Neural Networks & Deep Learning

Formulas

Neural networks are inspired by biology, but have long since become an independent mathematical field. Rosenblatt's perceptron (1957) is the simplest formal neuron that already reveals the main principle: linear transformation + nonlinear activation. Multilayer networks (MLP) made of such neurons...

Biological motivation: A neuron in the brain receives signals through dendrites, sums them in the cell body, and "fires" (spike) only when the threshold is exceeded. The perceptron models this: weighted sum of inputs + threshold activation function.

Perceptron: $y = \operatorname{sign}(w^\mathrm{T}x + b) = \operatorname{sign}(\sum_i w_ix_i + b)$, where $x \in \mathbb{R}^n$ is the input vector, $w \in \mathbb{R}^n$ are weights, $b$ is bias (threshold), sign is the sign function ($\pm1$). Geometrically: $w$ and $b$ define the hyperplane $w^\ma...

Perceptron convergence theorem (Rosenblatt, 1957): If classes are linearly separable, the update algorithm $w_{t+1} = w_t + y_ix_i$ (when there is an error on $(x_i, y_i)$) converges in a finite number of steps. The upper bound of iterations: $(R/\gamma)^2$, where $R = \max||x_i||$ is the radius,...

Formulas

Image processing has become a breakthrough application of neural networks. Convolutional networks (CNN, LeCun et al., 1989) leverage the local structure of images: features (edges, textures) are local and shift-invariant. CNNs revolutionized computer vision, surpassing humans on ImageNet in 2012.

A fully connected network for a 224×224×3 image: the first layer would have 224×224×3 = 150,528 inputs. With 1,000 neurons: 150M parameters — inefficient, does not scale.

2D Convolution: A kernel (filter) $K \in \mathbb{R}^{f\times f\times C_{in}}$ “slides” over the input image $X \in \mathbb{R}^{H\times W\times C_{in}}$:

$(X * K)[i,j] = \sum_{m=0}^{f-1} \sum_{n=0}^{f-1} \sum_{c=0}^{C_{in}-1} X[i+m, j+n, c] \cdot K[m, n, c] + b$

Neural networks with millions of parameters can "memorize" the training set completely — overfit. The gap between train and test error is the main enemy in practice. Regularization is a set of techniques that reduce overfitting without sacrificing model expressiveness.

Simple model: high bias, low variance. Complex model (DNN): low bias (can approximate everything), but high variance (sensitive to data).

Double descent: With modern overparameterized networks, the classical U-shaped tradeoff is broken — when the number of parameters ≫ n, error decreases again.

Penalty: L_reg = L + λ/2 · ||θ||² = L + λ/2 · Σᵢ θᵢ². SGD update: θ ← θ − α·∇L − αλθ = θ(1 − αλ) − α·∇L. "Weight decay": each step decreases weights by a factor of (1−αλ) — pulls towards zero.

Formulas

Training neural networks with dozens of layers is a non-trivial task. Vanishing gradient, exploding gradient, dead neurons, poor initialization—all these are practical problems that have required decades of research to solve.

Mechanism: During backpropagation through $L$ layers: $\dfrac{\partial L}{\partial x_0} = \left(\prod_l \dfrac{\partial a^l}{\partial a^{l-1}}\right) \cdot \dfrac{\partial L}{\partial a^L}$. Each multiplier $\dfrac{\partial a^l}{\partial a^{l-1}} = W^l \cdot \operatorname{diag}(\sigma'(z^l))$. If...

Sigmoid and tanh: $\sigma'(z) \leq 0.25$ (maximum at $z=0$). For $L$ layers: $||\dfrac{\partial L}{\partial x_0}|| \leq (0.25 \cdot \max||W^l||)^L \rightarrow 0$ if $\max||W^l|| < 4$. ReLU eliminates saturation problem: $\sigma'(z) = 1$ for $z>0$ — gradient passes without attenuation.

Gradient clipping: for explosion: $\nabla \leftarrow \nabla \cdot \min(1, \tau/||\nabla||)$. Limit the gradient's norm. Standard in RNN/LSTM. PyTorch: `torch.nn.utils.clip_grad_norm_(model.parameters(), max_norm=1.0)`.

Neural networks are "black boxes": they output predictions but do not explain their decisions. In high-stakes domains (medicine, finance, law), explainability is a legal requirement (GDPR Art. 22). Interpretability also helps debug models and detect systematic errors.

What has the CNN learned? One can directly study the filters $W^{(1)}$ of the first layer (dimensions 3×3×3 or 7×7×3): they often resemble Gabor edges. But deeper layers are high-dimensional and not directly interpretable.

Activation maps: for an image $x$, we visualize the feature map $a^l[k]$ (the $k$-th channel of layer $l$). Bright pixels = strong response to the $k$-th pattern.

$ w_k = \frac{1}{Z}\sum_{i,j} \frac{\partial y^c}{\partial A_{ij}^k} \quad \text{(global average pooling of the gradients)} $

Many real-world data have a graph structure: social networks (users + connections), molecules (atoms + bonds), road networks (intersections + roads), knowledge (entities + relations). Standard CNN and MLP ignore the structure — GNN preserve it.

Graph G = (V, E): n nodes, m edges. Each node v has features x_v ∈ ℝᵈ. Adjacency matrix A ∈ {0,1}^{n×n}. Task: prediction of the label for each node (node classification), edge (link prediction), or the entire graph (graph classification).

The main difference from ordinary data: there is no fixed order of nodes (graph is permutation-equivariant). The size of the graph may vary. It is necessary to account for the neighborhood structure.

The general principle of GNN is "message passing" between neighbors. Each node aggregates information from neighbors, updates its representation:

Recurrent neural networks are designed for processing sequences—text, time series, audio. They maintain “memory” through a hidden state passed through time. LSTM is a key invention from 1997 that enabled learning of long-term dependencies.

Breakdown: hₜ is the hidden state (memory of the past), xₜ is the input at time t, Wₕₕ is the “memory” matrix (links hₜ and hₜ₋₁), Wₓₕ is the input matrix, Wₕᵧ is the output matrix. The same weights W are used at every step (weight sharing over time).

Backpropagation Through Time (BPTT): To train an RNN, we unroll it over time and apply the chain rule: ∂L/∂Wₕₕ = Σₜ ∂L/∂hₜ · ∂hₜ/∂Wₕₕ. Gradient: ∂hₜ/∂hₛ = Π_{k=s}^{t-1} ∂hₖ₊₁/∂hₖ = Π_{k=s}^{t-1} Wₕₕᵀ diag(tanh'(zₖ)).

The product of (t−s) matrices → if ||Wₕₕᵀ diag(tanh'(zₖ))|| < 1: vanishing; if > 1: exploding. This is a problem for long sequences (T > 20–30).

Before neural networks can process text, words must be transformed into numerical representations. The evolution from one-hot vectors to Word2Vec, FastText, and BERT demonstrates how neural networks have "learned" to "understand" language. This is one of the most revolutionary trajectories in NLP.

One-hot: word i → a vector of zeros with a one at the i-th position. Dimension = |V| (vocabulary, 50K–200K). Problems: (1) All pairwise cosine distances = 0 — there is no semantic similarity. (2) Huge dimensionality. (3) Sparsity.

Distributional Hypothesis (Harris, 1954): words with similar context have similar meanings. "Bank" and "finance" often occur in the same context → are close. "Cat" and "dog" are close. This is the basis for Word2Vec.

Skip-gram: Given a word $w_t$ — predict the surrounding words $w_{t-2}, \ldots, w_{t-1}, w_{t+1}, \ldots, w_{t+2}$ (window $c=2$).

Formulas

Time series—financial data, IoT sensors, meteorological observations, energy consumption—require specialized architectures. Neural networks compete with classical methods (ARIMA, ES) and often outperform them when sufficient data volume is available.

Key properties: Autocorrelation (values depend on the past). Trend (long-term direction). Seasonality (periodic patterns: daily, weekly, yearly). Non-stationarity (statistics change over time: mean, variance).

Preprocessing: Normalization: $\hat{x}_t = (x_t − \mu)/\sigma$. Differencing (removes non-stationarity): $\Delta x_t = x_t − x_{t-1}$. Seasonal differencing: $x_t − x_{t-s}$. Log transformation: $\ln(x_t)$—stabilizes variance under multiplicative seasonality.

Train/test split: You cannot randomly shuffle! Only chronological splitting: train = first 80%, test = last 20%.

GAN (Goodfellow et al., 2014) is one of the most creative architectural patterns in the history of machine learning. Yann LeCun called them “the most interesting idea in machine learning in the past 20 years.” They spawned an entire era of synthetic media: realistic faces, deepfakes, AI artists.

Two agents: Generator G: z → x̂ (generates synthetic data from noise z). Discriminator D: x → [0,1] (distinguishes real data from synthetic).

Decoding: D maximizes the probability of correct classification of real (D(x)→1) and fake (D(G(z))→0) data. G minimizes the probability that D will detect the fake (D(G(z))→1).

Optimal D with fixed G: D*(x) = p_data(x)/(p_data(x) + p_G(x)). Substituting: max_D V = −log(4) + 2·JSD(p_data || p_G). JSD is the Jensen-Shannon divergence. At equilibrium: p_G = p_data → D* = 1/2 → V = −log(4).

Formulas

VAE and diffusion models are alternatives to GANs for data generation, with fundamentally different mathematical foundations. VAE is based on variational Bayesian inference; diffusion models are based on iterative denoising and connections with diffusion equations.

Problem: We want to train a generative model $p_\theta(x) = \int p_\theta(x|z) p(z) dz$, where $z$ is a latent code. The integral over all $z$ is intractable (high-dimensional integration).

$ \log p_\theta(x) \geq \mathbb{E}_{q_\phi(z|x)}[\log p_\theta(x|z)] - KL(q_\phi(z|x) \| p(z)) = \mathrm{ELBO}(\theta, \phi; x) $

Decoding the two terms: (1) $\mathbb{E}[\log p_\theta(x|z)]$ — reconstruction loss: how well the decoder reconstructs $x$ from $z$ — reconstruction quality. (2) $-KL(q_\phi(z|x)\|p(z))$ — KL-divergence between the posterior $q_\phi(z|x)$ and prior $p(z)=N(0,I)$ — regularization, pulls the posteri...

Generative models have gone far beyond academic tasks. They create real value in pharmaceuticals, materials science, data synthesis, media production, and climate science.

AlphaFold 2 (DeepMind, 2021): Revolutionary prediction of 3D protein structure from amino acid sequence. Architecture: Evoformer (transformer with “pair representation” of residue interactions) + Structure Module (explicit 3D positioning). Accuracy: error ≈ 0.96 Å — comparable to crystallographic...

Generative molecule discovery: VAE or normalizing flows in the space of SMILES strings. Candidate molecules: maximize binding to target, minimize toxicity, maximize solubility. Navigation in the latent space of molecules. Company Insilico Medicine: GAN-designed molecule against IPF (lung fibrosis...

Protein design (RFDiffusion, David Baker lab, 2023): Diffusion model for de novo protein design. Specifies desired function → generates structure → no amino acids yet → inverse folding (ProteinMPNN) → synthesis and test. 70% of designs function in the laboratory (vs 5% for predecessors).

Transformer (Vaswani et al., “Attention Is All You Need”, 2017) is a revolutionary architecture that eliminated recurrence in sequence processing. Self-attention allows each element to interact directly with any other, without recurrent bottlenecks. Transformers now dominate in NLP, vision, prote...

Intuition: When translating “The animal didn't cross the street because it was too tired” → it is necessary to realize that “it” = “animal”. An RNN “forgets” long-range dependencies. Attention allows you to explicitly “look” at needed positions when generating each token.

Q (queries) ∈ ℝ^{n×d_k}: “what we’re searching for”. K (keys) ∈ ℝ^{m×d_k}: “what is offered” at all positions. V (values) ∈ ℝ^{m×d_v}: “what we take” from the selected positions.

Steps: (1) QKᵀ ∈ ℝ^{n×m}: dot products of every query-key pair. Element [i,j]: how much position j “responds” to query at position i. (2) /√d_k: scaling prevents overly large values with high d_k (otherwise softmax saturates). (3) softmax: normalization into a distribution (rows sum to 1). (4) ·V...

GPT-4, Claude, Gemini, LLaMA — large language models (LLM) have become a transformative technology of the 2020s. Understanding their capabilities, training mechanisms, and limitations is a necessity for every AI specialist.

Interpretation: N — number of parameters, D — dataset size in tokens, C — compute budget in FLOPs. Loss decreases according to a power law as each factor increases — improvement is predictable and continuous. A practical conclusion: more data + larger model + more computation → better results.

Hoffmann-Chinchilla Law (DeepMind, 2022): Optimal ratio at fixed C: N∝D — parameters ≈ tokens (multiplied by a constant ≈ 20). GPT-3 (175B) was trained on 300B tokens → “undertrained”. Chinchilla (70B, 1.4T tokens) outperforms Gopher (280B, 300B tokens). Strategy: fewer parameters, more data → gr...

Inference efficiency: LLaMA-3-8B (8B parameters, 15T tokens): 90% of GPT-3 performance with 22× fewer parameters and open access.

Universal LLMs are strong across a broad range of tasks, but in specialized fields—medicine, law, finance, code—adaptation is required. Fine-tuning and RAG allow for the creation of domain experts without training from scratch.

Full Fine-tuning: we update all model parameters on domain data. GPT-3 (175B): fine-tuning cost ~$100K. Risk of "catastrophic forgetting": the model loses general knowledge. Justified only for highly specialized tasks with a large dataset.

LoRA (Low-Rank Adaptation, Hu et al., 2021): We add to each weight matrix W a low-rank update: W' = W + BA, where B ∈ ℝ^{d×r}, A ∈ ℝ^{r×k}, r ≪ min(d,k). We train only B and A (0.01–1% parameters). Initialization: B=0, A~N — initially, there are no changes. During inference: W' = W + BA — just ad...

QLoRA (Dettmers et al., 2023): LoRA + 4-bit NormalFloat quantization of the base model. Allows fine-tuning LLaMA-65B on a single A100 GPU (80GB). "Killer feature" for academic labs.