In this paper we extend temporal difference policy evaluation algorithms to performance criteria that include the variance of the cumulative reward. Such criteria are useful for risk management, and are important in domains such as finance and process control. We propose both TD(0) and LSTD(lambda) variants with linear function approximation, prove their convergence, and demonstrate their utility in a 4-dimensional continuous state space problem.

Decision making is a key feature of autonomous systems. It involves choosing optimally between different lines of action in various information contexts that range from perfectly knowing all aspects of the decision problem to having just partial knowledge about it. The physical context often includes other interacting autonomous systems, typically called agents. In this article, I focus on decision making in a multiagent context with partial information about the problem. Relevant research in this complex but realistic setting has converged around two complementary, general frameworks and also introduced myriad specializations on its way. I put the two frameworks, decentralized partially observable Markov decision process (Dec-POMDP) and the interactive partially observable Markov decision process (I-POMDP), in context and review the foundational algorithms for these frameworks, while briefly discussing the advances in their specializations. I conclude by examining the avenues that research pertaining to these frameworks is pursuing.

We describe the tradeoff between the performance in a visual recognition problem and the control authority that the agent can exercise on the sensing process. We focus on the problem of "visual search" of an object in an otherwise known and static scene, propose a measure of control authority, and relate it to the expected risk and its proxy (conditional entropy of the posterior density). We show this analytically, as well as empirically by simulation using the simplest known model that captures the phenomenology of image formation, including scaling and occlusions. We show that a "passive" agent given a training set can provide no guarantees on performance beyond what is afforded by the priors, and that an "omnipotent" agent, capable of infinite control authority, can achieve arbitrarily good performance (asymptotically). In between these limiting cases, the tradeoff can be characterized empirically.

We describe how the pre-training algorithm for Deep Boltzmann Machines (DBMs) is related to the pre-training algorithm for Deep Belief Networks and we show that under certain conditions, the pre-training procedure improves the variational lower bound of a two-hidden-layer DBM. Based on this analysis, we develop a different method of pre-training DBMs that distributes the modelling work more evenly over the hidden layers. Our results on the MNIST and NORB datasets demonstrate that the new pre-training algorithm allows us to learn better generative models.

Sum-product networks are a new deep architecture that can perform fast, exact inference on high-treewidth models. Only generative methods for training SPNs have been proposed to date. In this paper, we present the first discriminative training algorithms for SPNs, combining the high accuracy of the former with the representational power and tractability of the latter. We show that the class of tractable discriminative SPNs is broader than the class of tractable generative ones, and propose an efficient backpropagation-style algorithm for computing the gradient of the conditional log likelihood. Standard gradient descent suffers from the diffusion problem, but networks with many layers can be learned reliably using "hard" gradient descent, where marginal inference is replaced by MPE inference (i.e., inferring the most probable state of the non-evidence variables). The resulting updates have a simple and intuitive form. We test discriminative SPNs on standard image classification tasks. We obtain the best results to date on the CIFAR-10 dataset, using fewer features than prior methods with an SPN architecture that learns local image structure discriminatively. We also report the highest published test accuracy on STL-10 even though we only use the labeled portion of the dataset.

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.

We consider the problem of performing efficient sum-product computations in an online setting over a tree. A natural application of our methods is to compute the marginal distribution at a vertex in a tree-structured Markov random field. Belief propagation can be used to solve this problem, but requires time linear in the size of the tree, and is therefore too slow in an online setting where we are continuously receiving new data and computing individual marginals. With our method we aim to update the data and compute marginals in time that is no more than logarithmic in the size of the tree, and is often significantly less. We accomplish this via a hierarchical covering structure that caches previous local sum-product computations. Our contribution is threefold: we i) give a linear time algorithm to find an optimal hierarchical cover of a tree; ii) give a sum-productlike algorithm to efficiently compute marginals with respect to this cover; and iii) apply "i" and "ii" to find an efficient algorithm with a regret bound for the online allocation problem in a multi-task setting.

Sum-product networks are a new deep architecture that can perform fast, exact inference on high-treewidth models. Only generative methods for training SPNs have been proposed to date. In this paper, we present the first discriminative training algorithms for SPNs, combining the high accuracy of the former with the representational power and tractability of the latter. We show that the class of tractable discriminative SPNs is broader than the class of tractable generative ones, and propose an efficient backpropagation-style algorithm for computing the gradient of the conditional log likelihood. Standard gradient descent suffers from the diffusion problem, but networks with many layers can be learned reliably using "hard" gradient descent, where marginal inference is replaced by MPE inference (i.e., inferring the most probable state of the non-evidence variables). The resulting updates have a simple and intuitive form. We test discriminative SPNs on standard image classification tasks. We obtain the best results to date on the CIFAR-10 dataset, using fewer features than prior methods with an SPN architecture that learns local image structure discriminatively. We also report the highest published test accuracy on STL-10 even though we only use the labeled portion of the dataset.

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.

We consider the problem of performing efficient sum-product computations in an online setting over a tree. A natural application of our methods is to compute the marginal distribution at a vertex in a tree-structured Markov random field. Belief propagation can be used to solve this problem, but requires time linear in the size of the tree, and is therefore too slow in an online setting where we are continuously receiving new data and computing individual marginals. With our method we aim to update the data and compute marginals in time that is no more than logarithmic in the size of the tree, and is often significantly less. We accomplish this via a hierarchical covering structure that caches previous local sum-product computations. Our contribution is threefold: we i) give a linear time algorithm to find an optimal hierarchical cover of a tree; ii) give a sum-productlike algorithm to efficiently compute marginals with respect to this cover; and iii) apply "i" and "ii" to find an efficient algorithm with a regret bound for the online allocation problem in a multi-task setting.