We propose a novel boosting approach to multi-class classification problems, in which multiple classes are distinguished by a set of random projection matrices in essence. The approach uses random projections to alleviate the proliferation of binary classifiers typically required to perform multi-class classification. The result is a multi-class classifier with a single vector-valued parameter, irrespective of the number of classes involved. Two variants of this approach are proposed. The first method randomly projects the original data into new spaces, while the second method randomly projects the outputs of learned weak classifiers. These methods are not only conceptually simple but also effective and easy to implement. A series of experiments on synthetic, machine learning and visual recognition data sets demonstrate that our proposed methods compare favorably to existing multi-class boosting algorithms in terms of both the convergence rate and classification accuracy.
Learning a matrix of coefficients in a single optimization problem One learns a linear ensemble for each class. Given a test example, the label is predicted by @math @math Each row of the matrix @math corresponds to one of the classes. The sample is assigned to the class whose row has the largest value of the weighted combination. To learn the matrix @math , one can formulate the problem in the framework of multi-class maximum-margin learning. Shen and Hao show that the large-margin multi-class boosting can be implemented using column generation @cite_11 .
@cite_11
2962750290 2112788936 2963185791 2030628896
A matrix M: A × X → −1,1 corresponds to the following learning problem: An unknown element x ∈ X is chosen uniformly at random. A learner tries to learn x from a stream of samples, (a1, b1), (a2, b2) …, where for every i, ai ∈ A is chosen uniformly at random and bi = M(ai,x). Assume that k, l, r are such that any submatrix of M of at least 2−k · |A| rows and at least 2−l · |X| columns, has a bias of at most 2−r. We show that any learning algorithm for the learning problem corresponding to M requires either a memory of size at least Ω(k · l ), or at least 2Ω(r) samples. The result holds even if the learner has an exponentially small success probability (of 2−Ω(r)). In particular, this shows that for a large class of learning problems, any learning algorithm requires either a memory of size at least Ω((log|X|) · (log|A|)) or an exponential number of samples, achieving a tight Ω((log|X|) · (log|A|)) lower bound on the size of the memory, rather than a bound of Ω(min (log|X|)2,(log|A|)2 ) obtained in previous works by Raz [FOCS’17] and Moshkovitz and Moshkovitz [ITCS’18]. Moreover, our result implies all previous memory-samples lower bounds, as well as a number of new applications. Our proof builds on the work of Raz [FOCS’17] that gave a general technique for proving memory samples lower bounds. We consider packing LP's with m rows where all constraint coefficients are normalized to be in the unit interval. The n columns arrive in random order and the goal is to set the corresponding decision variables irrevocably when they arrive to obtain a feasible solution maximizing the expected reward. Previous (1−e)-competitive algorithms require the right-hand side of the LP to be @math , a bound that worsens with the number of columns and rows. However, the dependence on the number of columns is not required in the single-row case and known lower bounds for the general case are also independent of n. Our goal is to understand whether the dependence on n is required in the multi-row case, making it fundamentally harder than the single-row version. We refute this by exhibiting an algorithm which is (1−e)-competitive as long as the right-hand sides are @math . Our techniques refine previous PAC-learning based approaches which interpret the online decisions as linear classifications of the columns based on sampled dual prices. The key ingredient of our improvement comes from a non-standard covering argument together with the realization that only when the columns of the LP belong to few 1-d subspaces we can obtain small such covers; bounding the size of the cover constructed also relies on the geometry of linear classifiers. General packing LP's are handled by perturbing the input columns, which can be seen as making the learning problem more robust. Multiclass classification problems such as image annotation can involve a large number of classes. In this context, confusion between classes can occur, and single label classification may be misleading. We provide in the present paper a general device that, given an unlabeled dataset and a score function defined as the minimizer of some empirical and convex risk, outputs a set of class labels, instead of a single one. Interestingly, this procedure does not require that the unlabeled dataset explores the whole classes. Even more, the method is calibrated to control the expected size of the output set while minimizing the classification risk. We show the statistical optimality of the procedure and establish rates of convergence under the Tsybakov margin condition. It turns out that these rates are linear on the number of labels. We apply our methodology to convex aggregation of confidence sets based on the V-fold cross validation principle also known as the superlearning principle. We illustrate the numerical performance of the procedure on real data and demonstrate in particular that with moderate expected size, w.r.t. the number of labels, the procedure provides significant improvement of the classification risk. We present a framework based on convex optimization and spectral regularization to perform learning when feature observations are multidimensional arrays (tensors). We give a mathematical characterization of spectral penalties for tensors and analyze a unifying class of convex optimization problems for which we present a provably convergent and scalable template algorithm. We then specialize this class of problems to perform learning both in a transductive as well as in an inductive setting. In the transductive case one has an input data tensor with missing features and, possibly, a partially observed matrix of labels. The goal is to both infer the missing input features as well as predict the missing labels. For induction, the goal is to determine a model for each learning task to be used for out of sample prediction. Each training pair consists of a multidimensional array and a set of labels each of which corresponding to related but distinct tasks. In either case the proposed technique exploits precise low multilinear rank assumptions over unknown multidimensional arrays; regularization is based on composite spectral penalties and connects to the concept of Multilinear Singular Value Decomposition. As a by-product of using a tensor-based formalism, our approach allows one to tackle the multi-task case in a natural way. Empirical studies demonstrate the merits of the proposed methods.