We introduce the variational filtering EM algorithm, a simple, general-purpose method for performing variational inference in dynamical latent variable models using information from only past and present variables, i.e. filtering. The algorithm is derived from the variational objective in the filtering setting and consists of an optimization procedure at each time step. By performing each inference optimization procedure with an iterative amortized inference model, we obtain a computationally efficient implementation of the algorithm, which we call amortized variational filtering. We present experiments demonstrating that this general-purpose method improves inference performance across several recent deep dynamical latent variable models.

Estimating the structure of directed acyclic graphs (DAGs, also known as Bayesian networks) is a challenging problem since the search space of DAGs is combinatorial and scales superexponentially with the number of nodes. Existing approaches rely on various local heuristics for enforcing the acyclicity constraint. In this paper, we introduce a fundamentally different strategy: we formulate the structure learning problem as a purely continuous optimization problem over real matrices that avoids this combinatorial constraint entirely. This is achieved by a novel characterization of acyclicity that is not only smooth but also exact. The resulting problem can be efficiently solved by standard numerical algorithms, which also makes implementation effortless. The proposed method outperforms existing ones, without imposing any structural assumptions on the graph such as bounded treewidth or in-degree.

Bayesian optimization is a powerful tool for global optimization of expensive functions. One of its key components is the underlying probabilistic model used for the objective function f. In practice, however, it is often unclear how one should appropriately choose a model, especially when gathering data is expensive. In this work, we introduce a novel automated Bayesian optimization approach that dynamically selects promising models for explaining the observed data using Bayesian Optimization in the model space. Crucially, we account for the uncertainty in the choice of model; our method is capable of using multiple models to represent its current belief about f and subsequently using this information for decision making. We argue, and demonstrate empirically, that our approach automatically finds suitable models for the objective function, which ultimately results in more-efficient optimization.

Regression with group-sparsity penalty plays a central role in high-dimensional prediction problems. However, most existing methods require the group structure to be known a priori. In practice, this may be a too strong assumption, potentially hampering the effectiveness of the regularization method. To circumvent this issue, we present a method to estimate the group structure by means of a continuous bilevel optimization problem where the data is split into training and validation sets. Our approach relies on an approximation scheme where the lower level problem is replaced by a smooth dual forward-backward algorithm with Bregman distances. We provide guarantees regarding the convergence of the approximate procedure to the exact problem and demonstrate the well behaviour of the proposed method on synthetic experiments. Finally, a preliminary application to genes expression data is tackled with the purpose of unveiling functional groups.

We propose a new primal-dual homotopy smoothing algorithm for a linearly constrained convex program, where neither the primal nor the dual function has to be smooth or strongly convex. The best known iteration complexity solving such a non-smooth problem is $\mathcal{O}(\varepsilon^{-1})$. In this paper, we show that by leveraging a local error bound condition on the dual function, the proposed algorithm can achieve a better primal convergence time of $\mathcal{O}\l(\varepsilon^{-2/(2+\beta)}\log_2(\varepsilon^{-1})\r)$, where $\beta\in(0,1]$ is a local error bound parameter. As an example application, we show that the distributed geometric median problem, which can be formulated as a constrained convex program, has its dual function non-smooth but satisfying the aforementioned local error bound condition with $\beta=1/2$, therefore enjoying a convergence time of $\mathcal{O}\l(\varepsilon^{-4/5}\log_2(\varepsilon^{-1})\r)$. This result improves upon the $\mathcal{O}(\varepsilon^{-1})$ convergence time bound achieved by existing distributed optimization algorithms. Simulation experiments also demonstrate the performance of our proposed algorithm.

We propose a novel randomized first order optimization method---SEGA (SkEtched GrAdient method)---which progressively throughout its iterations builds a variance-reduced estimate of the gradient from random linear measurements (sketches) of the gradient provided at each iteration by an oracle. In each iteration, SEGA updates the current estimate of the gradient through a sketch-and-project operation using the information provided by the latest sketch, and this is subsequently used to compute an unbiased estimate of the true gradient through a random relaxation procedure. This unbiased estimate is then used to perform a gradient step. Unlike standard subspace descent methods, such as coordinate descent, SEGA can be used for optimization problems with a non-separable proximal term. We provide a general convergence analysis and prove linear convergence for strongly convex objectives. In the special case of coordinate sketches, SEGA can be enhanced with various techniques such as importance sampling, minibatching and acceleration, and its rate is up to a small constant factor identical to the best-known rate of coordinate descent.

We study the computational tractability of PAC reinforcement learning with rich observations. We present new provably sample-efficient algorithms for environments with deterministic hidden state dynamics and stochastic rich observations. These methods operate in an oracle model of computation -- accessing policy and value function classes exclusively through standard optimization primitives -- and therefore represent computationally efficient alternatives to prior algorithms that require enumeration. With stochastic hidden state dynamics, we prove that the only known sample-efficient algorithm, OLIVE, cannot be implemented in the oracle model. We also present several examples that illustrate fundamental challenges of tractable PAC reinforcement learning in such general settings.

In many real-world learning tasks, it is hard to directly optimize the true performance measures, meanwhile choosing the right surrogate objectives is also difficult. Under this situation, it is desirable to incorporate an optimization of objective process into the learning loop based on weak modeling of the relationship between the true measure and the objective. In this work, we discuss the task of objective adaptation, in which the learner iteratively adapts the learning objective to the underlying true objective based on the preference feedback from an oracle. We show that when the objective can be linearly parameterized, this preference based learning problem can be solved by utilizing the dueling bandit model. A novel sampling based algorithm DL^2M is proposed to learn the optimal parameter, which enjoys strong theoretical guarantees and efficient empirical performance. To avoid learning a hypothesis from scratch after each objective function update, a boosting based hypothesis adaptation approach is proposed to efficiently adapt any pre-learned element hypothesis to the current objective. We apply the overall approach to multi-label learning, and show that the proposed approach achieves significant performance under various multi-label performance measures.

In this paper we address the graph matching problem. Following the recent works of \cite{zaslavskiy2009path,Vestner2017} we analyze and generalize the idea of concave relaxations. We introduce the concepts of \emph{conditionally concave} and \emph{probably conditionally concave} energies on polytopes and show that they encapsulate many instances of the graph matching problem, including matching Euclidean graphs and graphs on surfaces. We further prove that local minima of probably conditionally concave energies on general matching polytopes (\eg, doubly stochastic) are with high probability extreme points of the matching polytope (\eg, permutations).

Risk management in dynamic decision problems is a primary concern in many fields, including financial investment, autonomous driving, and healthcare. The mean-variance function is one of the most widely used objective functions in risk management due to its simplicity and interpretability. Existing algorithms for mean-variance optimization are based on multi-time-scale stochastic approximation, whose learning rate schedules are often hard to tune, and have only asymptotic convergence proof. In this paper, we develop a model-free policy search framework for mean-variance optimization with finite-sample error bound analysis (to local optima). Our starting point is a reformulation of the original mean-variance function with its Legendre-Fenchel dual, from which we propose a stochastic block coordinate ascent policy search algorithm. Both the asymptotic convergence guarantee of the last iteration's solution and the convergence rate of the randomly picked solution are provided, and their applicability is demonstrated on several benchmark domains.