We study the problem of estimating multiple linear regression equations for the purpose of both prediction and variable selection. Following recent work on multi-task learning [2008], we assume that the regression vectors share the same sparsity pattern. This means that the set of relevant predictor variables is the same across the different equations. This assumption leads us to consider the Group Lasso as a candidate estimation method. We show that this estimator enjoys nice sparsity oracle inequalities and variable selection properties. The results hold under a certain restricted eigenvalue condition and a coherence condition on the design matrix, which naturally extend recent work in [2007], Lounici [2008]. In particular, in the multi-task learning scenario, in which the number of tasks can grow, we are able to remove completely the effect of the number of predictor variables in the bounds. Finally, we show how our results can be extended to more general noise distributions, of which we only require the variance to be finite.
In order to study the statistical properties of this estimator, it is useful to derive the optimality condition for a solution of the problem . Since the objective function in is convex, @math is a solution of if and only if @math (the @math -dimensional zero vector) belongs to the subdifferential of the objective function. In turn, this condition is equivalent to the requirement that @math where @math denotes the subdifferential (see, for example, @cite_4 for more information on convex analysis). Note that Thus, @math is a solution of if and only if
@cite_4
2081956522 2517669009 2604480575 2473549844
An algorithm for solving the problem: minimize @math (a convex function) subject to @math , @math , each @math a concave function, is presented. Specifically, the function [ P [ x,t,r_k ] f( x ) + r_k^ - 1 [ g_i ( x ) - t_i ] ^2 ] is minimized over all x, nonnegative t, for a strictly decreasing null sequence @math . This extends the work of T. Pietrzykowski [5]. It is proved that for every @math , there exists a finite point @math which minimizes P, and which solves the convex programming problem as @math . This algorithm is similar to the Sequential Unconstrained Minimization Technique (SUMT) [1] in that it solves the (Wolfe) dual programming problem [6]. It differs from SUMT in that (1) it approaches the optimum from the region of infeasibility (i.e., it is a relaxation technique), (2) it does not require a nonempty interior to the nonlinearly constrained region, (3) no separate feasibilit... In stochastic convex optimization the goal is to minimize a convex function @math over a convex set @math where @math is some unknown distribution and each @math in the support of @math is convex over @math . The optimization is commonly based on i.i.d. samples @math from @math . A standard approach to such problems is empirical risk minimization (ERM) that optimizes @math . Here we consider the question of how many samples are necessary for ERM to succeed and the closely related question of uniform convergence of @math to @math over @math . We demonstrate that in the standard @math setting of Lipschitz-bounded functions over a @math of bounded radius, ERM requires sample size that scales linearly with the dimension @math . This nearly matches standard upper bounds and improves on @math dependence proved for @math setting by Shalev- (2009). In stark contrast, these problems can be solved using dimension-independent number of samples for @math setting and @math dependence for @math setting using other approaches. We further show that our lower bound applies even if the functions in the support of @math are smooth and efficiently computable and even if an @math regularization term is added. Finally, we demonstrate that for a more general class of bounded-range (but not Lipschitz-bounded) stochastic convex programs an infinite gap appears already in dimension 2. We describe an @math -time algorithm for computing the exact probability that two @math -dimensional probabilistic point sets are linearly separable, for any fixed @math . A probabilistic point in @math -space is a normal point, but with an associated probability of existence; the existence probabilities of all points are independent. We also show that the @math -dimensional separability problem is equivalent to a @math -dimensional convex hull membership problem, which asks for the probability that a query point lies inside the convex hull of @math probabilistic points. Using this reduction, we improve the current best bound for the convex hull membership problem by a factor of @math . In addition, our algorithms can handle input degeneracies in which more than @math points may lie on a @math -dimensional subspace, thus resolving an open problem in 2013. Finally, we prove lower bounds for the separability problem via a reduction from the @math -SUM problem, which show in particular that our @math algorithms for @math -dimensional separability and @math -dimensional convex hull membership are nearly optimal. We consider the adversarial convex bandit problem and we build the first @math -time algorithm with @math -regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves @math -regret, and we show that a simple variant of this algorithm can be run in @math -time per step at the cost of an additional @math factor in the regret. These results improve upon the @math -regret and @math -time result of the first two authors, and the @math -regret and @math -time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve @math -regret, and moreover that this regret is unimprovable (the current best lower bound being @math and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order @math .