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.

Sum-product networks are a new deep architecture that can perform fast, exact inference onhigh-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 thatlearns 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.

Derivative Free Optimization (DFO) is attractive when the objective function's derivatives are not available and evaluations are costly. Moreover, if the function evaluations are noisy, then approximating gradients by finite differences is difficult. This paper gives quantitative lower bounds on the performance of DFO with noisy function evaluations, exposing a fundamental and unavoidable gap between optimization performance based on noisy evaluations versus noisy gradients. This challenges the conventional wisdom that the method of finite differences is comparable to a stochastic gradient. 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 only uses Boolean-valued function comparisons, rather than evaluations. This makes the algorithm useful in an even wider range of applications, including optimization based on paired comparisons from human subjects, for example. Remarkably, we show that regardless of whether DFO is based on noisy function evaluations or Boolean-valued function comparisons, the convergence rate is the same.

We propose a new stochastic gradient method for optimizing the sum of
 a finite set of smooth functions, where the sum is strongly convex.
 While standard stochastic gradient methods
 converge at sublinear rates for this problem, the proposed method incorporates a memory of previous gradient values in order to achieve a linear convergence 
rate. In a machine learning context, numerical experiments indicate that the new algorithm can dramatically outperform standard
 algorithms, both in terms of optimizing the training error and reducing the test error quickly.

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 $\sqrt{\dim}$ in convergence rate over traditional stochastic gradient methods, where $\dim$ is the dimension of the problem. We complement our algorithmic development with information-theoretic lower bounds on the minimax convergence rate of such problems, which show that our bounds are sharp with respect to all problem-dependent quantities: they cannot be improved by more than constant factors.

This paper describes gradient methods based on a scaled metric on the Grassmann manifold for low-rank matrix completion. The proposed methods significantly improve canonical gradient methods especially on ill-conditioned matrices, while maintaining established global convegence and exact recovery guarantees. A connection between a form of subspace iteration for matrix completion and the scaled gradient descent procedure is also established. The proposed conjugate gradient method based on the scaled gradient outperforms several existing algorithms for matrix completion and is competitive with recently proposed methods.

Although many variants of stochastic gradient descent have been proposed for large-scale convex optimization, most of them require projecting the solution at {\it each} iteration to ensure that the obtained solution stays within the feasible domain. For complex domains (e.g., positive semidefinite cone), the projection step can be computationally expensive, making stochastic gradient descent unattractive for large-scale optimization problems. We address this limitation by developing a novel stochastic gradient descent algorithm that does not need intermediate projections. Instead, only one projection at the last iteration is needed to obtain a feasible solution in the given domain. Our theoretical analysis shows that with a high probability, the proposed algorithms achieve an $O(1/\sqrt{T})$ convergence rate for general convex optimization, and an $O(\ln T/T)$ rate for strongly convex optimization under mild conditions about the domain and the objective function.

Stochastic Gradient Descent (SGD) is one of the simplest and most popular stochastic optimization methods. While it has already been theoretically studied for decades, the classical analysis usually required non-trivial smoothness assumptions, which do not apply to many modern applications of SGD with non-smooth objective functions such as support vector machines. In this paper, we investigate the performance of SGD without such smoothness assumptions, as well as a running average scheme to convert the SGD iterates to a solution with optimal optimization accuracy. In this framework, we prove that after T rounds, the suboptimality of the last SGD iterate scales as O(log(T)/\sqrt{T}) for non-smooth convex objective functions, and O(log(T)/T) in the non-smooth strongly convex case. To the best of our knowledge, these are the first bounds of this kind, and almost match the minimax-optimal rates obtainable by appropriate averaging schemes. We also propose a new and simple averaging scheme, which not only attains optimal rates, but can also be easily computed on-the-fly (in contrast, the suffix averaging scheme proposed in Rakhlin et al. (2011) is not as simple to implement). Finally, we provide some experimental illustrations.

This paper explores a new framework for reinforcement learning based on online convex optimization, in particular mirror descent and related algorithms. Mirror descent can be viewed as an enhanced gradient method, particularly suited to minimization of convex functions in highdimensional spaces. Unlike traditional gradient methods, mirror descent undertakes gradient updates of weights in both the dual space and primal space, which are linked together using a Legendre transform. Mirror descent can be viewed as a proximal algorithm where the distance generating function used is a Bregman divergence. A new class of proximal-gradient based temporal-difference (TD) methods are presented based on different Bregman divergences, which are more powerful than regular TD learning. Examples of Bregman divergences that are studied include p-norm functions, and Mahalanobis distance based on the covariance of sample gradients. A new family of sparse mirror-descent reinforcement learning methods are proposed, which are able to find sparse fixed points of an l1-regularized Bellman equation at significantly less computational cost than previous methods based on second-order matrix methods. An experimental study of mirror-descent reinforcement learning is presented using discrete and continuous Markov decision processes.

In this work we consider the stochastic minimization of nonsmooth convex loss functions, a central problem in machine learning. We propose a novel algorithm called Accelerated Nonsmooth Stochastic Gradient Descent (ANSGD), which exploits the structure of common nonsmooth loss functions to achieve optimal convergence rates for a class of problems including SVMs. It is the first stochastic algorithm that can achieve the optimal O(1/t) rate for minimizing nonsmooth loss functions (with strong convexity). The fast rates are confirmed by empirical comparisons, in which ANSGD significantly outperforms previous subgradient descent algorithms including SGD.