Inference in the presence of outliers is an important field of research as outliers are ubiquitous and may arise across a variety of problems and domains. Bayesian optimization is method that heavily relies on probabilistic inference. This allows outstanding sample efficiency because the probabilistic machinery provides a memory of the whole optimization process. However, that virtue becomes a disadvantage when the memory is populated with outliers, inducing bias in the estimation. In this paper, we present an empirical evaluation of Bayesian optimization methods in the presence of outliers. The empirical evidence shows that Bayesian optimization with robust regression often produces suboptimal results. We then propose a new algorithm which combines robust regression (a Gaussian process with Student-t likelihood) with outlier diagnostics to classify data points as outliers or inliers. By using an scheduler for the classification of outliers, our method is more efficient and has better convergence over the standard robust regression. Furthermore, we show that even in controlled situations with no expected outliers, our method is able to produce better results.

The dueling bandit is a learning framework wherein the feedback information in the learning process is restricted to a noisy comparison between a pair of actions. In this research, we address a dueling bandit problem based on a cost function over a continuous space. We propose a stochastic mirror descent algorithm and show that the algorithm achieves an $O(\sqrt{T\log T})$-regret bound under strong convexity and smoothness assumptions for the cost function. Subsequently, we clarify the equivalence between regret minimization in dueling bandit and convex optimization for the cost function. Moreover, when considering a lower bound in convex optimization, our algorithm is shown to achieve the optimal convergence rate in convex optimization and the optimal regret in dueling bandit except for a logarithmic factor.

This paper aims at theoretically and empirically comparing two standard optimization criteria for Reinforcement Learning: i) maximization of the mean value and ii) minimization of the Bellman residual. For that purpose, we place ourselves in the framework of policy search algorithms, that are usually designed to maximize the mean value, and derive a method that minimizes the residual $\|T_* v_\pi - v_\pi\|_{1,\nu}$ over policies. A theoretical analysis shows how good this proxy is to policy optimization, and notably that it is better than its value-based counterpart. We also propose experiments on randomly generated generic Markov decision processes, specifically designed for studying the influence of the involved concentrability coefficient. They show that the Bellman residual is generally a bad proxy to policy optimization and that directly maximizing the mean value is much better, despite the current lack of deep theoretical analysis. This might seem obvious, as directly addressing the problem of interest is usually better, but given the prevalence of (projected) Bellman residual minimization in value-based reinforcement learning, we believe that this question is worth to be considered.

Evolution sculpts both the body plans and nervous systems of agents together over time. In contrast, in AI and robotics, a robot's body plan is usually designed by hand, and control policies are then optimized for that fixed design. The task of simultaneously co-optimizing the morphology and controller of an embodied robot has remained a challenge. In psychology, the theory of embodied cognition posits that behavior arises from a close coupling between body plan and sensorimotor control, which suggests why co-optimizing these two subsystems is so difficult: most evolutionary changes to morphology tend to adversely impact sensorimotor control, leading to an overall decrease in behavioral performance. Here, we further examine this hypothesis and demonstrate a technique for "morphological innovation protection", which temporarily reduces selection pressure on recently morphologically-changed individuals, thus enabling evolution some time to "readapt" to the new morphology with subsequent control policy mutations. We show the potential for this method to avoid local optima and converge to similar highly fit morphologies across widely varying initial conditions, while sustaining fitness improvements further into optimization. While this technique is admittedly only the first of many steps that must be taken to achieve scalable optimization of embodied machines, we hope that theoretical insight into the cause of evolutionary stagnation in current methods will help to enable the automation of robot design and behavioral training -- while simultaneously providing a testbed to investigate the theory of embodied cognition.

The nonnegative matrix factorization is a widely used, flexible matrix decomposition, finding applications in biology, image and signal processing and information retrieval, among other areas. Here we present a related matrix factorization. A multi-objective optimization problem finds conical combinations of templates that approximate a given data matrix. The templates are chosen so that as far as possible only the initial data set can be represented this way. However, the templates are not required to be nonnegative nor convex combinations of the original data.

Embedding methods such as word embedding have become pillars for many applications containing discrete structures. Conventional embedding methods directly associate each symbol with a continuous embedding vector, which is equivalent to applying linear transformation based on "one-hot" encoding of the discrete symbols. Despite its simplicity, such approach yields number of parameters that grows linearly with the vocabulary size and can lead to overfitting. In this work we propose a much more compact K-way D-dimensional discrete encoding scheme to replace the "one-hot" encoding. In "KD encoding", each symbol is represented by a $D$-dimensional code, and each of its dimension has a cardinality of $K$. The final symbol embedding vector can be generated by composing the code embedding vectors. To learn the semantically meaningful code, we derive a relaxed discrete optimization technique based on stochastic gradient descent. By adopting the new coding system, the efficiency of parameterization can be significantly improved (from linear to logarithmic), and this can also mitigate the over-fitting problem. In our experiments with language modeling, the number of embedding parameters can be reduced by 97\% while achieving similar or better performance.

In order to explain the convexity of mathematical functions, think about a large drinking cup and a ball. Now, let the ball roll from the border of the cup. Obviously, the ball will roll down towards the bottom of the cup, then it will oscillate a bit, eventually landing towards the lowest point of the cup. This is called the minimum, and the mathematical function that describes the cup is a convex function. It turns out that the minimum, in this case, is also the global minimum.

In this paper, we study the general problem of optimizing a convex function $F(L)$ over the set of $p \times p$ matrices, subject to rank constraints on $L$. However, existing first-order methods for solving such problems either are too slow to converge, or require multiple invocations of singular value decompositions. On the other hand, factorization-based non-convex algorithms, while being much faster, require stringent assumptions on the \emph{condition number} of the optimum. In this paper, we provide a novel algorithmic framework that achieves the best of both worlds: asymptotically as fast as factorization methods, while requiring no dependency on the condition number. We instantiate our general framework for three important matrix estimation problems that impact several practical applications; (i) a \emph{nonlinear} variant of affine rank minimization, (ii) logistic PCA, and (iii) precision matrix estimation in probabilistic graphical model learning. We then derive explicit bounds on the sample complexity as well as the running time of our approach, and show that it achieves the best possible bounds for both cases. We also provide an extensive range of experimental results, and demonstrate that our algorithm provides a very attractive tradeoff between estimation accuracy and running time.

The kernel embedding algorithm is an important component for adapting kernel methods to large datasets. Since the algorithm consumes a major computation cost in the testing phase, we propose a novel teacher-learner framework of learning computation-efficient kernel embeddings from specific data. In the framework, the high-precision embeddings (teacher) transfer the data information to the computation-efficient kernel embeddings (learner). We jointly select informative embedding functions and pursue an orthogonal transformation between two embeddings. We propose a novel approach of constrained variational expectation maximization (CVEM), where the alternate direction method of multiplier (ADMM) is applied over a nonconvex domain in the maximization step. We also propose two specific formulations based on the prevalent Random Fourier Feature (RFF), the masked and blocked version of Computation-Efficient RFF (CERF), by imposing a random binary mask or a block structure on the transformation matrix. By empirical studies of several applications on different real-world datasets, we demonstrate that the CERF significantly improves the performance of kernel methods upon the RFF, under certain arithmetic operation requirements, and suitable for structured matrix multiplication in Fastfood type algorithms.

Rapidly growing product lines and services require a finer-granularity forecast that considers geographic locales. However the open question remains, how to assess the quality of a spatio-temporal forecast? In this manuscript we introduce a metric to evaluate spatio-temporal forecasts. This metric is based on an Opti- mal Transport (OT) problem. The metric we propose is a constrained OT objec- tive function using the Gini impurity function as a regularizer. We demonstrate through computer experiments both the qualitative and the quantitative charac- teristics of the Gini regularized OT problem. Moreover, we show that the Gini regularized OT problem converges to the classical OT problem, when the Gini regularized problem is considered as a function of {\lambda}, the regularization parame-ter. The convergence to the classical OT solution is faster than the state-of-the-art Entropic-regularized OT[Cuturi, 2013] and results in a numerically more stable algorithm.