Inverse classification is the process of manipulating an instance such that it is more likely to conform to a specific class. Past methods that address such a problem have shortcomings. Greedy methods make changes that are overly radical, often relying on data that is strictly discrete. Other methods rely on certain data points, the presence of which cannot be guaranteed. In this paper we propose a general framework and method that overcomes these and other limitations. The formulation of our method uses any differentiable classification function. We demonstrate the method by using Gaussian kernel SVMs. We constrain the inverse classification to occur on features that can actually be changed, each of which incurs an individual cost. We further subject such changes to fall within a certain level of cumulative change (budget). Our framework can also accommodate the estimation of features whose values change as a consequence of actions taken (indirectly changeable features). Furthermore, we propose two methods for specifying feature-value ranges that result in different algorithmic behavior. We apply our method, and a proposed sensitivity analysis-based benchmark method, to two freely available datasets: Student Performance, from the UCI Machine Learning Repository and a real-world cardiovascular disease dataset. The results obtained demonstrate the validity and benefits of our framework and method.
Framework refers to the constraints that govern recommendation feasibility. These are manifested in the literature as either @cite_11 @cite_23 @cite_5 or @cite_7 @cite_4 @cite_2 . Unconstrained problems lead to unrealistic recommendations that may also be very extreme (e.g., reduce your age by 30 years'). Constrained frameworks lead to more moderate and realistic recommendations. However, while @cite_7 @cite_4 focus on moderate objectives, they do not consider (1) what can cannot be changed, (2) how hard it might be to change and (3), cumulatively, how willing someone may be to make changes. In @cite_2 the authors consider (2), but do not consider (1) and (3). Additionally, in @cite_7 , the formulation of relies on data points which lie exactly on the separating hyperplane; there is not guarantee that such points exist in practice. In this work we propose a framework that considers (1), (2) and (3).
@cite_4 @cite_7 @cite_23 @cite_2 @cite_5 @cite_11
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We present a framework for robustly estimating registration between a 3D volume image and a 2D projection image and evaluate its precision and robustness in spine interventions for vertebral localization in the presence of anatomical deformation. The framework employs a normalized gradient information similarity metric and multi-start covariance matrix adaptation evolution strategy optimization with local-restarts, which provided improved robustness against deformation and content mismatch. The parallelized implementation allowed orders-of-magnitude acceleration in computation time and improved the robustness of registration via multi-start global optimization. Experiments involved a cadaver specimen and two CT datasets (supine and prone) and 36 C-arm fluoroscopy images acquired with the specimen in four positions (supine, prone, supine with lordosis, prone with kyphosis), three regions (thoracic, abdominal, and lumbar), and three levels of geometric magnification (1.7, 2.0, 2.4). Registration accuracy was evaluated in terms of projection distance error (PDE) between the estimated and true target points in the projection image, including 14 400 random trials (200 trials on the 72 registration scenarios) with initialization error up to ±200 mm and ±10°. The resulting median PDE was better than 0.1 mm in all cases, depending somewhat on the resolution of input CT and fluoroscopy images. The cadaver experiments illustrated the tradeoff between robustness and computation time, yielding a success rate of 99.993 in vertebral labeling (with 'success' defined as PDE <5 mm) using 1,718 664 ± 96 582 function evaluations computed in 54.0 ± 3.5 s on a mid-range GPU (nVidia, GeForce GTX690). Parameters yielding a faster search (e.g., fewer multi-starts) reduced robustness under conditions of large deformation and poor initialization (99.535 success for the same data registered in 13.1 s), but given good initialization (e.g., ±5 mm, assuming a robust initial run) the same registration could be solved with 99.993 success in 6.3 s. The ability to register CT to fluoroscopy in a manner robust to patient deformation could be valuable in applications such as radiation therapy, interventional radiology, and an assistant to target localization (e.g., vertebral labeling) in image-guided spine surgery. We propose a humanoid robot navigation planning framework that reuses previous experience to decrease planning time. The framework is intended for navigating complex unstructured environments using both palm and foot contacts. In a complex environment, discrete-search-based contact space planners trade-off between high branching factor and action flexibility. Although approaches such as weighted A∗, ARA∗ and ANA∗ could speed up the search by compromising on optimality, they can be very slow when the heuristic is inaccurate. In the proposed framework, an experience-retrieval module is added in parallel to ANA∗. This module collects previously-generated motion plans and clusters them based on contact pose similarity to form a motion plan library. To retrieve an appropriate plan from the library for a given environment, the framework uses a distance between the contact poses in the plan and environment surfaces. Candidate plans are then modified with local trajectory optimization until a plan fitting the query environment is found. Our experiments show that the proposed framework outperforms planning-from-scratch in success rate in unstructured environments by at least 28 and can navigate difficult environments such as rubble and narrow corridors. We present a general framework for defining nonmonotonic systems based on the notion of preferred maximal consistent subsets of the premises. This framework subsumes David Poole's THEORIST approach to default reasoning as a particular instance. A disadvantage of THEORIST is that it does not allow to represent priorities between defaults adequately (as distinct from blocking defaults in specific situations). We therefore propose two generalizations of Poole's system: in the first generalization several layers of possible hypotheses representing different degrees of reliability are introduced. In a second further generalization a partial ordering between premises is used to distinguish between more and less reliable formulas. In both approaches a formula is provable from a theory if it is possible to construct a consistent argument for it based on the most reliable hypotheses. This allows for a simple representation of priorities between defaults. The Necessary Maximality Principle for c.c.c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. The Necessary Maximality Principle for c.c.c. forcing, denoted 2mpccc(R), asserts that any statement about a real in a c.c.c. extension that could be- come true in a further c.c.c. extension and remain true in all subsequent c.c.c. extensions, is already true in the minimal extension containing the real. We show that this principle is equiconsistent with the existence of a weakly compact cardinal. The principle is one of a family of principles considered in (Ham03) (build- ing on ideas of (Cha00) and overlapping with independent work in (SV01)). MSC: 03E55, 03E40. Keywords: forcing axiom, ccc forcing, weakly compact cardinal. The first author is affiliated with the College of Staten Islandof CUNY and The CUNY Graduate Center, and his research has been supported by grants from the Research Foun- dation of CUNY and the National Science Foundation DMS-9970993. The research of the second author has been partially supported by National Science Foundation grant DMS-9970255. Most tasks require a person or an automated system to reason -- to reach conclusions based on available information. The framework of probabilistic graphical models, presented in this book, provides a general approach for this task. The approach is model-based, allowing interpretable models to be constructed and then manipulated by reasoning algorithms. These models can also be learned automatically from data, allowing the approach to be used in cases where manually constructing a model is difficult or even impossible. Because uncertainty is an inescapable aspect of most real-world applications, the book focuses on probabilistic models, which make the uncertainty explicit and provide models that are more faithful to reality. Probabilistic Graphical Models discusses a variety of models, spanning Bayesian networks, undirected Markov networks, discrete and continuous models, and extensions to deal with dynamical systems and relational data. For each class of models, the text describes the three fundamental cornerstones: representation, inference, and learning, presenting both basic concepts and advanced techniques. Finally, the book considers the use of the proposed framework for causal reasoning and decision making under uncertainty. The main text in each chapter provides the detailed technical development of the key ideas. Most chapters also include boxes with additional material: skill boxes, which describe techniques; case study boxes, which discuss empirical cases related to the approach described in the text, including applications in computer vision, robotics, natural language understanding, and computational biology; and concept boxes, which present significant concepts drawn from the material in the chapter. Instructors (and readers) can group chapters in various combinations, from core topics to more technically advanced material, to suit their particular needs. We consider the problem of using a large unlabeled sample to boost performance of a learning algorit,hrn when only a small set of labeled examples is available. In particular, we consider a problem setting motivated by the task of learning to classify web pages, in which the description of each example can be partitioned into two distinct views. For example, the description of a web page can be partitioned into the words occurring on that page, and the words occurring in hyperlinks t,hat point to that page. We assume that either view of the example would be sufficient for learning if we had enough labeled data, but our goal is to use both views together to allow inexpensive unlabeled data to augment, a much smaller set of labeled examples. Specifically, the presence of two distinct views of each example suggests strategies in which two learning algorithms are trained separately on each view, and then each algorithm’s predictions on new unlabeled examples are used to enlarge the training set of the other. Our goal in this paper is to provide a PAC-style analysis for this setting, and, more broadly, a PAC-style framework for the general problem of learning from both labeled and unlabeled data. We also provide empirical results on real web-page data indicating that this use of unlabeled examples can lead to significant improvement of hypotheses in practice. *This research was supported in part by the DARPA HPKB program under contract F30602-97-1-0215 and by NSF National Young investigator grant CCR-9357793. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. TO copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and or a fee. COLT 98 Madison WI USA Copyright ACM 1998 l-58113-057--0 98 7... 5.00 92 Tom Mitchell School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213-3891 mitchell+@cs.cmu.edu Compressive sensing (CS) is a framework whereby one performs n non-adaptive measurements to constitute an n-dimensional vector v, with v used to recover an m-dimensional approximation ^ u to a desired m-dimensional signal u, with n ? m; this is performed under the assumption that u is sparse in the basis represented by the matrix “, the columns of which define discrete basis vectors. It has been demonstrated that with appropriate design of the compressive measurements used to define v, the decompressive mapping v ! ^ u may be performed with error kui^ uk 2 having asymptotic properties (large n and m > n) analogous to those of the best adaptive transform-coding algorithm applied in the basis “. The mapping v ! ^ constitutes an inverse problem, often solved using ‘1 regularization or related techniques. In most previous research, if multiple compressive measurements fvigi=1;M are performed, each of the associated f^ uigi=1;M are recovered one at a time, independently. In many applications the M “tasks” defined by the mappings vi ! ^ ui are not statistically independent, and it may be possible to improve the performance of the inversion if statistical inter-relationships are exploited. In this paper we address this problem within a multi-task learning setting, wherein the mapping vi ! ^ ui for each task corresponds to inferring the parameters (here, wavelet coefficients) associated with the desired signal ui, and a shared prior is placed across all of the M tasks. In this multi-task learning framework data from all M tasks contribute toward inferring a posterior on the hyperparameters, and once the shared prior is thereby inferred, the data from each of the M individual tasks is then employed to estimate the task-dependent wavelet coefficients. An empirical Bayes procedure and fast inference algorithm is developed. Example results are presented on several data sets. Trajectory and kinematics of drawing movements are mutually constrained by functional relationships that reduce the degrees of freedom of the hand-arm system. Previous investigations of these relationships are extended here by considering their development in children between 5 and 12 years of age. Performances in a simple motor task—the continuous tracing of elliptic trajectories—demonstrate that both the phenomenon of isochrony (increase of the average movement velocity with the linear extent of the trajectory) and the so-called two-thirds power law (relation between tangential velocity and curvature) are qualitatively present already at the age of 5. The quantitative aspects of these regularities evolve with age, however, and steady-state adult performance is not attained even by the oldest children. The power-law formalism developed in previous reports is generalized to encompass these developmental aspects of the control of movement. Two general frameworks are currently available to conceptualize the motor-control problem. Broadly, the two frameworks differ in the answer that they give to the question "Where do form and structure come from?" According to the motor-program view (e.g., Schmidt, 1988), form and structure in a movement come from a central abstract representation of the intended result. Thus, making a movement entails a detailed mapping of this representation into an appropriate motor plan. The competing task-dynamic view (e.g., Kugler, 1989; Saltzman & Kelso, 1987) emphasizes instead the global morphogenetic power of the nonlinearities of the motor system. These two views need not be mutually exclusive. In fact, different classes of movements may require different conceptual frameworks for their analysis. At any rate (at least in the case of drawing movements considered here), an internal representation of the intended trajectory and kinematics must be available before execution. Consequently, much effort is being devoted to understanding how this internal model is translated into a set of motor commands. In this respect, one basic conceptual difficulty is because the geometry of the arm-forearm-hand system affords a large number of degrees of freedom. Provided that there are no constraints in the workspace, the only trajectory that matters in many manual Improving recommendation accuracy is the mostly focused target of recommendation systems, while it has been increasingly recognized that accuracy is not enough as the only quality criterion. More concepts have been proposed recently to augment the evaluation dimensions, such as similarity, diversity, long-tail, etc. Simultaneously considering multiple criteria leads to a multi-task recommendation. In this paper, a graph-based recommendation approach is proposed to effectively and flexibly trade-off among them. Our approach is considered based a 1st order Markovian graph with transition probabilities between user-item pairs. A "cost flow" concept is proposed over the graph, so that items with lower costs are stronger recommended to a user. The cost flows are formulated in a recursive dynamic form, whose stability is proved to be guaranteed by appropriately lower-bounding the transition costs. Furthermore, a mixture of transition costs is designed by combining three ingredients related to long-tail, focusing degree and similarity. To evaluate the ingredients, we propose an orthogonal-sparse-orthogonal nonnegative matrix tri-factorization model and an efficient multiplicative algorithm. Empirical experiments on real-world data show promising results of our approach, which could be regarded as a general framework for other affects if transition costs are designed in various ways. Many real-world domains can be represented as large node-link graphs: backbone Internet routers connect with 70,000 other hosts, mid-sized Web servers handle between 20,000 and 200,000 hyperlinked documents, and dictionaries contain millions of words defined in terms of each other. Computational manipulation of such large graphs is common, but previous tools for graph visualization have been limited to datasets of a few thousand nodes. Visual depictions of graphs and networks are external representations that exploit human visual processing to reduce the cognitive load of many tasks that require understanding of global or local structure. We assert that the two key advantages of computer-based systems for information visualization over traditional paper-based visual exposition are interactivity and scalability. We also argue that designing visualization software by taking the characteristics of a target user's task domain into account leads to systems that are more effective and scale to larger datasets than previous work. This thesis contains a detailed analysis of three specialized systems for the interactive exploration of large graphs, relating the intended tasks to the spatial layout and visual encoding choices. We present two novel algorithms for specialized layout and drawing that use quite different visual metaphors. The H3 system for visualizing the hyperlink structures of web sites scales to datasets of over 100,000 nodes by using a carefully chosen spanning tree as the layout backbone, 3D hyperbolic geometry for a Focus+Context view, and provides a fluid interactive experience through guaranteed frame rate drawing. The Constellation system features a highly specialized 2D layout intended to spatially encode domain-specific information for computational linguists checking the plausibility of a large semantic network created from dictionaries. The Planet Multicast system for displaying the tunnel topology of the Internet's multicast backbone provides a literal 3D geographic layout of arcs on a globe to help MBone maintainers find misconfigured long-distance tunnels. Each of these three systems provides a very different view of the graph structure, and we evaluate their efficacy for the intended task. We generalize these findings in our analysis of the importance of interactivity and specialization for graph visualization systems that are effective and scalable. The goal of sparse recovery is to recover a k-sparse signal x e Rn from (possibly noisy) linear measurements of the form y = Ax, where A e Rmxn describes the measurement process. Standard results in compressive sensing show that it is possible to recover the signal x from m = O(k log(n k)) measurements, and that this bound is tight. The framework of model-based compressive sensing [BCDH10] overcomes the lower bound and reduces the number of measurements further to O(k) by limiting the supports of x to a subset M of the (nk) possible supports. This has led to many measurement-efficient algorithms for a wide variety of signal models, including block-sparsity and tree-sparsity. Unfortunately, extending the framework to other, more general models has been stymied by the following obstacle: for the framework to apply, one needs an algorithm that, given a signal x, solves the following optimization problem exactly: [EQUATION] (here x[n] denotes the projection of x on coordinates not in Ω). However, an approximation algorithm for this optimization task is not sufficient. Since many problems of this form are not known to have exact polynomial-time algorithms, this requirement poses an obstacle for extending the framework to a richer class of models. In this paper, we remove this obstacle and show how to extend the model-based compressive sensing framework so that it requires only approximate solutions to the aforementioned optimization problems. Interestingly, our extension requires the existence of approximation algorithms for both the maximization and the minimization variants of the optimization problem. Further, we apply our framework to the Constrained Earth Mover's Distance (CEMD) model introduced in [SHI13], obtaining a sparse recovery scheme that uses significantly less than O(k log(n k)) measurements. This is the first non-trivial theoretical bound for this model, since the validation of the approach presented in [SHI13] was purely empirical. The result is obtained by designing a novel approximation algorithm for the maximization version of the problem and proving approximation guarantees for the minimization algorithm described in [SHI13]. Despite the availability of a huge amount of video data accompanied by descriptive texts, it is not always easy to exploit the information contained in natural language in order to automatically recognize video concepts. Towards this goal, in this paper we use textual cues as means of supervision, introducing two weakly supervised techniques that extend the Multiple Instance Learning (MIL) framework: the Fuzzy Sets Multiple Instance Learning (FSMIL) and the Probabilistic Labels Multiple Instance Learning (PLMIL). The former encodes the spatio-temporal imprecision of the linguistic descriptions with Fuzzy Sets, while the latter models different interpretations of each description's semantics with Probabilistic Labels, both formulated through a convex optimization algorithm. In addition, we provide a novel technique to extract weak labels in the presence of complex semantics, that consists of semantic similarity computations. We evaluate our methods on two distinct problems, namely face and action recognition, in the challenging and realistic setting of movies accompanied by their screenplays, contained in the COGNIMUSE database. We show that, on both tasks, our method considerably outperforms a state-of-the-art weakly supervised approach, as well as other baselines. The pairwise constraints specifying whether a pair of samples should be grouped together or not have been successfully incorporated into the conventional clustering methods such as k-means and spectral clustering for the performance enhancement. Nevertheless, the issue of pairwise constraints has not been well studied in the recently proposed maximum margin clustering (MMC), which extends the maximum margin framework in supervised learning for clustering and often shows a promising performance. This paper therefore proposes a pairwise constrained MMC algorithm. Based on the maximum margin idea in MMC, we propose a set of effective loss functions for discouraging the violation of given pairwise constraints. For the resulting optimization problem, we show that the original nonconvex problem in our approach can be decomposed into a sequence of convex quadratic program problems via constrained concave-convex procedure (CCCP). Subsequently, we present an efficient subgradient projection optimization method to solve each convex problem in the CCCP sequence. Experiments on a number of real-world data sets show that the proposed constrained MMC algorithm is scalable and outperforms the existing constrained MMC approach as well as the typical semi-supervised clustering counterparts. Let X denote the feature and Y the target. We consider domain adaptation under three possible scenarios: (1) the marginal PY changes, while the conditional PX Y stays the same (target shift), (2) the marginal PY is fixed, while the conditional PX Y changes with certain constraints (conditional shift), and (3) the marginal PY changes, and the conditional PX Y changes with constraints (generalized target shift). Using background knowledge, causal interpretations allow us to determine the correct situation for a problem at hand. We exploit importance reweighting or sample transformation to find the learning machine that works well on test data, and propose to estimate the weights or transformations by reweighting or transforming training data to reproduce the covariate distribution on the test domain. Thanks to kernel embedding of conditional as well as marginal distributions, the proposed approaches avoid distribution estimation, and are applicable for high-dimensional problems. Numerical evaluations on synthetic and real-world data sets demonstrate the effectiveness of the proposed framework. Decision making usually involves choosing among different courses of action over a broad range of time scales. For instance, a person planning a trip to a distant location makes high-level decisions regarding what means of transportation to use, but also chooses low-level actions, such as the movements for getting into a car. The problem of picking an appropriate time scale for reasoning and learning has been explored in artificial intelligence, control theory and robotics. In this dissertation we develop a framework that allows novel solutions to this problem, in the context of Markov Decision Processes (MDPs) and reinforcement learning. In this dissertation, we present a general framework for prediction, control and learning at multiple temporal scales. In this framework, temporally extended actions are represented by a way of behaving (a policy) together with a termination condition. An action represented in this way is called an option. Options can be easily incorporated in MDPs, allowing an agent to use existing controllers, heuristics for picking actions, or learned courses of action. The effects of behaving according to an option can be predicted using multi-time models, learned by interacting with the environment. In this dissertation we develop multi-time models, and we illustrate the way in which they can be used to produce plans of behavior very quickly, using classical dynamic programming or reinforcement learning techniques. The most interesting feature of our framework is that it allows an agent to work simultaneously with high-level and low-level temporal representations. The interplay of these levels can be exploited in order to learn and plan more efficiently and more accurately. We develop new algorithms that take advantage of this structure to improve the quality of plans, and to learn in parallel about the effects of many different options. A collaborative convex framework for factoring a data matrix X into a nonnegative product AS , with a sparse coefficient matrix S, is proposed. We restrict the columns of the dictionary matrix A to coincide with certain columns of the data matrix X, thereby guaranteeing a physically meaningful dictionary and dimensionality reduction. We use l1, ∞ regularization to select the dictionary from the data and show that this leads to an exact convex relaxation of l0 in the case of distinct noise-free data. We also show how to relax the restriction-to-X constraint by initializing an alternating minimization approach with the solution of the convex model, obtaining a dictionary close to but not necessarily in X. We focus on applications of the proposed framework to hyperspectral endmember and abundance identification and also show an application to blind source separation of nuclear magnetic resonance data. Many methods, including supervised and unsupervised algorithms, have been developed for extractive document summarization. Most supervised methods consider the summarization task as a two-class classification problem and classify each sentence individually without leveraging the relationship among sentences. The unsupervised methods use heuristic rules to select the most informative sentences into a summary directly, which are hard to generalize. In this paper, we present a Conditional Random Fields (CRF) based framework to keep the merits of the above two kinds of approaches while avoiding their disadvantages. What is more, the proposed framework can take the outcomes of previous methods as features and seamlessly integrate them. The key idea of our approach is to treat the summarization task as a sequence labeling problem. In this view, each document is a sequence of sentences and the summarization procedure labels the sentences by 1 and 0. The label of a sentence depends on the assignment of labels of others. We compared our proposed approach with eight existing methods on an open benchmark data set. The results show that our approach can improve the performance by more than 7.1 and 12.1 over the best supervised baseline and unsupervised baseline respectively in terms of two popular metrics F1 and ROUGE-2. Detailed analysis of the improvement is presented as well. We propose a novel approach based on Formal Concept Analysis for Topic Detection.Our proposal overcomes traditional problems of the clustering and classification techniques.We analyse the parameters involved in the process in a Twitter-based framework.We propose a topic selection methodology based on the stability concept.We overcome the state-of-the-art results for the task. The Topic Detection Task in Twitter represents an indispensable step in the analysis of text corpora and their later application in Online Reputation Management. Classification, clustering and probabilistic techniques have been traditionally applied, but they have some well-known drawbacks such as the need to fix the number of topics to be detected or the problem of how to integrate the prior knowledge of topics with the detection of new ones. This motivates the current work, where we present a novel approach based on Formal Concept Analysis (FCA), a fully unsupervised methodology to group similar content together in thematically-based topics (i.e., the FCA formal concepts) and to organize them in the form of a concept lattice. Formal concepts are conceptual representations based on the relationships between tweet terms and the tweets that have given rise to them. It allows, in contrast to other approaches in the literature, their clear interpretability. In addition, the concept lattice represents a formalism that describes the data, explores correlations, similarities, anomalies and inconsistencies better than other representations such as clustering models or graph-based representations. Our rationale is that these theoretical advantages may improve the Topic Detection process, making them able to tackle the problems related to the task. To prove this point, our FCA-based proposal is evaluated in the context of a real-life Topic Detection task provided by the Replab 2013 CLEF Campaign. To demonstrate the efficiency of the proposal, we have carried out several experiments focused on testing: (a) the impact of terminology selection as an input to our algorithm, (b) the impact of concept selection as the outcome of our algorithm, and; (c) the efficiency of the proposal to detect new and previously unseen topics (i.e., topic adaptation). An extensive analysis of the results has been carried out, proving the suitability of our proposal to integrate previous knowledge of prior topics without losing the ability to detect novel and unseen topics as well as improving the best Replab 2013 results. This paper deals with Bayesian selection of models that can be specified using inequality constraints among the model parameters. The concept of encompassing priors is introduced, that is, a prior distribution for an unconstrained model from which the prior distributions of the constrained models can be derived. It is shown that the Bayes factor for the encompassing and a constrained model has a very nice interpretation: it is the ratio of the proportion of the prior and posterior distribution of the encompassing model in agreement with the constrained model. It is also shown that, for a specific class of models, selection based on encompassing priors will render a virtually objective selection procedure. The paper concludes with three illustrative examples: an analysis of variance with ordered means; a contingency table analysis with ordered odds-ratios; and a multilevel model with ordered slopes. 1 Inequality constrained statistical models Researchers often have one or more (competing) theories about their field of research. Consider, for example, theories about the effect of behavioral therapy versus medication for children with an attention deficit disorder (ADD). Some researchers in this area believe medication is the only effective treatment for ADD, some believe strongly in behavioral therapy, and others may expect an additive effect of both therapies. To test or compare the plausibility of these theories they need to be translated into statistical models. Subsequently, empirical data can be used to determine which model is best. Inequality constraints on model parameters can be useful in the specification of statistical models. This paper deals with competing models that have the same parameter vector, but in one or more of the models parameters are subjected to inequality constraints. To continue the example, consider an experiment where children with ADD are randomly assigned to one of four conditions: no treatment (1), behavioral therapy (2), medication (3), and behavioral therapy plus medication (4). Let the outcome Building systems that autonomously create temporal abstractions from data is a key challenge in scaling learning and planning in reinforcement learning. One popular approach for addressing this challenge is the options framework (, 2000). However, only recently in (, 2017) was a policy gradient theorem derived for online learning of general purpose options in an end to end fashion. In this work, we extend previous work on this topic that only focuses on learning a two-level hierarchy including options and primitive actions to enable learning simultaneously at multiple resolutions in time. We achieve this by considering an arbitrarily deep hierarchy of options where high level temporally extended options are composed of lower level options with finer resolutions in time. We extend results from (, 2017) and derive policy gradient theorems for a deep hierarchy of options. Our proposed hierarchical option-critic architecture is capable of learning internal policies, termination conditions, and hierarchical compositions over options without the need for any intrinsic rewards or subgoals. Our empirical results in both discrete and continuous environments demonstrate the efficiency of our framework.