learning theory

view markdown*references: (1) Machine Learning - Tom Mitchell, (2) An Introduction to Computational Learning Theory - Kearns & Vazirani*

# evolution

- performance is correlation $Perf_D (h,c) = \sum h(x) \cdot c(x) \cdot P(x)$
- want $P(Perf_D(h,c) < Perf_D(c,c)-\epsilon) < \delta$

# sample problems

- ex: N marbles in a bag. How many draws with replacement needed before we draw all N marbles?
- write $P_i = \frac{N-(i-1)}{N}$ where i is number of distinct drawn marbles
- transition from i to i+1 is geometrically distributed with probability $P_i$
- mean times is sum of mean of each geometric

- in order to get probabilities of seeing all the marbles instead of just mean[# draws], want to use Markov’s inequailty

- write $P_i = \frac{N-(i-1)}{N}$ where i is number of distinct drawn marbles
- box full of 1e6 marbles
- if we have 10 evenly distributed classes of marbles, what is probability we identify all 10 classes of marbles after 100 draws?

# computational learning theory

- frameworks
- PAC
- mistake-bound - split into b processes which each fail with probability at most $\delta / b$

- questions
*sample complexity*- how many training examples needed to converge*computational complexity*- how much computational effort needed to converge*mistake bound*- how many training examples will learner misclassify before converging

- must define convergence based on some probability

## PAC - probably learning an approximately correct hypothesis - Mitchell

- want to learn C
- data X is sampled with Distribution D
- learner L considers set H of possible hypotheses

*true error*$err_d (h)$ of hypothesis h with respect to target concept c and distribution D is the probability that h will misclassify an instance drawn at random according to D.- $err_D(h) = \underset{x\in D}{Pr}[c(x) \neq h(x)]$

- getting $err_D(h)=0$ is infeasible
*PAC learnable*- consider concept class C defined over set of instances X of length n and a learner L using hypothesis space H- C is PAC-learnable by L using H if for all $c \in C$, distributions D over X, $\epsilon$ s.t. 0 < $\epsilon$ < 1/2 $\delta$ s.t. $0<\delta<1/2$, learner L will with probability at least $(1-\delta)$ output a hypothesis $h \in H$ s.t $err_D(h) \leq \epsilon$

*efficiently PAC learnable*-*time*that is polynomial in $1/\epsilon, 1/\delta, n, size(c )$*probably*- probability of failure bounded by some constant $\delta$*approximately correct*- err bounded by some constant $\epsilon$- assumes H contains hypothesis with artbitraily small error for every target concept in C

## sample complexity for finite hypothesis space - Mitchell

*sample complexity*- growth in the number of training examples required*consistent learner*- outputs hypotheses that perfectly fit training data whenever possible- outputs a hypothesis belonging to the version space

- consider hypothesis space H, target concept c, instance distribution $\mathcal{D}$, training examples D of c. The versions space $VS_{H,D}$ is
*$\epsilon$-exhaused*with respect to c and $\mathcal{D}$ if every hypothesis h in $VS_{H,D}$ has error less than $\epsilon$ with respect to c and $\mathcal{D}$: $(\forall h \in VS_{H,D}) err_\mathcal{D} (h) < \epsilon$

## rectangle learning game - Kearns

- data X is sampled with Distribution D
- simple soln: tightest-fit rectangle
- define region T so prob a draw misses T is $1-\epsilon /4$
- then, m draws miss with $(1-\epsilon /4)^m$
- choose m to satisfy $4(1-\epsilon/4)^m \leq \delta$

- then, m draws miss with $(1-\epsilon /4)^m$

## VC dimension

*VC dimension*measures*capacity*of a space of functions that can be learend by a statistical classification algorithm- let H be set of sets and C be a set
- $H \cap C := { h \cap C : \vert h \in H }$
- a set C is
*shattered*by H if $H \cap C$ contains all subsets of C - The VC dimension of $H$ is the largest integer $D$ such that there exists a set $C$ with cardinality $D$ that is shattered by $H$

- VC (Vapnic-Chervonenkis) dimension - if data is mapped into sufficiently high dimension, then samples will be linearly separable (N points, N-1 dims)
- VC dimension 0 -> hypothesis either always returns false or always returns true
*Sauer’s lemma*- let $d \geq 0, m \geq 1$, $H$ hypothesis space, VC-dim(H) = d. Then, $\Pi_H(m) \leq \phi (d,m)$- fundamental theorem of learning theory provides bound of m that guarantees learning: $m \geq [\frac{4}{\epsilon} \cdot (d \cdot ln(\frac{12}{\epsilon}) + ln(\frac{2}{\delta}))]$

# concept learning and the general-to-specific ordering

- definitions
*concept learning*- acquiring the definition of a general category given a sample of positive and negative training examples of the category- concept is boolean function that returns true for specific things
- can represent function as vector acceptable features, ?, or null (if any null, then entire vector is null)

*general hypothesis*- more generally true- general defines a partial ordering

- a hypothesis is
*consistent*with the training examples if it correctly classifies them - an example x
*satisfies*a hypothesis h if h(x) = 1

*find-S*- finding a maximally specific hypothesis- start with most specific possible
- generalize each time it fails to cover an observed positive training example
- flaws
- ignores negative examples

- if training data is perfect, then will get answer
- no errors
- there exists a hypothesis in H that describes target concept c

*version space*- set of all hypotheses consistent with the training examples*list-then-eliminate*- list all hypotheses and eliminate any that are inconsistent (slow)*candidate-elimination*- represent most general (G) and specific (S) members of version space- version space representation theorem - version space can be found from most general / specific version space members
- for positive examples
- make S more general
- fix G

- for negative examples
- fix S
- make G more specific

- in general, optimal query strategy is to generate instances that satisfy exactly half the hypotheses in the current version space

- testing?
- classify as positive if satisfies S
- classify as negative if doesn’t satisfy G

- inductive bias of candidate-elimination - target concept c is contained in H