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is learning what to do, it presents two main characteristics: - take actions that affects the state Difference from other ML -- no supervisor +- **no supervisor** - feedback may be delayed - time matters - agent action affects future decisions @@ -44,13 +44,6 @@ Learning online - we expect agents to get things wrong, to refine their understanding as they go - the world is not static, agents continuously encounter new situations -RL applications: -- self driving cars -- engineering -- healthcare -- news recommendation -- ... - Rewards - a reward is a scalar feedback signal (a number) - reward Rt indicates how well the agent is doing at step t @@ -63,12 +56,12 @@ communication in battery free environments - positive rewards if the queried device has new data - else negative -Challenge: +#### Challenges: - tradeoff between exploration and exploitation - to obtain a lot of reward a RL agent must prefer action that it tried in the past - but better actions may exist... So the agent has to exploit! -exploration vs exploitation dilemma: +##### exploration vs exploitation dilemma: - comes from incomplete information: we need to gather enough information to make best overall decisions while keeping the risk under control - exploitation: we take advanced of the best option we know - exploration: test new decisions @@ -108,6 +101,7 @@ one or more of these components - used to evaluate the goodness/badness of states - values are prediction of rewards - $V_\pi(s) = Ep[yRt+1 + y^2Rt+2 ... | St = s]$ + - better explained later - **Model:** - predicts what the environment will do next - may predict the resultant next state and/or the next reward @@ -124,103 +118,4 @@ back to the original problem: - positive when querying a device with new data - negative if it has no data - what to do if the device has lost data? - - state? - - -### Exploration vs exploitation trade-off -- Rewards evaluate actions taken -- evaluative feedback depends on the action taken -- no active exploration - -Let's consider a simplified version of an RL problem: K-armed bandit problem. -- K different options -- every time need to chose one -- maximize expected total reward over some time period -- analogy with slot machines - - the levers are the actions - - which level gives the highest reward? -- Formalization - - set of actions A (or "arms") - - reward function R that follows an unknown probability distributions - - only one state - - ... - -Example: doctor treatment -- doctor has 3 treatments (actions), each of them has a reward. -- for the doctor to decide which action to take is best, we must define the value of taking each action -- we call these values the action values (or action value function) -- action value: ... - -Each action has a reward defined by a probability distribution. -- the red treatment has a bernoulli probability -- the yellow treatment binomial -- the blue uniform -- the agent does not know the distributions! -- the estimated action for action a is the sum of rewards observed divided by the total time the action has been taken (add formula ...) - - 1predicate denotes the random variable (1 if true else 0) - -- greedy action: - - doctors assign the treatment they currently think is the best - - ... - - the greedy action is computed as the argmax of Q values - - greedy always exploits current knowledge -- epsilon-greedy: - - with a probability epsilon sometimes we explore - - 1-eps probability: we chose best greedy action - - eps probability: we chose random action - -exercises ... - -exercise 2: k-armed bandit problem. -K = 4 actions, denoted 1,2,3 and 4 -eps-greedy selection -initial Q estimantes = 0 for all a. - -Initial sequenze of actions and rewards is: -A1 = 1 R1 = 1 -A2 = 2 R2 = 2 -A3 = 2 R3 = 2 -A4 = 2 R4 = 2 -A5 = 3 R5 = 0 - ---- -step A1: action 1 selected. Q of action 1 is 1 -step A2: action 2 selected. Q(1) = 1, Q(2) = 1 -step A3: action 2 selected. Q(1) = 2, Q(2) = 1.5 -step A4: action 2. Q(1) = 1, Q(2) = 1.6 -step A5: action 3. Q(1) = 1, Q(2) = 1.6, Q(3) = 0 - -For sure A2 and A5 are epsilon cases, system didn't chose the one with highest Q value. -A3 and A4 can be both greedy and epsilon case. - -#### Incremental formula to estimate action-value -- to simplify notation we concentrate on a single action -- Ri denotes the reward received after the i(th) selection of this action. Qn denotes the estimate of its action value after it has been selected n-1 times (add Qn formula ...) -- given Qn and the reward Rn, the new average of rewards can be computed by (add formula with simplifications...) $Q_(n+1) = Q_{n} + \frac{1}{n}[Rn - Qn]$ - - NewEstimate <- OldEstimate + StepSize (Target - OldEstimate) - - Target - OldEstimate is the error - -Pseudocode for bandit algorithm: -``` -Initialize for a = 1 to k: - Q(a) = 0 - N(a) = 0 -Loop forever: - with probability 1-eps: - A = argmax_a(Q(a)) - else: - A = random action - R = bandit(A) # returns the reward of the action A - N(A) = N(A) + 1 - Q(A) = Q(A) + 1\N(A) * (R - Q(A)) -``` - -Nonstationary problem: rewards probabilities change over time. -- in the doctor example, a treatment may not be good in all conditions -- the agent (doctor) is unaware of the changes, he would like to adapt to it - -An option is to use a fixed step size. We remove the 1/n factor and add an $\alpha$ constant factor between 0 and 1. -And we get $Q_{n+1} = (1-\alpha)^{n}Q_1 + \sum_{i=1}^{n}{\alpha(1 - \alpha)^{(n-1)} R_i}$ - - -... ADD MISSING PART ... \ No newline at end of file + - state? \ No newline at end of file diff --git a/Autonomous Networking/notes/7.1 K-Armed bandit problem.md b/Autonomous Networking/notes/7.1 K-Armed bandit problem.md new file mode 100644 index 0000000..8a9b92d --- /dev/null +++ b/Autonomous Networking/notes/7.1 K-Armed bandit problem.md @@ -0,0 +1,122 @@ +### K-Armed bandit problem +- Rewards evaluate actions taken +- evaluative feedback depends on the action taken +- no active exploration + +Let's consider a simplified version of an RL problem: K-armed bandit problem. +- K different options +- every time need to chose one +- maximize expected total reward over some time period +- analogy with slot machines + - the levers are the actions + - which lever gives the highest reward? +- **Formalization** + - set of actions A (or "arms") + - reward function R that follows an unknown probability distributions + - only one state + - at each step t, agent selects an action A + - environment generates reward + - goal to maximize cumulative reward + +Example: doctor treatment +- doctor has 3 treatments (actions), each of them has a reward. +- for the doctor to decide which action to take is best, we must define the value of taking each action +- we call these values the action values (or action value function) +- **action value:** $$q_{*}=E[R_{t} \mid A_{t}=a]$$ +Each action has a reward defined by a probability distribution. +- the red treatment has a Bernoulli probability +- the yellow treatment binomial +- the blue uniform +- the agent does not know the distributions! +![[Pasted image 20241030165705.png]] + +- **the estimated action value Q** for action a is the sum of rewards observed divided by the total time the action has been taken + +- **greedy action:** + - doctors assign the treatment they currently think is the best + - greedy action is the action that currently has the largest estimated action value $$A_{t}=argmax(Q_{t}(a))$$ + - greedy always exploits current knowledge +- **epsilon-greedy:** + - with a probability epsilon sometimes we explore + - 1-eps probability: we chose best greedy action + - eps probability: we chose random action + +#### Exercise 1 +In ε-greedy action selection, for the case of two actions and ε=0.5, what is the probability that the greedy action is selected? +*We have two actions. The probability of selecting the greedy action is 50%. +But when the exploration happens, the greedy actions may be selected! +So, with 0.5 prob. we select greedy action. With 0.5 prob. we select random action, which can be both. So in the random case, we select the greedy action with 0.5 * 0.5 probability = 0.25. +Finally, we select the greedy action with 0.5 + 0.25 = 0.75 probability. + +#### Exercise 2 +Consider K-armed bandit problem. +K = 4 actions, denoted 1,2,3 and 4 +Agent uses eps-greedy action selection +initial Q estimantes is 0 for all actions: $$Q_{1}(a)=0$$ +Initial sequenze of actions and rewards is: +A1 = 1 R1 = 1 +A2 = 2 R2 = 2 +A3 = 2 R3 = 2 +A4 = 2 R4 = 2 +A5 = 3 R5 = 0 + +On some of those time steps, the epsilon case may have occurred, causing an action to be selected at random. On which time steps did this definitely occur? +On which time steps could this possibly have occurred? + +***Answer*** +to answer, we need to compute the action sequence + +in the table, Qa means Q value of the action a at current state! + +| steps | Q1 | Q2 | Q3 | Q4 | +| -------------- | --- | ---- | --- | --- | +| A1 \| action 1 | 1 | 0 | 0 | 0 | +| A2 \| action 2 | 1 | 1 | 0 | 0 | +| A3 \| action 2 | 1 | 1.5 | 0 | 0 | +| A4 \| action 2 | 1 | 1.66 | 0 | 0 | +| A5 \| action 3 | 1 | 1.66 | 0 | 0 | + + + +step A1: action 1 selected. Q of action 1 is 1 +step A2: action 2 selected. Q(1) = 1, Q(2) = 1 +step A3: action 2 selected. Q(1) = 2, Q(2) = 1.5 +step A4: action 2. Q(1) = 1, Q(2) = 1.6 +step A5: action 3. Q(1) = 1, Q(2) = 1.6, Q(3) = 0 + +For sure A2 and A5 are epsilon cases, system didn't chose the one with highest Q value. +A3 and A4 can be both greedy and epsilon case. + +#### Incremental formula to estimate action-value +- idea: compute incrementally the action values, to avoid doing it every time +- to simplify notation we concentrate on a single action on the next examples + - $R_{i}$ denotes the reward received after the i(th) selection of this action. + - $Q_{n}$ denotes the estimate of its action value after it has been selected n-1 times $$Q_{n}=\frac{R_{1}+R_{2}+\dots+R_{n-1}}{n-1}$$ +- given $Q_{n}$ and the reward Rn, the new average of rewards can be computed by $$Q_{n+1}=\frac{1}{n}\sum_{i=1}^nR_{i}$$ +General formula: NewEstimate <- OldEstimate + StepSize (Target - OldEstimate) $$Q_(n+1) = Q_{n} + \frac{1}{n}[Rn - Qn]$$Target - OldEstimate is the error + +Pseudocode for bandit algorithm: +``` +Initialize for a = 1 to k: + Q(a) = 0 + N(a) = 0 +Loop forever: + with probability 1-eps: + A = argmax_a(Q(a)) + else: + A = random action + R = bandit(A) # returns the reward of the action A + N(A) = N(A) + 1 + Q(A) = Q(A) + 1\N(A) * (R - Q(A)) +``` + +#### Nonstationary problem: +Rewards probabilities change over time. +- in the doctor example, a treatment may not be good in all conditions +- the agent (doctor) is unaware of the changes, he would like to adapt to it, maybe a treatment is good only on a specific season. + +An option is to use a fixed step size. We remove the 1/n factor and add an $\alpha$ constant factor between 0 and 1. +And we get $$Q_{n+1} = (1-\alpha)^{n}Q_1 + \sum_{i=1}^{n}{\alpha(1 - \alpha)^{(n-1)} R_i}$$ +#### Optimistic initial values +Initial action values can be used as a simple way to encourage exploration! +This way we can make the agent explore more at the beginning, and explore less after a while, this is cool! \ No newline at end of file diff --git a/Autonomous Networking/notes/8.md b/Autonomous Networking/notes/7.2 10 arm testbed - optimism in face of uncertainty.md similarity index 99% rename from Autonomous Networking/notes/8.md rename to Autonomous Networking/notes/7.2 10 arm testbed - optimism in face of uncertainty.md index 499ccc6..84be7ac 100644 --- a/Autonomous Networking/notes/8.md +++ b/Autonomous Networking/notes/7.2 10 arm testbed - optimism in face of uncertainty.md @@ -15,7 +15,7 @@ ![[Pasted image 20241025084755.png]] -.. add siled ... + ![[Pasted image 20241025084830.png]] #### Experiments diff --git a/Autonomous Networking/notes/9 Markov processes.md b/Autonomous Networking/notes/9 Markov processes.md new file mode 100644 index 0000000..40a263e --- /dev/null +++ b/Autonomous Networking/notes/9 Markov processes.md @@ -0,0 +1,132 @@ +MDPs are a classical formalization of sequential decision making, where actions influence not just immediate rewards, but also subsequent situations (states) and through those future rewards + +MDPs involve delayed rewards and the need to trade off immediate and delayed rewards + +Whereas in bandit we estimated the q*(a) of each action a, in MDPs we estimate the value q*(a,s) of each action a in each state s, or we estimate the value v*(s) of each state s given optimal action selection + + +- MDPs are meant to be a straightforward framing of the problem of learning from interaction to achieve a goal. +- The agent and environment interact at each of a sequence of discrete time steps, t = 0, 1, 2, 3, . . . +- At each timestep, the agent receives some representation of the environment state and on that basis selects an action +- One time step later, in part as a consequence of its action, the agent receives a numerical reward and finds itself in a new state + +- Markov decision processes formally describe an environment for reinforcement learning +- Where the environment is fully observable +- i.e. The current state completely characterises the process +- Almost all RL problems can be formalised as MDPs + - e.g. Bandits are MDPs with one state + +Markov property +“The future is independent of the past given the present” +![[Pasted image 20241030102226.png]]![[Pasted image 20241030102243.png]] + +- A Markov process (or markov chain) is a memoryless random process, i.e. a sequence of random states S1, S2, ... with the Markov property. + - S finite set of states + - P is a state transition probability matrix + - then $Pss′ = ℙ [St+1=s′ | St=s]$ + +Example +![[Pasted image 20241030102420.png]] +![[Pasted image 20241030102722.png]] + +#### Markov Reward Process +This is a Markov Process but we also have a reward function! We also have a discount factor. + + Markov Reward Process is a tuple ⟨S, P, R, γ⟩ + - S is a (finite) set of states + - P is a state transition probability matrix, $Pss′ = ℙ [ St+1=s′ | St=s ]$ + - R is a reward function, $Rs = 𝔼 [ R_{t+1} | St = s ]$ + - γ is a discount factor, $γ ∈ [0, 1]$ + +![[Pasted image 20241030103041.png]] + + ![[Pasted image 20241030103114.png]] +- The discount $γ ∈ [0, 1]$ is the present value of future rewards +- The value of receiving reward R after k + 1 time-steps is $γ^kR$ +- This values immediate reward above delayed reward + - γ close to 0 leads to ”short-sighted” evaluation + - γ close to 1 leads to ”far-sighted” evaluation + + Most Markov reward and decision processes are discounted. Why? + - mathematical convenience + - avoids infinite returns in cyclic Markov processes + - uncertainity about the future may not be fully represented + - if the reward is financial, immediate rewards may earn more interest than delayed rewards + - Animal/human behaviour shows preference for immediate rewards + - It is sometimes possible to use undiscounted Markov reward processess (gamma = 1) e.g. if all sequences terminate + +Value function +- The value function v(s) gives the long-term value of (being in) state s +- The state value function v(s) of an MRP is the expected return starting from state s $𝑉) = 𝔼 [𝐺𝑡 |𝑆𝑡 = 𝑠]$ + +![[Pasted image 20241030103519.png]] +![[Pasted image 20241030103706.png]] +is a prediction of the reward in next states + +![[Pasted image 20241030103739.png]] +![[Pasted image 20241030103753.png]] + +- The value function can be decomposed into two parts: + - immediate reward $R_{t+1}$ + - discounted value of successor state $γv(St+1)$ + +![[Pasted image 20241030103902.png]] + +#### Bellman Equation for MRPs +$v (s) = E [Rt+1 + v (St+1) | St = s]$ +![[Pasted image 20241030104056.png]] + +- Bellman equation averages over all the possibilities, weighting each by its probability of occurring +- The value of the start state must be equal the (discounted) value of the expected next state, plus the reward expected along the way + +![[Pasted image 20241030104229.png]] +4.3 is the reward I get exiting from the state (-2) plus the discount times the value of the next state + ecc. + + +![[Pasted image 20241030104451.png]] + +#### Solving the Bellman Equation +- The Bellman equation is a linear equation +- can be solved directly![[Pasted image 20241030104537.png]] +- complexity O(n^3) +- many iterative methods + - dynamic programming + - monte-carlo evaluation + - temporal-difference learning + +#### MDP +![[Pasted image 20241030104632.png]] + +![[Pasted image 20241030104722.png]] + +Before we had random probabilities, now we have actions to chose from. But how do we chose? +We have policies: a distribution over actions given the states: $$𝜋(a|s)= ℙ [ At=a | St=s ]$$ +- policy fully defines the behavior of the agent +- MDP policies depend on the current state (not the history) +- policies are stationary (time-independent, depend only on the state but not on the time) + + +##### Value function +The state-value function v𝜋(s) of an MDP is the expected return starting from state s, and then following policy 𝜋 $$v𝜋(s) = 𝔼𝜋 [ Gt | St=s ]$$ +The action-value function q 𝜋 (s,a) is the expected return starting from state s, taking action a, and then following policy 𝜋 $$q 𝜋(a|s)= 𝔼𝜋 [ Gt | St=s, At=a ]$$ +![[Pasted image 20241030105022.png]] + +- The state-value function can again be decomposed into immediate reward plus discounted value of successor state $$v\pi(s) = E\pi[Rt+1 + v⇡(St+1) | St = s]$$ +- The action-value function can similarly be decomposed $$q\pi(s, a) = E\pi [Rt+1 + q⇡(St+1, At+1) | St = s, At = a]$$ +![[Pasted image 20241030105148.png]]![[Pasted image 20241030105207.png]] +![[Pasted image 20241030105216.png]] +putting all together +(very important, remeber it) + +![[Pasted image 20241030105234.png]] + +![[Pasted image 20241030105248.png]] +as we can see, an action does not necessarily bring to a specific state. + + +Example: Gridworld +2 azioni che rimandano allo stesso stato, due azioni che vanno in uno stato diverso +![[Pasted image 20241030112555.png]] +![[Pasted image 20241030113041.png]] +![[Pasted image 20241030113135.png]] +C is low because from C I get reward of 0 everywhere I go. \ No newline at end of file diff --git a/Autonomous Networking/slides/9markovprocess.pdf b/Autonomous Networking/slides/9markovprocess.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b99989c00800d0064bba2fc56810761563a197a1 GIT binary patch literal 5952081 zcmeFZcUV(hw=Wt5q>J=UP(YMkq+=o=AR;2YgMjoVy#*qm^p1dn5TvR!=_n=iB0{A1 z4$>h(2!Vuf;`@EyK6~GNpXc1Y&;8?`=e`qW)?}@@#v0?d#$0Q(!2k4tx`>3R6eWN6 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zDWkVeBte7^#Q{LdlfM^*U*kgMlKkJ?bLaf4x% z*AG7zb)azpbGoa@FSXC0eB0&!fuADC>e zkyHE}47zTwp~(5Sm?G!8Kin?X`AuCzjVAAwHdD{F}mKtt&7#=I$eyY+KqA z3#_)dOl})fPh-W2=em3p6M9@@&wa=!ZM_RSOMW+PF<&%(x%X1>t+@+R`fo7rU)uV2 z#E6q;r|);kCCQ|AAN(~)i?+9ngsK2RirHnM&A|(Ond?m+)uh#dt$c2uX<6_+kJH-Otf6Xa7L5WM0$ zkt4j=a+GkXEhd9<#>fmv(PPQ^E*0ds?8@H@TJwQsSImYV>qr5E`d~6`x5qTIoyZTW zkbtGOTx)L~i=6XHu?I;?@SO8|z4m|kMF!U+lB57x;5Ec`Xlw^< z)Mi9o=M{`-*8bky1nl)kq+{LX^X2o0Uw-}g`Q7*5KfV9!FaP=YM0xA!>HUwt{r2T) z2IlF [!PDF|yellow] [[LEZIONE6_Face recognition2D.pdf#page=7&selection=6,0,6,17&color=yellow|LEZIONE6_Face recognition2D, p.7]] +> > Some consequences +> +> leggi le conseguenze + +Un'alternativa è rappresentare un'immagine nel feature space: +- Gabor filters +- discrete cosine transform (DCT) +- local binary pattern (LBP) operator +- fractal encoding + +#### PCA +possibile soluzione al curse of dimensionality: PCA. +Metodo statistico che consente di ridurre l'alta dimensionalità di uno spazio mappando i dati in un altro spazio con dimensionalità decisamente più piccola minimizzando la perdita di informazioni. +Vengono individuati dei nuovi assi, ortogonali tra loro, su cui proiettare i dati che però ne **massimizzano la varianza**. L'ortogonalità di questi ci permette di escludere componenti correlate tra loro che risultano quindi ridondanti. +Nello spazio delle immagini, i principali componenti sono ortogonali quando sono gli *autovettori della matrice di covarianza*. + +L'obiettivo è di eliminare features con varianza molto bassa, che quindi sono comuni tra tutti i sample e poco utili a discriminare classi. + +Calcoliamo l'optimal k +Dato un training set TS di m sample di n dimensioni, calcoliamo +- il vettore medio $\hat{x}$ del training set $$\hat{x}=\frac{1}{m}\sum_{i=1}^{m}x_{i}$$ +- matrice di covarianza C $$C=\frac{1}{m}\sum_{i=1}^m(x_{i}-\hat{x})(x_{i}-\hat{x})^T$$ +- la dimensione di C è (n x n) +- il nuovo spazio k-dimensionale è dato dalla matrice di proiezione dove le colonne sono i k autovettori di C, corrispondenti ai k autovalori di C. Plottando gli autovalori possiamo ottenere la varianza lungo gli autovettori. + +> [!PDF|red] [[LEZIONE6_Face recognition2D.pdf#page=14&selection=0,0,2,10&color=red|LEZIONE6_Face recognition2D, p.14]] +> > PCA and Eigenfaces +> +> esempio + +![[Pasted image 20241030142613.png]] +da questo otteniamo l'elemento medio: +![[Pasted image 20241030142635.png]] + +problema: bisogna avere tutte le facce ben centrate!! + +![[Pasted image 20241030142717.png]] +autovettori (completamente diverse da quelle di partenza se ci facciamo caso) +dobbiamo estrarre quelle + +Proiezione: eseguita moltiplicando la matrice di proiezione trasposta $𝜑_{k}^T$ per il vettore originale. +![[Pasted image 20241030142934.png]] +![[Pasted image 20241030143149.png]] + +##### Problemi di PCA +- mancanza di potere discriminativo + - la separazione degli autovettori dipende anche dalle differenze intra-classe + - ma il sistema è unsupervised learning +- in in presenza di variazioni PIE, il modello potrebbe usare quelle come componenti facciali e non essere quindi in grado di separare correttamente le classi + +Possibile soluzione: Fisherfaces, un'applicazione di FLD (Fisher's linear discriminant), spesso menzionato nel contesto del Linear Discriminant Analysis (LDA). + +### LDA +Alcuni dei problemi di PCA possono essere dati dal fatto che lavoriamo su dati in una maniera unsupervised (non supervisionata), ovvero senza tenere in considerazione le classi dei vari sample. +Una soluzione supervised è proposta da LDA, ovvero un metodo simile a PCA ma che minimizza la distanza intra-classe e cerca di massimizzare invece la distanza tra classi. + +Nell'immagine la riduzione è da 2 a 1 dimensioni. In alcuni casi tipo questo non basta ridurre da due a un asse, gli assi sono ruotati per massimizzare la separazione tra classi. L'importante è che le coordinate rimangano ortogonali tra loro +![[Pasted image 20241030144246.png]] +the best subspace is better cause we are able to separate class among the 1D axis. As we can see, in the worst one, we are definitely not able to. + +##### Possibile approccio +- partiamo, come nel PCA, da un set di m sample di n dimensioni. +- a differenza del PCA, splittiamo il set in base ai label che rappresentano le classi $PTS=\{P_{1},P_{2},{_{3},\dots,P_{s}}\}$, dove $P_{i}$ è una classe di cardinalità $m_{i}$ +- cerchiamo di ottenere uno scalare $y$ proiettando il sample $x$ su una linea, in modo che $y=w^Tx$ + - chiaramente x è un punto nell'immagine 2D + - y è il punto proiettato invece nel nuovo spazio 1D + +Consideriamo due classi $P_1$ e $P_{2}$, rispettivamente con $m_{1}$ e $m_{2}$ vettori ciascuna +- consideriamo i vettori medi di ogni classe nei due spazi: $$\mu_{i}=\frac{1}{m_{i}}\sum_{j=1}^{m_{i}}x_{j}$$ +- nel nuovo spazio si avrà (la tilde indica che è nel nuovo spazio) $$\tilde{}\mu_{i}=\frac{1}{m_{i}}\sum_{j=1}^{m_{i}}y_{j}=\frac{1}{m_{i}}\sum_{j=1}^{m_{i}}w^Tx_{j}=w^T\mu_{i}$$ +- possiamo scegliere la distanza dai vettori medi proiettati (quindi nel nuovo spazio) come una funzione che chiaramente vogliamo cercare di massimizzare $$J(w)=|\tilde{\mu_{1}}-\tilde{\mu_{2}}|=|w^T(\mu_{1}-\mu_{2})|$$ +![[Pasted image 20241031085606.png]] +in questo caso però notiamo come in un asse abbiamo un'ottima separabilità tra classi, nell'altro abbiamo una distanza maggiore tra i due vettori medi. +E mo come cazzo famo? Proviamo a cercare un metodo migliore + +##### Scattering matrices +Dobbiamo prendere in considerazione lo spread tra le classi. +Dobbiamo massimizzare il rapporto tra la varianza tra classi e la varianza della singola classe +- **varianza della singola classe** + - indica come i vettori sono distribuiti/dispersi rispetto al centro della stessa classe +- **varianza tra classi** + - indica come i centri sono distribuiti rispetto al centro di tutto (ok si potrebbe spiegare meglio ma so le 9:00 + +##### Approccio +- partiamo, come nel PCA, da un set di m sample di n dimensioni. +- a differenza del PCA, splittiamo il set in base ai label che rappresentano le classi $PTS=\{P_{1},P_{2},{_{3},\dots,P_{s}}\}$, dove $P_{i}$ è una classe di cardinalità $m_{i}$ +- per ogni classe $P_{i}$ calcoliamo il vettore medio e la "media dei medi"$$\mu_{i}=\frac{1}{m_{i}}\sum_{j=1}^{m_{i}}x_{j}$$$$\mu_{TS}=\frac{1}{m}\sum_{i=1}^{S}m_{i}\mu_{i}$$ +- calcoliamo la matrice di covarianza per la classe ${P_{i}}$ $$C_{i}=\frac{1}{m_{i}}\sum_{j=1}^{m_{i}}(x_{j}-\mu_{i})(x_{j}-\mu_{i})^T$$ +- ora possiamo calcolare scatter matrix whithin-class (della stessa classe) $$S_{W}=\sum_{i=1}^Sm_{i}C_{i}$$ +- scatter matrix tra classi $$S_{B}=\sum_{i=1}^Sm_{i}(\mu_{i}-\mu_{TS})(\mu_{i}-\mu_{TS})^T$$per due classi è anche definita come $$(\mu_{1}-\mu_{2})(\mu_{1}-\mu_{2})^T$$ +- le stesse definizioni sono valide anche nel nuovo spazio con $\tilde{S_{W}}$ e $\tilde{S_{B}}$ con $y=w^Tx$ e la dispersione (scatter) della classe proiettata $P_{i}$ è definito come $$\tilde{s_{i}}^2=\sum_{y \in P_{i}}(y-\tilde{\mu_{i}})^2$$ la somma di tutti gli $\tilde{s_{i}}^2$ (di tutte le classi) è la within-class scatter (dispersione tra classi) + +- adesso vogliamo massimizzare la differenza tra i vettori medi, normalizzati secondo una misura della dispersione della singola classe (within-class scatter) +- con due classi, è definita come la funzione lineare $w^Tx$ che massimizza la funzione $$J(w)=\frac{{|\tilde{\mu_{1}}-\tilde{\mu_{2}}|^2}}{\tilde{s_{1}}²+\tilde{s_{2}}^2}$$ +- cerchiamo una proiezione dove sample della stessa classe sono proiettati molto vicini tra loro, ma con i valori medi delle classi il più possibile lontani tra loro![[Pasted image 20241031091853.png]] + +### Feature spaces +- possiamo estrarre feature applicando filtri o trasformazioni alle immagini +- possiamo usare uno dei metodi appena descritti se le feature estratte hanno una dimensionalità alta +Quali sono le migliori feature da estrarre? Vediamo alcuni operatori + +##### Bubbles +- un osservatore umano può subito ipotizzare il genere, l'identità, l'età e l'espressione guardando una faccia +- l'esperimento consiste nel mostrare l'immagine con sopra un pattern a bolle e vedere se l'osservatore umano e un sistema biometrico riescono a classificarlo bene![[Pasted image 20241031100207.png]] +- alla fine dell'esperimento riusciamo a capire quali sono gli elementi (le features quindi) della faccia che vengono prese in considerazione da un essere umano o che un sistema biometrico utilizza per classificare + +#### Wavelet +provides a time-frequency representation of the signal. Fornisce tempo e frequenza contemporaneamente. +![[Pasted image 20241031102321.png]] + +#### Gabor filters +È un filtro lineare utilizzato in applicazioni di edge detection, analisi di texture e estrazione di feature, e si pensa che sia simile al sistema percettivo visivo di noi umani. Un Gabor filter 2D non è altro che un kernel Gaussiano modulato da un’onda piana sinusoidale. +> [!PDF|yellow] [[LEZIONE6_Face recognition2D.pdf#page=43&selection=0,2,11,7&color=yellow|LEZIONE6_Face recognition2D, p.43]] +> > An example of filters: Gabor filters +> +> guarda dalle slide + +![[Pasted image 20241031102640.png]] + +Un feature vector viene ricavato eseguendo varie convoluzioni con un insieme Gabor filter (Gabor filter bank) con diversi orientamenti e scale (nell'immagine sull'asse x troviamo gli orientamenti e su y le scale) + +Otteniamo quindi diversi filtri andando a cambiare i parametri: +- λ: frequenza, controlla lo spessore della striscia; più è grande più sarà spessa +- θ: orientamento, controlla la rotazione della striscia ed è espresso come un angolo +- γ: aspect ratio, controlla l’altezza della striscia; più è grande più l’altezza diminuirà +- σ: larghezza di banda, controlla la scala generale determinando anche il numero di strisce; aumentando questo valore avremo più strisce con meno distanza tra loro. + +Questa procedura, su immagini molto grandi, porta ad un’alta dimension- +alità; possibili soluzioni sono: +- Applicare il filtro su un subset di pixel, come ad esempio una griglia =⇒ dimensionalità ridotta, ma possibile perdità di informazioni salienti (dovuta a possibili rotazioni ad esempio) +- Applico il filtro su tutti i pixel, ma poi seleziono solo i punti con picchi di valori =⇒ problemi di allineamento + +###### Ma come facciamo la convoluzione? +- Griglia fissa + - possiamo perdere punti importanti +- Griglia definita a mano in modo che i vertici sono vicini alle feature importanti, come gli occhi + + +#### EBGM Elastic Bunch Graph Matching +Questo metodo fa uso di grafi, nello specifico abbiamo per ogni soggetto una collezione di grafi (per questo il nome “bunch”), uno per ogni posa, dove: +- i vertici (edges) sono labellati (pesati) con la distanza tra i nodi +- i nodi contengono un insieme di risultati di diversi gabor filter (in genere 5 differenti frequenze e 8 diversi orientamenti), conservati in questa struttura chiamata “Jet”, e sono posizionati in punti importanti come ad esempio naso, occhi e bocca +![[Pasted image 20241031104206.png]] +![[Pasted image 20241031104526.png]] + +oggi non lo usa nessuno. diff --git a/Biometric Systems/slides/Biometric_System___Notes.pdf b/Biometric Systems/slides/Biometric_System___Notes.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4c68fc13b1364c2e4fdac157bafd90c89a96d2ad GIT binary patch literal 882682 zcmeFYWpEummo03FnVIclrkI(TnVFfHnVFd#bBr;?%*@Qp5Hs5`T<3iZ|#6 zf4!&cluBy#YH78$T5Iti- zG%*G+eC7b`?2N4Ji~u@ufEIv>nS%qs!p;iN0no_-7#WxWbRqx-dPWv{CIFo*fa$ZE z^5+E%3@r2jUS4PuTjM{dfd5Co&yFzsA01Klur~qFsVkb8es;se*38))!0@NVq88T9 zCXSz}wSlvVu!)hK@n^MvQh5VM8z;aY7KH4qU2JTf04xAHc}Ejt3nOPcM*sukXRiQs zYM)Fn0GK}ipEO9=8k@L(7W^TRnV$KdB2&B1TQPmMMG-)!XlLjAd6w~?oYE(;0M2pI}-=Ji3u|cJ)03DFSjYZ 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