# New Support Vector Algorithms

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 LETTER Communicated by John Platt New Support Vector Algorithms Bernhard Schölkopf Alex J. Smola GMD FIRST, Berlin, Germany, and Department of Engineering, Australian National University, Canberra 0200, Australia Robert C. Williamson Department of Engineering, Australian National University, Canberra 0200, Australia Peter L. Bartlett RSISE, Australian National University, Canberra 0200, Australia We propose a new class of support vector algorithms for regression and classi cation. In these algorithms, a parameter º lets one effectively control the number of support vectors. While this can be useful in its own right, the parameterization has the additional bene t of enabling us to eliminate one of the other free parameters of the algorithm: the accuracy parameter e in the regression case, and the regularization constant C in the classi cation case. We describe the algorithms, give some theoretical results concerning the meaning and the choice of º, and report experimental results. 1 Introduction Support vector (SV) machines comprise a new class of learning algorithms, motivated by results of statistical learning theory (Vapnik, 1995). Originally developed for pattern recognition (Vapnik & Chervonenkis, 1974; Boser, Guyon, & Vapnik, 1992), they represent the decision boundary in terms of a typically small subset (Schölkopf, Burges, & Vapnik, 1995) of all training examples, called the support vectors. In order for this sparseness property to carry over to the case of SV Regression, Vapnik devised the so-called e-insensitive loss function, y f (x) e D maxf0, y f (x) eg, (1.1) which does not penalize errors below some e > 0, chosen a priori. His algorithm, which we will henceforth call e-svr, seeks to estimate functions, f (x) D (w x) C b, w, x 2 R N, b 2 R, (1.2) Present address: Microsoft Research, 1 Guildhall Street, Cambridge, U.K. Neural Computation 12, (2000) c 2000 Massachusetts Institute of Technology

2 1208 B. Schölkopf, A. J. Smola, R. C. Williamson, and P. L. Bartlett based on independent and identically distributed (i.i.d.) data, (x 1, y 1 ),..., (x`, y`) 2 R N R. (1.3) Here, R N is the space in which the input patterns live but most of the following also applies for inputs from a set X. The goal of the learning process is to nd a function f with a small risk (or test error), Z R[ f ] D l( f, x, y) dp(x, y), (1.4) where P is the probability measure, which is assumed to be responsible for the generation of the observations (see equation 1.3) and l is a loss function, for example, l( f, x, y) D ( f (x) y) 2, or many other choices (Smola & Schölkopf, 1998). The particular loss function for which we would like to minimize equation 1.4 depends on the speci c regression estimation problem at hand. This does not necessarily have to coincide with the loss function used in our learning algorithm. First, there might be additional constraints that we would like our regression estimation to satisfy, for instance, that it have a sparse representation in terms of the training data. In the SV case, this is achieved through the insensitive zone in equation 1.1. Second, we cannot minimize equation 1.4 directly in the rst place, since we do not know P. Instead, we are given the sample, equation 1.3, and we try to obtain a small risk by minimizing the regularized risk functional, 1 2 kwk2 C C R e emp [ f ]. (1.5) Here, kwk 2 is a term that characterizes the model complexity, R e emp[ f ] :D 1` `X y i f (x i ) e, (1.6) id1 measures the e-insensitive training error, and C is a constant determining the trade-off. In short, minimizing equation 1.5 captures the main insight of statistical learning theory, stating that in order to obtain a small risk, one needs to control both training error and model complexity that is, explain the data with a simple model. The minimization of equation 1.5 is equivalent to the following constrained optimization problem (see Figure 1): minimize t (w, ( ) ) D 1 2 kwk2 C C 1` `X (j i C j i ), (1.7) id1

3 New Support Vector Algorithms 1209 Figure 1: In SV regression, a desired accuracy e is speci ed a priori. It is then attempted to t a tube with radius e to the data. The trade-off between model complexity and points lying outside the tube (with positive slack variables j ) is determined by minimizing the expression 1.5. subject to ((w x i ) C b) y i e C j i (1.8) y i ((w x i ) C b) e C j i (1.9) j ( ) i 0. (1.10) Here and below, it is understood that i D 1,..., `, and that boldface Greek letters denote `-dimensional vectors of the corresponding variables; ( ) is a shorthand implying both the variables with and without asterisks. By using Lagrange multiplier techniques, one can show (Vapnik, 1995) that this leads to the following dual optimization problem. Maximize `X W(, ) D e (a i C a i ) C id1 `X (a i a i )y i id1 1 `X (a i a i )(a j a j )(x i x j ) (1.11) 2 i, jd1 subject to `X (a i a i ) D 0 (1.12) id1 a ( ) i 2 h i 0, C`. (1.13) The resulting regression estimates are linear; however, the setting can be generalized to a nonlinear one by using the kernel method. As we will use precisely this method in the next section, we shall omit its exposition at this point.

4 1210 B. Schölkopf, A. J. Smola, R. C. Williamson, and P. L. Bartlett To motivate the new algorithm that we shall propose, note that the parameter e can be useful if the desired accuracy of the approximation can be speci ed beforehand. In some cases, however, we want the estimate to be as accurate as possible without having to commit ourselves to a speci c level of accuracy a priori. In this work, we rst describe a modi cation of the e- SVR algorithm, called º-SVR, which automatically minimizes e. Following this, we present two theoretical results onº-svr concerning the connection to robust estimators (section 3) and the asymptotically optimal choice of the parameter º (section 4). Next, we extend the algorithm to handle parametric insensitivity models that allow taking into account prior knowledge about heteroscedasticity of the noise. As a bridge connecting this rst theoretical part of the article to the second one, we then present a de nition of a margin that both SV classi cation and SV regression algorithms maximize (section 6). In view of this close connection between both algorithms, it is not surprising that it is possible to formulate also a º-SV classi cation algorithm. This is done, including some theoretical analysis, in section 7. We conclude with experiments and a discussion. 2 º-SV Regression To estimate functions (see equation 1.2) from empirical data (see equation 1.3) we proceed as follows (Schölkopf, Bartlett, Smola, & Williamson, 1998). At each point x i, we allow an error of e. Everything above e is captured in slack variables j i ( ), which are penalized in the objective function via a regularization constant C, chosen a priori (Vapnik, 1995). The size of e is traded off against model complexity and slack variables via a constant º 0:! minimize t (w, ( ), e) D 1 `X 2 kwk2 C C ºe C ( (j i C j i 1` ) (2.1) id1 subject to ((w x i ) C b) y i e C j i (2.2) y i ((w x i ) C b) e C j i (2.3) j ( ) i 0, e 0. (2.4) For the constraints, we introduce multipliers a ( ) i, g ( ) i, b 0, and obtain the Lagrangian L(w, b, ( ), b, ( ), e, ( ) ) D 1 2 kwk2 C Cºe C C` `X `X (j i C j i ) be (g i j i C g i j i ) id1 id1

5 New Support Vector Algorithms 1211 `X a i (j i C y i (w x i ) b C e) id1 `X a i (j i C (w x i ) C b y i C e). (2.5) id1 To minimize the expression 2.1, we have to nd the saddle point of L that is, minimize over the primal variables w, e, b, j ( ) i and maximize over the dual variables a i ( ), b, g i ( ). Setting the derivatives with respect to the primal variables equal to zero yields four equations: w D X i (a i a i )x i (2.6) C º X i (a i C a i ) b D 0 (2.7) `X (a i a i ) D 0 (2.8) id1 C ` a( ) i g i ( ) D0. (2.9) In the SV expansion, equation 2.6, only those a i ( ) will be nonzero that correspond to a constraint, equations 2.2 or 2.3, which is precisely met; the corresponding patterns are called support vectors. This is due to the Karush- Kuhn-Tucker (KKT) conditions that apply to convex constrained optimization problems (Bertsekas, 1995). If we write the constraints as g(x i, y i ) 0, with corresponding Lagrange multipliers a i, then the solution satis es a i g(x i, y i ) D 0 for all i. Substituting the above four conditions into L leads to another optimization problem, called the Wolfe dual. Before stating it explicitly, we carry out one further modi cation. Following Boser et al. (1992), we substitute a kernel k for the dot product, corresponding to a dot product in some feature space related to input space via a nonlinear map W, k(x, y) D (W(x) W(y)). (2.10) By using k, we implicitly carry out all computations in the feature space that W maps into, which can have a very high dimensionality. The feature space has the structure of a reproducing kernel Hilbert space (Wahba, 1999; Girosi, 1998; Schölkopf, 1997) and hence minimization of kwk 2 can be understood in the context of regularization operators (Smola, Schölkopf, & Müller, 1998). The method is applicable whenever an algorithm can be cast in terms of dot products (Aizerman, Braverman, & Rozonoer, 1964; Boser et al., 1992; Schölkopf, Smola, & Müller, 1998). The choice of k is a research topic in its

6 1212 B. Schölkopf, A. J. Smola, R. C. Williamson, and P. L. Bartlett own right that we shall not touch here (Williamson, Smola, & Schölkopf, 1998; Schölkopf, Shawe-Taylor, Smola, & Williamson, 1999); typical choices include gaussian kernels, k(x, y) D exp( kx yk 2 /(2s 2 )) and polynomial kernels, k(x, y) D (x y) d (s > 0, d 2 N). Rewriting the constraints, noting that b, g i ( ) 0 do not appear in the dual, we arrive at the º-SVR optimization problem: for º 0, C > 0, `X maximize W( ( ) ) D (a i a i )y i id1 1 2 `X (a i a i )(a j a j )k(x i, x j ) (2.11) i, jd1 subject to `X (a i a i ) D0 (2.12) id1 h i a i ( ) 2 0, C` (2.13) `X (a i C a i ) C º. (2.14) id1 The regression estimate then takes the form (cf. equations 1.2, 2.6, and 2.10), f (x) D `X (a i a i )k(x i, x) C b, (2.15) id1 where b (and e) can be computed by taking into account that equations 2.2 and 2.3 (substitution of P j (a j a j )k(x j, x) for (w x) is understood; cf. equations 2.6 and 2.10) become equalities with j i ( ) D 0 for points with < C/`, respectively, due to the KKT conditions. Before we give theoretical results explaining the signi cance of the parameter º, the following observation concerning e is helpful. If º > 1, then e D 0, since it does not pay to increase e (cf. equation 2.1). If º 1, it can still happen that e D 0 for example, if the data are noise free and can be perfectly interpolated with a low-capacity model. The case e D 0, however, is not what we are interested in; it corresponds to plain L 1 -loss regression. We will use the term errors to refer to training points lying outside the tube 1 and the term fraction of errors or SVs to denote the relative numbers of 0 < a ( ) i 1 For N > 1, the tube should actually be called a slab the region between two parallel hyperplanes.

7 New Support Vector Algorithms 1213 errors or SVs (i.e., divided by `). In this proposition, we de ne the modulus of absolute continuity of a function f as the function 2 (d) D sup P i f (b i) f (a i ), where the supremum is taken over all disjoint intervals (a i, b i ) with a i < b i satisfying P i (b i a i ) < d. Loosely speaking, the condition on the conditional density of y given x asks that it be absolutely continuous on average. Proposition 1. Suppose º-SVR is applied to some data set, and the resulting e is nonzero. The following statements hold: i. º is an upper bound on the fraction of errors. ii. º is a lower bound on the fraction of SVs. iii. Suppose the data (see equation 1.3) were generated i.i.d. from a distribution P(x, y) D P(x)P(y x) with P(y x) continuous and the expectation of the modulus of absolute continuity of its density satisfying lim d!0 E2 (d) D 0. With probability 1, asymptotically, º equals both the fraction of SVs and the fraction of errors. Proof. Ad (i). The constraints, equations 2.13 and 2.14, imply that at most a fraction º of all examples can have a i ( ) D C/`. All examples with j i ( ) > 0 (i.e., those outside the tube) certainly satisfy a ( ) i D C/` (if not, a ( ) i could grow further to reduce j ( ) i ). Ad (ii). By the KKT conditions, e > 0 implies b D 0. Hence, equation 2.14 becomes an equality (cf. equation 2.7). 2 Since SVs are those examples for which 0 < a i ( ) C/`, the result follows (using a i a i D 0 for all i; Vapnik, 1995). Ad (iii). The strategy of proof is to show that asymptotically, the probability of a point is lying on the edge of the tube vanishes. The condition on P(y x) means that sup EP f (x) C t y < c x < d(c ) (2.16) f,t for some function d(c ) that approaches zero as c! 0. Since the class of SV regression estimates f has well-behaved covering numbers, we have (Anthony & Bartlett, 1999, chap. 21) that for all t,! Pr( sup P`( O f (x)c t y < c /2) < P( f (x)c t y < c ) > a < c 1 c ` 2, f 2 In practice, one can alternatively work with equation 2.14 as an equality constraint.

8 1214 B. Schölkopf, A. J. Smola, R. C. Williamson, and P. L. Bartlett where O P` is the sample-based estimate of P (that is, the proportion of points that satisfy f (x) yc t < c ), and c 1, c 2 may depend onc and a. Discretizing the values of t, taking the union bound, and applying equation 2.16 shows that the supremum over f and t of O P`( f (x) y C t D 0) converges to zero in probability. Thus, the fraction of points on the edge of the tube almost surely converges to 0. Hence the fraction of SVs equals that of errors. Combining statements i and ii then shows that both fractions converge almost surely to º. Hence, 0 º 1 can be used to control the number of errors (note that for º 1, equation 2.13 implies 2.14, since a i a i D 0 for all i (Vapnik, 1995)). Moreover, since the constraint, equation 2.12, implies that equation 2.14 is equivalent to P i a( ) i Cº/2, we conclude that proposition 1 actually holds for the upper and the lower edges of the tube separately, with º/2 each. (Note that by the same argument, the number of SVs at the two edges of the standard e-svr tube asymptotically agree.) A more intuitive, albeit somewhat informal, explanation can be given in terms of the primal objective function (see equation 2.1). At the point of the solution, note that if e > 0, we must have (w, e) D 0, that is, º C e emp D 0, hence º D e emp. This is greater than or equal to the fraction of errors, since the points outside the tube certainly do contribute to a change in R emp when e is changed. Points at the edge of the tube possibly might also contribute. This is where the inequality comes from. Note that this does not contradict our freedom to choose º > 1. In that case, e D 0, since it does not pay to increase e (cf. equation 2.1). Let us brie y discuss howº-svr relates to e-svr (see section 1). Both algorithms use the e-insensitive loss function, but º-SVR automatically computes e. In a Bayesian perspective, this automatic adaptation of the loss function could be interpreted as adapting the error model, controlled by the hyperparameter º. Comparing equation 1.11 (substitution of a kernel for the dot product is understood) and equation 2.11, we note that e-svr requires an additional term, e P` id1 (a i C a i ), which, for xed e > 0, encourages that some of the a i ( ) will turn out to be 0. Accordingly, the constraint (see equation 2.14), which appears inº-svr, is not needed. The primal problems, equations 1.7 and 2.1, differ in the term ºe. If º D 0, then the optimization can grow e arbitrarily large; hence zero empirical risk can be obtained even when all as are zero. In the following sense, º-SVR includes e-svr. Note that in the general case, using kernels, Nw is a vector in feature space. Proposition 2. If º-SVR leads to the solution Ne, Nw, b, N then e-svr with e set a priori to Ne, and the same value of C, has the solution Nw, b. N

9 New Support Vector Algorithms 1215 Proof. If we minimize equation 2.1, then x e and minimize only over the remaining variables. The solution does not change. 3 The Connection to Robust Estimators Using the e-insensitive loss function, only the patterns outside the e-tube enter the empirical risk term, whereas the patterns closest to the actual regression have zero loss. This, however, does not mean that it is only the outliers that determine the regression. In fact, the contrary is the case. Proposition 3 (resistance of SV regression). Using support vector regression with the e-insensitive loss function (see equation 1.1), local movements of target values of points outside the tube do not in uence the regression. Proof. Shifting y i locally does not change the status of (x i, y i ) as being a point outside the tube. Then the dual solution ( ) is still feasible; it satis es the constraints (the point still has a i ( ) D C/`). Moreover, the primal solution, withj i transformed according to the movement of y i, is also feasible. Finally, the KKT conditions are still satis ed, as still a i ( ) D C/`. Thus (Bertsekas, 1995), ( ) is still the optimal solution. The proof relies on the fact that everywhere outside the tube, the upper bound on the a i ( ) is the same. This is precisely the case if the loss function increases linearly ouside the e-tube (cf. Huber, 1981, for requirements on robust cost functions). Inside, we could use various functions, with a derivative smaller than the one of the linear part. For the case of the e-insensitive loss, proposition 3 implies that essentially, the regression is a generalization of an estimator for the mean of a random variable that does the following: Throws away the largest and smallest examples (a fractionº/2 of either category; in section 2, it is shown that the sum constraint, equation 2.12, implies that proposition 1 can be applied separately for the two sides, using º/2). Estimates the mean by taking the average of the two extremal ones of the remaining examples. This resistance concerning outliers is close in spirit to robust estimators like the trimmed mean. In fact, we could get closer to the idea of the trimmed mean, which rst throws away the largest and smallest points and then computes the mean of the remaining points, by using a quadratic loss inside the e-tube. This would leave us with Huber s robust loss function.

10 1216 B. Schölkopf, A. J. Smola, R. C. Williamson, and P. L. Bartlett Note, moreover, that the parameter º is related to the breakdown point of the corresponding robust estimator (Huber, 1981). Because it speci es the fraction of points that may be arbitrarily bad outliers, º is related to the fraction of some arbitrary distribution that may be added to a known noise model without the estimator failing. Finally, we add that by a simple modi cation of the loss function (White, 1994) weighting the slack variables ( ) above and below the tube in the target function, equation 2.1, by 2l and 2(1 l), withl 2 [0, 1] respectively one can estimate generalized quantiles. The argument proceeds as follows. Asymptotically, all patterns have multipliers at bound (cf. proposition 1). The l, however, changes the upper bounds in the box constraints applying to the two different types of slack variables to 2Cl /` and 2C(1 l) /`, respectively. The equality constraint, equation 2.8, then implies that (1 l) and l give the fractions of points (out of those which are outside the tube) that lie on the two sides of the tube, respectively. 4 Asymptotically Optimal Choice of º Using an analysis employing tools of information geometry (Murata, Yoshizawa, & Amari, 1994; Smola, Murata, Schölkopf, & Müller, 1998), we can derive the asymptotically optimal º for a given class of noise models in the sense of maximizing the statistical ef ciency. 3 Remark. Denote p a density with unit variance, 4 and P a family of noise models generated from P by P :D fp p D 1 s P( y s ), s > 0g. Moreover assume that the data were generated i.i.d. from a distribution p(x, y) D p(x)p(y f (x)) with p(y f (x)) continuous. Under the assumption that SV regression produces an estimate f O converging to the underlying functional dependency f, the asymptotically optimal º, for the estimation-of-locationparameter model of SV regression described in Smola, Murata, Schölkopf, & Müller (1998), is Z e º D 1 P(t) dt where e e :D argmin t 1 (P( t ) C P(t)) 2 1 Z t t P(t) dt (4.1) 3 This section assumes familiarity with some concepts of information geometry. A more complete explanation of the model underlying the argument is given in Smola, Murata, Schölkopf, & Müller (1998) and can be downloaded from rst.gmd.de. 4 is a prototype generating the class of densities. Normalization assumptions are made for ease of notation.

11 New Support Vector Algorithms 1217 To see this, note that under the assumptions stated above, the probability of a deviation larger than e, Prf y f O (x) > eg, will converge to Pr y f (x) > e ª Z D D 1 p(x)p(j ) dx dj frn[ e,e]g Z e e p(j ) dj. (4.2) This is also the fraction of samples that will (asymptotically) become SVs (proposition R 1, iii). Therefore an algorithm generating a fraction º D 1 e e p(j ) dj SVs will correspond to an algorithm with a tube of size e. The consequence is that given a noise model p(j ), one can compute the optimal e for it, and then, by using equation 4.2, compute the corresponding optimal value º. To this end, one exploits the linear scaling behavior between the standard deviation s of a distribution p and the optimal e. This result, established in Smola, Murata, Schölkopf, & Müller (1998) and Smola (1998), cannot be proved here; instead, we shall merely try to give a avor of the argument. The basic idea is to consider the estimation of a location parameter using the e-insensitive loss, with the goal of maximizing the statistical ef ciency. Using the Cramér-Rao bound and a result of Murata et al. (1994), the ef ciency is found to be e e D Q2 s GI. (4.3) Here, I is the Fisher information, while Q and G are information geometrical quantities computed from the loss function and the noise model. This means that one only has to consider distributions of unit variance, say, P, to compute an optimal value of º that holds for the whole class of distributions P. Using equation 4.3, one arrives at 1 e(e) / G Z Q 2 D 1 e 1 (P( e) C P(e)) 2 P(t) dt e. (4.4) The minimum of equation 4.4 yields the optimal choice of e, which allows computation of the corresponding º and thus leads to equation 4.1. Consider now an example: arbitrary polynomial noise models (/ e j p ) with unit variance can be written as P(j ) D 1 2 r C(3/p) C(1/p) p C 1/p exp ( ( r C(3/p) C(1/p) j! p! (4.5) where C denotes the gamma function. Table 1 shows the optimal value of º for different polynomial degrees. Observe that the more lighter-tailed

12 1218 B. Schölkopf, A. J. Smola, R. C. Williamson, and P. L. Bartlett Table 1: Optimal º for Various Degrees of Polynomial Additive Noise. Polynomial degree p Optimal º the distribution becomes, the smaller º are optimal that is, the tube width increases. This is reasonable as only for very long tails of the distribution (data with many outliers) it appears reasonable to use an early cutoff on the in uence of the data (by basically giving all data equal in uence via a i D C/`). The extreme case of Laplacian noise (º D 1) leads to a tube width of 0, that is, to L 1 regression. We conclude this section with three caveats: rst, we have only made an asymptotic statement; second, for nonzero e, the SV regression need not necessarily converge to the target f : measured using. e, many other functions are just as good as f itself; third, the proportionality between e and s has only been established in the estimation-of-location-parameter context, which is not quite SV regression. 5 Parametric Insensitivity Models We now return to the algorithm described in section 2. We generalized e-svr by estimating the width of the tube rather than taking it as given a priori. What we have so far retained is the assumption that the e-insensitive zone has a tube (or slab) shape. We now go one step further and use parametric models of arbitrary shape. This can be useful in situations where the noise is heteroscedastic, that is, where it depends on x. Let ff q ( ) g (here and below, q D 1,..., p is understood) be a set of 2p positive functions on the input space X. Consider the following quadratic program: for given º ( ) 1,..., º( ) p 0, minimize t (w, ( ), " ( ) ) D kwk 2 /2 0 1 px C (º q e q C º q e q ) C 1` `X (j i C j i ) A (5.1) qd1 id1 subject to ((w x i ) C b) y i X e q f q (x i ) C j i (5.2) q y i ((w x i ) C b) X e q f q (x i) C j i (5.3) q j ( ) i 0, e ( ) q 0. (5.4)

13 New Support Vector Algorithms 1219 A calculation analogous to that in section 2 shows that the Wolfe dual consists of maximizing the expression 2.11 subject to the constraints 2.12 and 2.13, and, instead of 2.14, the modi ed constraints, still linear in ( ), `X id1 a ( ) i f ( ) q (x i ) C º ( ) q. (5.5) In the experiments in section 8, we use a simpli ed version of this optimization problem, where we drop the termº q e q from the objective function, equation 5.1, and use e q and f q in equation 5.3. By this, we render the problem symmetric with respect to the two edges of the tube. In addition, we use p D 1. This leads to the same Wolfe dual, except for the last constraint, which becomes (cf. equation 2.14), `X (a i C a i )f(x i) C º. (5.6) id1 Note that the optimization problem of section 2 can be recovered by using the constant function f 1. 5 The advantage of this setting is that since the same º is used for both sides of the tube, the computation of e, b is straightforward: for instance, by solving a linear system, using two conditions as those described following equation Otherwise, general statements are harder to make; the linear system can have a zero determinant, depending on whether the functions f ( ) p, evaluated on the x i with 0 < a ( ) i < C/`, are linearly dependent. The latter occurs, for instance, if we use constant functions f ( ) 1. In this case, it is pointless to use two different values º, º ; for the constraint (see equation 2.12) then implies that both sums P` id1 a( ) i will be bounded by C minfº, º g. We conclude this section by giving, without proof, a generalization of proposition 1 to the optimization problem with constraint (see equation 5.6): Proposition 4. Suppose we run the above algorithm on a data set with the result that e > 0. Then º` i. is an upper bound on the fraction of errors. Pi f (x i) º` ii. is an upper bound on the fraction of SVs. Pi f (xi ) 5 Observe the similarity to semiparametric SV models (Smola, Frieß, & Schölkopf, 1999) where a modi cation of the expansion of f led to similar additional constraints. The important difference in the present setting is that the Lagrange multipliers a i and a i are treated equally and not with different signs, as in semiparametric modeling.

14 1220 B. Schölkopf, A. J. Smola, R. C. Williamson, and P. L. Bartlett iii. Suppose the data in equation 1.3 were generated i.i.d. from a distribution P(x, y) D P(x)P(y x) with P(y x) continuous and the expectation of its modulus of continuity satisfying lim d!0 E2 (d) D 0. With probability 1, asymptotically, the fractions of SVs and errors equal º ( R f(x) d QP(x)) 1, where QP is the asymptotic distribution of SVs over x. 6 Margins in Regression and Classi cation The SV algorithm was rst proposed for the case of pattern recognition (Boser et al., 1992), and then generalized to regression (Vapnik, 1995). Conceptually, however, one can take the view that the latter case is actually the simpler one, providing a posterior justi cation as to why we started this article with the regression case. To explain this, we will introduce a suitable de nition of a margin that is maximized in both cases. At rst glance, the two variants of the algorithm seem conceptually different. In the case of pattern recognition, a margin of separation between two pattern classes is maximized, and the SVs are those examples that lie closest to this margin. In the simplest case, where the training error is xed to 0, this is done by minimizing kwk 2 subject to y i ((w x i ) C b) 1 (note that in pattern recognition, the targets y i are in f 1g). In regression estimation, on the other hand, a tube of radius e is tted to the data, in the space of the target values, with the property that it corresponds to the attest function in feature space. Here, the SVs lie at the edge of the tube. The parameter e does not occur in the pattern recognition case. We will show how these seemingly different problems are identical (cf. also Vapnik, 1995; Pontil, Rifkin, & Evgeniou, 1999), how this naturally leads to the concept of canonical hyperplanes (Vapnik, 1995), and how it suggests different generalizations to the estimation of vector-valued functions. De nition 1 (e-margin). Let (E, k.k E ), (F, k.k F ) be normed spaces, and X ½ E. We de ne the e-margin of a function f : X! F as me( f ) :D inffkx yk E : x, y 2 X, k f (x) f (y)k F 2eg. (6.1) m e ( f ) can be zero, even for continuous functions, an example being f (x) D 1/x on X DR C. There, m e ( f ) D0 for all e > 0. Note that the e-margin is related (albeit not identical) to the traditional modulus of continuity of a function: given d > 0, the latter measures the largest difference in function values that can be obtained using points within a distance d in E. The following observations characterize the functions for which the margin is strictly positive. Lemma 1 (uniformly continuous functions). With the above notations, m e ( f ) is positive for all e > 0 if and only if f is uniformly continuous.

15 New Support Vector Algorithms 1221 Proof. By de nition of m e, we have k f (x) f (y)kf 2e H) kx yk E me( f ) (6.2) () kx yk E < me( f ) H) k f (x) f (y)k F < 2e, (6.3) that is, if m e ( f ) > 0, then f is uniformly continuous. Similarly, if f is uniformly continuous, then for each e > 0, we can nd a d > 0 such that k f (x) f (y)k F 2e implies kx yk E d. Since the latter holds uniformly, we can take the in mum to get m e ( f ) d > 0. We next specialize to a particular set of uniformly continuous functions. Lemma 2 (Lipschitz-continuous functions). If there exists some L > 0 such that for all x, y 2 X, k f (x) f (y)k F L kx yk E, then m e 2e L. Proof. Take the in mum over kx yk E k f (x) f (y)kf L 2e L. Example 1 (SV regression estimation). Suppose that E is endowed with a dot product (..) (generating the norm k.k E ). For linear functions (see equation 1.2) the margin takes the form m e ( f ) D 2e kwk. To see this, note that since f (x) f (y) D (w (x y)), the distance kx yk will be smallest given (w (x y)) D 2e, when x y is parallel to w (due to Cauchy-Schwartz), i.e. if x y D 2ew /kwk 2. In that case, kx yk D 2e /kwk. For xed e > 0, maximizing the margin hence amounts to minimizing kwk, as done in SV regression: in the simplest form (cf. equation 1.7 without slack variables j i ) the training on data (equation 1.3) consists of minimizing kwk 2 subject to f (x i ) y i e. (6.4) Example 2 (SV pattern recognition; see Figure 2). We specialize the setting of example 1 to the case where X D fx 1,..., x`g. Then m 1 ( f ) D 2 kwk is equal to the margin de ned for Vapnik s canonical hyperplane (Vapnik, 1995). The latter is a way in which, given the data set X, an oriented hyperplane in E can be uniquely expressed by a linear function (see equation 1.2) requiring that minf f (x) : x 2 X g D 1. (6.5) Vapnik gives a bound on the VC-dimension of canonical hyperplanes in terms of kwk. An optimal margin SV machine for pattern recognition can be constructed from data, (x 1, y 1 ),..., (x`, y`) 2 X f 1g (6.6)

16 1222 B. Schölkopf, A. J. Smola, R. C. Williamson, and P. L. Bartlett Figure 2: 1D toy problem. Separate x from o. The SV classi cation algorithm constructs a linear function f (x) D w x C b satisfying equation 6.5 (e D 1). To maximize the margin m e ( f ), one has to minimize w. as follows (Boser et al., 1992): minimize kwk 2 subject to y i f (x i ) 1. (6.7) The decision function used for classi cation takes the form f (x) D sgn((w x) C b). (6.8) The parameter e is super uous in pattern recognition, as the resulting decision function, f (x) D sgn((w x) C b), (6.9) will not change if we minimize kwk 2 subject to y i f (x i ) e. (6.10) Finally, to understand why the constraint (see equation 6.7) looks different from equation 6.4 (e.g., one is multiplicative, the other one additive), note that in regression, the points (x i, y i ) are required to lie within a tube of radius e, whereas in pattern recognition, they are required to lie outside the tube (see Figure 2), and on the correct side. For the points on the tube, we have 1 D y i f (x i ) D 1 f (x i ) y i. So far, we have interpreted known algorithms only in terms of maximizing m e. Next, we consider whether we can use the latter as a guide for constructing more general algorithms. Example 3 (SV regression for vector-valued functions). Assume E D R N. For linear functions f (x) DWx C b, with W being an N N matrix, and b 2 R N,

17 New Support Vector Algorithms 1223 we have, as a consequence of lemma 1, m e ( f ) 2e kwk, (6.11) where kwk is any matrix norm of W that is compatible (Horn & Johnson, 1985) with k.k E. If the matrix norm is the one induced by k.k E, that is, there exists a unit vector z 2 E such that kwzk E D kwk, then equality holds in To see the latter, we use the same argument as in example 1, setting x y D 2ez /kwk. q PN For the Hilbert-Schmidt norm kwk 2 D i, jd1 W2 ij, which is compatible with the vector norm k.k 2, the problem of minimizing kwk subject to separate constraints for each output dimension separates into N regression problems. In Smola, Williamson, Mika, & Schölkopf (1999), it is shown that one can specify invariance requirements, which imply that the regularizers act on the output dimensions separately and identically (i.e., in a scalar fashion). In particular, it turns out that under the assumption of quadratic homogeneity and permutation symmetry, the Hilbert-Schmidt norm is the only admissible one. 7 º-SV Classi cation We saw that º-SVR differs from e-svr in that it uses the parameters º and C instead of e and C. In many cases, this is a useful reparameterization of the original algorithm, and thus it is worthwhile to ask whether a similar change could be incorporated in the original SV classi cation algorithm (for brevity, we call it C-SVC). There, the primal optimization problem is to minimize (Cortes & Vapnik, 1995) subject to t(w, ) D 1 X 2 kwk2 C C` j i (7.1) i y i ((x i w) C b) 1 j i, j i 0. (7.2) The goal of the learning process is to estimate a function f (see equation 6.9) such that the probability of misclassi cation on an independent test set, the risk R[ f ], is small. 6 Here, the only parameter that we can dispose of is the regularization constant C. To substitute it by a parameter similar to the º used in the regression case, we proceed as follows. As a primal problem for º-SVC, we 6 Implicitly we make use of the f0, 1g loss function; hence the risk equals the probability of misclassi cation.

18 1224 B. Schölkopf, A. J. Smola, R. C. Williamson, and P. L. Bartlett consider the minimization of t (w,, r ) D 1 2 kwk2 ºr C 1` X j i (7.3) i subject to (cf. equation 6.10) y i ((x i w) C b) r j i, (7.4) j i 0, r 0. (7.5) For reasons we shall expain, no constant C appears in this formulation. To understand the role of r, note that for D 0, the constraint (see 7.4) simply states that the two classes are separated by the margin 2r /kwk (cf. example 2). To derive the dual, we consider the Lagrangian L(w,, b, r,,, d) D 1 2 kwk2 ºr C 1` X j i i X i (a i (y i ((x i w) C b) r C j i ) C b i j i ) dr, (7.6) using multipliers a i, b i, d 0. This function has to be mimimized with respect to the primal variables w,, b, r and maximized with respect to the dual variables,, d. To eliminate the former, we compute the corresponding partial derivatives and set them to 0, obtaining the following conditions: w D X i a i y i x i (7.7) a i C b i D1/`, 0 D X i a i y i, X a i d Dº. (7.8) i In the SV expansion (see equation 7.7), only those a i can be nonzero that correspond to a constraint (see 7.4) that is precisely met (KKT conditions; cf. Vapnik, 1995). Substituting equations 7.7 and 7.8 into L, using a i, b i, d 0, and incorporating kernels for dot products leaves us with the following quadratic optimization problem: maximize W( ) D 1 X a i a j y i y j k(x i, x j ) (7.9) 2 ij

19 New Support Vector Algorithms 1225 subject to 0 a i 1/` (7.10) 0 D X i a i y i (7.11) X a i º. (7.12) i The resulting decision function can be shown to take the form f (x) Dsgn( X! a i y i k(x, x i ) C b. (7.13) i Compared to the original dual (Boser et al., 1992; Vapnik, 1995), there are two differences. First, there is an additional constraint, 7.12, similar to the regression case, Second, the linear term P i a i of Boser et al. (1992) no longer appears in the objective function 7.9. This has an interesting consequence: 7.9 is now quadratically homogeneous in. It is straightforward to verify that one obtains exactly the same objective function if one starts with the primal function t (w,, r ) Dkwk 2 /2 C C ( ºr C (1/`) P i j i) (i.e., if one does use C), the only difference being that the constraints, 7.10 and 7.12 would have an extra factor C on the right-hand side. In that case, due to the homogeneity, the solution of the dual would be scaled by C; however, it is straightforward to see that the corresponding decision function will not change. Hence we may set C D 1. To compute b and r, we consider two sets S, of identical size s > 0, containing SVs x i with 0 < a i < 1 and y i D 1, respectively. Then, due to the KKT conditions, 7.4 becomes an equality with j i D0. Hence, in terms of kernels, b D 1 X X a j y j k(x, x j ), (7.14) 2s x2s C [S j 0 1 r D X X a j y j k(x, x j ) X X a j y j k(x, x j ) A. (7.15) 2s x2s C x2s j As in the regression case, the º parameter has a more natural interpretation than the one we removed, C. To formulate it, let us rst de ne the term margin error. By this, we denote points with j i > 0 that is, that are either errors or lie within the margin. Formally, the fraction of margin errors is j R r emp[ f ] :D 1` fi: y i f (x i ) < r g. (7.16)

20 1226 B. Schölkopf, A. J. Smola, R. C. Williamson, and P. L. Bartlett Here, f is used to denote the argument of the sgn in the decision function, equation 7.13, that is, f Dsgn ± f. We are now in a position to modify proposition 1 for the case of pattern recognition: Proposition 5. Suppose k is a real analytic kernel function, and we run º-SVC with k on some data with the result that r > 0. Then i. º is an upper bound on the fraction of margin errors. ii. º is a lower bound on the fraction of SVs. iii. Suppose the data (see equation 6.6) were generated i.i.d. from a distribution P(x, y) D P(x)P(y x) such that neither P(x, y D 1) nor P(x, y D 1) contains any discrete component. Suppose, moreover, that the kernel is analytic and non-constant. With probability 1, asymptotically, º equals both the fraction of SVs and the fraction of errors. Proof. Ad (i). By the KKT conditions, r > 0 impliesd D 0. Hence, inequality 7.12 becomes an equality (cf. equations 7.8). Thus, at most a fraction º of all examples can have a i D 1/`. All examples with j i > 0 do satisfy a i D1/` (if not, a i could grow further to reduce j i ). Ad (ii). SVs can contribute at most 1/`to the left-hand side of 7.12; hence there must be at least º` of them. Ad (iii). It follows from the condition on P(x, y) that apart from some set of measure zero (arising from possible singular components), the two class distributions are absolutely continuous and can be written as integrals over distribution functions. Because the kernel is analytic and nonconstant, it cannot be constant in any open set; otherwise it would be constant everywhere. Therefore, functions f constituting the argument of the sgn in the SV decision function (see equation 7.13) essentially functions in the class of SV regression functions) transform the distribution over x into distributions such that for all f, and all t 2 R, lim c!0 P( f (x) C t < c ) D 0. At the same time, we know that the class of these functions has well-behaved covering numbers; hence we get uniform convergence: for all c > 0, sup f P( f (x) C t < c ) OP`( f (x) C t < c ) converges to zero in probability, where O P` is the sample-based estimate of P (that is, the proportion of points that satisfy f (x) C t < c ). But then for all a > 0, lim c!0 lim`!1 P(sup f OP`( f (x) C t < c ) > a) D 0. Hence, sup f OP`( f (x) C t D 0) converges to zero in probability. Using t D r thus shows that almost surely the fraction of points exactly on the margin tends to zero; hence the fraction of SVs equals that of margin errors.

### Regression Using Support Vector Machines: Basic Foundations

Regression Using Support Vector Machines: Basic Foundations Technical Report December 2004 Aly Farag and Refaat M Mohamed Computer Vision and Image Processing Laboratory Electrical and Computer Engineering

### Introduction to Support Vector Machines. Colin Campbell, Bristol University

Introduction to Support Vector Machines Colin Campbell, Bristol University 1 Outline of talk. Part 1. An Introduction to SVMs 1.1. SVMs for binary classification. 1.2. Soft margins and multi-class classification.

### Support Vector Machines with Clustering for Training with Very Large Datasets

Support Vector Machines with Clustering for Training with Very Large Datasets Theodoros Evgeniou Technology Management INSEAD Bd de Constance, Fontainebleau 77300, France theodoros.evgeniou@insead.fr Massimiliano

### Support Vector Machines Explained

March 1, 2009 Support Vector Machines Explained Tristan Fletcher www.cs.ucl.ac.uk/staff/t.fletcher/ Introduction This document has been written in an attempt to make the Support Vector Machines (SVM),

### Section 1.1. Introduction to R n

The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

### Several Views of Support Vector Machines

Several Views of Support Vector Machines Ryan M. Rifkin Honda Research Institute USA, Inc. Human Intention Understanding Group 2007 Tikhonov Regularization We are considering algorithms of the form min

### Support Vector Machine. Tutorial. (and Statistical Learning Theory)

Support Vector Machine (and Statistical Learning Theory) Tutorial Jason Weston NEC Labs America 4 Independence Way, Princeton, USA. jasonw@nec-labs.com 1 Support Vector Machines: history SVMs introduced

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

### Support Vector Machines for Classification and Regression

UNIVERSITY OF SOUTHAMPTON Support Vector Machines for Classification and Regression by Steve R. Gunn Technical Report Faculty of Engineering, Science and Mathematics School of Electronics and Computer

### An Introduction to Machine Learning

An Introduction to Machine Learning L5: Novelty Detection and Regression Alexander J. Smola Statistical Machine Learning Program Canberra, ACT 0200 Australia Alex.Smola@nicta.com.au Tata Institute, Pune,

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### A Simple Introduction to Support Vector Machines

A Simple Introduction to Support Vector Machines Martin Law Lecture for CSE 802 Department of Computer Science and Engineering Michigan State University Outline A brief history of SVM Large-margin linear

### Metric Spaces. Chapter 7. 7.1. Metrics

Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

### PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION

PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION Introduction In the previous chapter, we explored a class of regression models having particularly simple analytical

### Support Vector Machine (SVM)

Support Vector Machine (SVM) CE-725: Statistical Pattern Recognition Sharif University of Technology Spring 2013 Soleymani Outline Margin concept Hard-Margin SVM Soft-Margin SVM Dual Problems of Hard-Margin

### Notes on Support Vector Machines

Notes on Support Vector Machines Fernando Mira da Silva Fernando.Silva@inesc.pt Neural Network Group I N E S C November 1998 Abstract This report describes an empirical study of Support Vector Machines

### Support Vector Machines

Support Vector Machines Charlie Frogner 1 MIT 2011 1 Slides mostly stolen from Ryan Rifkin (Google). Plan Regularization derivation of SVMs. Analyzing the SVM problem: optimization, duality. Geometric

### Statistical Machine Learning

Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes

### Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions

Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions

### Date: April 12, 2001. Contents

2 Lagrange Multipliers Date: April 12, 2001 Contents 2.1. Introduction to Lagrange Multipliers......... p. 2 2.2. Enhanced Fritz John Optimality Conditions...... p. 12 2.3. Informative Lagrange Multipliers...........

### Increasing for all. Convex for all. ( ) Increasing for all (remember that the log function is only defined for ). ( ) Concave for all.

1. Differentiation The first derivative of a function measures by how much changes in reaction to an infinitesimal shift in its argument. The largest the derivative (in absolute value), the faster is evolving.

### constraint. Let us penalize ourselves for making the constraint too big. We end up with a

Chapter 4 Constrained Optimization 4.1 Equality Constraints (Lagrangians) Suppose we have a problem: Maximize 5, (x 1, 2) 2, 2(x 2, 1) 2 subject to x 1 +4x 2 =3 If we ignore the constraint, we get the

### Tangent and normal lines to conics

4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### 2.3 Convex Constrained Optimization Problems

42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

### By W.E. Diewert. July, Linear programming problems are important for a number of reasons:

APPLIED ECONOMICS By W.E. Diewert. July, 3. Chapter : Linear Programming. Introduction The theory of linear programming provides a good introduction to the study of constrained maximization (and minimization)

### EC 6310: Advanced Econometric Theory

EC 6310: Advanced Econometric Theory July 2008 Slides for Lecture on Bayesian Computation in the Nonlinear Regression Model Gary Koop, University of Strathclyde 1 Summary Readings: Chapter 5 of textbook.

### Chapter 15 Introduction to Linear Programming

Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of

### Duality in General Programs. Ryan Tibshirani Convex Optimization 10-725/36-725

Duality in General Programs Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: duality in linear programs Given c R n, A R m n, b R m, G R r n, h R r : min x R n c T x max u R m, v R r b T

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### ME128 Computer-Aided Mechanical Design Course Notes Introduction to Design Optimization

ME128 Computer-ided Mechanical Design Course Notes Introduction to Design Optimization 2. OPTIMIZTION Design optimization is rooted as a basic problem for design engineers. It is, of course, a rare situation

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### Stochastic Inventory Control

Chapter 3 Stochastic Inventory Control 1 In this chapter, we consider in much greater details certain dynamic inventory control problems of the type already encountered in section 1.3. In addition to the

### Measuring Rationality with the Minimum Cost of Revealed Preference Violations. Mark Dean and Daniel Martin. Online Appendices - Not for Publication

Measuring Rationality with the Minimum Cost of Revealed Preference Violations Mark Dean and Daniel Martin Online Appendices - Not for Publication 1 1 Algorithm for Solving the MASP In this online appendix

### Introduction. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type 1 agents utility is given by

Introduction General equilibrium analysis looks at how multiple markets come into equilibrium simultaneously. With many markets, equilibrium analysis must take explicit account of the fact that changes

### Gaussian Processes in Machine Learning

Gaussian Processes in Machine Learning Carl Edward Rasmussen Max Planck Institute for Biological Cybernetics, 72076 Tübingen, Germany carl@tuebingen.mpg.de WWW home page: http://www.tuebingen.mpg.de/ carl

### 2DI36 Statistics. 2DI36 Part II (Chapter 7 of MR)

2DI36 Statistics 2DI36 Part II (Chapter 7 of MR) What Have we Done so Far? Last time we introduced the concept of a dataset and seen how we can represent it in various ways But, how did this dataset came

### Machine Learning and Pattern Recognition Logistic Regression

Machine Learning and Pattern Recognition Logistic Regression Course Lecturer:Amos J Storkey Institute for Adaptive and Neural Computation School of Informatics University of Edinburgh Crichton Street,

### Linear Threshold Units

Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear

### Density Level Detection is Classification

Density Level Detection is Classification Ingo Steinwart, Don Hush and Clint Scovel Modeling, Algorithms and Informatics Group, CCS-3 Los Alamos National Laboratory {ingo,dhush,jcs}@lanl.gov Abstract We

### 1.2 Solving a System of Linear Equations

1.. SOLVING A SYSTEM OF LINEAR EQUATIONS 1. Solving a System of Linear Equations 1..1 Simple Systems - Basic De nitions As noticed above, the general form of a linear system of m equations in n variables

### A Coefficient of Variation for Skewed and Heavy-Tailed Insurance Losses. Michael R. Powers[ 1 ] Temple University and Tsinghua University

A Coefficient of Variation for Skewed and Heavy-Tailed Insurance Losses Michael R. Powers[ ] Temple University and Tsinghua University Thomas Y. Powers Yale University [June 2009] Abstract We propose a

### 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is:

CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2π-periodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx

### 2.5 Gaussian Elimination

page 150 150 CHAPTER 2 Matrices and Systems of Linear Equations 37 10 the linear algebra package of Maple, the three elementary 20 23 1 row operations are 12 1 swaprow(a,i,j): permute rows i and j 3 3

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### Introduction to Machine Learning NPFL 054

Introduction to Machine Learning NPFL 054 http://ufal.mff.cuni.cz/course/npfl054 Barbora Hladká hladka@ufal.mff.cuni.cz Martin Holub holub@ufal.mff.cuni.cz Charles University, Faculty of Mathematics and

### Cost Minimization and the Cost Function

Cost Minimization and the Cost Function Juan Manuel Puerta October 5, 2009 So far we focused on profit maximization, we could look at a different problem, that is the cost minimization problem. This is

### 4.6 Null Space, Column Space, Row Space

NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear

### Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay

Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture - 17 Shannon-Fano-Elias Coding and Introduction to Arithmetic Coding

### Sequences and Convergence in Metric Spaces

Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the n-th term of s, and write {s

### Bootstrapping Big Data

Bootstrapping Big Data Ariel Kleiner Ameet Talwalkar Purnamrita Sarkar Michael I. Jordan Computer Science Division University of California, Berkeley {akleiner, ameet, psarkar, jordan}@eecs.berkeley.edu

### THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS

THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### A QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS

A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors

### 7 Gaussian Elimination and LU Factorization

7 Gaussian Elimination and LU Factorization In this final section on matrix factorization methods for solving Ax = b we want to take a closer look at Gaussian elimination (probably the best known method

### Random graphs with a given degree sequence

Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.

### Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem

### Linear Programming, Lagrange Multipliers, and Duality Geoff Gordon

lp.nb 1 Linear Programming, Lagrange Multipliers, and Duality Geoff Gordon lp.nb 2 Overview This is a tutorial about some interesting math and geometry connected with constrained optimization. It is not

### THE SVM APPROACH FOR BOX JENKINS MODELS

REVSTAT Statistical Journal Volume 7, Number 1, April 2009, 23 36 THE SVM APPROACH FOR BOX JENKINS MODELS Authors: Saeid Amiri Dep. of Energy and Technology, Swedish Univ. of Agriculture Sciences, P.O.Box

### IDENTIFICATION IN A CLASS OF NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS. Steven T. Berry and Philip A. Haile. March 2011 Revised April 2011

IDENTIFICATION IN A CLASS OF NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS By Steven T. Berry and Philip A. Haile March 2011 Revised April 2011 COWLES FOUNDATION DISCUSSION PAPER NO. 1787R COWLES FOUNDATION

### discuss how to describe points, lines and planes in 3 space.

Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position

### Chapter 3: The Multiple Linear Regression Model

Chapter 3: The Multiple Linear Regression Model Advanced Econometrics - HEC Lausanne Christophe Hurlin University of Orléans November 23, 2013 Christophe Hurlin (University of Orléans) Advanced Econometrics

### 24. The Branch and Bound Method

24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NP-complete. Then one can conclude according to the present state of science that no

### MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

### Properties of BMO functions whose reciprocals are also BMO

Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a non-negative BMO-function w, whose reciprocal is also in BMO, belongs to p> A p,and

### BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

### Ideal Class Group and Units

Chapter 4 Ideal Class Group and Units We are now interested in understanding two aspects of ring of integers of number fields: how principal they are (that is, what is the proportion of principal ideals

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### EMPIRICAL RISK MINIMIZATION FOR CAR INSURANCE DATA

EMPIRICAL RISK MINIMIZATION FOR CAR INSURANCE DATA Andreas Christmann Department of Mathematics homepages.vub.ac.be/ achristm Talk: ULB, Sciences Actuarielles, 17/NOV/2006 Contents 1. Project: Motor vehicle

### Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

### Moving Least Squares Approximation

Chapter 7 Moving Least Squares Approimation An alternative to radial basis function interpolation and approimation is the so-called moving least squares method. As we will see below, in this method the

### Exact Nonparametric Tests for Comparing Means - A Personal Summary

Exact Nonparametric Tests for Comparing Means - A Personal Summary Karl H. Schlag European University Institute 1 December 14, 2006 1 Economics Department, European University Institute. Via della Piazzuola

### LINEAR ALGEBRA W W L CHEN

LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,

### FIRST YEAR CALCULUS. Chapter 7 CONTINUITY. It is a parabola, and we can draw this parabola without lifting our pencil from the paper.

FIRST YEAR CALCULUS WWLCHENW L c WWWL W L Chen, 1982, 2008. 2006. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It It is is

### OPRE 6201 : 2. Simplex Method

OPRE 6201 : 2. Simplex Method 1 The Graphical Method: An Example Consider the following linear program: Max 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2

### Lecture 3: Linear methods for classification

Lecture 3: Linear methods for classification Rafael A. Irizarry and Hector Corrada Bravo February, 2010 Today we describe four specific algorithms useful for classification problems: linear regression,

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Monotonicity Hints. Abstract

Monotonicity Hints Joseph Sill Computation and Neural Systems program California Institute of Technology email: joe@cs.caltech.edu Yaser S. Abu-Mostafa EE and CS Deptartments California Institute of Technology

### Basics of Statistical Machine Learning

CS761 Spring 2013 Advanced Machine Learning Basics of Statistical Machine Learning Lecturer: Xiaojin Zhu jerryzhu@cs.wisc.edu Modern machine learning is rooted in statistics. You will find many familiar

### Representation of functions as power series

Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions

### Linear Systems. Singular and Nonsingular Matrices. Find x 1, x 2, x 3 such that the following three equations hold:

Linear Systems Example: Find x, x, x such that the following three equations hold: x + x + x = 4x + x + x = x + x + x = 6 We can write this using matrix-vector notation as 4 {{ A x x x {{ x = 6 {{ b General

### Summary of week 8 (Lectures 22, 23 and 24)

WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry

### Vector and Matrix Norms

Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

### A New Quantitative Behavioral Model for Financial Prediction

2011 3rd International Conference on Information and Financial Engineering IPEDR vol.12 (2011) (2011) IACSIT Press, Singapore A New Quantitative Behavioral Model for Financial Prediction Thimmaraya Ramesh

### A Study on the Comparison of Electricity Forecasting Models: Korea and China

Communications for Statistical Applications and Methods 2015, Vol. 22, No. 6, 675 683 DOI: http://dx.doi.org/10.5351/csam.2015.22.6.675 Print ISSN 2287-7843 / Online ISSN 2383-4757 A Study on the Comparison

### A Tutorial on Support Vector Machines for Pattern Recognition

c,, 1 43 () Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. A Tutorial on Support Vector Machines for Pattern Recognition CHRISTOPHER J.C. BURGES Bell Laboratories, Lucent Technologies

### CHAPTER 1 Splines and B-splines an Introduction

CHAPTER 1 Splines and B-splines an Introduction In this first chapter, we consider the following fundamental problem: Given a set of points in the plane, determine a smooth curve that approximates the

### We call this set an n-dimensional parallelogram (with one vertex 0). We also refer to the vectors x 1,..., x n as the edges of P.

Volumes of parallelograms 1 Chapter 8 Volumes of parallelograms In the present short chapter we are going to discuss the elementary geometrical objects which we call parallelograms. These are going to

### The Trimmed Iterative Closest Point Algorithm

Image and Pattern Analysis (IPAN) Group Computer and Automation Research Institute, HAS Budapest, Hungary The Trimmed Iterative Closest Point Algorithm Dmitry Chetverikov and Dmitry Stepanov http://visual.ipan.sztaki.hu

### Table 1: Summary of the settings and parameters employed by the additive PA algorithm for classification, regression, and uniclass.

Online Passive-Aggressive Algorithms Koby Crammer Ofer Dekel Shai Shalev-Shwartz Yoram Singer School of Computer Science & Engineering The Hebrew University, Jerusalem 91904, Israel {kobics,oferd,shais,singer}@cs.huji.ac.il

### Lectures 5-6: Taylor Series

Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

### CAPM, Arbitrage, and Linear Factor Models

CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose mean-variance e cient portfolios. By equating these investors

### Baum s Algorithm Learns Intersections of Halfspaces with respect to Log-Concave Distributions

Baum s Algorithm Learns Intersections of Halfspaces with respect to Log-Concave Distributions Adam R. Klivans 1, Philip M. Long 2, and Alex K. Tang 3 1 UT-Austin, klivans@cs.utexas.edu 2 Google, plong@google.com

### Separation Properties for Locally Convex Cones

Journal of Convex Analysis Volume 9 (2002), No. 1, 301 307 Separation Properties for Locally Convex Cones Walter Roth Department of Mathematics, Universiti Brunei Darussalam, Gadong BE1410, Brunei Darussalam

### 10. Proximal point method

L. Vandenberghe EE236C Spring 2013-14) 10. Proximal point method proximal point method augmented Lagrangian method Moreau-Yosida smoothing 10-1 Proximal point method a conceptual algorithm for minimizing