AISTATS 2019 论文列表

We prove that two popular linear contextual bandit algorithms, OFUL and Thompson Sampling, can be made efficient using Frequent Directions, a deterministic online sketching technique. More precisely, we show that a sketch of size $m$ allows a $\mathcal{O}(md)$ update time for both algorithms, as opposed to $\Omega(d^2)$ required by their non-sketched versions in general (where $d$ is the dimension of context vectors). This computational speedup is accompanied by regret bounds of order $(1+\varepsilon_m)^{3/2}d\sqrt{T}$ for OFUL and of order $\big((1+\varepsilon_m)d\big)^{3/2}\sqrt{T}$ for Thompson Sampling, where $\varepsilon_m$ is bounded by the sum of the tail eigenvalues not covered by the sketch. In particular, when the selected contexts span a subspace of dimension at most $m$, our algorithms have a regret bound matching that of their slower, non-sketched counterparts. Experiments on real-world datasets corroborate our theoretical results.

We generalize the log Gaussian Cox process (LGCP) framework to model multiple correlated point data jointly. The observations are treated as realizations of multiple LGCPs, whose log intensities are given by linear combinations of latent functions drawn from Gaussian process priors. The combination coefficients are also drawn from Gaussian processes and can incorporate additional dependencies. We derive closed-form expressions for the moments of the intensity functions and develop an efficient variational inference algorithm that is orders of magnitude faster than competing deterministic and stochastic approximations of multivariate LGCPs, coregionalization models, and multi-task permanental processes. Our approach outperforms these benchmarks in multiple problems, offering the current state of the art in modeling multivariate point processes.

Coordinate descent with random coordinate selection is the current state of the art for many large scale optimization problems. However, greedy selection of the steepest coordinate on smooth problems can yield convergence rates independent of the dimension $n$, requiring $n$ times fewer iterations. In this paper, we consider greedy updates that are based on subgradients for a class of non-smooth composite problems, including $L1$-regularized problems, SVMs and related applications. For these problems we provide (i) the first linear rates of convergence independent of $n$, and show that our greedy update rule provides speedups similar to those obtained in the smooth case. This was previously conjectured to be true for a stronger greedy coordinate selection strategy. Furthermore, we show that (ii) our new selection rule can be mapped to instances of maximum inner product search, allowing to leverage standard nearest neighbor algorithms to speed up the implementation. We demonstrate the validity of the approach through extensive numerical experiments.

We consider the problem of estimating a target vector by querying an unknown multi-output function which is stochastic and expensive to evaluate. Through sequential experimental design the aim is to minimize the squared Euclidean distance between the output of the function and the target vector. Applying standard single-objective Bayesian optimization to this problem is both wasteful, since individual output components are never observed, and imprecise since the predictive distribution for new inputs will be symmetric and have negative support. We address this issue by proposing a Gaussian process model that considers the individual function outputs and derive a distribution over the resulting 2-norm. Furthermore we derive computationally efficient acquisition functions and evaluate the resulting optimization framework on several synthetic problems and a real-world problem. The results demonstrate a significant improvement over Bayesian optimization based on both standard and warped Gaussian processes.

Bayesian experimental design involves the optimal allocation of resources in an experiment, with the aim of optimising cost and performance. For implicit models, where the likelihood is intractable but sampling from the model is possible, this task is particularly difficult and therefore largely unexplored. This is mainly due to technical difficulties associated with approximating posterior distributions and utility functions. We devise a novel experimental design framework for implicit models that improves upon previous work in two ways. First, we use the mutual information between parameters and data as the utility function, which has previously not been feasible. We achieve this by utilising Likelihood-Free Inference by Ratio Estimation (LFIRE) to approximate posterior distributions, instead of the traditional approximate Bayesian computation or synthetic likelihood methods. Secondly, we use Bayesian optimisation in order to solve the optimal design problem, as opposed to the typically used grid search or sampling-based methods. We find that this increases efficiency and allows us to consider higher design dimensions.

In some real world applications, acquiring covariates for classification can be cost-intensive and should be limited as much as possible. For example, in the medical setting, a doctor cannot just perform all possible types of tests to classify whether the patient has diabetes or not. The decision of classifying or acquiring more covariates before classifying is dependent on the costs of new covariates and the expected optimal cost of misclassification (Bayes risk). However, estimating the latter is a formidable task due to the estimation of a high dimensional probability density and intractable integrals. In this work, we show that for linear classifiers this task can be considerably simplified, leading to a one dimensional integral for which we propose an efficient approximation. Experimental results on three datasets show consistent improvements over previously proposed methods for cost-sensitive classification. We also demonstrate that our proposed Bayes risk estimation procedure can benefit from additional unlabeled data which can be helpful when only small amount of labeled data is available.

We demonstrate that in residual neural networks (ResNets) dynamical isometry is achievable irrespective of the activation function used. We do that by deriving, with the help of Free Probability and Random Matrix Theories, a universal formula for the spectral density of the input-output Jacobian at initialization, in the large network width and depth limit. The resulting singular value spectrum depends on a single parameter, which we calculate for a variety of popular activation functions, by analyzing the signal propagation in the artificial neural network. We corroborate our results with numerical simulations of both random matrices and ResNets applied to the CIFAR-10 classification problem. Moreover, we study consequences of this universal behavior for the initial and late phases of the learning processes. We conclude by drawing attention to the simple fact, that initialization acts as a confounding factor between the choice of activation function and the rate of learning. We propose that in ResNets this can be resolved based on our results by ensuring the same level of dynamical isometry at initialization.

We extend the existing framework of semi-implicit variational inference (SIVI) and introduce doubly semi-implicit variational inference (DSIVI), a way to perform variational inference and learning when both the approximate posterior and the prior distribution are semi-implicit. In other words, DSIVI performs inference in models where the prior and the posterior can be expressed as an intractable infinite mixture of some analytic density with a highly flexible implicit mixing distribution. We provide a sandwich bound on the evidence lower bound (ELBO) objective that can be made arbitrarily tight. Unlike discriminator-based and kernel-based approaches to implicit variational inference, DSIVI optimizes a proper lower bound on ELBO that is asymptotically exact. We evaluate DSIVI on a set of problems that benefit from implicit priors. In particular, we show that DSIVI gives rise to a simple modification of VampPrior, the current state-of-the-art prior for variational autoencoders, which improves its performance.

Several domains in AI need to represent the relational structure as well as model uncertainty. Markov Logic is a powerful formalism which achieves this by attaching weights to formulas in finite first-order logic. Though Markov Logic Networks (MLNs) have been used for a wide variety of applications, a significant challenge remains that weights do not generalize well when training domain sizes are different from those seen during testing. In particular, it has been observed that marginal probabilities tend to extremes in the limit of increasing domain sizes. As the first contribution of our work, we further characterize the distribution and show that marginal probabilities tend to a constant independent of weights and not always to extremes as was previously observed. As our second contribution, we present a principled solution to this problem by defining Domain-size Aware Markov Logic Networks (DA-MLNs) which can be seen as re-parameterizing the MLNs after taking domain size into consideration. For some simple but representative MLN formulas, we formally prove that probabilities defined by DA-MLNs are well behaved. On a practical side, DA-MLNs allow us to generalize the weights learned over small-sized training data to much larger domains. Experiments on three different benchmark MLNs show that our approach results in significant performance gains compared to existing methods.

We show that classical learning methods interpolating the training data can achieve optimal rates for the problems of nonparametric regression and prediction with square loss.

Submodular functions have applications throughout machine learning, but in many settings, we do not have direct access to the underlying function f. We focus on stochastic functions that are given as an expectation of functions over a distribution P. In practice, we often have only a limited set of samples f_i from P. The standard approach indirectly optimizes f by maximizing the sum of f_i. However, this ignores generalization to the true (unknown) distribution. In this paper, we achieve better performance on the actual underlying function f by directly optimizing a combination of bias and variance. Algorithmically, we accomplish this by showing how to carry out distributionally robust optimization (DRO) for submodular functions, providing efficient algorithms backed by theoretical guarantees which leverage several novel contributions to the general theory of DRO. We also show compelling empirical evidence that DRO improves generalization to the unknown stochastic submodular function.

Despite many algorithmic advances, our theoretical understanding of practical distributional reinforcement learning methods remains limited. One exception is Rowland et al. (2018)’s analysis of the C51 algorithm in terms of the Cramer distance, but their results only apply to the tabular setting and ignore C51’s use of a softmax to produce normalized distributions. In this paper we adapt the Cramer distance to deal with arbitrary vectors. From it we derive a new distributional algorithm which is fully Cramer-based and can be combined to linear function approximation, with formal guarantees in the context of policy evaluation. In allowing the model’s prediction to be any real vector, we lose the probabilistic interpretation behind the method, but otherwise maintain the appealing properties of distributional approaches. To the best of our knowledge, ours is the first proof of convergence of a distributional algorithm combined with function approximation. Perhaps surprisingly, our results provide evidence that Cramer-based distributional methods may perform worse than directly approximating the value function.

There are many problems in machine learning and data mining which are equivalent to selecting a non-redundant, high "quality" set of objects. Recommender systems, feature selection, and data summarization are among many applications of this. In this paper, we consider this problem as an optimization problem that seeks to maximize the sum of a sum-sum diversity function and a non-negative monotone submodular function. The diversity function addresses the redundancy, and the submodular function controls the predictive quality. We consider the problem in big data settings (in other words, distributed and streaming settings) where the data cannot be stored on a single machine or the process time is too high for a single machine. We show that a greedy algorithm achieves a constant factor approximation of the optimal solution in these settings. Moreover, we formulate the multi-label feature selection problem as such an optimization problem. This formulation combined with our algorithm leads to the first distributed multi-label feature selection method. We compare the performance of this method with centralized multi-label feature selection methods in the literature, and we show that its performance is comparable or in some cases is even better than current centralized multi-label feature selection methods.

In this paper, we present a sample distributed greedy pursuit method for non-convex sparse learning under cardinality constraint. Given the training samples uniformly randomly partitioned across multiple machines, the proposed method alternates between local inexact sparse minimization of a Newton-type approximation and centralized global results aggregation. Theoretical analysis shows that for a general class of convex functions with Lipschitze continues Hessian, the method converges linearly with contraction factor scaling inversely to the local data size; whilst the communication complexity required to reach desirable statistical accuracy scales logarithmically with respect to the number of machines for some popular statistical learning models. For nonconvex objective functions, up to a local estimation error, our method can be shown to converge to a local stationary sparse solution with sub-linear communication complexity. Numerical results demonstrate the efficiency and accuracy of our method when applied to large-scale sparse learning tasks including deep neural nets pruning

The transfer of knowledge from one policy to another is an important tool in Deep Reinforcement Learning. This process, referred to as distillation, has been used to great success, for example, by enhancing the optimisation of agents, leading to stronger performance faster, on harder domains. Despite the widespread use and conceptual simplicity of distillation, many different formulations are used in practice, and the subtle variations between them can often drastically change the performance and the resulting objective that is being optimised. In this work, we rigorously explore the entire landscape of policy distillation, comparing the motivations and strengths of each variant through theoretical and empirical analysis. Our results point to three distillation techniques, that are preferred depending on specifics of the task. Specifically a newly proposed expected entropy regularised distillation allows for quicker learning in a wide range of situations, while still guaranteeing convergence.

Variance reduction is a simple and effective technique that accelerates convex (or non-convex) stochastic optimization. Among existing variance reduction methods, SVRG and SAGA adopt unbiased gradient estimators and are the most popular variance reduction methods in recent years. Although various accelerated variants of SVRG (e.g., Katyusha and Acc-Prox-SVRG) have been proposed, the direct acceleration of SAGA still remains unknown. In this paper, we propose a directly accelerated variant of SAGA using a novel Sampled Negative Momentum (SSNM), which achieves the best known oracle complexity for strongly convex problems (with known strong convexity parameter). Consequently, our work fills the void of directly accelerated SAGA.

In this paper we develop the first algorithms for online submodular minimization that preserve differential privacy under full information feedback and bandit feedback. Our first result is in the full information setting, where the algorithm can observe the entire function after making its decision at each time step. We give an algorithm in this setting that is $\eps$-differentially private and achieves expected regret $\tilde{O}\left(\frac{n\sqrt{T}}{\eps}\right)$ over $T$ rounds for a collection of $n$ elements. Our second result is in the bandit setting, where the algorithm can only observe the cost incurred by its chosen set, and does not have access to the entire function. This setting is significantly more challenging due to the limited information. Our algorithm using bandit feedback is $\eps$-differentially private and achieves expected regret $\tilde{O}\left(\frac{n^{3/2}T^{2/3}}{\eps}\right)$.

Stochastic optimization techniques are standard in variational inference algorithms. These methods estimate gradients by approximating expectations with independent Monte Carlo samples. In this paper, we explore a technique that uses correlated, but more representative, samples to reduce estimator variance. Specifically, we show how to generate antithetic samples that match sample moments with the true moments of an underlying importance distribution. Combining a differentiable antithetic sampler with modern stochastic variational inference, we showcase the effectiveness of this approach for learning a deep generative model. An implementation is available at https://github.com/mhw32/antithetic-vae-public.

We study the detection problem of finding planted solutions in random instances of flat satisfiability problems, a generalization of boolean satisfiability formulas. We describe the properties of random instances of flat satisfiability, as well of the optimal rates of detection of the associated hypothesis testing problem. We also study the performance of an algorithmically efficient testing procedure. We introduce a modification of our model, the light planting of solutions, and show that it is as hard as the problem of learning parity with noise. This hints strongly at the difficulty of detecting planted flat satisfiability for a wide class of tests.