From c0edd4465b5697ad45763f1659a5ca297309a7a8 Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Wed, 25 Feb 2026 15:14:04 +0100 Subject: [PATCH 1/6] add LLM grammar check --- chapters/Lezione12.typ | 492 +++++++++++++++++++++++++++++++++++++++++ main.typ | 1 + 2 files changed, 493 insertions(+) create mode 100644 chapters/Lezione12.typ diff --git a/chapters/Lezione12.typ b/chapters/Lezione12.typ new file mode 100644 index 0000000..561cb14 --- /dev/null +++ b/chapters/Lezione12.typ @@ -0,0 +1,492 @@ +#import "../template.typ": * + += Transformer + +I *Transformer* sono una delle architetture più importanti nel deep learning moderno, introdotti da Vaswani et al. nel 2017 nell'articolo _"Attention Is All You Need"_. + +#informalmente()[ + Un Transformer prende un insieme di vettori in uno spazio di rappresentazione e li *trasforma* in un altro insieme di vettori: + - Stesso numero di vettori + - Stessa dimensionalità + - Nuovo spazio di rappresentazione più ricco + + L'obiettivo è produrre una rappresentazione interna più espressiva, adatta a compiti a valle (classificazione, generazione, ecc.). +] + +== Perché i Transformer sono importanti + +/ Attenzione come meccanismo centrale: il meccanismo di *attenzione* permette al modello di decidere su cosa concentrarsi. Non tutte le parti di un input sono ugualmente rilevanti. + +/ Scalabilità: i Transformer si mappano efficientemente su hardware parallelo massivo (GPU). Modelli con $10^12$ parametri mostrano *capacità emergenti*, talvolta descritte come primi segnali verso l'intelligenza artificiale generale (AGI). + +/ Versatilità: le architetture Transformer sono state estese con successo a: + - Testo (NLP) + - Immagini (Vision Transformer, ViT) + - Audio, video, DNA + - Modelli multimodali + +/ Apprendimento auto-supervisionato: i Transformer possono essere addestrati su dati non etichettati (es. testo grezzo da Internet), sfruttando la *scaling hypothesis*: aumentando la dimensione del modello e dei dati, le prestazioni continuano a migliorare. + +#nota()[ + Prima dei Transformer, i modelli di NLP più diffusi erano: + - *Bag of Words (BoW)*: ignora l'ordine delle parole, non può esprimere il contesto. + - *Reti Neurali Ricorrenti (RNN)*: processano il testo sequenzialmente, difficili da parallelizzare e soggette a dimenticare informazioni a lunga distanza. + + I Transformer superano tutti questi limiti grazie alla self-attention. +] + +=== Dal testo agli embedding + +Il testo viene prima convertito in vettori densi attraverso: ++ *Tokenizzazione*: il testo viene suddiviso in token (parole, sottoparole o caratteri). ++ *Vocabulary mapping*: ogni token riceve un identificatore intero univoco. ++ *Embedding layer*: ogni ID viene mappato a un vettore denso tramite una matrice appresa: + $ + mb(E) in RR^(K times D) + $ + dove $K$ è la dimensione del vocabolario e $D$ è la dimensione dell'embedding. + +#attenzione()[ + Un embedding *statico* è insufficiente: la parola "banca" ha significati diversi in contesti diversi ("banca del fiume" vs "conto in banca"). La *self-attention* permette a ogni token di aggiornare la propria rappresentazione guardando tutti gli altri token nella sequenza. +] + +== Self-Attention + +=== Che cos'è l'attenzione? + +L'attenzione è un meccanismo che permette al modello di combinare vettori di input con *pesi dipendenti dai dati*. + +Data una sequenza di $N$ vettori token: +$ + mb(x)_1, dots, mb(x)_N in RR^D +$ + +la matrice dei token è: +$ + mb(X) = mat(mb(x)_1^T; mb(x)_2^T; dots.v; mb(x)_N^T) in RR^(N times D) +$ + +Gli elementi $x_(n i)$ dei token sono chiamati *feature*. Il blocco fondamentale di un Transformer è una funzione: +$ + tilde(mb(X)) = "TransformerLayer"(mb(X)) +$ +con $tilde(mb(X)) in RR^(N times D)$ (stessa dimensionalità dell'input). + +=== Calcolo delle similarità + +L'attenzione inizia calcolando le *similarità dot-product*: +$ + s_(m n) = mb(x)_m^T mb(x)_n +$ + +In forma matriciale (*matrice di similarità*): +$ + mb(S) = mb(X) mb(X)^T in RR^(N times N) +$ + +Un prodotto scalare elevato $arrow.r.double$ alta similarità. + +=== Normalizzazione Softmax + +Le similarità grezze vengono convertite in *pesi di attenzione* tramite softmax (applicata per righe): +$ + a_(m n) = e^(s_(m n)) / (sum_j e^(s_(m j))) +$ + +In forma matriciale: +$ + mb(A) = "Softmax"(mb(X) mb(X)^T) +$ + +Questo garantisce che le righe di $mb(A)$ sommino a 1 e che $a_(m n) >= 0$. + +=== Output del layer Transformer + +L'output è una combinazione pesata di tutti i vettori value: +$ + mb(y)_m = sum_(n=1)^N a_(m n) mb(x)_n +$ + +In forma matriciale: +$ + mb(Y) = mb(A) mb(X) = "Softmax"[mb(X) mb(X)^T] mb(X) +$ + +#informalmente()[ + Ogni $mb(y)_m$ è una *media pesata* di tutti gli $mb(x)_n$. L'attenzione apprende quali token sono rilevanti per ciascun output. +] + +Questo processo è chiamato *self-attention* perché la stessa sequenza $mb(X)$ viene usata come: +- Query (cosa cerco) +- Key (cosa è disponibile) +- Value (l'informazione che recupero) + +=== Query, Key, Value: rendere l'attenzione apprendibile + +#nota()[ + *Analogia con il recupero di informazioni*: immagina di cercare un film online. Ogni film ha attributi descrittivi (vettore *key*) e un contenuto (vettore *value*). L'utente esprime le sue preferenze (vettore *query*). Il sistema confronta la query con tutte le key, trova la corrispondenza più vicina e restituisce il value. +] + +Nei Transformer si introducono *proiezioni apprendibili*: +$ + mb(Q) = mb(X) mb(W)^((q)), quad mb(W)^((q)) in RR^(D times D_k) \ + mb(K) = mb(X) mb(W)^((k)), quad mb(W)^((k)) in RR^(D times D_k) \ + mb(V) = mb(X) mb(W)^((v)), quad mb(W)^((v)) in RR^(D times D_v) +$ + +La matrice di similarità diventa: +$ + mb(S) = mb(Q) mb(K)^T +$ + +e l'output: +$ + mb(A) = "Softmax"(mb(Q) mb(K)^T), quad mb(Y) = mb(A) mb(V) +$ + +Questo permette al modello di *apprendere cosa significa similarità* e come rappresentare i value. + +=== Scaled Dot-Product Self-Attention + +#attenzione()[ + *Problema*: quando gli elementi dei vettori query e key hanno media 0 e varianza 1, il loro prodotto scalare ha varianza $D_k$. Un'alta varianza porta a input grandi per la softmax, che diventa molto piatta e produce *gradienti molto piccoli* (problema simile a sigmoid/tanh saturi). + + *Soluzione*: dividere il punteggio di similarità per la deviazione standard $sqrt(D_k)$: + $ + "Attention"(mb(Q), mb(K), mb(V)) = "Softmax"(mb(Q) mb(K)^T / sqrt(D_k)) mb(V) + $ + + Questo *stabilizza la softmax*, mantiene i gradienti sani e rende l'attenzione robusta all'aumentare della dimensionalità. +] + +Questa versione scalata è la *forma finale* della self-attention dot-product usata nei Transformer. + +== Multi-Head Attention + +Una singola testa di attenzione può modellare solo *un tipo di relazione* alla volta. Ma il linguaggio naturale (e altri dati) contiene molteplici pattern simultanei: +- struttura grammaticale +- ruoli semantici +- tempo verbale +- dipendenze a lunga distanza +- relazioni lessicali + +*Idea*: usare più teste di attenzione in parallelo, ognuna con le proprie proiezioni apprese: +$ + mb(Q)_h = mb(X) mb(W)_h^((q)), quad + mb(K)_h = mb(X) mb(W)_h^((k)), quad + mb(V)_h = mb(X) mb(W)_h^((v)) +$ + +Ogni testa calcola la scaled dot-product attention: +$ + mb(H)_h = "Attention"(mb(Q)_h, mb(K)_h, mb(V)_h) +$ + +Le teste vengono poi *concatenate* e proiettate con una trasformazione lineare finale: +$ + mb(Y)(mb(X)) = "Concat"(mb(H)_1, dots, mb(H)_H) mb(W)^((o)) +$ + +#nota()[ + *Dimensioni*: ogni testa produce $mb(H)_h in RR^(N times D_v)$. Con $H$ teste: + $ + "Concat"(mb(H)_1, dots, mb(H)_H) in RR^(N times (H D_v)) + $ + La proiezione finale $mb(W)^((o)) in RR^(H D_v times D)$ riporta al embedding dimension del modello. + + Scegliendo $D_v = D / H$, la matrice concatenata ha dimensione $(N times D)$, uguale all'input. Tutte le matrici $mb(W)_h^((q)), mb(W)_h^((k)), mb(W)_h^((v)}, mb(W)^((o))$ vengono apprese congiuntamente. +] + +#informalmente()[ + Le diverse teste imparano *modi diversi di attendere*. Il modello cattura così più tipi di struttura contemporaneamente, in modo analogo ai filtri multipli in un layer CNN. +] + +== Self-Attention e Cross-Attention + +Un layer di attenzione standard prende in input due sequenze $mb(X)$ e $mb(X)'$ e calcola: +$ + mb(Q) = mb(X) mb(W)^((q)), quad + mb(K) = mb(X)' mb(W)^((k)), quad + mb(V) = mb(X)' mb(W)^((v)) +$ +$ + mb(A) = "Softmax"(mb(Q) mb(K)^T / sqrt(D_k)), quad + mb(Y) = mb(A) mb(V) +$ + +/ Self-attention: $mb(X) = mb(X)'$ — la stessa sequenza è usata per query, key e value. +/ Cross-attention: $mb(X) != mb(X)'$ — query dalla sequenza di output, key e value dalla sequenza di input (encoder). + +#nota()[ + La *cross-attention* esiste solo nei Transformer encoder–decoder, come: + - Modelli di machine translation + - Modelli di summarization + - Modelli sequence-to-sequence (T5, BART) +] + +== Transformer Layer: Residui, LayerNorm e MLP + +Per migliorare la stabilità e l'efficienza del training, il blocco di multi-head attention è arricchito con: + +/ Connessione residua: garantisce che l'output mantenga la stessa forma dell'input $mb(X) in RR^(N times D)$. + +/ Layer normalization: applicata dopo il blocco di attenzione (*post-norm*): + $ + mb(Z) = "LayerNorm"[mb(Y)(mb(X)) + mb(X)] + $ + oppure prima (*pre-norm*): + $ + mb(Z) = mb(Y)(mb(X)_0) + mb(X), quad mb(X)_0 = "LayerNorm"(mb(X)) + $ + +/ MLP position-wise: l'output dell'attenzione è una combinazione lineare di vettori input. Per aumentare l'espressività, ogni token viene passato attraverso lo stesso MLP non-lineare: + - Post-norm: $tilde(mb(X)) = "LayerNorm"["MLP"(mb(Z)) + mb(Z)]$ + - Pre-norm: $tilde(mb(X)) = "MLP"(mb(Z)_0) + mb(Z), quad mb(Z)_0 = "LayerNorm"(mb(Z))$ + +#informalmente()[ + Un layer Transformer completo combina: + + *Multi-Head Attention → Residuo → Norm → MLP → Residuo → Norm* + + Più layer di questo tipo vengono impilati con parametri indipendenti per costruire reti profonde. +] + +== Positional Encoding + +#attenzione()[ + La self-attention tratta tutti i token di input in modo *identico* rispetto alla loro posizione: riordinare l'input riordina semplicemente l'output. Questo è un vantaggio per il parallelismo, ma un *problema per le sequenze* dove l'ordine porta significato. + + Esempio: "Il cibo era cattivo, non buono." e "Il cibo era buono, non cattivo." hanno gli stessi token ma significato opposto. +] + +=== Soluzione: aggiungere vettori posizionali + +Per ogni posizione $n$, si crea un vettore $mb(r)_n$ e si forma: +$ + mb(x)'_n = mb(x)_n + mb(r)_n +$ + +*Requisiti per un buon positional encoding*: +- Unico per ogni posizione +- Limitato (bounded) +- Generalizza a sequenze più lunghe di quelle viste durante il training +- Rappresenta la *distanza relativa*, non solo l'indice assoluto + +Le *funzioni sinusoidali a frequenze multiple* soddisfano tutte queste proprietà. Per un token alla posizione $n$, il vettore di codifica posizionale $mb(r)_n$ ha componenti: +$ + r_(n,i) = cases( + sin(L^(i \/ D) n) & "se" i "è pari", + cos(L^((i-1) \/ D) n) & "se" i "è dispari" + ) +$ + +#nota()[ + *Proprietà chiave*: + - Ogni dimensione codifica la posizione con una *lunghezza d'onda diversa*. + - Le dimensioni inferiori variano lentamente → catturano struttura a lungo raggio. + - Le dimensioni superiori variano rapidamente → catturano struttura fine. + - La combinazione codifica *univocamente* la posizione rimanendo bounded. + - Le posizioni relative possono essere inferite dalle *differenze di fase* dei sinusoidi. + + Mescolando sin e cos a frequenze scalate esponenzialmente, il modello ottiene una rappresentazione continua e liscia dell'ordine dei token che si generalizza oltre la lunghezza di training. +] + +== Reti Transformer + +Vaswani et al. (2017) proposero un modello senza operazioni convoluzionali né ricorrenti, composto *esclusivamente da layer di attenzione*. Il Transformer completo è composto da: + +- Un *encoder* che combina $N=6$ moduli, ognuno con un sottomodulo di multi-head attention e un MLP a un layer nascosto per token, con connessioni residue e layer normalization. +- Un *decoder* con struttura simile, ma con layer di attenzione *causale* (masked) e layer di cross-attention che attendono alle key e value finali dell'encoder. + +=== Transformer Encoder + +L'encoder processa una sequenza di input e produce una rappresentazione contestuale più ricca. + +*Input*: $mb(X) in RR^(N times D)$, con $N$ = numero di token e $D$ = dimensione dell'embedding. + +Ogni layer dell'encoder è composto da: ++ Multi-head self-attention ++ Connessione residua + LayerNorm ++ MLP position-wise ++ Connessione residua + LayerNorm + +Ogni layer preserva la dimensionalità: input $N times D arrow$ output $N times D$. + +// Nota: il diagramma testuale viene descritto narrativamente +// poiché richiederebbe asset grafici specifici del corso + +#informalmente()[ + L'encoder costruisce *rappresentazioni contestuali dei token*: + - Ogni token attende a tutti gli altri token. + - Le dipendenze a lungo raggio vengono catturate direttamente. + - Il significato delle parole diventa dipendente dal contesto. + + Esempio: in "L'animale era stanco perché aveva corso", il token "aveva" attende fortemente ad "animale". + + L'impilamento dei layer raffina progressivamente le rappresentazioni. L'output finale dell'encoder ha la stessa forma dell'input ma con *struttura semantica molto più ricca*. +] + +==== Pre-training dell'encoder: Masked Language Modeling + +Un sottoinsieme casuale di token (es. il 15%) viene sostituito con un token speciale `[MASK]`. Il modello viene addestrato a *predire i token mancanti* dato il contesto: + +#esempio()[ + `Sono andato [MASK] il fiume per raggiungere la banca.` + + Il modello deve predire la parola mascherata dal contesto circostante. Una volta addestrato, l'encoder può essere *fine-tuned* per task di classificazione usando il primo token di output (corrispondente a `[CLS]`). +] + +=== Transformer Decoder + +Il decoder genera sequenze di output *un token alla volta*. Un decoder-only Transformer viene usato come *modello generativo*. + +*Input*: +- Token generati in precedenza (shifted right) +- Rappresentazioni dell'encoder (per modelli encoder–decoder) + +Ogni layer del decoder contiene: ++ Masked multi-head self-attention ++ Connessione residua + LayerNorm ++ Cross-attention (encoder–decoder attention) ++ Connessione residua + LayerNorm ++ MLP position-wise ++ Connessione residua + LayerNorm + +/ Masked self-attention: garantisce che un token possa attendere *solo ai token precedenti* (maschera causale). Nessun accesso alle informazioni future. + +#esempio()[ + *GPT (Generative Pretrained Transformer)*: il decoder apprende le probabilità condizionali: + $ + p(x_n | x_1, dots, x_(n-1)) + $ + - Input: i primi $n-1$ token + - Output: distribuzione di probabilità sul token $x_n$ + - Si campiona $x_n$, si appende alla sequenza, si ripete. +] + +==== Cosa apprende il decoder + +Il decoder esegue *generazione autoregressiva*. Per ogni posizione $t$: +- Attende ai token generati in precedenza. +- Opzionalmente attende agli output dell'encoder (in task sequence-to-sequence). +- Predice la distribuzione di probabilità del token successivo. +- L'ultimo layer produce logit → softmax → probabilità del prossimo token. + +Questo abilita: generazione di testo, traduzione, sistemi di dialogo, generazione di codice. + +=== Encoder vs Decoder: confronto + +#table( + columns: (auto, 1fr, 1fr), + [*Caratteristica*], [*Encoder*], [*Decoder*], + [Ruolo principale], [Costruire rappresentazioni contestuali], [Generare sequenza di output], + [Tipo di attenzione], [Full self-attention], [Masked self-attention], + [Accesso a token futuri], [Sì], [No (maschera causale)], + [Cross-attention], [No], [Sì (attende l'output dell'encoder)], + [Usato in], [BERT, Vision Transformer], [GPT, modelli di traduzione], + [Output], [Feature interne ricche], [Predizione del prossimo token], +) + +#informalmente()[ + - *Encoder* = Capire + - *Decoder* = Generare + - *Transformer completo* = Encoder → Decoder +] + +== Esempio: Machine Translation + +#esempio()[ + - Input (inglese): _"The cat is sleeping."_ + - Output (olandese): _"De kat slaapt."_ + + *Step 1 — Encoding*: l'encoder mappa l'intera frase di input in una rappresentazione interna: + $ + mb(Z) = "Encoder"(x_1, dots, x_N) + $ + + *Step 2 — Decoding*: il decoder genera i token di output uno alla volta, condizionato su: + - I token di output generati in precedenza + - La rappresentazione codificata $mb(Z)$ + + Questo condizionamento è realizzato tramite *cross-attention*, dove: + - Query ($mb(Q)$): stato corrente del decoder (cosa sto cercando) + - Key ($mb(K)$): output dell'encoder (cosa è disponibile) + - Value ($mb(V)$): informazione che recupero dall'encoder + + *Training*: usa coppie sequenza input–output, ottimizzate con cross-entropy loss sui token generati. +] + +#nota()[ + Questa architettura è usata in: + - Traduzione automatica + - Summarization + - Speech-to-text + - Task sequence-to-sequence generali +] + +== Training: Synthetic Copy Dataset + +Per verificare il corretto funzionamento di un Transformer end-to-end, si usa spesso un task sintetico: il modello deve *copiare una sequenza*. + +=== Token speciali + +/ `PAD = 0`: padding +/ `BOS = 1`: beginning of sequence +/ `EOS = 2`: end of sequence +/ Token payload: $in {3, dots, K-1}$ + +=== Struttura di un campione + +Sia $S$ la lunghezza della sequenza sorgente. Un campione contiene: + +*Encoder input (source)*: +$ + "src" = [x_1, x_2, dots, x_S] +$ + +*Decoder input (shifted right)*: +$ + "decoder\_input" = ["BOS", x_1, x_2, dots, x_S] +$ + +*Label (next-token targets)*: +$ + "label" = [x_1, x_2, dots, x_S, "EOS"] +$ + +#informalmente()[ + Questo task sintetico è utile perché: + - Ha un obiettivo semplice, facile da debuggare. + - Conferma che masking, attenzione e loss siano cablati correttamente. + - La loss dovrebbe scendere rapidamente se l'architettura è corretta. +] + +=== Training loop (teacher forcing) + +Per ogni batch: ++ *Costruire le maschere* (padding + causale per il decoder): `src_mask`, `tgt_mask` ++ *Encoder* costruisce la memoria: $mb(E) = "encode"("src", "src\_mask")$ ++ *Decoder* predice tutti i time step in parallelo: $mb(D) = "decode"(mb(E), "src\_mask", "decoder\_input", "tgt\_mask")$ ++ *Proiezione* al vocabolario: $"logits" in RR^(B times T times K)$ ++ *Ottimizzazione* con cross-entropy su tutte le posizioni: + $ + cal(L) = sum_(b,t) "CE"("logits"_(b,t), "label"_(b,t)) + $ + +=== Greedy Decoding (inference) + ++ Inizia con $y_1 = "BOS"$ ++ Poi ripete: + - predice il prossimo token con $arg max$ sui logit + - appende il token alla sequenza + - si ferma quando viene generato `EOS` (o si raggiunge la lunghezza massima) + +Comportamento atteso: dopo il training, il modello genera +$ + ["BOS", x_1, dots, x_S, "EOS"] +$ +ovvero, apprende a copiare la sequenza di input. + +#nota()[ + Il *teacher forcing* durante il training significa che il decoder riceve i token di ground truth come input (anche se ha predetto il token sbagliato al passo precedente). Questo accelera la convergenza ma può causare una discrepanza tra training e inference (*exposure bias*). +] \ No newline at end of file diff --git a/main.typ b/main.typ index d95487a..f3fd26b 100644 --- a/main.typ +++ b/main.typ @@ -42,3 +42,4 @@ #include "chapters/Lezione-7.typ" #include "chapters/Lezione10.typ" #include "chapters/Lezione11.typ" +#include "chapters/Lezione12.typ" From 7542a6beca0e0c3120c948f0c0c176d1fd5b66d6 Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Wed, 25 Feb 2026 15:39:00 +0100 Subject: [PATCH 2/6] add img --- chapters/Lezione12.typ | 101 ++++++++++++++++++++++++++++++++++------- 1 file changed, 85 insertions(+), 16 deletions(-) diff --git a/chapters/Lezione12.typ b/chapters/Lezione12.typ index 561cb14..ff14e83 100644 --- a/chapters/Lezione12.typ +++ b/chapters/Lezione12.typ @@ -2,7 +2,7 @@ = Transformer -I *Transformer* sono una delle architetture più importanti nel deep learning moderno, introdotti da Vaswani et al. nel 2017 nell'articolo _"Attention Is All You Need"_. +I *Transformer* sono una delle architetture più importanti nel deep learning moderno, introdotti recentemente nel $2017$. #informalmente()[ Un Transformer prende un insieme di vettori in uno spazio di rappresentazione e li *trasforma* in un altro insieme di vettori: @@ -10,14 +10,13 @@ I *Transformer* sono una delle architetture più importanti nel deep learning mo - Stessa dimensionalità - Nuovo spazio di rappresentazione più ricco - L'obiettivo è produrre una rappresentazione interna più espressiva, adatta a compiti a valle (classificazione, generazione, ecc.). + L'obiettivo è produrre una rappresentazione interna più espressiva, adatta a diversi compiti (classificazione, generazione, ecc.). ] -== Perché i Transformer sono importanti +Essi si basano su un meccanismo fondamentale: *l'attenzione*. Tale oggetto, permette al modello di decidere su cosa concentrarsi. Non tutte le parti di un input sono ugualmente rilevanti. L'attenzione darà *pesi diversi* a diverse parti dell'input, permettendo al modello di focalizzarsi sulle informazioni più importanti per il compito. -/ Attenzione come meccanismo centrale: il meccanismo di *attenzione* permette al modello di decidere su cosa concentrarsi. Non tutte le parti di un input sono ugualmente rilevanti. - -/ Scalabilità: i Transformer si mappano efficientemente su hardware parallelo massivo (GPU). Modelli con $10^12$ parametri mostrano *capacità emergenti*, talvolta descritte come primi segnali verso l'intelligenza artificiale generale (AGI). +Le pricnipali caratteristiche dei Transformer sono: +/ Scalabilità: i Transformer si mappano efficientemente su hardware parallelo (GPU). Modelli con $10^12$ parametri mostrano *capacità emergenti*, talvolta descritte come primi segnali verso l'intelligenza artificiale generale (AGI). / Versatilità: le architetture Transformer sono state estese con successo a: - Testo (NLP) @@ -29,30 +28,100 @@ I *Transformer* sono una delle architetture più importanti nel deep learning mo #nota()[ Prima dei Transformer, i modelli di NLP più diffusi erano: - - *Bag of Words (BoW)*: ignora l'ordine delle parole, non può esprimere il contesto. + - *Bag of Words (BoW)*: ignora l'ordine delle parole, non può esprimere il contesto. In questa tecnica abbiamo un vocabolario fissato. Ogni parola è rappresentata da un indice intero, e una frase è rappresentata da un vettore di conteggio (o frequenza) delle parole presenti. Non cattura la struttura sintattica o semantica del testo. + - *Reti Neurali Ricorrenti (RNN)*: processano il testo sequenzialmente, difficili da parallelizzare e soggette a dimenticare informazioni a lunga distanza. I Transformer superano tutti questi limiti grazie alla self-attention. ] -=== Dal testo agli embedding +== Testo -> Embedding + +Una parte fondamentale di qualsiasi modello di NLP è la *rappresentazione del testo*. I Transformer non fanno eccezione. Il testo viene convertito in vettori densi (embedding) che catturano il significato semantico. -Il testo viene prima convertito in vettori densi attraverso: +I tranformer come prima cosa, presentano una fase di *tokenizzazione* e *embedding*: + *Tokenizzazione*: il testo viene suddiviso in token (parole, sottoparole o caratteri). -+ *Vocabulary mapping*: ogni token riceve un identificatore intero univoco. -+ *Embedding layer*: ogni ID viene mappato a un vettore denso tramite una matrice appresa: ++ *Vocabulary mapping*: ogni token riceve un identificatore intero univoco. La frase originale viene convertita in una sequenza di ID: + $ + x_1, x_2, dots, x_N in {0, dots, K-1} + $ ++ *Embedding layer*: ogni ID viene mappato a un vettore denso tramite una matrice $W$ appresa: $ - mb(E) in RR^(K times D) + W in RR^(K times D) $ - dove $K$ è la dimensione del vocabolario e $D$ è la dimensione dell'embedding. + dove $K$ è la dimensione del vocabolario e $D$ è la dimensione dell'embedding (features). Viene eseguita una semplice operazione di *lookup*, dove l'ID del token viene usato come indice per estrarre il corrispondente vettore di embedding dalla matrice $W$: -#attenzione()[ - Un embedding *statico* è insufficiente: la parola "banca" ha significati diversi in contesti diversi ("banca del fiume" vs "conto in banca"). La *self-attention* permette a ogni token di aggiornare la propria rappresentazione guardando tutti gli altri token nella sequenza. + L'*output* è una lista di vettori densi: + $ + y_1, y_2, dots, y_N in RR^D + $ + +#esempio()[ + Consideriamo un vocabolario di $K=5$ token e embedding di dimensione $D=4$. La matrice di embedding è: + + #align(center)[ + #cetz.canvas({ + import cetz.draw: * + + // Matrice W + let cell-width = 0.8 + let cell-height = 0.6 + + // Disegna la matrice W + for i in range(5) { + for j in range(4) { + let x = j * cell-width + let y = -i * cell-height + rect((x, y), (x + cell-width, y + cell-height), stroke: black) + content((x + cell-width/2, y + cell-height/2), text(size: 9pt, $w_(#i #j)$)) + } + } + + // Label W + content((-0.6, -1.2), $mb(W) =$) + + // Dimensioni + content((1.6, 1), text(size: 9pt, $D=4$)) + content((-0.6, -2.3), text(size: 9pt, $K=5$)) + + // Freccia e lookup + line((3.8, -1.2), (5.0, -1.2), mark: (end: "stealth")) + content((4.4, -0.9), text(size: 8pt, [lookup])) + + // Token ID + content((5.5, -1.2), text(size: 10pt, [ID = 2])) + + // Freccia verso vettore risultante + line((6.2, -1.2), (7.4, -1.2), mark: (end: "stealth")) + + // Vettore risultante (riga 2 della matrice) + for j in range(4) { + let x = 7.6 + j * cell-width + let y = -1.2 + cell-height/2 + rect((x, y - cell-height/2), (x + cell-width, y + cell-height/2), + stroke: black, fill: rgb("#e8f4f8")) + content((x + cell-width/2, y), text(size: 9pt, $w_(2 #j)$)) + } + + // Label vettore + content((9.5, -0.2), text(size: 9pt, $mb(y) in RR^4$)) + }) + ] + + Dato un token con ID = 2, l'embedding corrispondente è semplicemente la *terza riga* della matrice (indice 2): + $ + mb(y) = [w_(2 0), w_(2 1), w_(2 2), w_(2 3)]^T + $ + + Questa operazione è estremamente efficiente: non richiede moltiplicazioni, solo un accesso indicizzato alla memoria. ] + == Self-Attention -=== Che cos'è l'attenzione? +#attenzione()[ + Un embedding *statico* è insufficiente: la parola "banca" ha significati diversi in contesti diversi ("banca del fiume" vs "conto in banca"). La *self-attention* permette a ogni token di aggiornare la propria rappresentazione guardando tutti gli altri token nella sequenza. +] L'attenzione è un meccanismo che permette al modello di combinare vettori di input con *pesi dipendenti dai dati*. From 1be2d758be31f0f2c0be5d7ec07b43e1d5a49130 Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Wed, 25 Feb 2026 18:47:53 +0100 Subject: [PATCH 3/6] add img --- chapters/Lezione12.typ | 314 ++++++++++++++++++++++++++++++++++++----- 1 file changed, 276 insertions(+), 38 deletions(-) diff --git a/chapters/Lezione12.typ b/chapters/Lezione12.typ index ff14e83..4125cd1 100644 --- a/chapters/Lezione12.typ +++ b/chapters/Lezione12.typ @@ -53,7 +53,7 @@ I tranformer come prima cosa, presentano una fase di *tokenizzazione* e *embeddi L'*output* è una lista di vettori densi: $ - y_1, y_2, dots, y_N in RR^D + x_1, x_2, dots, x_N in RR^D $ #esempio()[ @@ -125,36 +125,41 @@ I tranformer come prima cosa, presentano una fase di *tokenizzazione* e *embeddi L'attenzione è un meccanismo che permette al modello di combinare vettori di input con *pesi dipendenti dai dati*. -Data una sequenza di $N$ vettori token: +I dati vengono processati come un *insieme di vettori* (token), ognuno con $D$ features. Data una sequenza di $N$ vettori token ($N$ = dimensione del contesto): $ - mb(x)_1, dots, mb(x)_N in RR^D + x_1, dots, x_N in RR^D $ - -la matrice dei token è: +La matrice dei token è: $ - mb(X) = mat(mb(x)_1^T; mb(x)_2^T; dots.v; mb(x)_N^T) in RR^(N times D) + mb(X) = mat( + x_(1,1), x_(1,2), dots.h, x_(1,D); + x_(2,1), x_(2,2), dots.h, x_(2,D); + dots.v, dots.v, dots.down, dots.v; + x_(N,1), x_(N,2), dots.h, x_(N,D); + ) = mat(mb(x)_1^T; mb(x)_2^T; dots.v; mb(x)_N^T) in underbrace(RR^(N times D), "N Tokens" times "D Features") $ -Gli elementi $x_(n i)$ dei token sono chiamati *feature*. Il blocco fondamentale di un Transformer è una funzione: +Gli elementi *$x_(n i)$* dei token (celle della matrice) sono chiamati *feature*. Il blocco fondamentale di un Transformer è una funzione: $ - tilde(mb(X)) = "TransformerLayer"(mb(X)) + tilde(X) = "TransformerLayer"(X) $ -con $tilde(mb(X)) in RR^(N times D)$ (stessa dimensionalità dell'input). +con $tilde(mb(X)) in RR^(N times D)$ (stessa dimensionalità dell'input). L'idea è quella di andare a concatenare *più layer* di questo tipo, ognuno con *parametri indipendenti*, per costruire una rete profonda che trasforma progressivamente la rappresentazione dei token. + +I parametri di ogni layer possono essere appresi tramite backpropagation, ottimizzando una loss task-specific (es. cross-entropy per classificazione o generazione). === Calcolo delle similarità -L'attenzione inizia calcolando le *similarità dot-product*: +L'attenzione inizia calcolando le *similarità dot-product* tra tutti i token. Per ogni coppia di token *$m$* e *$n$*, si calcola: $ s_(m n) = mb(x)_m^T mb(x)_n $ +il risultato $s_(m n)$ è un punteggio che indica quanto i due token sono *_affini_* o _simili_ tra loro. Un punteggio più alto indica una maggiore similarità. -In forma matriciale (*matrice di similarità*): +In forma matriciale (*matrice di similarità*) viene creata una matrice quadrata $N times N$, in modo da considerare tutte le coppie di token: $ mb(S) = mb(X) mb(X)^T in RR^(N times N) $ -Un prodotto scalare elevato $arrow.r.double$ alta similarità. - === Normalizzazione Softmax Le similarità grezze vengono convertite in *pesi di attenzione* tramite softmax (applicata per righe): @@ -167,32 +172,43 @@ $ mb(A) = "Softmax"(mb(X) mb(X)^T) $ -Questo garantisce che le righe di $mb(A)$ sommino a 1 e che $a_(m n) >= 0$. +#nota()[ + La softmax normalizza i punteggi di similarità, garantendo due proprietà chiave: + - *Sommatoria a 1*: Tutte le righe di $mb(A)$ sommano a 1, quindi i *pesi di attenzione* rappresentano una *distribuzione di probabilità sui token*: + $ + sum_(m=1)^N a_(m n) = 1, quad forall m in {1, dots, N} + $ + + - *Non negatività*: Tutti gli elementi $a_(m n) >= 0$. Questo significa che *ogni token contribuisce* positivamente *alla rappresentazione finale*, con un peso che riflette la sua rilevanza. +] === Output del layer Transformer L'output è una combinazione pesata di tutti i vettori value: $ - mb(y)_m = sum_(n=1)^N a_(m n) mb(x)_n + y_m = sum_(n=1)^N underbrace(mr(a_(m n)),"peso "\ "attenzione") underbrace(mb(w_n), "embedding" \ "token" n) $ In forma matriciale: $ - mb(Y) = mb(A) mb(X) = "Softmax"[mb(X) mb(X)^T] mb(X) + Y = mr(A) mb(X) = "Softmax"[mb(X) mb(X)^T] mb(X) $ #informalmente()[ Ogni $mb(y)_m$ è una *media pesata* di tutti gli $mb(x)_n$. L'attenzione apprende quali token sono rilevanti per ciascun output. ] -Questo processo è chiamato *self-attention* perché la stessa sequenza $mb(X)$ viene usata come: -- Query (cosa cerco) -- Key (cosa è disponibile) -- Value (l'informazione che recupero) +=== Key, Query e Value -=== Query, Key, Value: rendere l'attenzione apprendibile +Siccome il calcolo precedente $S = X X^T$ considera solamente la similarità tra embedding (*similarità vettoriale*), non permette al modello di *apprendere* cosa significa similarità o considerare la similarità semantica. -#nota()[ +Per questo motivo, nei Transformer si introducono *proiezioni lineari apprese* per query, key e value. Ogni token $x_n$ viene proiettato in tre spazi distinti: +- *Query* (cosa cerco) +- *Key* (cosa è disponibile) +- *Value* (l'informazione che recupero) + + +#informalmente()[ *Analogia con il recupero di informazioni*: immagina di cercare un film online. Ogni film ha attributi descrittivi (vettore *key*) e un contenuto (vettore *value*). L'utente esprime le sue preferenze (vettore *query*). Il sistema confronta la query con tutte le key, trova la corrispondenza più vicina e restituisce il value. ] @@ -203,33 +219,255 @@ $ mb(V) = mb(X) mb(W)^((v)), quad mb(W)^((v)) in RR^(D times D_v) $ -La matrice di similarità diventa: -$ - mb(S) = mb(Q) mb(K)^T -$ +Dove *$D_k$* e *$D_v$* sono le dimensioni delle query/key e value rispettivamente. Solitamente hanno una dimensione più piccola di $D$ per ridurre il costo computazionale. + +#attenzione()[ + $D_k$ e $D_q$ devono corrispondere per permettere il calcolo del prodotto scalare. $D_v$ può essere diverso, poiché rappresenta l'informazione che viene recuperata, non la similarità. Solitamente $D_v$ è molto simile a $D_k + D_v$ +] + +La *matrice di similarità* diventa: +$ + S = mr(Q) mr(K)^T +$ +Viene computato un dot-product tra ogni possibile di coppia di query e key, producendo una matrice $N times N$ di punteggi di similarità. + +Sucessivamente viene calcolata una somma pesata colonna per colonna. *L'output* finale è dato da: +$ + A = "Softmax"(mr(Q) mr(K)^T), quad mb(Y) = A mr(V) +$ +Dove $A$ avrà una dimensione $N times N$ e $mb(Y)$ avrà dimensione $N times D_v$ (scegliendo $D_v = D$ otteniamo $mb(Y) in R^(N times D)$). + +Graficamente le operazioni possono essere rappresentate come segue: +#align(center)[ + #cetz.canvas({ + import cetz.draw: * + + let box-width = 2.5 + let box-height = 0.6 + + // Input X + rect((0, 0), (box-width, box-height), stroke: black, fill: white, name: "X") + content((box-width/2, box-height/2), text(size: 11pt, weight: "bold", $mb(X) in R^(N times D)$)) + + // Matrici di peso W^(q), W^(k), W^(v) + let y1 = 1.8 + rect((-1.5, y1), (-0.3, y1 + box-height), stroke: blue.darken(20%), + fill: blue.lighten(70%), name: "Wq") + content((-0.9, y1 + box-height/2), text(size: 10pt, $mb(W)^((q))$)) + + rect((0.4, y1), (1.6, y1 + box-height), stroke: blue.darken(20%), + fill: blue.lighten(70%), name: "Wk") + content((1.0, y1 + box-height/2), text(size: 10pt, $mb(W)^((k))$)) + + rect((2.3, y1), (3.5, y1 + box-height), stroke: blue.darken(20%), + fill: blue.lighten(70%), name: "Wv") + content((2.9, y1 + box-height/2), text(size: 10pt, $mb(W)^((v))$)) + + // Frecce da X a W + line((box-width/2, box-height), (-0.9, y1), mark: (end: "stealth")) + content((-1.2, 1.2), text(size: 9pt, $mb(Q) in R^(D times D_q)$)) + + line((box-width/2, box-height), (1.0, y1), mark: (end: "stealth")) + content((1.3, 1.3), text(size: 9pt, $mb(K) in R^(D times D_k)$)) + + line((box-width/2, box-height), (2.9, y1), mark: (end: "stealth")) + content((3.3, 1.2), text(size: 9pt, $mb(V) in R^(D times D_v)$)) + + content((1.6, 3.4), text(size: 9pt, $in R^(N times N)$)) + + content((4.5, 2.1), text(size: 9pt, $W^v in R^(N times D_v)$)) + + // MatMul (Q * K^T) + let y2 = 3.2 + rect((-0.9, y2), (0.9, y2 + box-height), stroke: black, + fill: rgb("#f4c9a6"), radius: 0.15cm) + content((0, y2 + box-height/2), text(size: 10pt, [mat mul])) + + line((-0.9, y1 + box-height), (-0.5, y2), mark: (end: "stealth")) + line((1.0, y1 + box-height), (0.5, y2), mark: (end: "stealth")) + + // Scale + let y3 = 4.4 + rect((-0.9, y3), (0.9, y3 + box-height), stroke: black, + fill: rgb("#fff8dc"), radius: 0.15cm) + content((0, y3 + box-height/2), text(size: 10pt, [scale])) + + line((0, y2 + box-height), (0, y3), mark: (end: "stealth")) + + // Softmax + let y4 = 5.6 + rect((-0.9, y4), (0.9, y4 + box-height), stroke: black, + fill: rgb("#b8e6b8"), radius: 0.15cm) + content((0, y4 + box-height/2), text(size: 10pt, [softmax])) + + line((0, y3 + box-height), (0, y4), mark: (end: "stealth")) + + // MatMul finale (A * V) + let y5 = 6.8 + rect((-0.9, y5), (0.9, y5 + box-height), stroke: black, + fill: rgb("#f4c9a6"), radius: 0.15cm) + content((0, y5 + box-height/2), text(size: 10pt, [mat mul])) + + line((0, y4 + box-height), (0, y5), mark: (end: "stealth")) + + // Freccia curva da V a mat mul finale + line((2.9, y1 + box-height), (2.9, y5 + box-height/2), stroke: (dash: "dashed")) + line((2.9, y5 + box-height/2), (0.9, y5 + box-height/2), mark: (end: "stealth")) + + // Output Y + let y6 = 8.2 + rect((-0.5, y6), (0.5, y6 + box-height), stroke: black, fill: white) + content((0, y6 + box-height/2), text(size: 11pt, weight: "bold", $mb(Y)$)) + + line((0, y5 + box-height), (0, y6), mark: (end: "stealth")) + }) +] + +#nota[ + La matrice _value_ $V = W^v$, viene moltiplicata per la matrice di attenzione $A$ per produrre l'output finale $Y$, ovvero l'embedding arriccheto del contesto. Ogni riga di $Y$ è una combinazione pesata delle righe di $V$, con i pesi dati da $A$ (somma pesata vettoriale). + + Per ogni token $i$ (una riga della matrice output $Y$), il nuovo embedding $y_i$ è costruito prendendo un po' del valore di ogni token $j$ della sequenza, in base a quanto $i$ ha prestato attenzione a $j$: + $ + mb(y)i = a(i,1) v_1 + a(i,2) v_2 + dots + a(i,N) v_N + $ +] -e l'output: -$ - mb(A) = "Softmax"(mb(Q) mb(K)^T), quad mb(Y) = mb(A) mb(V) -$ -Questo permette al modello di *apprendere cosa significa similarità* e come rappresentare i value. === Scaled Dot-Product Self-Attention -#attenzione()[ - *Problema*: quando gli elementi dei vettori query e key hanno media 0 e varianza 1, il loro prodotto scalare ha varianza $D_k$. Un'alta varianza porta a input grandi per la softmax, che diventa molto piatta e produce *gradienti molto piccoli* (problema simile a sigmoid/tanh saturi). - *Soluzione*: dividere il punteggio di similarità per la deviazione standard $sqrt(D_k)$: + +$mr("Problema")$: Se gli elementi dei vettori query e key hanno media $0$ e varianza $1$, il loro prodotto scalare ha varianza $D_k$ (le varianze si sommano). Una varianza troppo alta porta i numeri in un intervallo molto ampio. + +Un'alta varianza porta a input grandi per la softmax, che diventa molto piatta e produce *gradienti molto piccoli* (funzione esponenziale). + +La $mg("soluzione")$ è scalare i punteggi di similarità dividendo per $sqrt(D_k)$, che mantiene i valori in un intervallo più gestibile: $ - "Attention"(mb(Q), mb(K), mb(V)) = "Softmax"(mb(Q) mb(K)^T / sqrt(D_k)) mb(V) + "Attention"(Q, K, V) = "Softmax"((Q K^T) / sqrt(D_k)) V $ - Questo *stabilizza la softmax*, mantiene i gradienti sani e rende l'attenzione robusta all'aumentare della dimensionalità. +Questa operazione permette di *stabilizza la softmax*, rendendo l'attenzione robusta all'aumentare della dimensionalità. + +#align(center)[ + #cetz.canvas(length: 1cm, { + import cetz.draw: * + + // Funzione gaussiana + let gaussian(x, sigma) = { + calc.pow(calc.e, -calc.pow(x, 2) / (2 * calc.pow(sigma, 2))) / (sigma * calc.sqrt(2 * calc.pi)) + } + + // Funzione per disegnare una curva gaussiana + let draw-gaussian(origin, sigma, scale, color, fill-tails: false) = { + let points = () + let x-min = -3.0 + let x-max = 3.0 + let steps = 100 + let dx = (x-max - x-min) / steps + + for i in range(steps + 1) { + let x = x-min + i * dx + let y = gaussian(x, sigma) * scale + points.push((origin.at(0) + x, origin.at(1) + y)) + } + + // Disegna la curva + line(..points, stroke: (paint: color, thickness: 1.5pt)) + + // Riempi le code se richiesto + if fill-tails { + // Coda sinistra + let left-points = ((origin.at(0) - 3, origin.at(1)),) + for i in range(steps + 1) { + let x = x-min + i * dx + if x < -1.5 { + let y = gaussian(x, sigma) * scale + left-points.push((origin.at(0) + x, origin.at(1) + y)) + } + } + left-points.push((origin.at(0) - 1.5, origin.at(1))) + line(..left-points, stroke: none, fill: red.lighten(70%), close: true) + + // Coda destra + let right-points = ((origin.at(0) + 1.5, origin.at(1)),) + for i in range(steps + 1) { + let x = x-min + i * dx + if x >= 1.5 { + let y = gaussian(x, sigma) * scale + right-points.push((origin.at(0) + x, origin.at(1) + y)) + } + } + right-points.push((origin.at(0) + 3, origin.at(1))) + line(..right-points, stroke: none, fill: red.lighten(70%), close: true) + } + } + + // --- Grafico sinistra: Without Scaling --- + set-origin((-8, 0)) + + // Titolo + content((0, 3.8), text(size: 10pt, weight: "bold", [Senza Scaling])) + + // Assi + line((-3.2, 0), (3.2, 0), stroke: (paint: gray, thickness: 0.8pt)) + line((0, 0), (0, 3), stroke: (paint: gray, thickness: 0.8pt), mark: (end: ">")) + + // Tick marks + for x in (-2, -1, 0, 1, 2) { + line((x, -0.1), (x, 0.1), stroke: (thickness: 0.8pt)) + content((x, -0.4), text(size: 8pt, str(x))) + } + + // Gaussiana larga con code evidenziate + draw-gaussian((0, 0), 1.5, 6, black, fill-tails: true) + + // Label sotto + content((0, -1.5), text(size: 8pt)[ + Variance = $D_k$. Large dot products push\ + softmax into regions with #text(fill: red)[vanishing gradients]. + ]) + + // Freccia verso l'alto + line((0, 3.5), (0, 3.1), mark: (start: "stealth"), stroke: (thickness: 1.2pt)) + + // --- Box centrale: Formula --- + set-origin((4, 0)) + + rect((-1.8, 1.2), (1.8, 2.2), stroke: (paint: blue.darken(20%), thickness: 1.2pt), + fill: blue.lighten(90%), radius: 0.1cm) + content((0, 1.7), text(size: 10pt, $"Score" = (mb(Q) mb(K)^T) / sqrt(D_k)$)) + + // --- Grafico destra: With Scaling --- + set-origin((4, 0)) + + // Titolo + content((0, 3.8), text(size: 10pt, weight: "bold", [ Con Scaling])) + + // Freccia verso l'alto + line((0, 3.5), (0, 3.1), mark: (start: "stealth"), stroke: (thickness: 1.2pt)) + + // Assi + line((-3.2, 0), (3.2, 0), stroke: (paint: gray, thickness: 0.8pt)) + line((0, 0), (0, 3), stroke: (paint: gray, thickness: 0.8pt), mark: (end: ">")) + + // Tick marks + for x in (-2, -1, 0, 1, 2) { + line((x, -0.1), (x, 0.1), stroke: (thickness: 0.8pt)) + content((x, -0.4), text(size: 8pt, str(x))) + } + + // Gaussiana stretta (più alta e concentrata) + draw-gaussian((0, 0), 0.7, 6, black) + + // Label sotto + content((0, -1.5), text(size: 8pt)[ + Variance = 1. Dividing by $sqrt(D_k)$\ + stabilizes the training dynamics. + ]) + }) ] -Questa versione scalata è la *forma finale* della self-attention dot-product usata nei Transformer. - == Multi-Head Attention Una singola testa di attenzione può modellare solo *un tipo di relazione* alla volta. Ma il linguaggio naturale (e altri dati) contiene molteplici pattern simultanei: From 709a6e89f2e759ff68fc4c8921d5921a3fa4d604 Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Wed, 25 Feb 2026 19:04:07 +0100 Subject: [PATCH 4/6] Update --- chapters/Lezione12.typ | 34 +++++++++++++++++++++++----------- 1 file changed, 23 insertions(+), 11 deletions(-) diff --git a/chapters/Lezione12.typ b/chapters/Lezione12.typ index 4125cd1..e4aa1a9 100644 --- a/chapters/Lezione12.typ +++ b/chapters/Lezione12.typ @@ -477,35 +477,47 @@ Una singola testa di attenzione può modellare solo *un tipo di relazione* alla - dipendenze a lunga distanza - relazioni lessicali -*Idea*: usare più teste di attenzione in parallelo, ognuna con le proprie proiezioni apprese: +*Idea*: usare più teste di attenzione in parallelo, ognuna con le proprie proiezioni apprese. Fissando il numero di teste $H$, ogni testa $h$ calcola le proprie query, key e value: $ - mb(Q)_h = mb(X) mb(W)_h^((q)), quad - mb(K)_h = mb(X) mb(W)_h^((k)), quad - mb(V)_h = mb(X) mb(W)_h^((v)) + Q_h = X W_h^((q)), quad + K_h = X W_h^((k)), quad + V_h = X W_h^((v)) $ Ogni testa calcola la scaled dot-product attention: $ - mb(H)_h = "Attention"(mb(Q)_h, mb(K)_h, mb(V)_h) + mb(H)_h = "Attention"(Q_h, K_h, V_h) $ + Le teste vengono poi *concatenate* e proiettate con una trasformazione lineare finale: $ - mb(Y)(mb(X)) = "Concat"(mb(H)_1, dots, mb(H)_H) mb(W)^((o)) + Y (X) = "Concat"(mb(H)_1, dots, mb(H)_H) W^((o)) $ +Quello che accade è che ogni testa impara a focalizzarsi su *aspetti diversi* della sequenza, catturando così una varietà di relazioni e pattern contemporaneamente. I _cambiamenti_ proposti da ogni testa vengono combinati tra di loro per produrre un embedding finale più ricco e informativo. #nota()[ - *Dimensioni*: ogni testa produce $mb(H)_h in RR^(N times D_v)$. Con $H$ teste: + Ognuna delle $H$ teste produce una matrice di attenzione $H_h$: + $ + mb(H)_h in RR^(N times D_v) + $ + Le matrici di tutte le teste vengono concatenate lungo la dimensione delle features, producendo una matrice di dimensione $N times (H D_v)$: $ "Concat"(mb(H)_1, dots, mb(H)_H) in RR^(N times (H D_v)) $ - La proiezione finale $mb(W)^((o)) in RR^(H D_v times D)$ riporta al embedding dimension del modello. + La proiezione finale $W^((o)) in RR^(H D_v times D)$ riporta al embedding dimension del modello. Il risultato $Y in N times D$, uguale all'input, permettendo di concatenare più layer Transformer. + + - Scegliendo $D_v = D / H$, la matrice concatenata ha dimensione $(N times D)$, uguale all'input. Tutte le matrici $mb(W)_h^((q)), mb(W)_h^((k)), mb(W)_h^((v)}, mb(W)^((o))$ vengono apprese congiuntamente. + Scegliendo *$D_v = D / H$*, la matrice concatenata ha dimensione $(N times D)$, uguale all'input. ] -#informalmente()[ - Le diverse teste imparano *modi diversi di attendere*. Il modello cattura così più tipi di struttura contemporaneamente, in modo analogo ai filtri multipli in un layer CNN. +#attenzione()[ + Tutte le matrici: + $ + W_h^((q)), W_h^((k)), W_h^((v)) in RR^(D times D_v) "e" W^((o)) in RR^(H D_v times D) + $ + vengono *apprese congiuntamente*. ] == Self-Attention e Cross-Attention From e452b8d3f3673d39babe06c42a31911d8fa981fa Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Thu, 26 Feb 2026 12:27:28 +0100 Subject: [PATCH 5/6] update --- assets/image-1.png | Bin 0 -> 118724 bytes assets/image-2.png | Bin 0 -> 217065 bytes assets/image.png | Bin 0 -> 197885 bytes chapters/Lezione12.typ | 370 +++++++++++++++++++++++++---------------- 4 files changed, 224 insertions(+), 146 deletions(-) create mode 100644 assets/image-1.png create mode 100644 assets/image-2.png create mode 100644 assets/image.png diff --git a/assets/image-1.png b/assets/image-1.png new file mode 100644 index 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Large dot products push\ + Variance = $D_k$. Large dot products push\ softmax into regions with #text(fill: red)[vanishing gradients]. ]) - + // Freccia verso l'alto line((0, 3.5), (0, 3.1), mark: (start: "stealth"), stroke: (thickness: 1.2pt)) - + // --- Box centrale: Formula --- set-origin((4, 0)) - - rect((-1.8, 1.2), (1.8, 2.2), stroke: (paint: blue.darken(20%), thickness: 1.2pt), - fill: blue.lighten(90%), radius: 0.1cm) + + rect( + (-1.8, 1.2), + (1.8, 2.2), + stroke: (paint: blue.darken(20%), thickness: 1.2pt), + fill: blue.lighten(90%), + radius: 0.1cm, + ) content((0, 1.7), text(size: 10pt, $"Score" = (mb(Q) mb(K)^T) / sqrt(D_k)$)) - + // --- Grafico destra: With Scaling --- set-origin((4, 0)) - + // Titolo content((0, 3.8), text(size: 10pt, weight: "bold", [ Con Scaling])) - + // Freccia verso l'alto line((0, 3.5), (0, 3.1), mark: (start: "stealth"), stroke: (thickness: 1.2pt)) - + // Assi line((-3.2, 0), (3.2, 0), stroke: (paint: gray, thickness: 0.8pt)) line((0, 0), (0, 3), stroke: (paint: gray, thickness: 0.8pt), mark: (end: ">")) - + // Tick marks for x in (-2, -1, 0, 1, 2) { line((x, -0.1), (x, 0.1), stroke: (thickness: 0.8pt)) content((x, -0.4), text(size: 8pt, str(x))) } - + // Gaussiana stretta (più alta e concentrata) draw-gaussian((0, 0), 0.7, 6, black) - + // Label sotto content((0, -1.5), text(size: 8pt)[ Variance = 1. Dividing by $sqrt(D_k)$\ @@ -497,7 +494,7 @@ $ Quello che accade è che ogni testa impara a focalizzarsi su *aspetti diversi* della sequenza, catturando così una varietà di relazioni e pattern contemporaneamente. I _cambiamenti_ proposti da ogni testa vengono combinati tra di loro per produrre un embedding finale più ricco e informativo. #nota()[ - Ognuna delle $H$ teste produce una matrice di attenzione $H_h$: + Ognuna delle $H$ teste produce una matrice di attenzione $H_h$: $ mb(H)_h in RR^(N times D_v) $ @@ -513,54 +510,110 @@ Quello che accade è che ogni testa impara a focalizzarsi su *aspetti diversi* d ] #attenzione()[ - Tutte le matrici: + Tutte le matrici: $ W_h^((q)), W_h^((k)), W_h^((v)) in RR^(D times D_v) "e" W^((o)) in RR^(H D_v times D) - $ + $ vengono *apprese congiuntamente*. ] == Self-Attention e Cross-Attention -Un layer di attenzione standard prende in input due sequenze $mb(X)$ e $mb(X)'$ e calcola: +Un layer di attenzione standard prende in input due sequenze $mb(X)$ e $mr(X)'$ (ovvero due matrici da $N$ token) e calcola: $ - mb(Q) = mb(X) mb(W)^((q)), quad - mb(K) = mb(X)' mb(W)^((k)), quad - mb(V) = mb(X)' mb(W)^((v)) + Q = mb(X) W^((q)), quad + K = mr(X)' W^((k)), quad + V = mr(X)' W^((v)) $ $ - mb(A) = "Softmax"(mb(Q) mb(K)^T / sqrt(D_k)), quad - mb(Y) = mb(A) mb(V) + A = "Softmax"((Q K^T) / sqrt(D_k)), quad + Y = A V $ -/ Self-attention: $mb(X) = mb(X)'$ — la stessa sequenza è usata per query, key e value. -/ Cross-attention: $mb(X) != mb(X)'$ — query dalla sequenza di output, key e value dalla sequenza di input (encoder). +/ Self-attention: $mb(X) = mr(X)'$ — la stessa sequenza è usata per query, key e value. +/ Cross-attention: $mb(X) != mr(X)'$ — vengono utilizzate due sequenze di vettori diverse: + - *Query* ($Q$): Viene dal Decoder (dalla frase che sto traducendo/generando ora). -#nota()[ - La *cross-attention* esiste solo nei Transformer encoder–decoder, come: - - Modelli di machine translation - - Modelli di summarization - - Modelli sequence-to-sequence (T5, BART) + - *Key* ($K$) & *Value* ($V$): Vengono dall'Encoder (dalla frase originale completa che ho già elaborato). + + +#align(center)[ + #table( + columns: 3, + [-], [*Self-Attention*], [*Cross-Attention*], + [_input_], + [ + - La Self attention opera su una singola sequenza di input + + - tipicamente usata nell'*encoder*, dove la seuqnza di input e la sorgente o il testo in ingresso + ], + [ + - Due sequenze differenti in input. Una sequenza sorgenete e una target + + - Usata principalmente nel *decoder*. La sequenza sorgente è il contesto (Key, Value), mentre quella target è la sequenza che sta venendo generata (Query) + ], + + [_Scopo_], + [ + La self attention permette di catturare relazioni *interne* alla sequenza di input. Aiuta il modello a capire il contesto e dipendenze a lungo termine tra elementi della sequenza. + ], + [ + La cross-attention permette al modello di concentrarsi su parti differenti della sequenza sorgente per generare correttamente ogni elemento della sequenza target.\ + Cattura come gli elementi della sequenza sorgenete sono relazionati alla seqeunza target. Permette di generare *output contestualmente rilevanti*. + ], + + [_Uso_], + [ + Ogni token della sequenza presta attenzione ad ogni altra parola della sequenza (matrice $N times N$). Permette di creare degli embedding che considerano il *contesto della frase* in cui sono inseriti. + ], + [ + Permette al decoder di osservare la sequenza sorgente mentre viene generato ogni token della sequenza destinazione. Assicura che la *sequenza generata* sia coerente e *contestualmente accurata*. + ], + ) +] + +#esempio()[ + Supponiamo di voler tradurre _The black cat eats_ in _Il gatto nero mangia_. + + - *Self-Attention* (Encoder - Inglese): Analizzi la frase inglese. Capisci che _black_ si riferisce a _cat_. Ha un'idea chiara del concetto. + + - *Self-Attention* (Decoder - Italiano): Hai scritto _Il_. Ora devi scrivere _gatto_. La self-attention ti dice che dopo un articolo serve un sostantivo. Ma quale? + + - *Cross-Attention* (Il Ponte): Il Decoder lancia una Query: _Cerco il soggetto della frase_. Guardi gli appunti presi sulla frase inglese (Keys dell'Encoder). Trovi un match forte sulla parola _cat_. Prelevi il significato di _cat_(Value) e scrivi _gatto_. ] == Transformer Layer: Residui, LayerNorm e MLP Per migliorare la stabilità e l'efficienza del training, il blocco di multi-head attention è arricchito con: -/ Connessione residua: garantisce che l'output mantenga la stessa forma dell'input $mb(X) in RR^(N times D)$. +/ Connessione residua: garantisce che l'output mantenga la stessa forma dell'input $X in RR^(N times D)$. / Layer normalization: applicata dopo il blocco di attenzione (*post-norm*): $ - mb(Z) = "LayerNorm"[mb(Y)(mb(X)) + mb(X)] + "TMP" = underbrace(Y(X), "nuovi attention"\ "embedding") + underbrace(X, "embedding"\ "iniziali")\ + Z = "LeyerNorm"("TMP") $ oppure prima (*pre-norm*): $ - mb(Z) = mb(Y)(mb(X)_0) + mb(X), quad mb(X)_0 = "LayerNorm"(mb(X)) + Z = Y(X_0) + X, quad X_0 = "LayerNorm"(X) $ + #attenzione()[ + Considerando un tranformer con un solo layer, $X$ rappresenta gli embedding iniziali, altrimenti $X$ è l'output del layer precedente $i-1$ + ] + / MLP position-wise: l'output dell'attenzione è una combinazione lineare di vettori input. Per aumentare l'espressività, ogni token viene passato attraverso lo stesso MLP non-lineare: - - Post-norm: $tilde(mb(X)) = "LayerNorm"["MLP"(mb(Z)) + mb(Z)]$ - - Pre-norm: $tilde(mb(X)) = "MLP"(mb(Z)_0) + mb(Z), quad mb(Z)_0 = "LayerNorm"(mb(Z))$ + - Post-norm: $tilde(X) = "LayerNorm"["MLP"(Z) + Z]$ + - Pre-norm: $tilde(X) = "MLP"(Z_0) + Z, quad Z_0 = "LayerNorm"(Z)$ + +Viene introdotto un MLP per le seguenti ragioni: +- L'attenzione è principalmente un'operazione lineare (somma pesata). Senza l'attivazione non-lineare (ReLU o GeLU) nell'MLP, l'intera rete collasserebbe in una singola, enorme matrice lineare, incapace di apprendere pattern complessi. + +- Solitamente, lo strato intermedio dell'MLP è molto più largo dell'embedding. Questa espansione proietta il token in uno spazio molto più vasto, permettendo di "sbrogliare" le caratteristiche (features) e analizzarle con più risoluzione, per poi ricomprimerle. + +//aggiungi disegno + + #informalmente()[ Un layer Transformer completo combina: @@ -573,20 +626,18 @@ Per migliorare la stabilità e l'efficienza del training, il blocco di multi-hea == Positional Encoding #attenzione()[ - La self-attention tratta tutti i token di input in modo *identico* rispetto alla loro posizione: riordinare l'input riordina semplicemente l'output. Questo è un vantaggio per il parallelismo, ma un *problema per le sequenze* dove l'ordine porta significato. + La self-attention tratta tutti i token di input in modo *identico* rispetto alla loro posizione: riordinare l'input riordina semplicemente l'output. Questo è un vantaggio per il parallelismo, ma un *problema per le sequenze* dove l'ordine è significante. - Esempio: "Il cibo era cattivo, non buono." e "Il cibo era buono, non cattivo." hanno gli stessi token ma significato opposto. + Esempio: _Il cibo era cattivo, non buono_ e _Il cibo era buono, non cattivo_ hanno gli stessi token ma significato opposto. ] -=== Soluzione: aggiungere vettori posizionali - -Per ogni posizione $n$, si crea un vettore $mb(r)_n$ e si forma: +La soluzione è codificare nell'embedding anche la posizione assoluto o relativa all'interno della frase di ogni token. Per ogni posizione $n$, si crea un vettore $r_n in R^D$ e si forma: $ - mb(x)'_n = mb(x)_n + mb(r)_n + x'_n = x_n + r_n $ *Requisiti per un buon positional encoding*: -- Unico per ogni posizione +- *Unico per ogni posizione* - Limitato (bounded) - Generalizza a sequenze più lunghe di quelle viste durante il training - Rappresenta la *distanza relativa*, non solo l'indice assoluto @@ -594,34 +645,63 @@ $ Le *funzioni sinusoidali a frequenze multiple* soddisfano tutte queste proprietà. Per un token alla posizione $n$, il vettore di codifica posizionale $mb(r)_n$ ha componenti: $ r_(n,i) = cases( - sin(L^(i \/ D) n) & "se" i "è pari", - cos(L^((i-1) \/ D) n) & "se" i "è dispari" + sin(n / (L^(i/D))) & "se" i "è pari", + cos(n/(L^((i-1)/D))) & "se" i "è dispari" ) $ +Dove: +- $D$ = numero di dimensioni del modello +- $i$ = Indice della dimensione interna al vettore $i in {0,dots,D}$ +- $n$ = posizione del token all'interno della frase -#nota()[ - *Proprietà chiave*: - - Ogni dimensione codifica la posizione con una *lunghezza d'onda diversa*. - - Le dimensioni inferiori variano lentamente → catturano struttura a lungo raggio. - - Le dimensioni superiori variano rapidamente → catturano struttura fine. - - La combinazione codifica *univocamente* la posizione rimanendo bounded. - - Le posizioni relative possono essere inferite dalle *differenze di fase* dei sinusoidi. - - Mescolando sin e cos a frequenze scalate esponenzialmente, il modello ottiene una rappresentazione continua e liscia dell'ordine dei token che si generalizza oltre la lunghezza di training. +#attenzione()[ + La formula del Positional Encoding dipende esclusivamente da due variabili: + - La posizione $n$ = $"pos"$ + - L'indice della dimensione ($i$) + + *Non* dipende dai token in input. Il vettore per la posizione $3$ sarà sempre lo stesso, indipendentemente dalla parola in poszione $3$. + + Il totale di vettori posizionali calcolati sarà: + $ + "Frase più lunga" times D + $ +] + +#align(center)[ + #image("/assets/image.png", width: 70%) ] +Nel grafico: +- $p_1, dots, p_3$: sono le posizioni dei token +- $i=0, dots, i=4$: sono le dimensionalità dei vettori $D$. +- I vettori $r_n$ saranno le colonne. + +Man mano che l'indice $i$ aumenta, l'onda si _allarga_ (il periodo aumenta). Le prime dimensioni del vettore cambiano valore molto rapidamente da una parola all'altra, mentre le ultime dimensioni cambiano in modo molto più graduale. + +Questa combinazione crea un *pattern unico e irripetibile* per ogni posizione $"pos"$, che il modello può riconoscere facilmente. + + +Grafico interattivo #link("https://erdem.pl/2021/05/understanding-positional-encoding-in-transformers") + == Reti Transformer Vaswani et al. (2017) proposero un modello senza operazioni convoluzionali né ricorrenti, composto *esclusivamente da layer di attenzione*. Il Transformer completo è composto da: - Un *encoder* che combina $N=6$ moduli, ognuno con un sottomodulo di multi-head attention e un MLP a un layer nascosto per token, con connessioni residue e layer normalization. -- Un *decoder* con struttura simile, ma con layer di attenzione *causale* (masked) e layer di cross-attention che attendono alle key e value finali dell'encoder. + +- Un *decoder* con struttura simile, ma con layer di attenzione *causale* (masked) e layer di cross-attention che attendono alle key e value finali dell'encoder: + - Key e Value vengono dall'encoder + - Query proviene dal decoder + +#align(center)[ + #image("/assets/image-1.png", width: 55%) +] === Transformer Encoder L'encoder processa una sequenza di input e produce una rappresentazione contestuale più ricca. -*Input*: $mb(X) in RR^(N times D)$, con $N$ = numero di token e $D$ = dimensione dell'embedding. +*Input*: $X in RR^(N times D)$, con $N$ = numero di token e $D$ = dimensione dell'embedding. Ogni layer dell'encoder è composto da: + Multi-head self-attention @@ -631,19 +711,17 @@ Ogni layer dell'encoder è composto da: Ogni layer preserva la dimensionalità: input $N times D arrow$ output $N times D$. -// Nota: il diagramma testuale viene descritto narrativamente -// poiché richiederebbe asset grafici specifici del corso +#align(center)[ + #image("/assets/image-2.png", width: 70%) +] -#informalmente()[ - L'encoder costruisce *rappresentazioni contestuali dei token*: - - Ogni token attende a tutti gli altri token. - - Le dipendenze a lungo raggio vengono catturate direttamente. - - Il significato delle parole diventa dipendente dal contesto. +L'encoder costruisce *rappresentazioni contestuali dei token*: +- Ogni token attende a tutti gli altri token. +- Le dipendenze a lungo raggio vengono catturate direttamente. +- Il significato delle parole diventa dipendente dal contesto. - Esempio: in "L'animale era stanco perché aveva corso", il token "aveva" attende fortemente ad "animale". +La concatenazione dei layer raffina progressivamente le rappresentazioni. L'output finale dell'encoder ha la stessa forma dell'input ma con *struttura semantica molto più ricca*. - L'impilamento dei layer raffina progressivamente le rappresentazioni. L'output finale dell'encoder ha la stessa forma dell'input ma con *struttura semantica molto più ricca*. -] ==== Pre-training dell'encoder: Masked Language Modeling @@ -808,4 +886,4 @@ ovvero, apprende a copiare la sequenza di input. #nota()[ Il *teacher forcing* durante il training significa che il decoder riceve i token di ground truth come input (anche se ha predetto il token sbagliato al passo precedente). Questo accelera la convergenza ma può causare una discrepanza tra training e inference (*exposure bias*). -] \ No newline at end of file +] From 7888f6e6332af08922ea5b5e99f7a9cb9be55677 Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Thu, 26 Feb 2026 15:28:05 +0100 Subject: [PATCH 6/6] update --- chapters/Lezione12.typ | 65 +++++++++++++++++++++++++++++++++++------- 1 file changed, 55 insertions(+), 10 deletions(-) diff --git a/chapters/Lezione12.typ b/chapters/Lezione12.typ index 36263d6..fb06cfe 100644 --- a/chapters/Lezione12.typ +++ b/chapters/Lezione12.typ @@ -749,7 +749,7 @@ Ogni layer del decoder contiene: + MLP position-wise + Connessione residua + LayerNorm -/ Masked self-attention: garantisce che un token possa attendere *solo ai token precedenti* (maschera causale). Nessun accesso alle informazioni future. +/ Masked self-attention: garantisce che un token possa attendere *solo ai token precedenti* (maschera causale). Nessun accesso alle informazioni future, altrimenti il modello non imparerebbe niente. #esempio()[ *GPT (Generative Pretrained Transformer)*: il decoder apprende le probabilità condizionali: @@ -761,15 +761,13 @@ Ogni layer del decoder contiene: - Si campiona $x_n$, si appende alla sequenza, si ripete. ] -==== Cosa apprende il decoder - Il decoder esegue *generazione autoregressiva*. Per ogni posizione $t$: - Attende ai token generati in precedenza. - Opzionalmente attende agli output dell'encoder (in task sequence-to-sequence). - Predice la distribuzione di probabilità del token successivo. - L'ultimo layer produce logit → softmax → probabilità del prossimo token. -Questo abilita: generazione di testo, traduzione, sistemi di dialogo, generazione di codice. +Tipologie di task: generazione di testo, traduzione, sistemi di dialogo, generazione di codice. === Encoder vs Decoder: confronto @@ -825,6 +823,8 @@ Questo abilita: generazione di testo, traduzione, sistemi di dialogo, generazion Per verificare il corretto funzionamento di un Transformer end-to-end, si usa spesso un task sintetico: il modello deve *copiare una sequenza*. +L'idea è che se il modello non riesce a ricopiare una sequenza in modo corretto allora sono presenti dei bug nel codice. + === Token speciali / `PAD = 0`: padding @@ -840,17 +840,25 @@ Sia $S$ la lunghezza della sequenza sorgente. Un campione contiene: $ "src" = [x_1, x_2, dots, x_S] $ +La nostra sequenza di token. *Decoder input (shifted right)*: $ - "decoder\_input" = ["BOS", x_1, x_2, dots, x_S] + "decoder_input" = ["BOS", x_1, x_2, dots, x_S] $ +In modo tale che il decoder sappia da dove deve iniziare a generare la sequenza, in questo caso dal token `BOS`. *Label (next-token targets)*: $ "label" = [x_1, x_2, dots, x_S, "EOS"] $ +#esempio()[ + Data $S = ["BOS", C,I,A,O]$: + - Il modello parte a leggere `BOS` è deve prevedere `C` e cosi via + - Alla fine leggendo `BOS, C, I, A, O` deve prevedere `EOS` +] + #informalmente()[ Questo task sintetico è utile perché: - Ha un obiettivo semplice, facile da debuggare. @@ -860,21 +868,58 @@ $ === Training loop (teacher forcing) +Supponiamo di avere un batch di $B$ frasi: +- Ogni frase è lunga $T$ +- Vocabolario di $K$ parole + Per ogni batch: -+ *Costruire le maschere* (padding + causale per il decoder): `src_mask`, `tgt_mask` -+ *Encoder* costruisce la memoria: $mb(E) = "encode"("src", "src\_mask")$ -+ *Decoder* predice tutti i time step in parallelo: $mb(D) = "decode"(mb(E), "src\_mask", "decoder\_input", "tgt\_mask")$ -+ *Proiezione* al vocabolario: $"logits" in RR^(B times T times K)$ ++ *Costruire le maschere*: + - `src_mask`(Machera dell'encorder): Serve solo per il padding. Se una frase è più corta di $T$ viene riempita con dei token speciali ``. Tali posizioni verrano ignorate dall'encoder. + + - `tgt_mask`(Maschera del decoder): Combina il padding e la maschera casuale (matrice triangolare). Non permette al decoder di guardare i token successivi durante le predizione. + ++ *Encoder* costruisce la memoria: + $ + mb(E) = "encode"("src", "src_mask") + $ + L'input sono i token originali. Il risultato è una matrice densa $E$ che contiene gli embeddings. + ++ *Decoder* predice tutti i time step in parallelo: + + $ + D = "decode"(mb(E), "src_mask", "decoder_input", "tgt_mask") + $ + #nota()[ + Il decoder *non* genera una parola alla volta, ma gli viene data in pasto l'intera sequenza sfalsata `decoder_input`. Grazie alla `tgt_mask` viene calcolata l'attenzione parola per parola, ogni volta mascherando cioò che non è il contesto. Il calcolo avviene *contemporaneamente*. + ] + L'output è una matrice $D$ contenete le rappresentazioni finali dei decoder per ogni singola posizione della frase. + ++ *Proiezione* al vocabolario: $"logits" in RR^(B times T times K)$. Il Decoder restituisce dei vettori di dimensione $D$. Siccome vogliamo delle parole dobbiamo proiettarle nello spazio più ampio ovvero $K$ + + + + *Ottimizzazione* con cross-entropy su tutte le posizioni: $ cal(L) = sum_(b,t) "CE"("logits"_(b,t), "label"_(b,t)) $ + La loss: + - Prende i punteggi generati al passo precedente per la posizione $t$. + + - Guarda qual era la vera parola successiva. + + - Se il modello aveva dato un punteggio alto alla parola giusta, la loss (l'errore) scende. Se aveva dato un punteggio alto a una parola sbagliata, la loss sale. + + - La sommatoria $sum_(b,t)$ significa semplicemente che calcoliamo questo errore per *ogni parola* ($t$) *in ogni frase* ($b$) del nostro batch, e facciamo una media. + +$L$ è l'errore totale. Da qui parte la backpropagation per aggiornare i pesi. + + === Greedy Decoding (inference) + Inizia con $y_1 = "BOS"$ + Poi ripete: - - predice il prossimo token con $arg max$ sui logit + - predice il prossimo token con $arg max$ sui logit (predizione next token più alta) - appende il token alla sequenza - si ferma quando viene generato `EOS` (o si raggiunge la lunghezza massima)