Machine Learning - DISCo

More documents

Recommendations

Info

form a < x < b, where a and b may be any real constants. What is VC(H)? To answer this question, we must find the largest subset of X that can be shattered by H. Consider a particular subset containing two distinct instances, say S = {3.1,5.7}. Can S be shattered by H? Yes. For example, the four hypotheses (1 < x < 2), (1 < x < 4), (4 < x < 7), and (1 < x < 7) will do. Together, they represent each of the four dichotomies over S, covering neither instance, either one of the instances, and both of the instances, respectively. Since we have found a set of size two that can be shattered by H, we know the VC dimension of H is at least two. Is there a set of size three that can be shattered? Consider a set S = (xo, xl, x2} containing three arbitrary instances. Without loss of generality, assume xo < xl < x2. Clearly this set cannot be shattered, because the dichotomy that includes xo and x2, but not XI, cannot be represented by a single closed interval. Therefore, no subset S of size three can be shattered, and VC(H) = 2. Note here that H is infinite, but VC(H) finite. Next consider the set X of instances corresponding to points on the x, y plane (see Figure 7.4). Let H be the set of all linear decision surfaces in the plane. In other words, H is the hypothesis space corresponding to a single perceptron unit with two inputs (see Chapter 4 for a general discussion of perceptrons). What is the VC dimension of this H? It is easy to see that any two distinct points in the plane can be shattered by H, because we can find four linear surfaces that include neither, either, or both points. What about sets of three points? As long as the points are not colinear, we will be able to find 23 linear surfaces that shatter them. Of course three colinear points cannot be shattered (for the same reason that the three points on the real line could not be shattered in the previous example). What is VC(H) in this case-two or three? It is at least three. The definition of VC dimension indicates that if we find any set of instances of size d that can be shattered, then VC(H) 2 d. To show that VC(H) < d, we must show that no set of size d can be shattered. In this example, no sets of size four can be shattered, so VC(H) = 3. More generally, it can be shown that the VC dimension of linear decision surfaces in an r dimensional space (i.e., the VC dimension of a perceptron with r inputs) is r + 1. As one final example, suppose each instance in X is described by the conjunction of exactly three boolean literals, and suppose that each hypothesis in H is described by the conjunction of up to three boolean literals. What is VC(H)? We FIGURE 7.4 The VC dimension for linear decision surfaces in the x, y plane is 3. (a) A set of three points that can be shattered using linear decision surfaces. (b) A set of three that cannot be shattered.
can show that it is at least 3, as follows. Represent each instance by a 3-bit string corresponding to the values of each of its three literals 11, 12, and 13. Consider the following set of three instances: This set of three instances can be shattered by H, because a hypothesis can be constructed for any desired dichotomy as follows: If the dichotomy is to exclude instancei, add the literal -li to the hypothesis. For example, suppose we wish to include instance2, but exclude instance1 and instance3. Then we use the hypothesis -Il A -I3. This argument easily extends from three features to n. Thus, the VC dimension for conjunctions of n boolean literals is at least n. In fact, it is exactly n, though showing this is more difficult, because it requires demonstrating that no set of n + 1 instances can be shattered. i 7.4.3 Sample Complexity and the VC Dimension Earlier we considered the question "How many randomly drawn training examples suffice to probably approximately learn any target concept in C?' (i.e., how many examples suffice to €-exhaust the version space with probability (1 - a)?). Using VC(H) as a measure for the complexity of H, it is possible to derive an alternative answer to this question, analogous to the earlier bound of Equation (7.2). This new bound (see Blumer et al. 1989) is Note that just as in the bound from Equation (7.2), the number of required training examples m grows logarithmically in 118. It now grows log times linear in 116, rather than linearly. Significantly, the In I HI term in the earlier bound has now been replaced by the alternative measure of hypothesis space complexity, VC(H) (recall VC(H) I log2 I H I). Equation (7.7) provides an upper bound on the number of training examples sufficient to probably approximately learn any target concept in C, for any desired t and a. It is also possible to obtain a lower bound, as summarized in the following theorem (see Ehrenfeucht et al. 1989). Theorem 7.3. Lower bound on sample complexity. Consider any concept class C such that VC(C) 2 2, any learner L, and any 0 < E < $, and 0 < S < &. Then there exists a distribution 23 and target concept in C such that if L observes fewer examples than then with probability at least 6, L outputs a hypothesis h having errorD(h) > E.
Page 2 and 3:
Machine Learning Tom M. Mitchell Pr
Page 4 and 5:
xvi PREFACE A third principle that
Page 13 and 14:
CHAPTER INTRODUCTION Ever since com
Page 15 and 16:
CHAPTER 1 INTRODUCITON 3 0 Learning
Page 17 and 18:
CHAFTlB 1 INTRODUCTION 5 that allow
Page 19 and 20:
1.2.2 Choosing the Target Function
Page 21 and 22:
CHAPTER I INTRODUCTION 9 x5: the nu
Page 23 and 24:
Thus, we seek the weights, or equiv
Page 25 and 26:
could involve creating board positi
Page 27 and 28:
the available training examples. Th
Page 29 and 30:
0 Chapter 10 covers algorithms for
Page 31 and 32:
ination of board features of your c
Page 33 and 34:
CHAFER 2 CONCEm LEARNING AND THE GE
Page 35 and 36:
When learning the target concept, t
Page 37 and 38:
CHAPTER 2 CONCEPT LEARNING AND THE
Page 39 and 40:
is needed. In the general case, as
Page 41 and 42:
2.5 VERSION SPACES AND THE CANDIDAT
Page 43 and 44:
{ 1 FIGURE 2.3 A version space w
Page 45 and 46:
CHAPTER 2 CONCEET LEARNJNG AND THE
Page 47 and 48:
C H m R 2 CONCEPT LEARNING AND THE
Page 49 and 50:
CH.4PTF.R 2 CONCEFT LEARNING AND TH
Page 51 and 52:
Page 53 and 54:
the power set of X? In general, the
Page 55 and 56:
CHAFI%R 2 CONCEPT LEARNING AND THE
Page 57 and 58:
CHAPTER 2 CONCEPT. LEARNING AND THE
Page 59 and 60:
CHAFTER 2 CONCEPT LEARNING AND THE
Page 61 and 62:
Page 63 and 64:
Hirsh, H. (1991). Theoretical under
Page 65 and 66:
CHAPTER 3 DECISION TREE LEARNING 53
Page 67 and 68:
CHAPTER 3 DECISION TREE LEARMNG 55
Page 69 and 70:
Page 71 and 72:
High wx CHAPTER 3 DECISION TREE LEA
Page 73 and 74:
{Dl, D2, ..., Dl41 P+S-I Which attr
Page 75 and 76:
3.6 INDUCTIVE BIAS IN DECISION TREE
Page 77 and 78:
3.6.2 Why Prefer Short Hypotheses?
Page 79 and 80:
easonable strategy, in fact it can
Page 81 and 82:
Although the first of these approac
Page 83 and 84:
multiple ways, then averaging the r
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
CHAFER 3 DECISION TREE LEARNING 77
Page 91 and 92:
C m 3 DECISION TREE LEARNING 79 Fay
Page 93 and 94:
CHAPTER ARTIFICIAL NEURAL NETWORKS
Page 95 and 96:
at normal speeds on public highways
Page 97 and 98:
~t is also applicable to problems f
Page 99 and 100:
FIGURE 43 A perceptron. A single pe
Page 101 and 102:
this example. Notice the training r
Page 103 and 104:
Gradient descent search determines
Page 105 and 106:
- - CHAF'l'ER 4 ARTIFICIAL NEURAL N
Page 107 and 108:
C H m R 4 ARTIFICIAL NEURAL NETWORK
Page 109 and 110:
CHAPTER 4 ARTIFICIAL NEURAL NETWORK
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
and combining this with Equations (
Page 117 and 118:
Page 119 and 120:
Inputs Outputs Input 10000000 0 100
Page 121 and 122:
FIGURE 4.8 Learning the 8 x 3 x 8 N
Page 123 and 124:
Page 125 and 126:
30 x 32 resolution input images lef
Page 127 and 128:
left, (0,1,0,O) to encode a face lo
Page 129 and 130:
close to zero, as the bright face w
Page 131 and 132:
4.8.2 Alternative Error Minimizatio
Page 133 and 134:
Page 135 and 136:
BACKPROPAGATION searches the space
Page 137 and 138:
4.6. Explain informally why the del
Page 139 and 140:
Mitchell, T. M., & Thrun, S. B. (19
Page 141 and 142:
the impact of possible pruning step
Page 143 and 144:
Here the notation Pr denotes that t
Page 145 and 146:
a A random variable can be viewed a
Page 147 and 148:
5.3.2 The Binomial Distribution A g
Page 149 and 150:
In case the random variable Y is go
Page 151 and 152:
which is wide enough to contain 95%
Page 153 and 154:
intervals for discrete-valued hypot
Page 155 and 156:
Then as n + co, the distribution go
Page 157 and 158:
we might observe this difference in
Page 159 and 160:
1. Partition the available data Do
Page 161 and 162:
perfectly fits the form of the abov
Page 163 and 164:
CHAFER 5 EVALUATING HYPOTHESES 151
Page 165 and 166:
Geman, S., Bienenstock, E., & Dours
Page 167 and 168:
CHAFER 6 BAYESIAN LEARNING 155 U Fo
Page 169 and 170:
CHAPTER 6 BAYESIAN LEARNING 157 As
Page 171 and 172:
. - Product rule: probability P(A A
Page 173 and 174:
CHAPTER 6 BAYESIAN LEARNING 161 Rec
Page 175 and 176:
CHAPTER 6 BAYESIAN LEARNING 163 of
Page 177 and 178: CHAPTER 6 BAYESIAN LEARNING 165 a l
Page 179 and 180: CHAPTER 6 BAYESIAN LEARNING 167 the
Page 181 and 182: CHAPTER 6 BAYESIAN LEARNING 169 doe
Page 183 and 184: CIUPlER 6 BAYESIAN LEARNING 171 sub
Page 185 and 186: CHAPTER 6 BAYESIAN LEARNING 173 0 T
Page 187 and 188: CHAFER 6 BAYESIAN LEARNING 175 the
Page 189 and 190: 6.9 NAIVE BAYES CLASSIFIER CHAPTJZR
Page 191 and 192: CHAETER 6 BAYESIAN LEARNING 179 Sim
Page 193 and 194: -a- CHAPTER 6 BAYESIAN LEARNING 181
Page 195 and 196: CHAPTER 6 BAYESIAN LEARNING 183 Exa
Page 197 and 198: CHAPTER 6 BAYESIAN LEARNING 185 In
Page 199 and 200: C H m R 6 BAYESIAN LEARNING 187 Fir
Page 201 and 202: CHAPTER 6 BAYESIAN LEARNING 189 For
Page 203 and 204: CHAPTER 6 BAYESIAN LEARNING 191 ord
Page 205 and 206: CHAPTER 6 BAYESIAN LEARNING 193 max
Page 207 and 208: CHAPTER 6 BAYESIAN LEARNING 195 Ste
Page 209 and 210: 6.13 SUMMARY AND FURTHER READING Th
Page 211 and 212: CHAPTER 6 BAYESIAN LEARNING 199 pro
Page 213 and 214: CHAPTER COMPUTATIONAL LEARNING THEO
Page 215 and 216: CHAPTER 7 COMPUTATIONAL LEARNING TH
Page 217 and 218: C Instance space X Where c and h di
Page 219 and 220: usually more concerned with the num
Page 221 and 222: are drawn. Haussler (1988) provides
Page 223 and 224: CHAPTER 7 COMPUTATIONAL LEARNING TH
Page 225 and 226: C that contains every teachable con
Page 227: Instance space X FIGURE 73 A set of
Page 231 and 232: The following theorem bounds the VC
Page 233 and 234: FIND-S: C CHAPTER 7 COMPUTATIONAL L
Page 235 and 236: We define the optimal mistake bound
Page 237 and 238: log' Rearranging terms yields which
Page 239 and 240: Current research on computational l
Page 241 and 242: (b) You now draw a new set of 100 i
Page 243 and 244: CHAPTER 8 INSTANCE-BASED LEARNING 2
Page 247 and 248: (i.e., on all axes in the Euclidean
Page 249 and 250: 8.3.1 Locally Weighted Linear Regre
Page 251 and 252: C H m R 8 INSTANCE-BASED LEARNING 2
Page 253 and 254: CHAPTER 8 INSTANCEBASED LEARNING 24
Page 255 and 256: CHAPTER 8 INSTANCE-BASED LEARMNG 24
Page 259 and 260: A thorough discussion of radial bas
Page 261 and 262: CHAPTER GENETIC ALGORITHMS Genetic
Page 263 and 264: Fitness: A function that assigns an
Page 265 and 266: can then be represented by the foll
Page 267 and 268: the crossover point n is chosen at
Page 269 and 270: value of the target attribute c. Th
Page 271 and 272: In the above experiment, the two ne
Page 273 and 274: The second step above follows from
Page 275 and 276: FIGURE 9.2 Crossover operation appl
Page 277 and 278: 0 (DU x y) (do until), which execut
Page 279 and 280:
9.6.2 Baldwin Effect Although Lamar
Page 281 and 282:
iteration, the most fit members of
Page 283 and 284:
Belew, R. K., & Mitchell, M. (Eds.)
Page 285 and 286:
Tumey, P. D., Whitley, D., & Anders
Page 287 and 288:
As an example of first-order rule s
Page 289 and 290:
10.2.1 General to Specific Beam Sea
Page 291 and 292:
A few remarks on the LEARN-ONE-RULE
Page 293 and 294:
decisions regarding preconditions o
Page 295 and 296:
10.4 LEARNING FIRST-ORDER RULES In
Page 297 and 298:
Every well-formed expression is com
Page 299 and 300:
eralize the current disjunctive hyp
Page 301 and 302:
Here let us also make the closed wo
Page 303 and 304:
In the case of noise-free training
Page 305 and 306:
(e.g., neural network) fits the dat
Page 307 and 308:
C : KnowMaterial v -Study C: KnowMa
Page 309 and 310:
CHAPTER 10 LEARNING SETS OF RULES 2
Page 311 and 312:
In contrast, the generate-and-test
Page 313 and 314:
which enable the user to specify th
Page 315 and 316:
Wrobel (1992) use rule schemata in
Page 317 and 318:
De Raedt, L., & Bruynooghe, M. (199
Page 319 and 320:
CHAPTER ANALYTICAL LEARNING Inducti
Page 321 and 322:
What is interesting about this ches
Page 323 and 324:
Given: rn Instance space X: Each in
Page 325 and 326:
theories to automatically improve p
Page 327 and 328:
Explanation: Training Example: FIGU
Page 329 and 330:
FIGURE 11.3 Computing the weakest p
Page 331 and 332:
example that is not yet covered by
Page 333 and 334:
contain no description of such a pr
Page 335 and 336:
additional detail that must be cons
Page 337 and 338:
CHAPTER 11 ANALYTICAL LEARNING 325
Page 339 and 340:
explaining to itself the reason for
Page 341 and 342:
that fits the learner's prior knowl
Page 343 and 344:
Height (Joe, Short) Height(Sue, Sho
Page 345 and 346:
CHAF'TER 11 ANALYTICAL LEARNING 333
Page 347 and 348:
CHAPTER 12 COMBINING INDUCTIVE AND
Page 349 and 350:
CHAF'TER 12 COMBINING INDUCTIVE AND
Page 351 and 352:
12.2.2 Hypothesis Space Search CAAP
Page 353 and 354:
- KBANN(Domain-Theory, Training_Exa
Page 355 and 356:
Page 357 and 358:
Page 359 and 360:
CHAF'TER 12 COMBINING INDUCTIVE AND
Page 361 and 362:
Page 363 and 364:
Hypothesis Space Hypotheses that ma
Page 365 and 366:
CHAFER 12 COMBINING INDUCTIVE AND A
Page 367 and 368:
CHAPTER 12 COMBmG INDUCTIVE AND ANA
Page 369 and 370:
Page 371 and 372:
Page 373 and 374:
Hypothesis Space 1 Hypotheses thatf
Page 375 and 376:
y first order Horn clauses to learn
Page 377 and 378:
Craven, M. W., & Shavlik, J. W. (19
Page 379 and 380:
CHAPTER REINFORCEMENT LEARNING Rein
Page 381 and 382:
has or does not have prior knowledg
Page 383 and 384:
CHAPTER 13 REINFORCEMENT LEARNING 3
Page 385 and 386:
Page 387 and 388:
CHAPTER 13 REINFORCEMENT LEARNJNG 3
Page 389 and 390:
Page 391 and 392:
Page 393 and 394:
Page 395 and 396:
While Q learning and related reinfo
Page 397 and 398:
a neural network reinforcement lear
Page 399 and 400:
CHAFER 13 REINFORCEMENT LEARNING 38
Page 401 and 402:
CHaPrER 13 REINFORCEMENT LEARNING 3
Page 403 and 404:
APPENDIX NOTATION Below is a summar
Page 405 and 406:
INDEXES
Page 407 and 408:
in KBANN algorithm, 343-344 optimiz
Page 409 and 410:
leave-one-out, 235 in neural networ
Page 411 and 412:
extensions to, 258-259 ID5R algorit
Page 413 and 414:
of LMS algorithm, 64 of ROTE-LEARNE
Page 415 and 416:
prediction of probabilities with, 1
Page 417 and 418:
Probability distribution, 133. See
Page 419 and 420:
Split infomation, 73-74 Squashing f
show all

Machine Learning - DISCo

Create successful ePaper yourself

Delete template?

Save as template?