We consider the problem of global optimization of an unknown non-convex smooth function with noisy zeroth-order feedback. We propose a local minimax framework to study the fundamental difficulty of optimizing smooth functions with adaptive function evaluations. We show that for functions with fast growth around their global minima, carefully designed optimization algorithms can identify a near global minimizer with many fewer queries than worst-case global minimax theory predicts. For the special case of strongly convex and smooth functions, our implied convergence rates match the ones developed for zeroth-order convex optimization problems. On the other hand, we show that in the worst case no algorithm can converge faster than the minimax rate of estimating an unknown functions in linf-norm. Finally, we show that non-adaptive algorithms, although optimal in a global minimax sense, do not attain the optimal local minimax rate.

The design of neural network architectures is an important component for achieving state-of-the-art performance with machine learning systems across a broad array of tasks. Much work has endeavored to design and build architectures automatically through clever construction of a search space paired with simple learning algorithms. Recent progress has demonstrated that such meta-learning methods may exceed scalable human-invented architectures on image classification tasks. An open question is the degree to which such methods may generalize to new domains. In this work we explore the construction of meta-learning techniques for dense image prediction focused on the tasks of scene parsing, person-part segmentation, and semantic image segmentation. Constructing viable search spaces in this domain is challenging because of the multi-scale representation of visual information and the necessity to operate on high resolution imagery. Based on a survey of techniques in dense image prediction, we construct a recursive search space and demonstrate that even with efficient random search, we can identify architectures that outperform human-invented architectures and achieve state-of-the-art performance on three dense prediction tasks including 82.7% on Cityscapes (street scene parsing), 71.3% on PASCAL-Person-Part (person-part segmentation), and 87.9% on PASCAL VOC 2012 (semantic image segmentation). Additionally, the resulting architecture is more computationally efficient, requiring half the parameters and half the computational cost as previous state of the art systems.

We study the problem of maximizing deep submodular functions (DSFs) subject to a matroid constraint. DSFs are an expressive class of submodular functions that include, as strict subfamilies, the facility location, weighted coverage, and sums of concave composed with modular functions. We use a strategy similar to the continuous greedy approach, but we show that the multilinear extension of any DSF has a natural and computationally attainable concave relaxation that we can optimize using gradient ascent. Our results show a guarantee of $\max_{0<\delta<1}(1-\epsilon-\delta-e^{-\delta^2\Omega(k)})$ with a running time of $O(\nicefrac{n^2}{\epsilon^2})$ plus time for pipage rounding to recover a discrete solution, where $k$ is the rank of the matroid constraint. This bound is often better than the standard $1-1/e$ guarantee of the continuous greedy algorithm, but runs much faster. Our bound also holds even for fully curved ($c=1$) functions where the guarantee of $1-c/e$ degenerates to $1-1/e$ where $c$ is the curvature of $f$. We perform computational experiments that support our theoretical results.

Neyshabur and Srebro proposed SIMPLE-LSH, which is the state-of-the-art hashing based algorithm for maximum inner product search (MIPS). We found that the performance of SIMPLE-LSH, in both theory and practice, suffers from long tails in the 2-norm distribution of real datasets. We propose NORM-RANGING LSH, which addresses the excessive normalization problem caused by long tails by partitioning a dataset into sub-datasets and building a hash index for each sub-dataset independently. We prove that NORM-RANGING LSH achieves lower query time complexity than SIMPLE-LSH under mild conditions. We also show that the idea of dataset partitioning can improve another hashing based MIPS algorithm. Experiments show that NORM-RANGING LSH probes much less items than SIMPLE-LSH at the same recall, thus significantly benefiting MIPS based applications.

We consider a generalization of mixed regression where the response is an additive combination of several mixture components. Standard mixed regression is a special case where each response is generated from exactly one component. Typical approaches to the mixture regression problem employ local search methods such as Expectation Maximization (EM) that are prone to spurious local optima. On the other hand, a number of recent theoretically-motivated \emph{Tensor-based methods} either have high sample complexity, or require the knowledge of the input distribution, which is not available in most of practical situations. In this work, we study a novel convex estimator \emph{MixLasso} for the estimation of generalized mixed regression, based on an atomic norm specifically constructed to regularize the number of mixture components. Our algorithm gives a risk bound that trades off between prediction accuracy and model sparsity without imposing stringent assumptions on the input/output distribution, and can be easily adapted to the case of non-linear functions. In our numerical experiments on mixtures of linear as well as nonlinear regressions, the proposed method yields high-quality solutions in a wider range of settings than existing approaches.

Beam search is widely used for approximate decoding in structured prediction problems. Models often use a beam at test time but ignore its existence at train time, and therefore do not explicitly learn how to use the beam. We develop an unifying meta-algorithm for learning beam search policies using imitation learning. In our setting, the beam is part of the model, and not just an artifact of approximate decoding. Our meta-algorithm captures existing learning algorithms and suggests new ones. It also lets us show novel no-regret guarantees for learning beam search policies.

We introduce a new RL problem where the agent is required to generalize to a previously-unseen environment characterized by a subtask graph which describes a set of subtasks and their dependencies. Unlike existing hierarchical multitask RL approaches that explicitly describe what the agent should do at a high level, our problem only describes properties of subtasks and relationships among them, which requires the agent to perform complex reasoning to find the optimal subtask to execute. To solve this problem, we propose a neural subtask graph solver (NSGS) which encodes the subtask graph using a recursive neural network embedding. To overcome the difficulty of training, we propose a novel non-parametric gradient-based policy, graph reward propagation, to pre-train our NSGS agent and further finetune it through actor-critic method. The experimental results on two 2D visual domains show that our agent can perform complex reasoning to find a near-optimal way of executing the subtask graph and generalize well to the unseen subtask graphs. In addition, we compare our agent with a Monte-Carlo tree search (MCTS) method showing that our method is much more efficient than MCTS, and the performance of NSGS can be further improved by combining it with MCTS.

Stimulus-driven brain-computer interfaces (BCIs), such as the P300 speller, rely on using a sequence of sensory stimuli to elicit specific neural responses as control signals, while a user attends to relevant target stimuli that occur within the sequence. In current BCIs, the stimulus presentation schedule is typically generated in a pseudo-random fashion. Given the non-stationarity of brain electrical signals, a better strategy could be to adapt the stimulus presentation schedule in real-time by selecting the optimal stimuli that will maximize the signal-to-noise ratios of the elicited neural responses and provide the most information about the user's intent based on the uncertainties of the data being measured. However, the high-dimensional stimulus space limits the development of algorithms with tractable solutions for optimized stimulus selection to allow for real-time decision-making within the stringent time requirements of BCI processing. We derive a simple analytical solution of an information-based objective function for BCI stimulus selection by transforming the high-dimensional stimulus space into a one-dimensional space that parameterizes the objective function - the prior probability mass of the stimulus under consideration, irrespective of its contents. We demonstrate the utility of our adaptive stimulus selection algorithm in improving BCI performance with results from simulation and real-time human experiments.

Active search is a learning paradigm for actively identifying as many members of a given class as possible. A critical target scenario is high-throughput screening for scientific discovery, such as drug or materials discovery. In these settings, specialized instruments can often evaluate \emph{multiple} points simultaneously; however, all existing work on active search focuses on sequential acquisition. We bridge this gap, addressing batch active search from both the theoretical and practical perspective. We first derive the Bayesian optimal policy for this problem, then prove a lower bound on the performance gap between sequential and batch optimal policies: the ``cost of parallelization.'' We also propose novel, efficient batch policies inspired by state-of-the-art sequential policies, and develop an aggressive pruning technique that can dramatically speed up computation. We conduct thorough experiments on data from three application domains: a citation network, material science, and drug discovery, testing all proposed policies (14 total) with a wide range of batch sizes. Our results demonstrate that the empirical performance gap matches our theoretical bound, that nonmyopic policies usually significantly outperform myopic alternatives, and that diversity is an important consideration for batch policy design.

Learning the minimum/maximum mean among a finite set of distributions is a fundamental sub-problem in planning, game tree search and reinforcement learning. We formalize this learning task as the problem of sequentially testing how the minimum mean among a finite set of distributions compares to a given threshold. We develop refined non-asymptotic lower bounds, which show that optimality mandates very different sampling behavior for a low vs high true minimum. We show that Thompson Sampling and the intuitive Lower Confidence Bounds policy each nail only one of these cases. We develop a novel approach that we call Murphy Sampling. Even though it entertains exclusively low true minima, we prove that MS is optimal for both possibilities. We then design advanced self-normalized deviation inequalities, fueling more aggressive stopping rules. We complement our theoretical guarantees by experiments showing that MS works best in practice.