We consider the problem of learning classifiers for structurally incomplete data, where some objects have a subset of features inherently absent due to complex relationships between the features. The common approach for handling missingfeatures is to begin with a preprocessing phase that completes the missing features, and then use a standard classification procedure. In this paper we show how incomplete data can be classified directly without any completion of the missing features using a max-margin learning framework. Weformulate this task using a geometrically-inspired objective function, and discuss two optimization approaches: The linearly separable case is written as a set of convex feasibility problems, and the non-separable case has a non-convex objective that we optimize iteratively. By avoiding the pre-processing phase in which the data is completed, these approaches offer considerable computational savings. More importantly, we show that by elegantly handlingcomplex patterns of missing values, our approach is both competitive with other methods when the values are missing at random and outperforms them when the missing values have nontrivial structure. We demonstrate our results on two real-world problems: edge prediction in metabolic pathways, and automobile detection in natural images.

In general, the problem of computing a maximum a posteriori (MAP) assignment in a Markov random eld (MRF) is computationally intractable. However, in certain subclasses of MRF, an optimal or close-to-optimal assignment can be found very ef ciently using combinatorial optimization algorithms: certain MRFs with mutual exclusion constraints can be solved using bipartite matching, and MRFs with regular potentials can be solved using minimum cut methods. However, these solutions do not apply to the many MRFs that contain such tractable components as sub-networks, but also other non-complying potentials.

We present a method for learning a low-dimensional representation which is shared across a set of multiple related tasks. The method builds upon the wellknown 1-normregularization problem using a new regularizer which controls the number of learned features common for all the tasks. We show that this problem is equivalent to a convex optimization problem and develop an iterative algorithm for solving it. The algorithm has a simple interpretation: it alternately performs a supervised and an unsupervised step, where in the latter step we learn commonacross-tasks representationsand in the former step we learn task-specific functions using these representations. We report experiments on a simulated and a real data set which demonstrate that the proposed method dramatically improves the performance relativeto learning each task independently. Our algorithm can also be used, as a special case, to simply select - not learn - a few common features across the tasks.

We describe Hidden Semi-Markov Support Vector Machines (SHM SVMs), an extension of HM SVMs to semi-Markov chains. This allows us to predict segmentations ofsequences based on segment-based features measuring properties such as the length of the segment. We propose a novel technique to partition the problem into sub-problems. The independently obtained partial solutions can then be recombined in an efficient way, which allows us to solve label sequence learning problemswith several thousands of labeled sequences. We have tested our algorithm for predicting gene structures, an important problem in computational biology. Results on a well-known model organism illustrate the great potential of SHM SVMs in computational biology.

Planning in partially observable domains is a notoriously difficult problem. However, inmany real-world scenarios, planning can be simplified by decomposing the task into a hierarchy of smaller planning problems. Several approaches have been proposed to optimize a policy that decomposes according to a hierarchy specified a priori. In this paper, we investigate the problem of automatically discovering the hierarchy. More precisely, we frame the optimization of a hierarchical policy as a non-convex optimization problem that can be solved with general nonlinear solvers, a mixed-integer nonlinear approximation or a form of bounded hierarchical policyiteration. By encoding the hierarchical structure as variables of the optimization problem, we can automatically discover a hierarchy. Our method is flexible enough to allow any parts of the hierarchy to be specified based on prior knowledge while letting the optimization discover the unknown parts. It can also discover hierarchical policies, including recursive policies, that are more compact (potentially infinitely fewer parameters) and often easier to understand given the decomposition induced by the hierarchy.

Many computer aided diagnosis (CAD) problems can be best modelled as a multiple-instance learning (MIL) problem with unbalanced data: i.e., the training data typically consists of a few positive bags, and a very large number of negative instances.Existing MIL algorithms are much too computationally expensive for these datasets. We describe CH, a framework for learning a Convex Hull representation of multiple instances that is significantly faster than existing MIL algorithms. Our CH framework applies to any standard hyperplane-based learning algorithm, and for some algorithms, is guaranteed to find the global optimal solution. Experimentalstudies on two different CAD applications further demonstrate that the proposed algorithm significantly improves diagnostic accuracy when compared toboth MIL and traditional classifiers. Although not designed for standard MIL problems (which have both positive and negative bags and relatively balanced datasets),comparisons against other MIL methods on benchmark problems also indicate that the proposed method is competitive with the state-of-the-art.

Computation of a satisfactory control policy for a Markov decision process when the parameters of the model are not exactly known is a problem encountered in many practical applications. The traditional robust approach is based on a worstcase analysisand may lead to an overly conservative policy. In this paper we consider thetradeoff between nominal performance and the worst case performance over all possible models. Based on parametric linear programming, we propose a method that computes the whole set of Pareto efficient policies in the performancerobustness planewhen only the reward parameters are subject to uncertainty. In the more general case when the transition probabilities are also subject to error, we show that the strategy with the "optimal" tradeoff might be non-Markovian and hence is in general not tractable.

We describe and analyze an algorithmic framework for online classification where each online trial consists of multiple prediction tasks that are tied together. We tackle the problem of updating the online hypothesis by defining a projection problem in which each prediction task corresponds to a single linear constraint. These constraints are tied together through a single slack parameter. We then introduce ageneral method for approximately solving the problem by projecting simultaneously and independently on each constraint which corresponds to a prediction sub-problem,and then averaging the individual solutions. We show that this approach constitutes a feasible, albeit not necessarily optimal, solution for the original projection problem. We derive concrete simultaneous projection schemes and analyze them in the mistake bound model. We demonstrate the power of the proposed algorithm in experiments with online multiclass text categorization. Our experiments indicate that a combination of class-dependent features with the simultaneous projection method outperforms previously studied algorithms.

We describe a nonnegative variant of the "Sparse PCA" problem. The goal is to create a low dimensional representation from a collection of points which on the one hand maximizes the variance of the projected points and on the other uses only parts of the original coordinates, and thereby creating a sparse representation. Whatdistinguishes our problem from other Sparse PCA formulations is that the projection involves only nonnegative weights of the original coordinates -- a desired quality in various fields, including economics, bioinformatics and computer vision.Adding nonnegativity contributes to sparseness, where it enforces a partitioning of the original coordinates among the new axes. We describe a simple yetefficient iterative coordinate-descent type of scheme which converges to a local optimum of our optimization criteria, giving good results on large real world datasets.

In this article we propose a method that can deal with certain combinatorial reinforcement learning tasks. We demonstrate the approach in the popular Ms. Pac-Man game. We define a set of high-level observation and action modules, from which rule-based policies are constructed automatically. In these policies, actions are temporally extended, and may work concurrently. The policy of the agent is encoded by a compact decision list. The components of the list are selected from a large pool of rules, which can be either hand-crafted or generated automatically. A suitable selection of rules is learnt by the cross-entropy method, a recent global optimization algorithm that fits our framework smoothly. Cross-entropy-optimized policies perform better than our hand-crafted policy, and reach the score of average human players. We argue that learning is successful mainly because (i) policies may apply concurrent actions and thus the policy space is sufficiently rich, (ii) the search is biased towards low-complexity policies and therefore, solutions with a compact description can be found quickly if they exist.