While finding the exact solution for the MAP inference problem is intractable for many real-world tasks, MAP LP relaxations have been shown to be very effective in practice. However, the most efficient methods that perform block coordinate descent can get stuck in sub-optimal points as they are not globally convergent. In this work we propose to augment these algorithms with an ɛ-descent approach and present a method to efficiently optimize for a descent direction in the subdifferential using a margin-based formulation of the Fenchel-Young duality theorem. Furthermore, the presented approach provides a methodology to construct a primal optimal solution from its dual optimal counterpart. We demonstrate the efficiency of the presented approach on spin glass models and protein interaction problems and show that our approach outperforms state-of-the-art solvers.

This paper provides lower bounds on the convergence rate of Derivative Free Optimization (DFO) with noisy function evaluations, exposing a fundamental and unavoidable gap between the performance of algorithms with access to gradients and those with access to only function evaluations. However, there are situations in which DFO is unavoidable, and for such situations we propose a new DFO algorithm that is proved to be near optimal for the class of strongly convex objective functions. A distinctive feature of the algorithm is that it uses only Boolean-valued function comparisons, rather than function evaluations. This makes the algorithm useful in an even wider range of applications, such as optimization based on paired comparisons from human subjects, for example. We also show that regardless of whether DFO is based on noisy function evaluations or Boolean-valued function comparisons, the convergence rate is the same.

We consider derivative-free algorithms for stochastic optimization problems that use only noisy function values rather than gradients, analyzing their finite-sample convergence rates. We show that if pairs of function values are available, algorithms that use gradient estimates based on random perturbations suffer a factor of at most d in convergence rate over traditional stochastic gradient methods, where d is the problem dimension. We complement our algorithmic development with information-theoretic lower bounds on the minimax convergence rateof such problems, which show that our bounds are sharp withrespect to all problemdependent quantities: they cannot be improved by more than constant factors.

The expected return is a widely used objective in decision making under uncertainty. Many algorithms, such as value iteration, have been proposed to optimize it. In risk-aware settings, however, the expected return is often not an appropriate objective to optimize. We propose a new optimization objective for risk-aware planning and show that it has desirable theoretical properties. We also draw connections to previously proposed objectives for risk-aware planing: minmax, exponential utility, percentile and mean minus variance. Our method applies to an extended class of Markov decision processes: we allow costs to be stochastic as long as they are bounded. Additionally, we present an efficient algorithm for optimizing the proposed objective. Synthetic and real-world experiments illustrate the effectiveness of our method, at scale.

A new result in convex analysis on the calculation of proximity operators in certain scaled norms is derived. We describe efficient implementations of the proximity calculation for a useful class of functions; the implementations exploit the piece-wise linear nature of the dual problem. The second part of the paper applies the previous result to acceleration of convex minimization problems, and leads to an elegant quasi-Newton method. The optimization method compares favorably against state-of-the-art alternatives. The algorithm has extensive applications including signal processing, sparse recovery and machine learning and classification.

We introduce a class of discrete divergences on sets (equivalently binary vectors) that we call the submodular-Bregman divergences. We consider two kinds, defined either from tight modular upper or tight modular lower bounds of a submodular function. We show that the properties of these divergences are analogous to the (standard continuous) Bregman divergence. We demonstrate how they generalize many useful divergences, including the weighted Hamming distance, squared weighted Hamming, weighted precision, recall, conditional mutual information, and a generalized KL-divergence on sets. We also show that the generalized Bregman divergence on the Lovász extension of a submodular function, which we call the Lovász-Bregman divergence, is a continuous extension of a submodular Bregman divergence. We point out a number of applications, and in particular show that a proximal algorithm defined through the submodular Bregman divergence provides a framework for many mirror-descent style algorithms related to submodular function optimization. We also show that a generalization of the k-means algorithm using the Lovász Bregman divergence is natural in clustering scenarios where ordering is important. A unique property of this algorithm is that computing the mean ordering is extremely efficient unlike other order based distance measures.

The partition function plays a key role in probabilistic modeling including conditional random fields, graphical models, and maximum likelihood estimation. To optimize partition functions, this article introduces a quadratic variational upper bound. This inequality facilitates majorization methods: optimization of complicated functions through the iterative solution of simpler sub-problems. Such bounds remain efficient to compute even when the partition function involves a graphical model (with small tree-width) or in latent likelihood settings. For large-scale problems, low-rank versions of the bound are provided and outperform LBFGS as well as first-order methods. Several learning applications are shown and reduce to fast and convergent update rules. Experimental results show advantages over state-of-the-art optimization methods.

We present a simplex algorithm for linear programming in a linear classification formulation. The paramount complexity parameter in linear classification problems is called the margin. We prove that for margin values of practical interest our simplex variant performs a polylogarithmic number of pivot steps in the worst case, and its overall running time is near linear. This is in contrast to general linear programming, for which no sub-polynomial pivot rule is known.

We study the scalability of consensus-based distributed optimization algorithms by considering two questions: How many processors should we use for a given problem, and how often should they communicate when communication is not free? Central to our analysis is a problem-specific value r which quantifies the communication/computation tradeoff. We show that organizing the communication among nodes as a k-regular expander graph [1] yields speedups, while when all pairs of nodes communicate (as in a complete graph), there is an optimal number of processors that depends on r. Surprisingly, a speedup can be obtained, in terms of the time to reach a fixed level of accuracy, by communicating less and less frequently as the computation progresses. Experiments on a real cluster solving metric learning and non-smooth convex minimization tasks demonstrate strong agreement between theory and practice.

We uncover relations between robust MDPs and risk-sensitive MDPs. The objective of a robust MDP is to minimize a function, such as the expectation of cumulative cost, for the worst case when the parameters have uncertainties. The objective of a risk-sensitive MDP is to minimize a risk measure of the cumulative cost when the parameters are known. We show that a risk-sensitive MDP of minimizing the expected exponential utility is equivalent to a robust MDP of minimizing the worst-case expectation with a penalty for the deviation of the uncertain parameters from their nominal values, which is measured with the Kullback-Leibler divergence. We also show that a risk-sensitive MDP of minimizing an iterated risk measure that is composed of certain coherent risk measures is equivalent to a robust MDP of minimizing the worst-case expectation when the possible deviations of uncertain parameters from their nominal values are characterized with a concave function.