We study a risk-constrained version of the stochastic shortest path (SSP) problem, where the risk measure considered is Conditional Value-at-Risk (CVaR). We propose two algorithms that obtain a locally risk-optimal policy by employing four tools: stochastic approximation, mini batches, policy gradients and importance sampling. Both the algorithms incorporate a CVaR estimation procedure, along the lines of Bardou et al. [2009], which in turn is based on Rockafellar-Uryasev's representation for CVaR and utilize the likelihood ratio principle for estimating the gradient of the sum of one cost function (objective of the SSP) and the gradient of the CVaR of the sum of another cost function (in the constraint of SSP). The algorithms differ in the manner in which they approximate the CVaR estimates/necessary gradients - the first algorithm uses stochastic approximation, while the second employ mini-batches in the spirit of Monte Carlo methods. We establish asymptotic convergence of both the algorithms. Further, since estimating CVaR is related to rare-event simulation, we incorporate an importance sampling based variance reduction scheme into our proposed algorithms.

Information spreads across social and technological networks, but often the network structures are hidden from us and we only observe the traces left by the diffusion processes, called cascades. Can we recover the hidden network structures from these observed cascades? What kind of cascades and how many cascades do we need? Are there some network structures which are more difficult than others to recover? Can we design efficient inference algorithms with provable guarantees? Despite the increasing availability of cascade data and methods for inferring networks from these data, a thorough theoretical understanding of the above questions remains largely unexplored in the literature. In this paper, we investigate the network structure inference problem for a general family of continuous-time diffusion models using an $l_1$-regularized likelihood maximization framework. We show that, as long as the cascade sampling process satisfies a natural incoherence condition, our framework can recover the correct network structure with high probability if we observe $O(d^3 \log N)$ cascades, where $d$ is the maximum number of parents of a node and $N$ is the total number of nodes. Moreover, we develop a simple and efficient soft-thresholding inference algorithm, which we use to illustrate the consequences of our theoretical results, and show that our framework outperforms other alternatives in practice.

We consider the infinite-horizon discounted optimal control problem formalized by Markov Decision Processes. We focus on several approximate variations of the Policy Iteration algorithm: Approximate Policy Iteration, Conservative Policy Iteration (CPI), a natural adaptation of the Policy Search by Dynamic Programming algorithm to the infinite-horizon case (PSDP$_\infty$), and the recently proposed Non-Stationary Policy iteration (NSPI(m)). For all algorithms, we describe performance bounds, and make a comparison by paying a particular attention to the concentrability constants involved, the number of iterations and the memory required. Our analysis highlights the following points: 1) The performance guarantee of CPI can be arbitrarily better than that of API/API($\alpha$), but this comes at the cost of a relative---exponential in $\frac{1}{\epsilon}$---increase of the number of iterations. 2) PSDP$_\infty$ enjoys the best of both worlds: its performance guarantee is similar to that of CPI, but within a number of iterations similar to that of API. 3) Contrary to API that requires a constant memory, the memory needed by CPI and PSDP$_\infty$ is proportional to their number of iterations, which may be problematic when the discount factor $\gamma$ is close to 1 or the approximation error $\epsilon$ is close to $0$; we show that the NSPI(m) algorithm allows to make an overall trade-off between memory and performance. Simulations with these schemes confirm our analysis.

Much recent research has been conducted in the area of Bayesian learning, particularly with regard to the optimization of hyper-parameters via Gaussian process regression. The methodologies rely chiefly on the method of maximizing the expected improvement of a score function with respect to adjustments in the hyper-parameters. In this work, we present a novel algorithm that exploits notions of confidence intervals and uncertainties to enable the discovery of the best optimal within a targeted region of the parameter space. We demonstrate the efficacy of our algorithm with respect to machine learning problems and show cases where our algorithm is competitive with the method of maximizing expected improvement.

In crowd labeling, a large amount of unlabeled data instances are outsourced to a crowd of workers. Workers will be paid for each label they provide, but the labeling requester usually has only a limited amount of the budget. Since data instances have different levels of labeling difficulty and workers have different reliability, it is desirable to have an optimal policy to allocate the budget among all instance-worker pairs such that the overall labeling accuracy is maximized. We consider categorical labeling tasks and formulate the budget allocation problem as a Bayesian Markov decision process (MDP), which simultaneously conducts learning and decision making. Using the dynamic programming (DP) recurrence, one can obtain the optimal allocation policy. However, DP quickly becomes computationally intractable when the size of the problem increases. To solve this challenge, we propose a computationally efficient approximate policy, called optimistic knowledge gradient policy. Our MDP is a quite general framework, which applies to both pull crowdsourcing marketplaces with homogeneous workers and push marketplaces with heterogeneous workers. It can also incorporate the contextual information of instances when they are available. The experiments on both simulated and real data show that the proposed policy achieves a higher labeling accuracy than other existing policies at the same budget level.

Extracting latent low-dimensional structure from high-dimensional data is of paramount importance in timely inference tasks encountered with `Big Data' analytics. However, increasingly noisy, heterogeneous, and incomplete datasets as well as the need for {\em real-time} processing of streaming data pose major challenges to this end. In this context, the present paper permeates benefits from rank minimization to scalable imputation of missing data, via tracking low-dimensional subspaces and unraveling latent (possibly multi-way) structure from \emph{incomplete streaming} data. For low-rank matrix data, a subspace estimator is proposed based on an exponentially-weighted least-squares criterion regularized with the nuclear norm. After recasting the non-separable nuclear norm into a form amenable to online optimization, real-time algorithms with complementary strengths are developed and their convergence is established under simplifying technical assumptions. In a stationary setting, the asymptotic estimates obtained offer the well-documented performance guarantees of the {\em batch} nuclear-norm regularized estimator. Under the same unifying framework, a novel online (adaptive) algorithm is developed to obtain multi-way decompositions of \emph{low-rank tensors} with missing entries, and perform imputation as a byproduct. Simulated tests with both synthetic as well as real Internet and cardiac magnetic resonance imagery (MRI) data confirm the efficacy of the proposed algorithms, and their superior performance relative to state-of-the-art alternatives.

The intent of this research is to generate a set of non-dominated policies from which one of two agents (the leader) can select a most preferred policy to control a dynamic system that is also affected by the control decisions of the other agent (the follower). The problem is described by an infinite horizon, partially observed Markov game (POMG). At each decision epoch, each agent knows: its past and present states, its past actions, and noise corrupted observations of the other agent's past and present states. The actions of each agent are determined at each decision epoch based on these data. The leader considers multiple objectives in selecting its policy. The follower considers a single objective in selecting its policy with complete knowledge of and in response to the policy selected by the leader. This leader-follower assumption allows the POMG to be transformed into a specially structured, partially observed Markov decision process (POMDP). This POMDP is used to determine the follower's best response policy. A multi-objective genetic algorithm (MOGA) is used to create the next generation of leader policies based on the fitness measures of each leader policy in the current generation. Computing a fitness measure for a leader policy requires a value determination calculation, given the leader policy and the follower's best response policy. The policies from which the leader can select a most preferred policy are the non-dominated policies of the final generation of leader policies created by the MOGA. An example is presented that illustrates how these results can be used to support a manager of a liquid egg production process (the leader) in selecting a sequence of actions to best control this process over time, given that there is an attacker (the follower) who seeks to contaminate the liquid egg production process with a chemical or biological toxin.

Conditional gradient methods are old and well studied optimization algorithms. Their origin dates at least to the 50's and the Frank-Wolfe algorithm for quadratic programming [18] but they apply to much more general optimization problems. General formulations of conditional gradient algorithms have been studied in the past and various convergence properties of these algorithms have been proven. Moreover, such algorithms have found application in many fields, such as optimal control, statistics, signal processing, computational geometry and machine learning. Currently, interest in conditional gradient methods is undergoing a revival because of their computational advantages when applied to certain large scale optimization problems. Such examples are regularization problems involving sparsity or low rank constraints, which appear in many widely used methods in machine learning. Inspired by such algorithms, in this chapter we study a first-order method for solving certain convex optimization problems. We focus on problems of the form min {f(x) g(Ax) ω(x): x H}. 1 over a real Hilbert space H. We assume that f is a convex function with Hölder continuous gradient, g a Lipschitz continuous convex function, A a bounded linear operator and ω a convex function defined over a bounded domain.

In a sequential auction with multiple bidding agents, it is highly challenging to determine the ordering of the items to sell in order to maximize the revenue due to the fact that the autonomy and private information of the agents heavily influence the outcome of the auction. The main contribution of this paper is two-fold. First, we demonstrate how to apply machine learning techniques to solve the optimal ordering problem in sequential auctions. We learn regression models from historical auctions, which are subsequently used to predict the expected value of orderings for new auctions. Given the learned models, we propose two types of optimization methods: a black-box best-first search approach, and a novel white-box approach that maps learned models to integer linear programs (ILP) which can then be solved by any ILP-solver. Although the studied auction design problem is hard, our proposed optimization methods obtain good orderings with high revenues. Our second main contribution is the insight that the internal structure of regression models can be efficiently evaluated inside an ILP solver for optimization purposes. To this end, we provide efficient encodings of regression trees and linear regression models as ILP constraints. This new way of using learned models for optimization is promising. As the experimental results show, it significantly outperforms the black-box best-first search in nearly all settings.

We propose a variable metric framework for minimizing the sum of a self-concordant function and a possibly non-smooth convex function, endowed with an easily computable proximal operator. We theoretically establish the convergence of our framework without relying on the usual Lipschitz gradient assumption on the smooth part. An important highlight of our work is a new set of analytic step-size selection and correction procedures based on the structure of the problem. We describe concrete algorithmic instances of our framework for several interesting applications and demonstrate them numerically on both synthetic and real data.