From a81165c3a9872a96e48c8db5887b38f081a87b3c Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Fri, 6 Feb 2026 10:43:19 +0100 Subject: [PATCH 01/25] add lezione 8 raw note --- chapters/Lezione8.typ | 156 ++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 156 insertions(+) create mode 100644 chapters/Lezione8.typ diff --git a/chapters/Lezione8.typ b/chapters/Lezione8.typ new file mode 100644 index 0000000..aca9772 --- /dev/null +++ b/chapters/Lezione8.typ @@ -0,0 +1,156 @@ +#import "../template.typ": * += Lezione 8 + +I modelli sono in grado di catturare cose al di la dell'osservabile. + +Funzione di attivazione: Sono funzioni che hanno due asintoti (da + infinito a - infitio). Per ogni valore x tendono ad essere in saturazione tra -1 e 1. La parte centrale è "meno convinta". Servono per arrivare a definire delle regioni non lineari (curvature dello spazio) in modo per distinguere gruppi di dati in base alla categoria di appartenenza. In uno stesso spazio di embedding devo creare delle superfici di separazione. Nella regione centrale il modello ha incertezza. +- Tangente iperbolica + +//aggiungere immagine + +RELU (molto usata): Approssimazione universale basata sulla RELU. Data una funzione $f in C([a,b],R)$. Una funzione può essere approssimata come una combinazioni lineari di RELU: +- Relu = $max(o, x)$. Ogni volta che la x è negativa prendo 0. $R(w x+b)$ dipende dal valore della retta $w x+ b$. Se applicco una trasformazione alla variabile indipendete la funzione reagisce in modi diversi +- Prendo tante ReLu e vado a coprire la mia funzione $f$ qualsiasi. Ottento una curva spezzata. +- Vado a combianre la funzione rossa e blue assime, voglio che la seconda sia più vicina possibile alla funzione che voglio approssimare (grigia). + +In generale più funzioni prendo meglio approssimo, con un certo costo di complicazione. + +Tutto quello che abbiamo bisogno sono le funzioni di attivzioni. Limitazioni: +- il modello può essere molto complesso +- più tempo di training e maggiore quantità di dati per avere una accuratezza accetabile + +Il teorema possiamo generalizzarlo come per ogni funzione continua $g:[0,1]^D -> R$ è possibile approssimarla con un livello nascosto (percetrone): $W in R^(K times D) - b in R^k . omega in R^K$. Aimentanto $K$ ovvero le unità nascoste abbiamo più possibilità di imparare, con un costo maggiore. + +Il vantaggio è che avere dei modelli profondi rispetto a modelli flat è che le reti neurali gerarchiche (diversi livelli di profondità) sono in grado di estrarre più informazoni dai dati. Vengono sfruttati dei bias induttivi (lo decidiamo noi, bias) per apprendere pattern complessi. + +Di solito si vede che nei primi livelli capiscono dei pattern molto basilari (edge delle immagini ad esempio) ma non capiscono l'immagine nel suo complesso. Sono gli ultimi livelli che lo fanno. + +Il learning è una stima di parametri (sempre più efficaci)usando questi bias che mettiamo noi (embedding e livelli intermidi) per favorie l'apprendimento di una rappresentazione (embedding) gerarchici. + +== Pythorc nn framework + +Usermeo il modulo `torch.nn` essenziale per costruire modelli deep. +Troviamo come moduli: +- layer: tutti i vari _livelli_. Esiston ricorrenti, lineari, convoluzionali ecc. +- Funzione di attivazione. +- Livelli di normalizzazione +- Funzione di perdità o loss. + +Quando creaiamo un nuovo modello è costruire una sotto-classe di `nn.Module`. Un modulo o layer vengono costruti in Pythorc come: +- Costruttore: definisce i layer in cui vogliamo suddividere il nostro modello +- Implementa un metodo forward. Dice come far fluire i dati attraverso i modelli e come utilizzarli. + +Esiste un medoto built in in python come $"__call__"$ sono all'interno della classe e non vengono chiamati esplicitamente (tipo il costruttore). Il metodo call è un metodo che viene chiamato per far si che sia un wrapper per il forward, fa delle cose prima e dopo e in mezzo chiama il metodo forward. +Quando chiamiamo +```py + def modelNN() + pass + model=ModelNN() +``` +In realtà costruiamo un modello da una certa classe (usa il metodo built in `__init__`). Quando mandiamo il modello in esecuzione, applichiamo il forward con parametro $x$ `y=model(x)`.\ + +#nota()[ + Se chiamassi il metodo come `y=model.forward(x)` mancano delle untià funzionali. +] + +Una trasformazione lineare Solitamente viene scritta come una trasformazione affine: +$ + y = A overline(x)+overline(b) +$ +Dovne A è una matrice. Moltiplico le righe di A per il vettore colonna (x) e poi le sommo. +Se applico la trasposizoone ad entrambi i membri ottengo: +$ + y & = (A overline(x))^t+overline(b)^t \ + y & = overline(x)^t A^t + overline(b)^t +$ +Equivale a scrivere (usando una prorpeità e per essere più efficiente da un punto di vista della rappresentazione): +$ + y = overline(x)W^t+overline(b) +$ +Tuttavia dobbiamo definire: +- Input (batch_size, in_features) +- Output(batch_size, out_features) +Le features differiscono in quanto stanno nelle dimensioni della matrice $W$ e del vettore bias $b$ +```py + layer = nn.Linear(in_features=5, out_features=3) + x = torch.random(2,5) #batch_size =2 + y = layer(x) + y.shape # torch.Size([2,3]) +``` +L'importante è che il tensore matchi sulle `in_features`. Il risultato dipende dal numero di `out_features`. + +Uso tipico con layer: +```py +model = nn.Sequential( + nn.linear(784,128) + nn.ReLu() + nn.linear(128,10) +) #compatibili, output matcha con input +``` +Alla x vengono applicati in sequena i layer che ho definito. + +=== Inizializzazione dei pesi + +All'inizio la matrice dei pesi (che dovrà essere imparata) è inizializzata casualmente. Esiste una branca che studia come inizializzare i parametri, da dove si parte dipende poi come il modello evolve. +`nn.init.xavier_normal(layer.weight)` permette di creare una matrice inizializzata secondo una certa distribuzione. + +//guardare image classifier + +Funzioni di loss. + +=== Mean Square loss +Calcola la differenza quadratica elemento ad elemento. +$ + L_"MSE" = 1/N sum_(i=1)^N (y_i-hat(y)_i)^2 +$ +Pensalizza se gli elementi sono molto differenti tra di loro. + +=== L1Loss +Misura la differenza media assoluta. più sensible agli outlier. Usata in regressione. + +#nota()[ + Son funzioni entrambe convesse, essenziali per la minimiazzaione. +] + +=== Binary cross-entropy +$ + L_"BCE" = - 1/N sum_(i=1)^N [y_i log(hat(y)_i)+(1-y_i)log(1-hat(y)_i)] +$ +L'entropia misura il tasso di sopresa. Se prendiamo un enciclopedia qual'è la probabilità di ogni singola lettera dell'alfabeto. Per una sorgente come l'enciclipedia misura come sono distrubite le lettere, quando lego una parola lettera per lettera è la sopresa nel leggere la prossima. +Ci dice quanto una distibuzione di probabilità abbia più o meno caos. Date due distribuzioni di probabilità $P_i$ e $P_j$. Quando ho $P_j = 1/k$ con $k$ numero simboli ho il massimo del caos. + +In questo caso ho un classificatore binario $0 "e" 1$. Chidiamo al modello di sputare fuori un valore che sia in [0,1]. Idialmente il modello restituisce il modello restituisce a 0 quando la pool label è molto vicina a 0. + +Per ongi elemento del dataset $i$, se la verità è $0$ la prima parte viene cancellata e vale la seconda. Quando la verità è $1$ la seconda parte della formula viene persa. La loss penalizza gli errori in base alla label che mi interessa (concetto entropico). + +== Classificazione multi classe + +Se facessi classificazione multiclasse e non binaria ho parecchi valori (logits) usciti dal modello. Applico il softmax e applico la negative log-likehood. + +Per ogni valore sputa una distribuzione di probabilità ognunga per ogni classe che abbiamo, ognuna rappresenta la probabilità che $x$ appartenga alla classe $k$. + +I logits sono dei valori a caso. Per riportarli nelle probabilità uso la funzione softmax + +Soft max restitusice una somma di probabilità con somma 1. Se prendo un dataset $D=[(x_i,y_i)]_(i=1)^n$ con label $y_i$ e dato $x_i$. Se prendo un modello $overline(y_i) = theta(M_(theta(x_i)))$ se è binario $y_i = in {0,1}$ allora il modello $theta$ appartiene a [0,1]. + +la verosimiglianza del modello è data da: +$ + P(D| theta) = product overline(y_i)^(1-overline(y_i))^(1-y_i) +$ +La probabità è molto alta quando azzecco tutto. Di solito si usa la log-likehood la produttoria si trasforma in somma (produttoria molto brutta se N è molto grande non sforo rappresentazione e se ho molte predizioni sbagliate è un problema): +$ + log(P(..)) = sum_(i=1)^N y_i log overline(y_i) ... +$ +Io voglio il miglior $theta$ che spiega in modo migliroe il dataset $D$ (applicando bias). Otteniamo così il miglir mdoello ovvero il miglior theta per D. + +//aggiugnere formula softmax +Nella softmax vado a definire la cross-entropy loss: +$ + L = -1/N .... +$ +Adesso ho che $y_i = {1,2,dots,K}$ dove $K$ è il numero di classi. la mia log-likehood diventa: +$ + P(D | theta) = -1/N sum_(k=1)^N log p(y_k | x_k) = -1/N sum_(k=1)^N log(y). +$ +i k valori che devo rappresentare li esprimo in one hote encoding (solo un 1 in posizione $j$). facendo $log overline(y)_(i,j)$ selezione solo quello di indice $j$. La maschera seleziona solo la classe $j$ e cstruisco una log-likehood che tiene conto del fatto che per l'esemplare $i$-esimo sto considerando la $j$-esima classe che mi da il valore.\ +$overline(y)$ è grande come $y_i$ e $overline(y)_i = (alpha, .. alpha_k)$ è la somma di k valori dove si spera che a spiccare sia quello di posto $j$ ovvero la classe corretta. From dcf4c7fb85d989e8e55f20be218d09db2abf0bac Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Mon, 9 Feb 2026 16:13:23 +0100 Subject: [PATCH 02/25] add link --- main.typ | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/main.typ b/main.typ index 4b7cf32..d982dd6 100644 --- a/main.typ +++ b/main.typ @@ -35,4 +35,5 @@ #include "chapters/Lezione1-Introduzione.typ" #include "chapters/Lezione2-Gerarchia-Thread.typ" #include "chapters/Lezione3-Warp.typ" -#include "chapters/Lezione4.typ" \ No newline at end of file +#include "chapters/Lezione4.typ" +#include "chapters/Lezione8.typ" \ No newline at end of file From 37d397049a6efb9b61f4dfac419ce38f42198694 Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Mon, 9 Feb 2026 16:26:12 +0100 Subject: [PATCH 03/25] ai grammar check --- chapters/Lezione8.typ | 324 ++++++++++++++++++++++++++++++++++-------- 1 file changed, 268 insertions(+), 56 deletions(-) diff --git a/chapters/Lezione8.typ b/chapters/Lezione8.typ index aca9772..e898981 100644 --- a/chapters/Lezione8.typ +++ b/chapters/Lezione8.typ @@ -1,92 +1,304 @@ #import "../template.typ": * -= Lezione 8 += Intro Deep Learning -I modelli sono in grado di catturare cose al di la dell'osservabile. +== Funzioni di Attivazione -Funzione di attivazione: Sono funzioni che hanno due asintoti (da + infinito a - infitio). Per ogni valore x tendono ad essere in saturazione tra -1 e 1. La parte centrale è "meno convinta". Servono per arrivare a definire delle regioni non lineari (curvature dello spazio) in modo per distinguere gruppi di dati in base alla categoria di appartenenza. In uno stesso spazio di embedding devo creare delle superfici di separazione. Nella regione centrale il modello ha incertezza. -- Tangente iperbolica +I modelli di deep learning sono in grado di catturare informazioni *al di là dell'osservabile*, grazie all'uso delle funzioni di attivazione che introducono non-linearità nel modello. -//aggiungere immagine +=== Caratteristiche delle Funzioni di Attivazione -RELU (molto usata): Approssimazione universale basata sulla RELU. Data una funzione $f in C([a,b],R)$. Una funzione può essere approssimata come una combinazioni lineari di RELU: -- Relu = $max(o, x)$. Ogni volta che la x è negativa prendo 0. $R(w x+b)$ dipende dal valore della retta $w x+ b$. Se applicco una trasformazione alla variabile indipendete la funzione reagisce in modi diversi -- Prendo tante ReLu e vado a coprire la mia funzione $f$ qualsiasi. Ottento una curva spezzata. -- Vado a combianre la funzione rossa e blue assime, voglio che la seconda sia più vicina possibile alla funzione che voglio approssimare (grigia). +Le *funzioni di attivazione* presentano caratteristiche specifiche: +- Hanno tipicamente *due asintoti* (da $+infinity$ a $-infinity$) +- Per ogni valore $x$ tendono ad essere in *saturazione* tra $-1$ e $1$ (o $0$ e $1$ a seconda della funzione) +- La *parte centrale* rappresenta la regione di incertezza del modello -In generale più funzioni prendo meglio approssimo, con un certo costo di complicazione. +#nota()[ + Le funzioni di attivazione servono per definire *regioni non lineari* nello spazio, creando curvature che permettono di separare gruppi di dati in base alla loro categoria di appartenenza. In uno stesso spazio di embedding, creano superfici di separazione complesse. +] -Tutto quello che abbiamo bisogno sono le funzioni di attivzioni. Limitazioni: -- il modello può essere molto complesso -- più tempo di training e maggiore quantità di dati per avere una accuratezza accetabile +#esempio()[ + *Tangente iperbolica* ($tanh$): Una delle funzioni di attivazione classiche, definita come: + $ + tanh(x) = (e^x - e^(-x))/(e^x + e^(-x)) + $ + Ha output nel range $[-1, 1]$ con saturazione agli estremi e maggiore sensibilità al centro. +] -Il teorema possiamo generalizzarlo come per ogni funzione continua $g:[0,1]^D -> R$ è possibile approssimarla con un livello nascosto (percetrone): $W in R^(K times D) - b in R^k . omega in R^K$. Aimentanto $K$ ovvero le unità nascoste abbiamo più possibilità di imparare, con un costo maggiore. +=== ReLU (Rectified Linear Unit) -Il vantaggio è che avere dei modelli profondi rispetto a modelli flat è che le reti neurali gerarchiche (diversi livelli di profondità) sono in grado di estrarre più informazoni dai dati. Vengono sfruttati dei bias induttivi (lo decidiamo noi, bias) per apprendere pattern complessi. +La *ReLU* è una delle funzioni di attivazione più utilizzate nel deep learning moderno. È definita come: +$ + "ReLU"(x) = max(0, x) +$ -Di solito si vede che nei primi livelli capiscono dei pattern molto basilari (edge delle immagini ad esempio) ma non capiscono l'immagine nel suo complesso. Sono gli ultimi livelli che lo fanno. +#nota()[ + La ReLU restituisce $0$ quando $x < 0$ e $x$ quando $x >= 0$. Applicando una trasformazione lineare si ottiene: $R(w x + b)$ che dipende dal valore della retta $w x + b$. +] -Il learning è una stima di parametri (sempre più efficaci)usando questi bias che mettiamo noi (embedding e livelli intermidi) per favorie l'apprendimento di una rappresentazione (embedding) gerarchici. +==== Teorema di Approssimazione Universale con ReLU -== Pythorc nn framework +Data una funzione continua $f in C([a,b], RR)$, è possibile *approssimala come combinazione lineare di ReLU*. -Usermeo il modulo `torch.nn` essenziale per costruire modelli deep. -Troviamo come moduli: -- layer: tutti i vari _livelli_. Esiston ricorrenti, lineari, convoluzionali ecc. -- Funzione di attivazione. -- Livelli di normalizzazione -- Funzione di perdità o loss. +Il principio di funzionamento è il seguente: ++ Prendiamo tante ReLU con diverse traslazioni e scalature ++ Combinandole opportunamente, otteniamo una *curva spezzata* che approssima $f$ ++ Più ReLU utilizziamo, migliore è l'approssimazione (ma aumenta la complessità) -Quando creaiamo un nuovo modello è costruire una sotto-classe di `nn.Module`. Un modulo o layer vengono costruti in Pythorc come: -- Costruttore: definisce i layer in cui vogliamo suddividere il nostro modello -- Implementa un metodo forward. Dice come far fluire i dati attraverso i modelli e come utilizzarli. +#esempio()[ + Combinando una funzione ReLU "rossa" e una "blu" con opportuni pesi, possiamo creare una funzione risultante che si avvicina alla funzione target (grigia) che vogliamo approssimare. +] -Esiste un medoto built in in python come $"__call__"$ sono all'interno della classe e non vengono chiamati esplicitamente (tipo il costruttore). Il metodo call è un metodo che viene chiamato per far si che sia un wrapper per il forward, fa delle cose prima e dopo e in mezzo chiama il metodo forward. -Quando chiamiamo -```py - def modelNN() - pass - model=ModelNN() -``` -In realtà costruiamo un modello da una certa classe (usa il metodo built in `__init__`). Quando mandiamo il modello in esecuzione, applichiamo il forward con parametro $x$ `y=model(x)`.\ +#attenzione()[ + *Limitazioni*: + - Il modello può diventare *molto complesso* con molte unità + - Richiede *più tempo di training* e *maggiore quantità di dati* per raggiungere un'accuratezza accettabile + - Trade-off tra capacità di approssimazione e costo computazionale +] + +==== Generalizzazione del Teorema + +Il teorema può essere generalizzato: per ogni funzione continua $g:[0,1]^D -> RR$ è possibile approssimarla con *un singolo livello nascosto* (percettrone) utilizzando: +- $W in RR^(K times D)$ (matrice dei pesi) +- $b in RR^K$ (vettore di bias) +- $omega in RR^K$ (pesi di output) #nota()[ - Se chiamassi il metodo come `y=model.forward(x)` mancano delle untià funzionali. + Aumentando $K$ (ovvero il numero di *unità nascoste*), abbiamo più possibilità di apprendere funzioni complesse, ma con un costo computazionale maggiore. ] -Una trasformazione lineare Solitamente viene scritta come una trasformazione affine: +=== Modelli Profondi vs Modelli Flat + +Il vantaggio principale dei *modelli profondi* (deep) rispetto ai modelli superficiali (flat) è che le *reti neurali gerarchiche* con diversi livelli di profondità sono in grado di estrarre *più informazioni strutturate* dai dati. + +==== Rappresentazioni Gerarchiche + +I modelli profondi sfruttano *bias induttivi* (scelti dal progettista) per apprendere pattern sempre più complessi attraverso i livelli: + +#esempio()[ + *Visione artificiale*: + - *Primi livelli*: catturano pattern basilari come edge, angoli, texture + - *Livelli intermedi*: combinano i pattern basilari in strutture più complesse (parti di oggetti) + - *Ultimi livelli*: comprendono l'immagine nel suo complesso, riconoscendo oggetti e scene +] + +#nota()[ + Il *learning* è una stima di parametri sempre più efficaci, utilizzando bias induttivi (embedding e livelli intermedi) per favorire l'apprendimento di *rappresentazioni gerarchiche* dei dati. +] + +== PyTorch nn Framework + +Utilizziamo il modulo `torch.nn`, essenziale per costruire modelli di deep learning in PyTorch. + +=== Componenti Principali + +Il framework `torch.nn` fornisce diversi moduli fondamentali: + +/ Layer: Tutti i vari *livelli* della rete. Esistono layer ricorrenti, lineari, convoluzionali, ecc. +/ Funzioni di attivazione: ReLU, Sigmoid, Tanh, Softmax, ecc. +/ Livelli di normalizzazione: BatchNorm, LayerNorm, ecc. +/ Funzioni di perdita (loss): MSE, CrossEntropy, BCE, ecc. + +=== Costruzione di un Modello Custom + +Quando creiamo un nuovo modello, dobbiamo costruire una *sotto-classe di `nn.Module`*. La struttura base è: + +```python +class ModelNN(nn.Module): + def __init__(self): + super().__init__() + # Definizione dei layer + + def forward(self, x): + # Definizione del flusso dei dati + return output + +``` +=== Layer Lineari (Trasformazioni Affini) + +Una *trasformazione lineare* in PyTorch è implementata come una *trasformazione affine*: $ - y = A overline(x)+overline(b) + y = A overline(x) + overline(b) $ -Dovne A è una matrice. Moltiplico le righe di A per il vettore colonna (x) e poi le sommo. -Se applico la trasposizoone ad entrambi i membri ottengo: + +dove $A$ è una matrice e $overline(b)$ è il vettore di bias. + +==== Rappresentazione Matriciale + +Moltiplicando le righe di $A$ per il vettore colonna $overline(x)$ e sommando il bias, otteniamo l'output. + +Se applichiamo la trasposizione a entrambi i membri: $ - y & = (A overline(x))^t+overline(b)^t \ + y & = (A overline(x))^t + overline(b)^t \ y & = overline(x)^t A^t + overline(b)^t $ -Equivale a scrivere (usando una prorpeità e per essere più efficiente da un punto di vista della rappresentazione): + +Equivalentemente (e più efficientemente per la rappresentazione in PyTorch): $ - y = overline(x)W^t+overline(b) + y = overline(x) W^t + overline(b) $ -Tuttavia dobbiamo definire: -- Input (batch_size, in_features) -- Output(batch_size, out_features) -Le features differiscono in quanto stanno nelle dimensioni della matrice $W$ e del vettore bias $b$ -```py + +#nota()[ + *Dimensioni dei tensori*: + - *Input*: `(batch_size, in_features)` + - *Output*: `(batch_size, out_features)` + + Le features differiscono in base alle dimensioni della matrice $W in RR^("out_features" times "in_features")$ e del vettore bias $b in RR^("out_features")$. +] + +#esempio()[ + Creazione e utilizzo di un layer lineare: + ```python layer = nn.Linear(in_features=5, out_features=3) - x = torch.random(2,5) #batch_size =2 + x = torch.randn(2, 5) # batch_size = 2 y = layer(x) - y.shape # torch.Size([2,3]) -``` -L'importante è che il tensore matchi sulle `in_features`. Il risultato dipende dal numero di `out_features`. + print(y.shape) # torch.Size([2, 3]) + ``` -Uso tipico con layer: -```py + L'importante è che il tensore corrisponda sulle `in_features` (dimensione 5). Il risultato avrà dimensione determinata da `out_features` (dimensione 3). +] + +==== Composizione Sequenziale di Layer + +Un uso tipico prevede la composizione di più layer in sequenza: + +```python model = nn.Sequential( + nn.Linear(784, 128), # Input layer + nn.ReLU(), # Activation + nn.Linear(128, 10) # Output layer +) +``` + +#nota()[ + I layer devono essere *compatibili*: l'output di un layer deve corrispondere all'input del layer successivo. Nell'esempio: $784 -> 128 -> 128 -> 10$. +] + +Quando passiamo un input `x` al modello, vengono applicati in sequenza tutti i layer definiti +$ + y & = (A overline(x))^t+overline(b)^t \ + y & = overline(x)^t A^t + overline(b)^t +$ +Equivale a scrivere (usanPesi + +All'inizio del training, la *matrice dei pesi* (che dovrà essere appresa) viene *inizializzata casualmente*. + +#attenzione()[ + L'inizializzazione dei parametri è fondamentale per il successo del training. Esiste un'intera branca di ricerca che studia come inizializzare i parametri: *da dove si parte influenza fortemente come il modello evolve* durante l'addestramento. +] + +#esempio()[ + *Xavier Normad Error (MSE) Loss* + + Calcola la *differenza quadratica* tra valori predetti e valori reali, elemento per elemento: + $ + L_"MSE" = 1/N sum_(i=1)^N (y_i - hat(y)_i)^2 + $ +] + +*Proprietà*: +- Penalizza fortemente quando gli elementi sono molto differenti tra loro (errori grandi pesano quadraticamente) +- Sensibile agli outlier +- Usata principalmente in problemi di *regressione* + +=== L1 Loss (Mean Absolute Error) + +Misura la *Cross-Entropy (BCE)* + +La *Binary Cross-Entropy* è utilizzata per problemi di *classificazione binaria*: +$ + L_"BCE" = -1/N sum_(i=1)^N [y_i log(hat(y)_i) + (1-y_i) log(1-hat(y)_i)] +$ + +==== Concetto di Entropia + +L'*entropia* è una misura informazionale che quantifica il "tasso di sorpresa" o l'incertezza di una distribuzione di probabilità. + +#esempio()[Multiclasse] + +Nella *classificazione multiclasse* (più di 2 classi), il modello produce diversi valori (chiamati *logits*) in uscita, uno per ogni classe. Per convertirli in probabilità e calcolare la loss, usiamo *Softmax* e *Negative Log-Likelihood*. + +=== Softmax + +La funzione *Softmax* converte i logits in una distribuzione di probabilità: +$ + "softmax"(z_i) = (e^(z_i))/(sum_(j=1)^K e^(z_j)) +$ + +dove $K$ è il numero di classi. + +#nota()[ + La Softmax garantisce che: + - Ogni output sia $in [0,1]$ + - La somma di tutte le probabilità sia uguale a $1$ + - Ogni valore rappresenta $P(y = k | x)$, la probabilità che $x$ appartenga alla classe $k$ +] + +=== Verosimiglianza e Log-Likelihood + +Dato un dataset $D = {(x_i, y_i)}_(i=1)^N$ con label $y_i$ e input $x_i$, consideriamo un modello $hat(y)_i = M_theta(x_i)$. + +==== Caso Binario (Richiamo) + +Per un problema binario $y_i in {0, 1}$ con modello $theta in [0,1]$, la *verosimiglianza* (likelihood) è: +$ + P(D | theta) = product_(i=1)^N hat(y)_i^(y_i) (1-hat(y)_i)^(1-y_i) +$ + +#nota()[ + La probabilità è molto alta quando il modello predice correttamente tutti gli esempi. +] + +Usiamo la *log-likelihood* perché: +- La produttoria diventa somma (più stabile numericamente) +- Evitiamo problemi di underflow con $N$ grande +- È più facile da ottimizzare + +$ + log P(D | theta) = sum_(i=1)^N [y_i log hat(y)_i + (1-y_i) log(1-hat(y)_i)] +$ + +==== Caso Multiclasse + +Per la classificazione multiclasse con $y_i in {1, 2, dots, K}$, la *log-likelihood* diventa: +$ + log P(D | theta) = sum_(i=1)^N log p(y_i | x_i) +$ + +=== Cross-Entropy Loss per Classificazione Multiclasse + +La *Cross-Entropy Loss* è definita come la *Negative Log-Likelihood*: +$ + L_"CE" = -1/N sum_(i=1)^N log p(y_i | x_i) = -1/N sum_(i=1)^N log hat(y)_(i,j) +$ + +dove $j$ è l'indice della classe corretta per l'esempio $i$-esimo. + +==== One-Hot Encoding + +Le label $y_i in {1, 2, dots, K}$ vengono rappresentate in *one-hot encoding*: +- Un vettore di dimensione $K$ con tutti $0$ tranne un $1$ in posizione $j$ (classe corretta) +- Esempio: per $K=3$ e classe $2$: $y = [0, 1, 0]$ + +#esempio()[ + Supponiamo $K = 3$ classi e l'esempio $i$-esimo ha label vera $j = 2$: + - Il modello produce: $hat(y)_i = (alpha_1, alpha_2, alpha_3)$ dopo softmax + - La loss considera solo: $log hat(y)_(i,2) = log alpha_2$ + - Idealmente vogliamo $alpha_2 approx 1$ e $alpha_1, alpha_3 approx 0$ +] + +#nota()[ + Il *one-hot encoding* agisce come una *maschera* che seleziona solo la probabilità predetta per la classe corretta $j$. La log-likelihood tiene conto solo del valore $hat(y)_(i,j)$ dove $j$ è la classe vera dell'esempio $i$-esimo. +] + +#attenzione()[ + *Obiettivo*: vogliamo trovare il miglior $theta$ (parametri del modello) che massimizza la log-likelihood (o equivalentemente minimizza la negative log-likelihood). Questo corrisponde a trovare il modello che spiega meglio il dataset $D$, tenendo conto dei bias induttivi che abbiamo introdotto nell'architettura. +] nn.linear(784,128) nn.ReLu() nn.linear(128,10) -) #compatibili, output matcha con input -``` + compatibili, output matcha con input + + Alla x vengono applicati in sequena i layer che ho definito. === Inizializzazione dei pesi From d73aa1e54a971319d3c27f180cc73df718a93cd8 Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Mon, 9 Feb 2026 16:40:53 +0100 Subject: [PATCH 04/25] update intro --- chapters/Lezione8.typ | 21 +++++++++++++++++---- 1 file changed, 17 insertions(+), 4 deletions(-) diff --git a/chapters/Lezione8.typ b/chapters/Lezione8.typ index e898981..df583d6 100644 --- a/chapters/Lezione8.typ +++ b/chapters/Lezione8.typ @@ -1,6 +1,19 @@ #import "../template.typ": * = Intro Deep Learning +Possiamo vedere un modello di deep learning come una _scatola nera_ il cui compito è apprendere un set di *parametri $Theta$* (detti anche iper-parametri).\ +Dati: +- Osservazioni ${(mr(x_i),mb(y_i))}$, dove: + - *$mr(x_i)$* sono ciò che ha predetto il modello + - *$mb(y_i)$* è la _label_ corretta +- Un modello parametrico: $f(x, Theta)$ + +La fase di apprendimento (learning) consiste nel trovare il set di parametri $theta^*$ tale che: +$ + theta^* = arg min_(Theta) mr(L)(f(x, Theta), y) +$ +Dove $mr(L)$ è una *funzione di loss*. + == Funzioni di Attivazione I modelli di deep learning sono in grado di catturare informazioni *al di là dell'osservabile*, grazie all'uso delle funzioni di attivazione che introducono non-linearità nel modello. @@ -293,10 +306,10 @@ Le label $y_i in {1, 2, dots, K}$ vengono rappresentate in *one-hot encoding*: #attenzione()[ *Obiettivo*: vogliamo trovare il miglior $theta$ (parametri del modello) che massimizza la log-likelihood (o equivalentemente minimizza la negative log-likelihood). Questo corrisponde a trovare il modello che spiega meglio il dataset $D$, tenendo conto dei bias induttivi che abbiamo introdotto nell'architettura. ] - nn.linear(784,128) - nn.ReLu() - nn.linear(128,10) - compatibili, output matcha con input +nn.linear(784,128) +nn.ReLu() +nn.linear(128,10) +compatibili, output matcha con input Alla x vengono applicati in sequena i layer che ho definito. From e1e83b62877d1e38b432141a0952e6165ecf9847 Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Mon, 9 Feb 2026 16:46:51 +0100 Subject: [PATCH 05/25] add nlp note --- chapters/Lezione8.typ | 793 +++++++++++++++++++++++++++++++++++++----- 1 file changed, 713 insertions(+), 80 deletions(-) diff --git a/chapters/Lezione8.typ b/chapters/Lezione8.typ index df583d6..a73dacf 100644 --- a/chapters/Lezione8.typ +++ b/chapters/Lezione8.typ @@ -48,6 +48,59 @@ $ La ReLU restituisce $0$ quando $x < 0$ e $x$ quando $x >= 0$. Applicando una trasformazione lineare si ottiene: $R(w x + b)$ che dipende dal valore della retta $w x + b$. ] +/* +#figura( + ```python + import cetz.plot + + cetz.canvas({ + import cetz.draw: * + + cetz.plot.plot( + size: (10, 6), + x-label: $x$, + y-label: $f(x)$, + x-tick-step: 1, + y-tick-step: 0.5, + x-min: -3, + x-max: 3, + y-min: -0.5, + y-max: 3, + { + // ReLU + cetz.plot.add( + ((x,) => (calc.max(0, x),)), + domain: (-3, 3), + samples: 100, + style: (stroke: (paint: mo, thickness: 2pt)), + label: [ReLU] + ) + + // Tanh (per confronto) + cetz.plot.add( + ((x,) => (calc.tanh(x),)), + domain: (-3, 3), + samples: 100, + style: (stroke: (paint: mb, thickness: 1.5pt, dash: "dashed")), + label: [Tanh] + ) + + // Sigmoid (per confronto) + cetz.plot.add( + ((x,) => (1 / (1 + calc.exp(-x)),)), + domain: (-3, 3), + samples: 100, + style: (stroke: (paint: mr, thickness: 1.5pt, dash: "dotted")), + label: [Sigmoid] + ) + } + ) + }) + ``` + caption: [Confronto tra funzioni di attivazione: ReLU (arancione), Tanh (blu tratteggiato) e Sigmoid (rosso punteggiato)] +) +*/ + ==== Teorema di Approssimazione Universale con ReLU Data una funzione continua $f in C([a,b], RR)$, è possibile *approssimala come combinazione lineare di ReLU*. @@ -126,6 +179,54 @@ class ModelNN(nn.Module): return output ``` +/* +#figura( + ```python + import fletcher as fl + + fl.diagram( + node-stroke: 1pt, + spacing: (20mm, 8mm), + edge-stroke: 1.5pt, + mark-scale: 70%, + { + let layers = ( + (pos: (0, 0), label: [Input\n$mb(x)$], fill: mb), + (pos: (0, 1), label: [Linear\n$W_1 mb(x) + mb(b)_1$], fill: mo), + (pos: (0, 2), label: [BatchNorm], fill: mg), + (pos: (0, 3), label: [ReLU\n$max(0, dot)$], fill: mr), + (pos: (0, 4), label: [Dropout\n$p=0.5$], fill: mp), + (pos: (0, 5), label: [Linear\n$W_2 dot + mb(b)_2$], fill: mo), + (pos: (0, 6), label: [Softmax], fill: mg), + (pos: (0, 7), label: [Output\n$hat(mb(y))$], fill: mb), + ) + + for (i, layer) in layers.enumerate() { + fl.node( + layer.pos, + layer.label, + shape: rect, + corner-radius: 5pt, + fill: gradient.linear(white, layer.fill, angle: 45deg), + width: 30mm, + height: 10mm + ) + + if i < layers.len() - 1 { + fl.edge(layer.pos, layers.at(i + 1).pos, "->", stroke: 2pt) + } + } + + // Annotazioni + fl.node((1.5, 1.5), [Trasformazione\nlineare], stroke: none, fill: none) + fl.node((1.5, 3.5), [Non-linearità], stroke: none, fill: none) + fl.node((1.5, 4), [Regolarizzazione], stroke: none, fill: none) + } + ) + ``` + caption: [Architettura tipica di una rete neurale feedforward con BatchNorm, ReLU e Dropout] +) +*/ === Layer Lineari (Trasformazioni Affini) Una *trasformazione lineare* in PyTorch è implementata come una *trasformazione affine*: @@ -199,36 +300,85 @@ All'inizio del training, la *matrice dei pesi* (che dovrà essere appresa) viene L'inizializzazione dei parametri è fondamentale per il successo del training. Esiste un'intera branca di ricerca che studia come inizializzare i parametri: *da dove si parte influenza fortemente come il modello evolve* durante l'addestramento. ] -#esempio()[ - *Xavier Normad Error (MSE) Loss* +== Funzioni di Loss - Calcola la *differenza quadratica* tra valori predetti e valori reali, elemento per elemento: - $ - L_"MSE" = 1/N sum_(i=1)^N (y_i - hat(y)_i)^2 - $ -] +Le *funzioni di loss* (o funzioni di costo) quantificano l'errore tra le predizioni del modello e i valori reali. Durante il training, l'obiettivo è minimizzare questa funzione. + +=== Mean Squared Error (MSE) Loss + +Calcola la *differenza quadratica media* tra valori predetti e valori reali: +$ + L_"MSE" = 1/N sum_(i=1)^N (y_i - hat(y)_i)^2 +$ *Proprietà*: -- Penalizza fortemente quando gli elementi sono molto differenti tra loro (errori grandi pesano quadraticamente) -- Sensibile agli outlier +- Penalizza fortemente gli errori grandi (peso quadratico) +- Molto sensibile agli *outlier* - Usata principalmente in problemi di *regressione* +- Funzione *convessa*, essenziale per l'ottimizzazione -=== L1 Loss (Mean Absolute Error) +#esempio()[ + Se $y_i = 5$ e $hat(y)_i = 3$, l'errore per questo esempio è $(5-3)^2 = 4$. + Se $y_i = 5$ e $hat(y)_i = 1$, l'errore diventa $(5-1)^2 = 16$ (4 volte maggiore pur avendo raddoppiato la distanza). +] + +=== Mean Absolute Error (MAE) Loss o L1 Loss + +Misura la *differenza assoluta media* tra predizioni e valori reali: +$ + L_"MAE" = 1/N sum_(i=1)^N |y_i - hat(y)_i| +$ + +*Proprietà*: +- Meno sensibile agli outlier rispetto a MSE +- Penalizza linearmente gli errori +- Funzione *convessa* +- Usata in problemi di *regressione* quando si vuole robustezza agli outlier -Misura la *Cross-Entropy (BCE)* +=== Binary Cross-Entropy (BCE) Loss -La *Binary Cross-Entropy* è utilizzata per problemi di *classificazione binaria*: +La *Binary Cross-Entropy* è utilizzata per problemi di *classificazione binaria* ($y_i in {0,1}$): $ L_"BCE" = -1/N sum_(i=1)^N [y_i log(hat(y)_i) + (1-y_i) log(1-hat(y)_i)] $ +#nota()[ + Il modello deve produrre valori $hat(y)_i in [0,1]$, tipicamente ottenuti con una funzione *Sigmoid* all'output. +] + ==== Concetto di Entropia L'*entropia* è una misura informazionale che quantifica il "tasso di sorpresa" o l'incertezza di una distribuzione di probabilità. -#esempio()[Multiclasse] +#nota()[ + *Interpretazione*: L'entropia misura quanto è "caotica" una distribuzione. + - *Alta entropia*: distribuzione uniforme (massima incertezza) + - *Bassa entropia*: distribuzione concentrata su pochi valori (bassa incertezza) +] + +#esempio()[ + Consideriamo la distribuzione delle lettere in un testo italiano: + - Se tutte le lettere hanno la stessa probabilità $P = 1/21$: *alta entropia* (massima sorpresa) + - Se alcune lettere sono molto frequenti (es. 'e', 'a', 'i'): *bassa entropia* (minore sorpresa) + + Quando leggiamo una parola lettera per lettera, l'entropia quantifica la nostra "sorpresa" nel vedere la lettera successiva. +] + +==== Funzionamento della BCE + +Nella BCE per classificazione binaria: +- Quando $y_i = 1$ (label vera): vogliamo $hat(y)_i approx 1$, quindi minimizziamo $-log hat(y)_i$ +- Quando $y_i = 0$ (label vera): vogliamo $hat(y)_i approx 0$, quindi minimizziamo $-log(1 - hat(y)_i)$ + +La loss *penalizza fortemente* predizioni sbagliate con alta confidenza. -Nella *classificazione multiclasse* (più di 2 classi), il modello produce diversi valori (chiamati *logits*) in uscita, uno per ogni classe. Per convertirli in probabilità e calcolare la loss, usiamo *Softmax* e *Negative Log-Likelihood*. +== Classificazione Multiclasse + +Nella *classificazione multiclasse* (più di 2 classi), il modello produce diversi valori (chiamati *logits*) in uscita, uno per ogni classe $k in {1, 2, dots, K}$. + +#nota()[ + I *logits* sono valori reali arbitrari (non normalizzati) prodotti dall'ultimo layer lineare del modello. Per convertirli in probabilità valide, applichiamo la funzione *Softmax*. +] === Softmax @@ -237,13 +387,29 @@ $ "softmax"(z_i) = (e^(z_i))/(sum_(j=1)^K e^(z_j)) $ -dove $K$ è il numero di classi. +dove $K$ è il numero di classi e $z_i$ è il logit per la classe $i$. #nota()[ La Softmax garantisce che: - Ogni output sia $in [0,1]$ - - La somma di tutte le probabilità sia uguale a $1$ + - La somma di tutte le probabilità sia uguale a $1$: $sum_(i=1)^K "softmax"(z_i) = 1$ - Ogni valore rappresenta $P(y = k | x)$, la probabilità che $x$ appartenga alla classe $k$ + - I logits più alti producono probabilità maggiori (enfasi esponenziale) +] + +#esempio()[ + Dato un vettore di logits per 3 classi: $z = [2.0, 1.0, 0.1]$ + + Calcolo Softmax: + - $e^(2.0) approx 7.39$ + - $e^(1.0) approx 2.72$ + - $e^(0.1) approx 1.11$ + - Somma: $7.39 + 2.72 + 1.11 = 11.22$ + + Probabilità: + - $P(y=1|x) = 7.39/11.22 approx 0.66$ + - $P(y=2|x) = 2.72/11.22 approx 0.24$ + - $P(y=3|x) = 1.11/11.22 approx 0.10$ ] === Verosimiglianza e Log-Likelihood @@ -252,130 +418,597 @@ Dato un dataset $D = {(x_i, y_i)}_(i=1)^N$ con label $y_i$ e input $x_i$, consid ==== Caso Binario (Richiamo) -Per un problema binario $y_i in {0, 1}$ con modello $theta in [0,1]$, la *verosimiglianza* (likelihood) è: +Per un problema binario $y_i in {0, 1}$ con modello parametrizzato da $theta$ che produce $hat(y)_i = M_theta (x_i) in [0,1]$, la *verosimiglianza* (likelihood) del dataset $D$ dato il modello è: $ P(D | theta) = product_(i=1)^N hat(y)_i^(y_i) (1-hat(y)_i)^(1-y_i) $ #nota()[ - La probabilità è molto alta quando il modello predice correttamente tutti gli esempi. + *Interpretazione*: + - Quando $y_i = 1$: contributo $hat(y)_i$ (vogliamo $hat(y)_i$ vicino a 1) + - Quando $y_i = 0$: contributo $(1-hat(y)_i)$ (vogliamo $hat(y)_i$ vicino a 0) + - La verosimiglianza è alta quando il modello predice correttamente tutti gli esempi ] -Usiamo la *log-likelihood* perché: -- La produttoria diventa somma (più stabile numericamente) -- Evitiamo problemi di underflow con $N$ grande -- È più facile da ottimizzare +==== Perché usiamo la Log-Likelihood? + +Usiamo la *log-likelihood* per diversi motivi pratici e numerici: + ++ *Stabilità numerica*: La produttoria diventa somma + $ + log P(D | theta) = log product_(i=1)^N p_i = sum_(i=1)^N log p_i + $ + ++ *Evitare underflow*: Con $N$ grande, prodotto di probabilità $< 1$ tende rapidamente a 0 ++ *Ottimizzazione più semplice*: Derivate della somma sono più facili da calcolare + ++ *Interpretazione additiva*: Contributi indipendenti di ogni esempio + +#esempio()[ + *Senza log*: $0.9 times 0.8 times 0.7 times dots times 0.6$ con 100 termini $->$ underflow + + *Con log*: $log 0.9 + log 0.8 + log 0.7 + dots + log 0.6$ $->$ stabile +] + +La log-likelihood per il caso binario diventa: $ log P(D | theta) = sum_(i=1)^N [y_i log hat(y)_i + (1-y_i) log(1-hat(y)_i)] $ ==== Caso Multiclasse -Per la classificazione multiclasse con $y_i in {1, 2, dots, K}$, la *log-likelihood* diventa: +Per la classificazione multiclasse con $y_i in {1, 2, dots, K}$, il modello produce un vettore di probabilità $hat(mb(y))_i = (hat(y)_(i,1), hat(y)_(i,2), dots, hat(y)_(i,K))$ dove $hat(y)_(i,k) = P(y_i = k | x_i)$. + +La *verosimiglianza* diventa: $ - log P(D | theta) = sum_(i=1)^N log p(y_i | x_i) + P(D | theta) = product_(i=1)^N P(y_i | x_i) = product_(i=1)^N hat(y)_(i,y_i) +$ + +Dove $hat(y)_(i,y_i)$ indica la probabilità assegnata alla classe corretta per l'esempio $i$-esimo. + +La *log-likelihood* diventa: +$ + log P(D | theta) = sum_(i=1)^N log P(y_i | x_i) = sum_(i=1)^N log hat(y)_(i,y_i) $ === Cross-Entropy Loss per Classificazione Multiclasse -La *Cross-Entropy Loss* è definita come la *Negative Log-Likelihood*: +La *Cross-Entropy Loss* (o *Categorical Cross-Entropy*) è definita come la *Negative Log-Likelihood* normalizzata: $ - L_"CE" = -1/N sum_(i=1)^N log p(y_i | x_i) = -1/N sum_(i=1)^N log hat(y)_(i,j) + L_"CE" = -1/N sum_(i=1)^N log P(y_i | x_i) = -1/N sum_(i=1)^N log hat(y)_(i,y_i) $ -dove $j$ è l'indice della classe corretta per l'esempio $i$-esimo. +dove $y_i in {1, 2, dots, K}$ è l'indice della classe corretta per l'esempio $i$-esimo e $hat(y)_(i,y_i)$ è la probabilità predetta per quella classe. + +#nota()[ + *Minimizzare la Cross-Entropy* equivale a *massimizzare la log-likelihood*: + $ + min_theta L_"CE" equiv max_theta log P(D | theta) + $ + + Cerchiamo quindi i parametri $theta^*$ che meglio spiegano il dataset $D$, considerando i bias induttivi dell'architettura scelta. +] ==== One-Hot Encoding -Le label $y_i in {1, 2, dots, K}$ vengono rappresentate in *one-hot encoding*: -- Un vettore di dimensione $K$ con tutti $0$ tranne un $1$ in posizione $j$ (classe corretta) -- Esempio: per $K=3$ e classe $2$: $y = [0, 1, 0]$ +Le label $y_i in {1, 2, dots, K}$ vengono spesso rappresentate in *one-hot encoding* per facilitare il calcolo: +- Un vettore $mb(y)_i in RR^K$ con tutti $0$ tranne un $1$ in posizione $y_i$ (classe corretta) +- Esempio: per $K=3$ e classe $y_i = 2$: $mb(y)_i = [0, 1, 0]$ + +*Formula alternativa con one-hot encoding*: +$ + L_"CE" = -1/N sum_(i=1)^N sum_(k=1)^K mb(y)_(i,k) log hat(y)_(i,k) +$ + +Dove: +- $mb(y)_(i,k) = cases(1 "se" k = y_i, 0 "altrimenti")$ +- La somma interna seleziona automaticamente solo la classe corretta #esempio()[ - Supponiamo $K = 3$ classi e l'esempio $i$-esimo ha label vera $j = 2$: - - Il modello produce: $hat(y)_i = (alpha_1, alpha_2, alpha_3)$ dopo softmax - - La loss considera solo: $log hat(y)_(i,2) = log alpha_2$ - - Idealmente vogliamo $alpha_2 approx 1$ e $alpha_1, alpha_3 approx 0$ + Supponiamo $K = 3$ classi e l'esempio $i$-esimo ha label vera $y_i = 2$: + + *One-hot encoding*: $mb(y)_i = [0, 1, 0]$ + + *Output Softmax*: $hat(mb(y))_i = [0.1, 0.7, 0.2]$ (dopo normalizzazione) + + *Calcolo loss*: + $ + L_i = -(0 times log 0.1 + 1 times log 0.7 + 0 times log 0.2) = -log 0.7 approx 0.357 + $ + + La loss considera solo: $-log hat(y)_(i,2) = -log 0.7$ + + *Obiettivo*: vogliamo $hat(y)_(i,2) approx 1$ e $hat(y)_(i,1), hat(y)_(i,3) approx 0$ ] #nota()[ - Il *one-hot encoding* agisce come una *maschera* che seleziona solo la probabilità predetta per la classe corretta $j$. La log-likelihood tiene conto solo del valore $hat(y)_(i,j)$ dove $j$ è la classe vera dell'esempio $i$-esimo. + Il *one-hot encoding* agisce come una *maschera* che seleziona solo la probabilità predetta per la classe corretta. Questa rappresentazione: + - Facilita il calcolo dei gradienti durante il backpropagation + - Rende uniforme il trattamento di tutte le classi + - Permette un'implementazione vettoriale efficiente ] #attenzione()[ - *Obiettivo*: vogliamo trovare il miglior $theta$ (parametri del modello) che massimizza la log-likelihood (o equivalentemente minimizza la negative log-likelihood). Questo corrisponde a trovare il modello che spiega meglio il dataset $D$, tenendo conto dei bias induttivi che abbiamo introdotto nell'architettura. + *Obiettivo finale del training*: + + Vogliamo trovare i parametri ottimali $theta^*$ che: + $ + theta^* = arg max_theta log P(D | theta) equiv arg min_theta L_"CE" + $ + + Questo corrisponde a trovare il modello che: + - *Spiega meglio* il dataset $D$ osservato + - Sfrutta i *bias induttivi* introdotti nell'architettura (layer, attivazioni, profondità) + - *Generalizza* bene su dati non visti (validation/test set) ] -nn.linear(784,128) -nn.ReLu() -nn.linear(128,10) -compatibili, output matcha con input +== Training e Ottimizzazione -Alla x vengono applicati in sequena i layer che ho definito. +=== Inizializzazione dei Pesi -=== Inizializzazione dei pesi +All'inizio del training, i *parametri del modello* (pesi e bias) devono essere inizializzati. L'inizializzazione è *fondamentale* per il successo dell'addestramento. -All'inizio la matrice dei pesi (che dovrà essere imparata) è inizializzata casualmente. Esiste una branca che studia come inizializzare i parametri, da dove si parte dipende poi come il modello evolve. -`nn.init.xavier_normal(layer.weight)` permette di creare una matrice inizializzata secondo una certa distribuzione. +#attenzione()[ + Una cattiva inizializzazione può: + - Far convergere il modello a minimi locali subottimali + - Causare *vanishing/exploding gradients* + - Rallentare significativamente il training + - Impedire completamente l'apprendimento +] -//guardare image classifier +==== Strategie di Inizializzazione -Funzioni di loss. +*Inizializzazione casuale semplice*: Non consigliata! +- Valori troppo grandi: gradienti esplodono +- Valori troppo piccoli: gradienti svaniscono +- Tutti uguali (es. zero): simmetria non rotta, neuroni imparano la stessa cosa -=== Mean Square loss -Calcola la differenza quadratica elemento ad elemento. -$ - L_"MSE" = 1/N sum_(i=1)^N (y_i-hat(y)_i)^2 -$ -Pensalizza se gli elementi sono molto differenti tra di loro. +*Inizializzazioni avanzate*: + +#esempio()[ + *Xavier/Glorot Initialization* (per tanh, sigmoid): + + I pesi di un layer con $n_"in"$ input e $n_"out"$ output sono inizializzati da: + $ + W tilde cal(N)(0, sigma^2) "dove" sigma = sqrt(2/(n_"in" + n_"out")) + $ + + In PyTorch: + ```python + nn.init.xavier_normal_(layer.weight) + # oppure uniforme: + nn.init.xavier_uniform_(layer.weight) + ``` +] + +#esempio()[ + *He Initialization* (per ReLU e varianti): + + Ottimizzata per funzioni di attivazione ReLU: + $ + W tilde cal(N)(0, sigma^2) "dove" sigma = sqrt(2/n_"in") + $ -=== L1Loss -Misura la differenza media assoluta. più sensible agli outlier. Usata in regressione. + In PyTorch: + ```python + nn.init.kaiming_normal_(layer.weight, mode='fan_in', nonlinearity='relu') + ``` +] #nota()[ - Son funzioni entrambe convesse, essenziali per la minimiazzaione. + *Perché queste inizializzazioni funzionano?* + + Mantengono la *varianza delle attivazioni* e dei *gradienti* costante attraverso i layer: + - Evitano il vanishing gradient (segnale troppo piccolo) + - Evitano l'exploding gradient (segnale troppo grande) + - Permettono un flusso stabile di informazione avanti e indietro ] -=== Binary cross-entropy -$ - L_"BCE" = - 1/N sum_(i=1)^N [y_i log(hat(y)_i)+(1-y_i)log(1-hat(y)_i)] -$ -L'entropia misura il tasso di sopresa. Se prendiamo un enciclopedia qual'è la probabilità di ogni singola lettera dell'alfabeto. Per una sorgente come l'enciclipedia misura come sono distrubite le lettere, quando lego una parola lettera per lettera è la sopresa nel leggere la prossima. -Ci dice quanto una distibuzione di probabilità abbia più o meno caos. Date due distribuzioni di probabilità $P_i$ e $P_j$. Quando ho $P_j = 1/k$ con $k$ numero simboli ho il massimo del caos. +==== Inizializzazione dei Bias -In questo caso ho un classificatore binario $0 "e" 1$. Chidiamo al modello di sputare fuori un valore che sia in [0,1]. Idialmente il modello restituisce il modello restituisce a 0 quando la pool label è molto vicina a 0. +I bias sono tipicamente inizializzati a *zero*: +```python +nn.init.zeros_(layer.bias) +``` -Per ongi elemento del dataset $i$, se la verità è $0$ la prima parte viene cancellata e vale la seconda. Quando la verità è $1$ la seconda parte della formula viene persa. La loss penalizza gli errori in base alla label che mi interessa (concetto entropico). +In alcuni casi specifici: +- LSTM/GRU: bias dei gate di forget inizializzati a 1 +- Output layer: bias può essere inizializzato in base alle frequenze delle classi -== Classificazione multi classe +=== Processo di Training -Se facessi classificazione multiclasse e non binaria ho parecchi valori (logits) usciti dal modello. Applico il softmax e applico la negative log-likehood. +Il processo di training di una rete neurale segue questi passi: -Per ogni valore sputa una distribuzione di probabilità ognunga per ogni classe che abbiamo, ognuna rappresenta la probabilità che $x$ appartenga alla classe $k$. ++ *Forward pass*: calcolo delle predizioni attraverso i layer + $ + hat(mb(y)) = f(mb(x); theta) + $ -I logits sono dei valori a caso. Per riportarli nelle probabilità uso la funzione softmax ++ *Calcolo della loss*: valutazione dell'errore + $ + L = L(hat(mb(y)), mb(y)) + $ -Soft max restitusice una somma di probabilità con somma 1. Se prendo un dataset $D=[(x_i,y_i)]_(i=1)^n$ con label $y_i$ e dato $x_i$. Se prendo un modello $overline(y_i) = theta(M_(theta(x_i)))$ se è binario $y_i = in {0,1}$ allora il modello $theta$ appartiene a [0,1]. ++ *Backward pass*: calcolo dei gradienti tramite backpropagation + $ + nabla_theta L + $ + ++ *Aggiornamento parametri*: ottimizzazione (es. SGD, Adam) + $ + theta <- theta - eta nabla_theta L + $ -la verosimiglianza del modello è data da: +/* +#figura( + ```python + import fletcher as fl + + fl.diagram( + node-stroke: 1pt, + node-fill: gradient.linear(..mo.colors), + spacing: (15mm, 10mm), + edge-stroke: 1pt, + { + let (input, forward, loss, backward, update, output) = ((0,0), (1,0), (2,0), (2,1), (1,1), (0,1)) + + fl.node(input, [Input \ $mb(x), mb(y)$], corner-radius: 5pt, extrude: (0, 3)) + fl.node(forward, [Forward \ Pass], corner-radius: 5pt) + fl.node(loss, [Loss \ $L$], corner-radius: 5pt, fill: mr) + fl.node(backward, [Backward \ Pass], corner-radius: 5pt) + fl.node(update, [Update \ $theta$], corner-radius: 5pt, fill: mg) + + fl.edge(input, forward, "->", [Batch]) + fl.edge(forward, loss, "->", [$hat(mb(y))$]) + fl.edge(loss, backward, "->", [Gradiente]) + fl.edge(backward, update, "->", [$nabla_theta L$]) + fl.edge(update, forward, "->", [Parametri\naggiornati], label-pos: 0.3) + } + ) + ``` + caption: [Ciclo di training di una rete neurale: forward pass, calcolo loss, backward pass, aggiornamento parametri] +) +*/ +#nota()[ + PyTorch automatizza il calcolo dei gradienti (punto 3) tramite *autograd*, permettendo di concentrarsi sulla definizione del modello e della loss. +] + +=== Ottimizzatori + +Gli *ottimizzatori* implementano algoritmi per aggiornare i parametri $theta$ in base ai gradienti calcolati. + +==== Stochastic Gradient Descent (SGD) + +L'ottimizzatore più semplice: $ - P(D| theta) = product overline(y_i)^(1-overline(y_i))^(1-y_i) + theta_(t+1) = theta_t - eta nabla_theta L(theta_t) $ -La probabità è molto alta quando azzecco tutto. Di solito si usa la log-likehood la produttoria si trasforma in somma (produttoria molto brutta se N è molto grande non sforo rappresentazione e se ho molte predizioni sbagliate è un problema): + +dove $eta$ è il *learning rate*. + +```python +optimizer = torch.optim.SGD(model.parameters(), lr=0.01) +``` + +*Variante con momentum*: $ - log(P(..)) = sum_(i=1)^N y_i log overline(y_i) ... + v_(t+1) & = gamma v_t + eta nabla_theta L(theta_t) \ + theta_(t+1) & = theta_t - v_(t+1) $ -Io voglio il miglior $theta$ che spiega in modo migliroe il dataset $D$ (applicando bias). Otteniamo così il miglir mdoello ovvero il miglior theta per D. -//aggiugnere formula softmax -Nella softmax vado a definire la cross-entropy loss: +```python +optimizer = torch.optim.SGD(model.parameters(), lr=0.01, momentum=0.9) +``` + +#nota()[ + Il *momentum* aiuta ad accelerare la convergenza e superare minimi locali accumulando una "velocità" nella direzione dei gradienti passati. +] + +==== Adam (Adaptive Moment Estimation) + +Uno degli ottimizzatori più usati, combina: +- *Momentum*: media mobile dei gradienti +- *RMSprop*: adatta il learning rate per ogni parametro + +```python +optimizer = torch.optim.Adam(model.parameters(), lr=0.001, betas=(0.9, 0.999)) +``` + +*Vantaggi di Adam*: +- Convergenza veloce +- Poco sensibile alla scelta del learning rate iniziale +- Adatta automaticamente il learning rate per ogni parametro +- Funziona bene nella maggior parte dei casi + +#esempio()[ + *Setup tipico per training*: + ```python + model = MyModel() + criterion = nn.CrossEntropyLoss() + optimizer = torch.optim.Adam(model.parameters(), lr=0.001) + + for epoch in range(num_epochs): + for batch_x, batch_y in dataloader: + # Forward pass + outputs = model(batch_x) + loss = criterion(outputs, batch_y) + + # Backward pass + optimizer.zero_grad() # Reset gradienti + loss.backward() # Calcolo gradienti + optimizer.step() # Aggiornamento parametri + ``` +] + +==== Altri Ottimizzatori + +*AdamW*: Variante di Adam con weight decay corretto +```python +optimizer = torch.optim.AdamW(model.parameters(), lr=0.001, weight_decay=0.01) +``` + +*RMSprop*: Adatta il learning rate usando media mobile dei gradienti al quadrato +```python +optimizer = torch.optim.RMSprop(model.parameters(), lr=0.01) +``` + +#attenzione()[ + *Scelta dell'ottimizzatore*: + - *Adam/AdamW*: scelta sicura per la maggior parte dei problemi + - *SGD con momentum*: può generalizzare meglio con tuning accurato + - *RMSprop*: buono per RNN e problemi con gradienti variabili +] + +=== Learning Rate e Scheduling + +Il *learning rate* $eta$ è uno degli iperparametri più importanti: +- Troppo alto: divergenza, oscillazioni +- Troppo basso: convergenza lenta, minimi locali + +==== Learning Rate Scheduling + +Modificare il learning rate durante il training: + +*Step Decay*: Riduce $eta$ ogni $N$ epoche +```python +scheduler = torch.optim.lr_scheduler.StepLR(optimizer, step_size=30, gamma=0.1) +``` + +*Cosine Annealing*: Riduce $eta$ seguendo una curva coseno +```python +scheduler = torch.optim.lr_scheduler.CosineAnnealingLR(optimizer, T_max=100) +``` + +*ReduceLROnPlateau*: Riduce $eta$ quando la loss smette di migliorare +```python +scheduler = torch.optim.lr_scheduler.ReduceLROnPlateau(optimizer, mode='min', + factor=0.5, patience=5) +``` + +#esempio()[ + *Uso dello scheduler*: + ```python + for epoch in range(num_epochs): + train_one_epoch(model, optimizer, criterion, train_loader) + val_loss = validate(model, criterion, val_loader) + + # Aggiorna learning rate + scheduler.step(val_loss) # per ReduceLROnPlateau + # oppure scheduler.step() per altri scheduler + ``` +] + +== Regolarizzazione e Prevenzione dell'Overfitting + +Le tecniche di *regolarizzazione* aiutano il modello a *generalizzare* meglio su dati non visti, prevenendo l'*overfitting*. + +=== Weight Decay (L2 Regularization) + +Aggiunge un termine di penalizzazione alla loss per limitare la magnitudine dei pesi: $ - L = -1/N .... + L_"total" = L_"task" + lambda/2 sum_i theta_i^2 $ -Adesso ho che $y_i = {1,2,dots,K}$ dove $K$ è il numero di classi. la mia log-likehood diventa: + +dove $lambda$ è il *coefficiente di regolarizzazione*. + +```python +optimizer = torch.optim.Adam(model.parameters(), lr=0.001, weight_decay=0.01) +``` + +#nota()[ + *Effetto*: pesi più piccoli rendono il modello meno sensibile a piccole variazioni nell'input, migliorando la generalizzazione. +] + +=== Dropout + +Durante il training, *disattiva casualmente* una frazione $p$ di neuroni: +- Ogni neurone ha probabilità $p$ di essere "spento" (output = 0) +- A ogni iterazione, diversi neuroni sono attivi +- Durante l'inferenza, tutti i neuroni sono attivi (con scaling) + +/* +```python +class ModelWithDropout(nn.Module): + def __init__(self): + super().__init__() + self.fc1 = nn.Linear(784, 256) + self.dropout1 = nn.Dropout(p=0.5) # 50% dropout + self.fc2 = nn.Linear(256, 128) + self.dropout2 = nn.Dropout(p=0.3) # 30% dropout + self.fc3 = nn.Linear(128, 10) + + def forward(self, x): + x = F.relu(self.fc1(x)) + x = self.dropout1(x) # Applicato solo in training + x = F.relu(self.fc2(x)) + x = self.dropout2(x) + return self.fc3(x) +``` +*/ + +#nota()[ + *Perché funziona?* + - Impedisce ai neuroni di co-adattarsi troppo + - Equivale a fare ensemble di tante reti diverse + - Riduce l'overfitting forzando ridondanza +] + +#attenzione()[ + Ricordarsi di mettere il modello in modalità corretta: + ```python + model.train() # Training: dropout attivo + model.eval() # Inferenza: dropout disattivo + ``` +] + +=== Batch Normalization + +Normalizza le attivazioni di ogni layer durante il training: $ - P(D | theta) = -1/N sum_(k=1)^N log p(y_k | x_k) = -1/N sum_(k=1)^N log(y). + hat(x) = (x - mu_cal(B))/sqrt(sigma_cal(B)^2 + epsilon) $ -i k valori che devo rappresentare li esprimo in one hote encoding (solo un 1 in posizione $j$). facendo $log overline(y)_(i,j)$ selezione solo quello di indice $j$. La maschera seleziona solo la classe $j$ e cstruisco una log-likehood che tiene conto del fatto che per l'esemplare $i$-esimo sto considerando la $j$-esima classe che mi da il valore.\ -$overline(y)$ è grande come $y_i$ e $overline(y)_i = (alpha, .. alpha_k)$ è la somma di k valori dove si spera che a spiccare sia quello di posto $j$ ovvero la classe corretta. + +dove $mu_cal(B)$ e $sigma_cal(B)$ sono media e varianza del batch. + +```python +class ModelWithBatchNorm(nn.Module): + def __init__(self): + super().__init__() + self.fc1 = nn.Linear(784, 256) + self.bn1 = nn.BatchNorm1d(256) + self.fc2 = nn.Linear(256, 128) + self.bn2 = nn.BatchNorm1d(128) + self.fc3 = nn.Linear(128, 10) + + def forward(self, x): + x = self.bn1(F.relu(self.fc1(x))) + x = self.bn2(F.relu(self.fc2(x))) + return self.fc3(x) +``` + +*Vantaggi*: +- Permette learning rate più alti +- Riduce la dipendenza dall'inizializzazione +- Ha effetto regolarizzante +- Accelera la convergenza + +#nota()[ + *Ordine tipico dei layer*: + ``` + Linear -> BatchNorm -> Activation (ReLU) -> Dropout + ``` + + (Anche se l'ordine BatchNorm/Activation è dibattuto) +] + +=== Early Stopping + +Interrompe il training quando la *validation loss* smette di migliorare: + +```python +best_val_loss = float('inf') +patience = 10 # Numero di epoche senza miglioramento +patience_counter = 0 + +for epoch in range(max_epochs): + train_loss = train_one_epoch(...) + val_loss = validate(...) + + if val_loss < best_val_loss: + best_val_loss = val_loss + patience_counter = 0 + # Salva il modello migliore + torch.save(model.state_dict(), 'best_model.pth') + else: + patience_counter += 1 + + if patience_counter >= patience: + print(f"Early stopping at epoch {epoch}") + break +``` + +#nota()[ + *Previene overfitting* interrompendo prima che il modello cominci a memorizzare i dati di training. +] + +=== Data Augmentation + +Per problemi di visione, aumenta artificialmente il dataset: +- Rotazioni, traslazioni, flip +- Crop casuali +- Variazioni di colore/contrasto +- Mixup, CutMix (tecniche più avanzate) + +```python +from torchvision import transforms + +train_transform = transforms.Compose([ + transforms.RandomHorizontalFlip(), + transforms.RandomRotation(10), + transforms.ColorJitter(brightness=0.2, contrast=0.2), + transforms.ToTensor(), + transforms.Normalize(mean=[0.485, 0.456, 0.406], + std=[0.229, 0.224, 0.225]) +]) +``` + +== Monitoraggio e Debugging + +=== Metriche di Valutazione + +Oltre alla loss, monitoriamo metriche interpretabili: + +*Classificazione*: +- *Accuracy*: percentuale di predizioni corrette +- *Precision, Recall, F1-score*: per classi sbilanciate +- *Confusion Matrix*: errori per classe + +*Regressione*: +- *MAE, RMSE*: errori medi +- *R²*: varianza spiegata + +```python +def compute_accuracy(outputs, labels): + _, predictions = torch.max(outputs, dim=1) + correct = (predictions == labels).sum().item() + return correct / labels.size(0) +``` + +=== Curve di Apprendimento + +Visualizzare training e validation loss/accuracy: + +#nota()[ + *Interpretazione delle curve*: + - *Underfitting*: train e val loss entrambe alte + - *Overfitting*: train loss bassa, val loss alta e crescente + - *Buon fit*: train e val loss entrambe basse e vicine +] + +=== Problemi Comuni + +*Loss non diminuisce*: +- Learning rate troppo basso o alto +- Inizializzazione sbagliata +- Bug nel codice (gradienti non calcolati) +- Normalizzazione dati mancante + +*Loss diventa NaN*: +- Learning rate troppo alto (gradienti esplodono) +- Divisione per zero o log(0) +- Overflow numerico + +*Overfitting*: +- Modello troppo complesso per i dati +- Dati di training insufficienti +- Mancano tecniche di regolarizzazione + +#attenzione()[ + *Debugging checklist*: + + Verifica shape dei tensori + + Controlla che i gradienti siano calcolati (`loss.backward()`) + + Verifica che l'ottimizzatore aggiorni i parametri + + Testa su un batch piccolo (overfit intenzionale) + + Visualizza le attivazioni e i gradienti +] From fec7183085c083198234578d706869b8714c3dff Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Mon, 9 Feb 2026 17:01:08 +0100 Subject: [PATCH 06/25] add first part --- chapters/Lezione8.typ | 68 +++++++++++++++++++++++++++++++------------ 1 file changed, 49 insertions(+), 19 deletions(-) diff --git a/chapters/Lezione8.typ b/chapters/Lezione8.typ index a73dacf..14d3f7e 100644 --- a/chapters/Lezione8.typ +++ b/chapters/Lezione8.typ @@ -14,6 +14,36 @@ $ $ Dove $mr(L)$ è una *funzione di loss*. +La predizione di una rete neurale ($x$) è quindi data da: +$ + x -> f(x, mr(W), mb(b))\ + Theta = underbrace(mr(W), "matrice"\ "pesi"),underbrace(mb(b), "bias") +$ + +La fase di *training* consiste nei seguenti passaggi (in loop): +- *forward pass*: vengono computate le predizioni, per un certo batch (sotto-insieme) di dati +- *loss*: le predizioni vengono comparate con la *ground truth* +- *backpropagation*: vengono computati i gradienti +- *update parameters*: vengono aggiornati il set di pesi $W$ del modello, in modo da minimizzare la loss. + +#informalmente()[ + Quello che accade è che dato un input $x$, esso subisce varie _trasformazioni_ ovvero passa per diversi *layer nascosti* del modello. Ogni layer nascosto è formato da una fila di neuoroni, ciascuno con la propria funzione di attivazione. La predizione finale si ottiene nel seguente modo: + $ + mg(o) = mr(tanh)(mb(W_(h))(dots mr(tanh)(mb(w_(2))(mr(tanh)(mb(w_1) mo(x) + mb(b_1))+mb(b_2)))dots +mb(b_n))) + $ + Dove: + - $mo(x)$ è l'input + - $mg(o)$ è la predizione del modello + - $mr(tanh)$ è la funzione di attivazione + - $mb(W_h)$ e $mb(b_h)$ sono i _pesi_ ai vari livelli che il modello apprende +] + + + + + + + == Funzioni di Attivazione I modelli di deep learning sono in grado di catturare informazioni *al di là dell'osservabile*, grazie all'uso delle funzioni di attivazione che introducono non-linearità nel modello. @@ -52,10 +82,10 @@ $ #figura( ```python import cetz.plot - + cetz.canvas({ import cetz.draw: * - + cetz.plot.plot( size: (10, 6), x-label: $x$, @@ -75,7 +105,7 @@ $ style: (stroke: (paint: mo, thickness: 2pt)), label: [ReLU] ) - + // Tanh (per confronto) cetz.plot.add( ((x,) => (calc.tanh(x),)), @@ -84,7 +114,7 @@ $ style: (stroke: (paint: mb, thickness: 1.5pt, dash: "dashed")), label: [Tanh] ) - + // Sigmoid (per confronto) cetz.plot.add( ((x,) => (1 / (1 + calc.exp(-x)),)), @@ -183,7 +213,7 @@ class ModelNN(nn.Module): #figura( ```python import fletcher as fl - + fl.diagram( node-stroke: 1pt, spacing: (20mm, 8mm), @@ -200,23 +230,23 @@ class ModelNN(nn.Module): (pos: (0, 6), label: [Softmax], fill: mg), (pos: (0, 7), label: [Output\n$hat(mb(y))$], fill: mb), ) - + for (i, layer) in layers.enumerate() { fl.node( - layer.pos, - layer.label, + layer.pos, + layer.label, shape: rect, corner-radius: 5pt, fill: gradient.linear(white, layer.fill, angle: 45deg), width: 30mm, height: 10mm ) - + if i < layers.len() - 1 { fl.edge(layer.pos, layers.at(i + 1).pos, "->", stroke: 2pt) } } - + // Annotazioni fl.node((1.5, 1.5), [Trasformazione\nlineare], stroke: none, fill: none) fl.node((1.5, 3.5), [Non-linearità], stroke: none, fill: none) @@ -644,7 +674,7 @@ Il processo di training di una rete neurale segue questi passi: #figura( ```python import fletcher as fl - + fl.diagram( node-stroke: 1pt, node-fill: gradient.linear(..mo.colors), @@ -652,13 +682,13 @@ Il processo di training di una rete neurale segue questi passi: edge-stroke: 1pt, { let (input, forward, loss, backward, update, output) = ((0,0), (1,0), (2,0), (2,1), (1,1), (0,1)) - + fl.node(input, [Input \ $mb(x), mb(y)$], corner-radius: 5pt, extrude: (0, 3)) fl.node(forward, [Forward \ Pass], corner-radius: 5pt) fl.node(loss, [Loss \ $L$], corner-radius: 5pt, fill: mr) fl.node(backward, [Backward \ Pass], corner-radius: 5pt) fl.node(update, [Update \ $theta$], corner-radius: 5pt, fill: mg) - + fl.edge(input, forward, "->", [Batch]) fl.edge(forward, loss, "->", [$hat(mb(y))$]) fl.edge(loss, backward, "->", [Gradiente]) @@ -837,7 +867,7 @@ class ModelWithDropout(nn.Module): self.fc2 = nn.Linear(256, 128) self.dropout2 = nn.Dropout(p=0.3) # 30% dropout self.fc3 = nn.Linear(128, 10) - + def forward(self, x): x = F.relu(self.fc1(x)) x = self.dropout1(x) # Applicato solo in training @@ -880,7 +910,7 @@ class ModelWithBatchNorm(nn.Module): self.fc2 = nn.Linear(256, 128) self.bn2 = nn.BatchNorm1d(128) self.fc3 = nn.Linear(128, 10) - + def forward(self, x): x = self.bn1(F.relu(self.fc1(x))) x = self.bn2(F.relu(self.fc2(x))) @@ -898,7 +928,7 @@ class ModelWithBatchNorm(nn.Module): ``` Linear -> BatchNorm -> Activation (ReLU) -> Dropout ``` - + (Anche se l'ordine BatchNorm/Activation è dibattuto) ] @@ -914,7 +944,7 @@ patience_counter = 0 for epoch in range(max_epochs): train_loss = train_one_epoch(...) val_loss = validate(...) - + if val_loss < best_val_loss: best_val_loss = val_loss patience_counter = 0 @@ -922,7 +952,7 @@ for epoch in range(max_epochs): torch.save(model.state_dict(), 'best_model.pth') else: patience_counter += 1 - + if patience_counter >= patience: print(f"Early stopping at epoch {epoch}") break @@ -948,7 +978,7 @@ train_transform = transforms.Compose([ transforms.RandomRotation(10), transforms.ColorJitter(brightness=0.2, contrast=0.2), transforms.ToTensor(), - transforms.Normalize(mean=[0.485, 0.456, 0.406], + transforms.Normalize(mean=[0.485, 0.456, 0.406], std=[0.229, 0.224, 0.225]) ]) ``` From 34ec235ddc7e0388240d5e45d0b0f2fcb992c9b8 Mon Sep 17 00:00:00 2001 From: LucaCorra02 Date: Mon, 9 Feb 2026 18:41:16 +0100 Subject: [PATCH 07/25] update activation func --- assets/linear-classifier.png | Bin 0 -> 1904604 bytes assets/mlp.png | Bin 0 -> 3121253 bytes chapters/Lezione8.typ | 122 +++++++++++++++++------------------ 3 files changed, 60 insertions(+), 62 deletions(-) create mode 100644 assets/linear-classifier.png create mode 100644 assets/mlp.png diff --git a/assets/linear-classifier.png b/assets/linear-classifier.png new file mode 100644 index 0000000000000000000000000000000000000000..255824dfc29905a689c45714810d535412040c0d GIT binary patch literal 1904604 zcmd?RWl)??_b7-3hu{nrEI0!U?#`gWb%GD>o*==U!3hC^yE}xS34y^axCVDiumAz> zrEejf8~5iiGqm9}^uh z@*FKv4DmqolvUQnM0^4;t)h{T=#Z4$GOY?9XG?5tC@Q(KQ!?MK|x{x5i2^UT$Z7hAE%dRX?9_h)rH!83* 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