We provide two fundamental results on the population (infinite-sample) likelihood function of Gaussian mixture models with M \geq 3 components. Our first main result shows that the population likelihood function has bad local maxima even in the special case of equally-weighted mixtures of well-separated and spherical Gaussians. We prove that the log-likelihood value of these bad local maxima can be arbitrarily worse than that of any global optimum, thereby resolving an open question of Srebro (2007). Our second main result shows that the EM algorithm (or a first-order variant of it) with random initialization will converge to bad critical points with probability at least 1-e {-\Omega(M)} . We further establish that a first-order variant of EM will not converge to strict saddle points almost surely, indicating that the poor performance of the first-order method can be attributed to the existence of bad local maxima rather than bad saddle points.

We present the first general purpose framework for marginal maximum a posteriori estimation of probabilistic program variables. By using a series of code transformations, the evidence of any probabilistic program, and therefore of any graphical model, can be optimized with respect to an arbitrary subset of its sampled variables. To carry out this optimization, we develop the first Bayesian optimization package to directly exploit the source code of its target, leading to innovations in problem-independent hyperpriors, unbounded optimization, and implicit constraint satisfaction; delivering significant performance improvements over prominent existing packages.

We introduce a new machine learning approach for image segmentation that uses a neural network to model the conditional energy of a segmentation given an image. Our approach, combinatorial energy learning for image segmentation (CELIS) places a particular emphasis on modeling the inherent combinatorial nature of dense image segmentation problems. We propose efficient algorithms for learning deep neural networks to model the energy function, and for local optimization of this energy in the space of supervoxel agglomerations. We extensively evaluate our method on a publicly available 3-D microscopy dataset with 25 billion voxels of ground truth data. On an 11 billion voxel test set, we find that our method improves volumetric reconstruction accuracy by more than 20% as compared to two state-of-the-art baseline methods: graph-based segmentation of the output of a 3-D convolutional neural network trained to predict boundaries, as well as a random forest classifier trained to agglomerate supervoxels that were generated by a 3-D convolutional neural network.

Sequence-to-sequence models have achieved impressive results on various tasks. However, they are unsuitable for tasks that require incremental predictions to be made as more data arrives or tasks that have long input sequences and output sequences. This is because they generate an output sequence conditioned on an entire input sequence. In this paper, we present a Neural Transducer that can make incremental predictions as more input arrives, without redoing the entire computation. Unlike sequence-to-sequence models, the Neural Transducer computes the next-step distribution conditioned on the partially observed input sequence and the partially generated sequence.

Influence maximization in social networks has typically been studied in the context of contagion models and irreversible processes. In this paper, we consider an alternate model that treats individual opinions as spins in an Ising system at dynamic equilibrium. We formalize the \textit{Ising influence maximization} problem, which has a natural physical interpretation as maximizing the magnetization given a budget of external magnetic field. Under the mean-field (MF) approximation, we present a gradient ascent algorithm that uses the susceptibility to efficiently calculate local maxima of the magnetization, and we develop a number of sufficient conditions for when the MF magnetization is concave and our algorithm converges to a global optimum. We apply our algorithm on random and real-world networks, demonstrating, remarkably, that the MF optimal external fields (i.e., the external fields which maximize the MF magnetization) exhibit a phase transition from focusing on high-degree individuals at high temperatures to focusing on low-degree individuals at low temperatures.

We analyze stochastic algorithms for optimizing nonconvex, nonsmooth finite-sum problems, where the nonsmooth part is convex. Surprisingly, unlike the smooth case, our knowledge of this fundamental problem is very limited. For example, it is not known whether the proximal stochastic gradient method with constant minibatch converges to a stationary point. To tackle this issue, we develop fast stochastic algorithms that provably converge to a stationary point for constant minibatches. Furthermore, using a variant of these algorithms, we obtain provably faster convergence than batch proximal gradient descent.

We present Cyclades, a general framework for parallelizing stochastic optimization algorithms in a shared memory setting. Cyclades is asynchronous during model updates, and requires no memory locking mechanisms, similar to Hogwild!-type algorithms. Unlike Hogwild!, Cyclades introduces no conflicts during parallel execution, and offers a black-box analysis for provable speedups across a large family of algorithms. Due to its inherent cache locality and conflict-free nature, our multi-core implementation of Cyclades consistently outperforms Hogwild!-type algorithms on sufficiently sparse datasets, leading to up to 40% speedup gains compared to Hogwild!, and up to 5\times gains over asynchronous implementations of variance reduction algorithms.

Variational methods that rely on a recognition network to approximate the posterior of directed graphical models offer better inference and learning than previous methods. Recent advances that exploit the capacity and flexibility in this approach have expanded what kinds of models can be trained. However, as a proposal for the posterior, the capacity of the recognition network is limited, which can constrain the representational power of the generative model and increase the variance of Monte Carlo estimates. To address these issues, we introduce an iterative refinement procedure for improving the approximate posterior of the recognition network and show that training with the refined posterior is competitive with state-of-the-art methods. The advantages of refinement are further evident in an increased effective sample size, which implies a lower variance of gradient estimates.

In this paper, we consider a non-convex loss-minimization problem of learning Supervised PageRank models, which can account for features of nodes and edges. We propose gradient-based and random gradient-free methods to solve this problem. Our algorithms are based on the concept of an inexact oracle and unlike the state-of-the-art gradient-based method we manage to provide theoretically the convergence rate guarantees for both of them. Finally, we compare the performance of the proposed optimization methods with the state of the art applied to a ranking task.

A fundamental problem in cognitive neuroscience is how humans make decisions, act, and behave in relation to other humans. Here we adopt the hypothesis that when we are in an interactive social setting, our brains perform Bayesian inference of the intentions and cooperativeness of others using probabilistic representations. We employ the framework of partially observable Markov decision processes (POMDPs) to model human decision making in a social context, focusing specifically on the volunteer's dilemma in a version of the classic Public Goods Game. We show that the POMDP model explains both the behavior of subjects as well as neural activity recorded using fMRI during the game. The decisions of subjects can be modeled across all trials using two interpretable parameters.