arXiv subjects
Publications and source records attributed to Mark Schmidt.
In this note, we present a new averaging technique for the projected stochastic subgradient method. By using a weighted average with a weight of t+1 for each iterate w_t at iteration t, we obtain the convergence rate of O(1/t) with both an easy proof and an easy implementation. The new scheme is compared empirically to existing techniques, with similar performance behavior.
We outline a representation for discrete multivariate distributions in terms of interventional potential functions that are globally normalized. This representation can be used to model the effects of interventions, and the independence properties encoded in this model can be represented as a directed graph that allows cycles. In addition to discussing inference and sampling with this representation, we give an exponential family parametrization that allows parameter estimation to be stated as a convex optimization problem; we also give a convex relaxation of the task of simultaneous parameter and structure learning using group l1-regularization. The model is evaluated on simulated data and intracellular flow cytometry data.
Recently it has become popular to learn sparse Gaussian graphical models (GGMs) by imposing l1 or group l1,2 penalties on the elements of the precision matrix. Thispenalized likelihood approach results in a tractable convex optimization problem. In this paper, we reinterpret these results as performing MAP estimation under a novel prior which we call the group l1 and l1,2 positivedefinite matrix distributions. This enables us to build a hierarchical model in which the l1 regularization terms vary depending on which group the entries are assigned to, which in turn allows us to learn block structured sparse GGMs with unknown group assignments. Exact inference in this hierarchical model is intractable, due to the need to compute the normalization constant of these matrix distributions. However, we derive upper bounds on the partition functions, which lets us use fast variational inference (optimizing a lower bound on the joint posterior). We show that on two real world data sets (motion capture and financial data), our method which infers the block structure outperforms a method that uses a fixed block structure, which in turn outperforms baseline methods that ignore block structure.
We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function using proximal-gradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity operator with respect to the non-smooth term. We show that both the basic proximal-gradient method and the accelerated proximal-gradient method achieve the same convergence rate as in the error-free case, provided that the errors decrease at appropriate rates.Using these rates, we perform as well as or better than a carefully chosen fixed error level on a set of structured sparsity problems.
We present alpha-expansion beta-shrink moves, a simple generalization of the widely-used alpha-beta swap and alpha-expansion algorithms for approximate energy minimization. We show that in a certain sense, these moves dominate both alpha-beta-swap and alpha-expansion moves, but unlike previous generalizations the new moves require no additional assumptions and are still solvable in polynomial-time. We show promising experimental results with the new moves, which we believe could be used in any context where alpha-expansions are currently employed.