attention. Run $h$ heads in parallel with separate $W_q^{(i)},W_k^{(i)},W_v^{(i)}$, concatenate outputs, project with $W_o$. Each head learns a different relational pattern.

Transformer block. Pre-norm variant, the modern default:

x = x + MHA(LN(x))            # self-attention sub-layer
x = x + MLP(LN(x))            # feed-forward sub-layer, typically 4x width

Positional information. Attention is permutation-equivariant; inject order via sinusoidal embeddings, learned absolute positions, or — in modern LLMs — Rotary Position Embeddings (RoPE) applied to $Q$ and $K$.

Causal vs bidirectional. Decoder-only (GPT-style) masks future positions — used for autoregressive generation. Encoder (BERT-style) sees both directions — used for embedding / classification.

Generative Models

Autoencoders & VAEs

An autoencoder learns an identity through a bottleneck, $\mathbf{x}\approx g(f(\mathbf{x}))$. A Variational AE places a latent prior $p(\mathbf{z})$ and an inference network $q_{\varphi }(\mathbf{z}\mid \mathbf{x})$, maximising the ELBO:

$$\log p_{\theta }(\mathbf{x})\ge {\mathbb{E}}_{q_{\varphi }(\mathbf{z}\mid \mathbf{x})}[\log p_{\theta }(\mathbf{x}\mid \mathbf{z})]-D_{\mathrm{KL}}(q_{\varphi }(\mathbf{z}\mid \mathbf{x})\Vert p(\mathbf{z})).$$

Trained by reparameterisation: $\mathbf{z}={\mathbf{\mu }}_{\varphi }(\mathbf{x})+{\mathbf{\sigma }}_{\varphi }(\mathbf{x})\odot \mathbf{\epsilon }$ with $\mathbf{\epsilon }\sim \mathcal{N}(0,I)$.

Generative Adversarial Networks

Minimax game between generator $G$ and discriminator $D$:

$$\underset{G}{\min }\underset{D}{\max }{\mathbb{E}}_{\mathbf{x}\sim p_{\text{data}}}[\log D(\mathbf{x})]+{\mathbb{E}}_{\mathbf{z}\sim p_z}[\log (1-D(G(\mathbf{z})))].$$

Non-saturating generator loss, Wasserstein GAN with gradient penalty, spectral normalisation — all are stabilisations around this same objective.

Diffusion models

Forward process adds Gaussian noise: ${\mathbf{x}}_t=\sqrt{{\bar{\alpha }}_t}{\mathbf{x}}_0+\sqrt{1-{\bar{\alpha }}_t}\mathbf{\epsilon }$. A network ${\mathbf{\epsilon }}_{\theta }({\mathbf{x}}_t,t)$ is trained to predict the noise; sampling reverses the chain. Simple loss:

$$\mathcal{L}={\mathbb{E}}_{t,{\mathbf{x}}_0,\mathbf{\epsilon }}\left[\Vert \mathbf{\epsilon }-{\mathbf{\epsilon }}_{\theta }({\mathbf{x}}_t,t){\Vert }^2\right].$$

Variants: score-matching / SDE formulation, DDIM for deterministic sampling, classifier-free guidance for conditional generation.

Quantitative research inherits the full ML toolkit but operates under constraints foreign to most ML papers: low signal-to-noise, non-stationarity, non-i.i.d. samples, and economic costs for every false positive. The techniques below address these constraints directly.

Time Series

Stationarity

A process is (weakly) stationary if its mean and autocovariance are time-invariant. Most classical methods assume stationarity; in finance, price series are non-stationary, but returns are approximately so.

ARMA(p,q)

$X_t={\sum }_{i=1}^p{\varphi }_i{X}_{t-i}+{\epsilon }_t+{\sum }_{j=1}^q{\theta }_j{\epsilon }_{t-j}$. AR$(p)$: last $p$ lags; MA$(q)$: last $q$ innovations.

ARIMA(p,d,q)

Difference the series $d$ times to achieve stationarity, then fit ARMA.

GARCH(1,1)

${\sigma }_t^2=\omega +\alpha {\epsilon }_{t-1}^2+\beta {\sigma }_{t-1}^2$. Captures volatility clustering — the single most robust empirical feature of financial returns.

Cointegration

Non-stationary $X_t,Y_t$ may have a stationary linear combination $Y_t-\beta X_t$. Foundation of pairs / statistical-arbitrage strategies.

Autocorrelation & tests

ACF $\rho (k)=\mathrm{Corr}(X_t,X_{t-k})$; PACF is the partial autocorrelation at lag $k$ controlling for intermediate lags. Ljung–Box tests jointly if the first $k$ autocorrelations are zero. ADF / KPSS test (non-)stationarity.

State-space models & the Kalman filter

Linear Gaussian state-space model:

$${\mathbf{x}}_t=F{\mathbf{x}}_{t-1}+{\mathbf{w}}_t,{\mathbf{w}}_t\sim \mathcal{N}(0,Q);{\mathbf{y}}_t=H{\mathbf{x}}_t+{\mathbf{v}}_t,{\mathbf{v}}_t\sim \mathcal{N}(0,R).$$

Kalman recursion alternates predict and update. It is the optimal linear estimator and, under Gaussian noise, the optimal estimator full stop. Used in quant for dynamic hedge ratios, time-varying betas, and signal smoothing.

Portfolio Theory & Risk

Mean–variance optimization

Minimum-variance portfolio with target return ${\mu }^{\star }$:

$$\underset{\mathbf{w}}{\min }\frac{1}{2}{\mathbf{w}}^{\top }\Sigma \mathbf{w}\text{s.t.}{\mathbf{w}}^{\top }1=1,{\mathbf{w}}^{\top }\mathbf{\mu }={\mu }^{\star }.$$

Closed-form solution via Lagrange multipliers. Tangency (max-Sharpe) portfolio: ${\mathbf{w}}^{\star }\propto {\Sigma }^{-1}(\mathbf{\mu }-r_{f}1)$.

Risk & performance metrics

Covariance estimation

The sample covariance $\hat{\Sigma }$ is noisy when $p\gtrsim n$ and inverting it (as in MV optimization) amplifies that noise. Remedies: Ledoit–Wolf shrinkage ${\hat{\Sigma }}_{\mathrm{LW}}=(1-\lambda )\hat{\Sigma }+\lambda \mu I$; factor-model covariance $\Sigma =B{\Sigma }_f{B}^{\top }+D$; minimum-variance with no short-sell constraints acts as implicit regularization.

Backtesting: Hazards & Hygiene

• Look-ahead bias. Using data not yet available at the trade decision time. Easily hidden in resampling, normalization, or target encodings computed on the full sample.
• Survivorship bias. Universe includes only firms that exist today, not the dead ones. Systematically inflates returns.
• Selection bias. Data-mining a strategy from a large search space without correcting for multiplicity. See deflated Sharpe ratio and the Bailey–López de Prado corrections.
• Transaction costs & slippage. A signal that is alpha gross of costs can easily be zero or negative net of them.
• Regime / non-stationarity. A backtest spanning one regime cannot validate a strategy in another. Always report performance per sub-period.
• Purged / embargoed cross-validation. Remove observations whose label windows overlap training data; embargo a gap after each test fold to prevent leakage of serial correlation.

Time-series cross-validation

Forward-chaining expanding-window CV respects causality. For overlapping labels (e.g. forward $h$-day returns), apply purging — remove training samples whose label window intersects the test window — and embargo — a buffer of length $h$ after the test set.

ML for Alpha — What Actually Matters

• Signal-to-noise is the enemy. Daily returns have SNR orders of magnitude lower than images or text. Capacity of the model must match the information in the data — enormous networks overfit noise.
• Prefer rank-based and robust losses. Squared error is dominated by outliers. Quantile, Huber, and rank-based losses (Spearman surrogates) typically generalise better for return prediction.
• Feature stability beats peak accuracy. A signal that works moderately across sub-periods is more valuable than one that is excellent in one and useless in another.
• Combine linear + non-linear. Linear factors remain the bulk of explanatory power; trees / NNs add interactions. Stack or residualise.
• Bet sizing is orthogonal to signal. Kelly (or a fractional Kelly) links expected edge and variance to position size. A good signal sized poorly is a loss-making strategy.
• Evaluate economic utility, not just statistical fit. Out-of-sample Sharpe, turnover, capacity, drawdown, and sensitivity to costs — in that order.

Each exercise is labelled by topic and difficulty (◆ introductory, ◆◆ intermediate, ◆◆◆ advanced). Solutions are hidden by default — attempt the problem before revealing the worked argument. The point of these is not the answer itself but the chain of reasoning; they are chosen to exercise canonical techniques you will reuse for the rest of your career.