We study a general adversarial online learning problem, in which we are given a decision set X' in a reflexive Banach space X and a sequence of reward vectors in the dual space of X. At each iteration, we choose an action from X', based on the observed sequence of previous rewards. Our goal is to minimize regret, defined as the gap between the realized reward and the reward of the best fixed action in hindsight. Using results from infinite dimensional convex analysis, we generalize the method of Dual Averaging (or Follow the Regularized Leader) to our setting and obtain upper bounds on the worst-case regret that generalize many previous results. Under the assumption of uniformly continuous rewards, we obtain explicit regret bounds in a setting where the decision set is the set of probability distributions on a compact metric space S. Importantly, we make no convexity assumptions on either the set S or the reward functions. We also prove a general lower bound on the worst-case regret for any online algorithm. We then apply these results to the problem of learning in repeated two-player zero-sum games on compact metric spaces. In doing so, we first prove that if both players play a Hannan-consistent strategy, then with probability 1 the empirical distributions of play weakly converge to the set of Nash equilibria of the game. We then show that, under mild assumptions, Dual Averaging on the (infinite-dimensional) space of probability distributions indeed achieves Hannan-consistency.

Semidefinite programs (SDP's) can be solved in polynomial time by interior point methods, but scalability can be an issue. To address this shortcoming, over a decade ago, Burer and Monteiro proposed to solve SDP's with few equality constraints via rank-restricted, non-convex surrogates. Remarkably, for some applications, local optimization methods seem to converge to global optima of these non-convex surrogates reliably. Although some theory supports this empirical success, a complete explanation of it remains an open question. In this paper, we consider a class of SDP's which includes applications such as max-cut, community detection in the stochastic block model, robust PCA, phase retrieval and synchronization of rotations. We show that the low-rank Burer-Monteiro formulation of SDP's in that class almost never has any spurious local optima.

In multi-step learning, where a final learning task is accomplished via a sequence of intermediate learning tasks, the intuition is that successive steps or levels transform the initial data into representations more and more ``suited to the final learning task. A related principle arises in transfer-learning where Baxter (2000) proposed a theoretical framework to study how learning multiple tasks transforms the inductive bias of a learner. The most widespread multi-step learning approach is semi-supervised learning with two steps: unsupervised, then supervised. Several authors (Castelli-Cover, 1996; Balcan-Blum, 2005; Niyogi, 2008; Ben-David et al, 2008; Urner et al, 2011) have analyzed SSL, with Balcan-Blum (2005) proposing a version of the PAC learning framework augmented by a ``compatibility function to link concept class and unlabeled data distribution. We propose to analyze SSL and other multi-step learning approaches, much in the spirit of Baxter's framework, by defining a learning problem generatively as a joint statistical model on $X \times Y$.

Online optimization covers problems such as online resource allocation, online bipartite matching, adwords (a central problem in e-commerce and advertising), and adwords with separable concave returns. We analyze the worst case competitive ratio of two primal-dual algorithms for a class of online convex (conic) optimization problems that contains the previous examples as special cases defined on the positive orthant. We derive a sufficient condition on the objective function that guarantees a constant worst case competitive ratio (greater than or equal to $\frac{1}{2}$) for monotone objective functions. We provide new examples of online problems on the positive orthant % and the positive semidefinite cone that satisfy the sufficient condition. We show how smoothing can improve the competitive ratio of these algorithms, and in particular for separable functions, we show that the optimal smoothing can be derived by solving a convex optimization problem. This result allows us to directly optimize the competitive ratio bound over a class of smoothing functions, and hence design effective smoothing customized for a given cost function.

Despite the success of kernel-based nonparametric methods, kernel selection still requires considerable expertise, and is often described as a "black art." We present a sophisticated method for automatically searching for an appropriate kernel from an infinite space of potential choices. Previous efforts in this direction have focused on traversing a kernel grammar, only examining the data via computation of marginal likelihood. Our proposed search method is based on Bayesian optimization in model space, where we reason about model evidence as a function to be maximized. We explicitly reason about the data distribution and how it induces similarity between potential model choices in terms of the explanations they can offer for observed data. In this light, we construct a novel kernel between models to explain a given dataset. Our method is capable of finding a model that explains a given dataset well without any human assistance, often with fewer computations of model evidence than previous approaches, a claim we demonstrate empirically.

Medical researchers are coming to appreciate that many diseases are in fact complex, heterogeneous syndromes composed of subpopulations that express different variants of a related complication. Longitudinal data extracted from individual electronic health records (EHR) offer an exciting new way to study subtle differences in the way these diseases progress over time. In this paper, we focus on answering two questions that can be asked using these databases of longitudinal EHR data. First, we want to understand whether there are individuals with similar disease trajectories and whether there are a small number of degrees of freedom that account for differences in trajectories across the population. Second, we want to understand how important clinical outcomes are associated with disease trajectories. To answer these questions, we propose the Disease Trajectory Map (DTM), a novel probabilistic model that learns low-dimensional representations of sparse and irregularly sampled longitudinal data. We propose a stochastic variational inference algorithm for learning the DTM that allows the model to scale to large modern medical datasets. To demonstrate the DTM, we analyze data collected on patients with the complex autoimmune disease, scleroderma. We find that DTM learns meaningful representations of disease trajectories and that the representations are significantly associated with important clinical outcomes.

Experience constantly shapes neural circuits through a variety of plasticity mechanisms. While the functional roles of some plasticity mechanisms are well-understood, it remains unclear how changes in neural excitability contribute to learning. Here, we develop a normative interpretation of intrinsic plasticity (IP) as a key component of unsupervised learning. We introduce a novel generative mixture model that accounts for the class-specific statistics of stimulus intensities, and we derive a neural circuit that learns the input classes and their intensities. We will analytically show that inference and learning for our generative model can be achieved by a neural circuit with intensity-sensitive neurons equipped with a specific form of IP. Numerical experiments verify our analytical derivations and show robust behavior for artificial and natural stimuli. Our results link IP to non-trivial input statistics, in particular the statistics of stimulus intensities for classes to which a neuron is sensitive. More generally, our work paves the way toward new classification algorithms that are robust to intensity variations.

We propose to prune a random forest (RF) for resource-constrained prediction. We first construct a RF and then prune it to optimize expected feature cost & accuracy. We pose pruning RFs as a novel 0-1 integer program with linear constraints that encourages feature re-use. We establish total unimodularity of the constraint set to prove that the corresponding LP relaxation solves the original integer program. We then exploit connections to combinatorial optimization and develop an efficient primal-dual algorithm, scalable to large datasets. In contrast to our bottom-up approach, which benefits from good RF initialization, conventional methods are top-down acquiring features based on their utility value and is generally intractable, requiring heuristics. Empirically, our pruning algorithm outperforms existing state-of-the-art resource-constrained algorithms.

Through the introduction of a diffusion-convolution operation, we show how diffusion-based representations can be learned from graph-structured data and used as an effective basis for node classification. DCNNs have several attractive qualities, including a latent representation for graphical data that is invariant under isomorphism, as well as polynomial-time prediction and learning that can be represented as tensor operations and efficiently implemented on a GPU. Through several experiments with real structured datasets, we demonstrate that DCNNs are able to outperform probabilistic relational models and kernel-on-graph methods at relational node classification tasks.

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. Overall, our results highlight the necessity of careful initialization when using the EM algorithm in practice, even when applied in highly favorable settings.