Understanding Machine Learning: From Theory to Algorithms
21.4 The Online Perceptron Algorithm 301 theorem21.15 The Online Gradient Descent algorithm enjoys the following regret bound fo ...
302 Online Learning w∈Rd}. In Section 9.1.2 we have presented the batch version of the Perceptron, which aims to solve the ERM p ...
21.4 The Online Perceptron Algorithm 303 function and we can takevt= 0. Otherwise, it is easy to verify thatvt=−ytxt is in∂ft(w( ...
304 Online Learning theorem21.16 Suppose that the Perceptron algorithm runs on a sequence (x 1 ,y 1 ),...,(xT,yT)and letR= maxt‖ ...
21.6 Bibliographic Remarks 305 21.6 Bibliographic Remarks The Standard Optimal Algorithm was derived by the seminal work of Lit- ...
306 Online Learning Show that if the regret ofAon each period of 2mrounds is at mostα √ 2 m, then the total regret is at most √ ...
22 Clustering Clustering is one of the most widely used techniques for exploratory data anal- ysis. Across all disciplines, from ...
308 Clustering In contrast, a clustering method that emphasizes not having far-away points share the same cluster (e.g., the 2-m ...
Clustering 309 This phenomenon is not just artificial but occurs in real applications. A given set of objects can be clustered i ...
310 Clustering In the following we survey some of the most popular clustering methods. In the last section of this chapter we re ...
22.2 k-Means and Other Cost Minimization Clusterings 311 b c d e a {a} {b} {c} {d} {e} {b,c} {d,e} {b,c,d,e} {a,b,c,d,e} The sin ...
312 Clustering parameter. In practice, it is often up to the user of the clustering algorithm to choose the parameterkthat is mo ...
22.2 k-Means and Other Cost Minimization Clusterings 313 An example where such an objective makes sense is thefacility location ...
314 Clustering Proof To simplify the notation, let us use the shorthandG(C 1 ,...,Ck) for the k-means objective, namely, G(C 1 , ...
22.3 Spectral Clustering 315 22.3 Spectral Clustering Often, a convenient way to represent the relationships between points in a ...
316 Clustering definition22.2 (Unnormalized Graph Laplacian) Theunnormalized graph Laplacianis them×mmatrixL=D−WwhereDis a diago ...
22.4 Information Bottleneck* 317 22.3.3 Unnormalized Spectral Clustering Unnormalized Spectral Clustering Input:W∈Rm,m ; Number ...
318 Clustering parameter, and the minimization is over all possible probabilistic assignments of points to clusters. Intuitively ...
22.5 A High Level View of Clustering 319 Consistency (Co)Ifdandd′ are dissimilarity functions overX, such that for everyx,y∈ X, ...
320 Clustering Alternatively, one can relax the Consistency property. For example, say that two clusteringsC = (C 1 ,...Ck) andC ...
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