Motivated by some applications in signal processing and machine learning, we consider two convex optimization problems where, given a cone $K$, a norm $\|\cdot\|$ and a smooth convex function $f$, we want either 1) to minimize the norm over the intersection of the cone and a level set of $f$, or 2) to minimize over the cone the sum of $f$ and a multiple of the norm. We focus on the case where (a) the dimension of the problem is too large to allow for interior point algorithms, (b) $\|\cdot\|$ is "too complicated" to allow for computationally cheap Bregman projections required in the first-order proximal gradient algorithms. On the other hand, we assume that {it is relatively easy to minimize linear forms over the intersection of $K$ and the unit $\|\cdot\|$-ball}. Motivating examples are given by the nuclear norm with $K$ being the entire space of matrices, or the positive semidefinite cone in the space of symmetric matrices, and the Total Variation norm on the space of 2D images. We discuss versions of the Conditional Gradient algorithm capable to handle our problems of interest, provide the related theoretical efficiency estimates and outline some applications.

Many data sets contain rich information about objects, as well as pairwise relations between them. For instance, in networks of websites, scientific papers, and other documents, each node has content consisting of a collection of words, as well as hyperlinks or citations to other nodes. In order to perform inference on such data sets, and make predictions and recommendations, it is useful to have models that are able to capture the processes which generate the text at each node and the links between them. In this paper, we combine classic ideas in topic modeling with a variant of the mixed-membership block model recently developed in the statistical physics community. The resulting model has the advantage that its parameters, including the mixture of topics of each document and the resulting overlapping communities, can be inferred with a simple and scalable expectation-maximization algorithm. We test our model on three data sets, performing unsupervised topic classification and link prediction. For both tasks, our model outperforms several existing state-of-the-art methods, achieving higher accuracy with significantly less computation, analyzing a data set with 1.3 million words and 44 thousand links in a few minutes.

A primary motivation for reasoning under uncertainty is to derive decisions in the face of inconclusive evidence. However, Shafer's theory of belief functions, which explicitly represents the underconstrained nature of many reasoning problems, lacks a formal procedure for making decisions. Clearly, when sufficient information is not available, no theory can prescribe actions without making additional assumptions. Faced with this situation, some assumption must be made if a clearly superior choice is to emerge. In this paper we offer a probabilistic interpretation of a simple assumption that disambiguates decision problems represented with belief functions. We prove that it yields expected values identical to those obtained by a probabilistic analysis that makes the same assumption. In addition, we show how the decision analysis methodology frequently employed in probabilistic reasoning can be extended for use with belief functions.

We introduce a graceful approach to probabilistic inference called bounded conditioning. Bounded conditioning monotonically refines the bounds on posterior probabilities in a belief network with computation, and converges on final probabilities of interest with the allocation of a complete resource fraction. The approach allows a reasoner to exchange arbitrary quantities of computational resource for incremental gains in inference quality. As such, bounded conditioning holds promise as a useful inference technique for reasoning under the general conditions of uncertain and varying reasoning resources. The algorithm solves a probabilistic bounding problem in complex belief networks by breaking the problem into a set of mutually exclusive, tractable subproblems and ordering their solution by the expected effect that each subproblem will have on the final answer. We introduce the algorithm, discuss its characterization, and present its performance on several belief networks, including a complex model for reasoning about problems in intensive-care medicine.

Considerable attention has been given to the problem of non-monotonic reasoning in a belief function framework. Earlier work (M. Ginsberg) proposed solutions introducing meta-rules which recognized conditional independencies in a probabilistic sense. More recently an e-calculus formulation of default reasoning (J. Pearl) shows that the application of Dempster's rule to a non-monotonic situation produces erroneous results. This paper presents a new belief function interpretation of the problem which combines the rules in a way which is more compatible with probabilistic results and respects conditions of independence necessary for the application of Dempster's combination rule. A new general framework for combining conflicting evidence is also proposed in which the normalization factor becomes modified. This produces more intuitively acceptable results.

This paper describes a generalization of previous methods for constructing tree-structured belief network with hidden variables. The major new feature of the described method is the ability to produce a tree decomposition even when there are errors in the correlation data among the input variables. This is an important extension of existing methods since the correlational co efficients usually cannot be measured with precision. The technique involves using a greedy search algorithm that locally minimizes an error function.

A. Abstract Control Strategies for hierachical treelike probabilistic inference networks are formulated and investigated. Strategies that utilize staged look-ahead and temporary focus on subgoals are formalized and refined using the Depth Vector concept that serves as a tool for defining the'virtual tree' regarded by the control strategy. The concept is illustrated by four types of control strategies for three-level trees that are characterized according to their Depth Vector, and according to the way they consider intermediate nodes and the role that they let these nodes play. INFERENTl is a computerized inference system written in Prolog, which provides tools for exercising a variety of control strategies. The system also provides tools for simulating test data and for comparing the relative average performance under different strategies.

Bayesian networks provide a probabilistic semantics for qualitative assertions about likelihood. A qualitative reasoner based on an algebra over these assertions can derive further conclusions about the influence of actions. While the conclusions are much weaker than those computed from complete probability distributions, they are still valuable for suggesting potential actions, eliminating obviously inferior plans, identifying important tradeoffs, and explaining probabilistic models.

(l) I have enough evidence to render the sentence S probable. (la) So, relative to what I know, it is rational of me to believe S. (2) Now that I have more evidence, S may no longer be probable. (2a) So now, relative to what I know, it is not rational of me to believe S. These seem a perfectly ordinary, common sense, pair of situations. Generally and vaguely, I take them to embody what I shall call probabilistic inference. This form of inference is clearly non-monotonic. Relatively few people have taken this form of inference, based on high probability, to serve as a foundation for non-monotonic logic or for a logical or defeasible inference. There are exceptions: Jane Nutter [16] thinks that sometimes probability has something to do with non-monotonic reasoning. Judea Pearl [ 17] has recently been exploring the possibility. There are any number of people whom one might call probability enthusiasts who feel that probability provides all the answers by itself, with no need of help from logic. Cheeseman [1], Henrion [5] and others think it useful to look at a distribution of probabilities over a whole algebra of statements, to update that distribution in the light of new evidence, and to use the latest updated distribution of probability over the algebra as a basis for planning and decision making. A slightly weaker form of this approach is captured by Nilsson [15], where one assumes certain probabilities for certain statements, and infers the probabilities, or constraints on the probabilities of other statement. None of this corresponds to what I call probabilistic inference. All of the inference that is taking place, either in Bayesian updating, or in probabilistic logic, is strictly deductive. Deductive inference, particularly that concerned with the distribution of classical probabilities or chances, is of great importance. But this is not to say that there is no important role for what earlier logicians have called "ampliative" or "inductive" or "scientific" inference, in which the conclusion goes beyond the premises, asserts more than do the premises. This depends on what David Israel [6] has called "real rules of inference". It is characteristic of any such logic or inference procedure that it can go wrong: that statements accepted at one point may be rejected at a later point. Research underlying the results reported here has been partially supported by the Signals Warfare Center of the United States Army.

We propose a novel general algorithm LHAC that efficiently uses second-order information to train a class of large-scale l1-regularized problems. Our method executes cheap iterations while achieving fast local convergence rate by exploiting the special structure of a low-rank matrix, constructed via quasi-Newton approximation of the Hessian of the smooth loss function. A greedy active-set strategy, based on the largest violations in the dual constraints, is employed to maintain a working set that iteratively estimates the complement of the optimal active set. This allows for smaller size of subproblems and eventually identifies the optimal active set. Empirical comparisons confirm that LHAC is highly competitive with several recently proposed state-of-the-art specialized solvers for sparse logistic regression and sparse inverse covariance matrix selection.