learning theory

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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
• 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
  1. PAC
  2. mistake-bound - split into b processes which each fail with probability at most $\delta / b$
• questions
  1. sample complexity - how many training examples needed to converge
  2. computational complexity - how much computational effort needed to converge
  3. 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 $4(1-\epsilon/4)^m \leq \delta$

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
    1. no errors
    2. 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