NeurIPS 2018 Papers

Neural models operating over structured spaces such as knowledge graphs require a continuous embedding of the discrete elements of this space (such as entities) as well as the relationships between them. Relational embeddings with high expressivity, however, have high model complexity, making them computationally difficult to train. We propose a new family of embeddings for knowledge graphs that interpolate between a method with high model complexity and one, namely Holographic embeddings (HolE), with low dimensionality and high training efficiency. This interpolation, termed HolEx, is achieved by concatenating several linearly perturbed copies of original HolE. We formally characterize the number of perturbed copies needed to provably recover the full entity-entity or entity-relation interaction matrix, leveraging ideas from Haar wavelets and compressed sensing. In practice, using just a handful of Haar-based or random perturbation vectors results in a much stronger knowledge completion system. On the Freebase FB15K dataset, HolEx outperforms originally reported HolE by 14.7\% on the HITS@10 metric, and the current path-based state-of-the-art method, PTransE, by 4\% (absolute).

Stochastic gradient descent (SGD) remains the method of choice for deep learning, despite the limitations arising for ill-behaved objective functions. In cases where it could be estimated, the natural gradient has proven very effective at mitigating the catastrophic effects of pathological curvature in the objective function, but little is known theoretically about its convergence properties, and it has yet to find a practical implementation that would scale to very deep and large networks. Here, we derive an exact expression for the natural gradient in deep linear networks, which exhibit pathological curvature similar to the nonlinear case. We provide for the first time an analytical solution for its convergence rate, showing that the loss decreases exponentially to the global minimum in parameter space. Our expression for the natural gradient is surprisingly simple, computationally tractable, and explains why some approximations proposed previously work well in practice. This opens new avenues for approximating the natural gradient in the nonlinear case, and we show in preliminary experiments that our online natural gradient descent outperforms SGD on MNIST autoencoding while sharing its computational simplicity.

Recent milestones in equilibrium computation, such as the success of Libratus, show that it is possible to compute strong solutions to two-player zero-sum games in theory and practice. This is not the case for games with more than two players, which remain one of the main open challenges in computational game theory. This paper focuses on zero-sum games where a team of players faces an opponent, as is the case, for example, in Bridge, collusion in poker, and many non-recreational applications such as war, where the colluders do not have time or means of communicating during battle, collusion in bidding, where communication during the auction is illegal, and coordinated swindling in public. The possibility for the team members to communicate before game play—that is, coordinate their strategies ex ante—makes the use of behavioral strategies unsatisfactory. The reasons for this are closely related to the fact that the team can be represented as a single player with imperfect recall. We propose a new game representation, the realization form, that generalizes the sequence form but can also be applied to imperfect-recall games. Then, we use it to derive an auxiliary game that is equivalent to the original one. It provides a sound way to map the problem of finding an optimal ex-ante-correlated strategy for the team to the well-understood Nash equilibrium-finding problem in a (larger) two-player zero-sum perfect-recall game. By reasoning over the auxiliary game, we devise an anytime algorithm, fictitious team-play, that is guaranteed to converge to an optimal coordinated strategy for the team against an optimal opponent, and that is dramatically faster than the prior state-of-the-art algorithm for this problem.

We propose a metalearning approach for learning gradient-based reinforcement learning (RL) algorithms. The idea is to evolve a differentiable loss function, such that an agent, which optimizes its policy to minimize this loss, will achieve high rewards. The loss is parametrized via temporal convolutions over the agent's experience. Because this loss is highly flexible in its ability to take into account the agent's history, it enables fast task learning. Empirical results show that our evolved policy gradient algorithm (EPG) achieves faster learning on several randomized environments compared to an off-the-shelf policy gradient method. We also demonstrate that EPG's learned loss can generalize to out-of-distribution test time tasks, and exhibits qualitatively different behavior from other popular metalearning algorithms.

We propose a population-based Evolutionary Stochastic Gradient Descent (ESGD) framework for optimizing deep neural networks. ESGD combines SGD and gradient-free evolutionary algorithms as complementary algorithms in one framework in which the optimization alternates between the SGD step and evolution step to improve the average fitness of the population. With a back-off strategy in the SGD step and an elitist strategy in the evolution step, it guarantees that the best fitness in the population will never degrade. In addition, individuals in the population optimized with various SGD-based optimizers using distinct hyper-parameters in the SGD step are considered as competing species in a coevolution setting such that the complementarity of the optimizers is also taken into account. The effectiveness of ESGD is demonstrated across multiple applications including speech recognition, image recognition and language modeling, using networks with a variety of deep architectures.

Deep Reinforcement Learning (DRL) algorithms have been successfully applied to a range of challenging control tasks. However, these methods typically suffer from three core difficulties: temporal credit assignment with sparse rewards, lack of effective exploration, and brittle convergence properties that are extremely sensitive to hyperparameters. Collectively, these challenges severely limit the applicability of these approaches to real world problems. Evolutionary Algorithms (EAs), a class of black box optimization techniques inspired by natural evolution, are well suited to address each of these three challenges. However, EAs typically suffer from high sample complexity and struggle to solve problems that require optimization of a large number of parameters. In this paper, we introduce Evolutionary Reinforcement Learning (ERL), a hybrid algorithm that leverages the population of an EA to provide diversified data to train an RL agent, and reinserts the RL agent into the EA population periodically to inject gradient information into the EA. ERL inherits EA's ability of temporal credit assignment with a fitness metric, effective exploration with a diverse set of policies, and stability of a population-based approach and complements it with off-policy DRL's ability to leverage gradients for higher sample efficiency and faster learning. Experiments in a range of challenging continuous control benchmarks demonstrate that ERL significantly outperforms prior DRL and EA methods.

Deterministic neural nets have been shown to learn effective predictors on a wide range of machine learning problems. However, as the standard approach is to train the network to minimize a prediction loss, the resultant model remains ignorant to its prediction confidence. Orthogonally to Bayesian neural nets that indirectly infer prediction uncertainty through weight uncertainties, we propose explicit modeling of the same using the theory of subjective logic. By placing a Dirichlet distribution on the class probabilities, we treat predictions of a neural net as subjective opinions and learn the function that collects the evidence leading to these opinions by a deterministic neural net from data. The resultant predictor for a multi-class classification problem is another Dirichlet distribution whose parameters are set by the continuous output of a neural net. We provide a preliminary analysis on how the peculiarities of our new loss function drive improved uncertainty estimation. We observe that our method achieves unprecedented success on detection of out-of-distribution queries and endurance against adversarial perturbations.

Information theoretic quantities play an important role in various settings in machine learning, including causality testing, structure inference in graphical models, time-series problems, feature selection as well as in providing privacy guarantees. A key quantity of interest is the mutual information and generalizations thereof, including conditional mutual information, multivariate mutual information, total correlation and directed information. While the aforementioned information quantities are well defined in arbitrary probability spaces, existing estimators employ a $\Sigma H$ method, which can only work in purely discrete space or purely continuous case since entropy (or differential entropy) is well defined only in that regime. In this paper, we define a general graph divergence measure ($\mathbb{GDM}$), generalizing the aforementioned information measures and we construct a novel estimator via a coupling trick that directly estimates these multivariate information measures using the Radon-Nikodym derivative. These estimators are proven to be consistent in a general setting which includes several cases where the existing estimators fail, thus providing the only known estimators for the following settings: (1) the data has some discrete and some continuous valued components (2) some (or all) of the components themselves are discrete-continuous \textit{mixtures} (3) the data is real-valued but does not have a joint density on the entire space, rather is supported on a low-dimensional manifold. We show that our proposed estimators significantly outperform known estimators on synthetic and real datasets.

We consider the problem of estimating how well a model class is capable of fitting a distribution of labeled data. We show that it is often possible to accurately estimate this ``learnability'' even when given an amount of data that is too small to reliably learn any accurate model. Our first result applies to the setting where the data is drawn from a $d$-dimensional distribution with isotropic covariance, and the label of each datapoint is an arbitrary noisy function of the datapoint. In this setting, we show that with $O(\sqrt{d})$ samples, one can accurately estimate the fraction of the variance of the label that can be explained via the best linear function of the data. We extend these techniques to a binary classification, and show that the prediction error of the best linear classifier can be accurately estimated given $O(\sqrt{d})$ labeled samples. For comparison, in both the linear regression and binary classification settings, even if there is no noise in the labels, a sample size linear in the dimension, $d$, is required to \emph{learn} any function correlated with the underlying model. We further extend our estimation approach to the setting where the data distribution has an (unknown) arbitrary covariance matrix, allowing these techniques to be applied to settings where the model class consists of a linear function applied to a nonlinear embedding of the data. We demonstrate the practical viability of our approaches on synthetic and real data. This ability to estimate the explanatory value of a set of features (or dataset), even in the regime in which there is too little data to realize that explanatory value, may be relevant to the scientific and industrial settings for which data collection is expensive and there are many potentially relevant feature sets that could be collected.

In this paper, we study the problem of escaping from saddle points in smooth nonconvex optimization problems subject to a convex set $\mathcal{C}$. We propose a generic framework that yields convergence to a second-order stationary point of the problem, if the convex set $\mathcal{C}$ is simple for a quadratic objective function. Specifically, our results hold if one can find a $\rho$-approximate solution of a quadratic program subject to $\mathcal{C}$ in polynomial time, where $\rho<1$ is a positive constant that depends on the structure of the set $\mathcal{C}$. Under this condition, we show that the sequence of iterates generated by the proposed framework reaches an $(\epsilon,\gamma)$-second order stationary point (SOSP) in at most $\mathcal{O}(\max\{\epsilon^{-2},\rho^{-3}\gamma^{-3}\})$ iterations. We further characterize the overall complexity of reaching an SOSP when the convex set $\mathcal{C}$ can be written as a set of quadratic constraints and the objective function Hessian has a specific structure over the convex $\mathcal{C}$. Finally, we extend our results to the stochastic setting and characterize the number of stochastic gradient and Hessian evaluations to reach an $(\epsilon,\gamma)$-SOSP.

The Equalized Odds (for short, EO) is one of the most popular measures of discrimination used in the supervised learning setting. It ascertains fairness through the balance of the misclassification rates (false positive and negative) across the protected groups -- e.g., in the context of law enforcement, an African-American defendant who would not commit a future crime will have an equal opportunity of being released, compared to a non-recidivating Caucasian defendant. Despite this noble goal, it has been acknowledged in the literature that statistical tests based on the EO are oblivious to the underlying causal mechanisms that generated the disparity in the first place (Hardt et al. 2016). This leads to a critical disconnect between statistical measures readable from the data and the meaning of discrimination in the legal system, where compelling evidence that the observed disparity is tied to a specific causal process deemed unfair by society is required to characterize discrimination. The goal of this paper is to develop a principled approach to connect the statistical disparities characterized by the EO and the underlying, elusive, and frequently unobserved, causal mechanisms that generated such inequality. We start by introducing a new family of counterfactual measures that allows one to explain the misclassification disparities in terms of the underlying mechanisms in an arbitrary, non-parametric structural causal model. This will, in turn, allow legal and data analysts to interpret currently deployed classifiers through causal lens, linking the statistical disparities found in the data to the corresponding causal processes. Leveraging the new family of counterfactual measures, we develop a learning procedure to construct a classifier that is statistically efficient, interpretable, and compatible with the basic human intuition of fairness. We demonstrate our results through experiments in both real (COMPAS) and synthetic datasets.

We examine a class of stochastic deep learning models with a tractable method to compute information-theoretic quantities. Our contributions are three-fold: (i) We show how entropies and mutual informations can be derived from heuristic statistical physics methods, under the assumption that weight matrices are independent and orthogonally-invariant. (ii) We extend particular cases in which this result is known to be rigorously exact by providing a proof for two-layers networks with Gaussian random weights, using the recently introduced adaptive interpolation method. (iii) We propose an experiment framework with generative models of synthetic datasets, on which we train deep neural networks with a weight constraint designed so that the assumption in (i) is verified during learning. We study the behavior of entropies and mutual information throughout learning and conclude that, in the proposed setting, the relationship between compression and generalization remains elusive.

Entropy estimation is one of the prototypical problems in distribution property testing. To consistently estimate the Shannon entropy of a distribution on $S$ elements with independent samples, the optimal sample complexity scales sublinearly with $S$ as $\Theta(\frac{S}{\log S})$ as shown by Valiant and Valiant \cite{Valiant--Valiant2011}. Extending the theory and algorithms for entropy estimation to dependent data, this paper considers the problem of estimating the entropy rate of a stationary reversible Markov chain with $S$ states from a sample path of $n$ observations. We show that \begin{itemize} \item Provided the Markov chain mixes not too slowly, \textit{i.e.}, the relaxation time is at most $O(\frac{S}{\ln^3 S})$, consistent estimation is achievable when $n \gg \frac{S^2}{\log S}$. \item Provided the Markov chain has some slight dependency, \textit{i.e.}, the relaxation time is at least $1+\Omega(\frac{\ln^2 S}{\sqrt{S}})$, consistent estimation is impossible when $n \lesssim \frac{S^2}{\log S}$. \end{itemize} Under both assumptions, the optimal estimation accuracy is shown to be $\Theta(\frac{S^2}{n \log S})$. In comparison, the empirical entropy rate requires at least $\Omega(S^2)$ samples to be consistent, even when the Markov chain is memoryless. In addition to synthetic experiments, we also apply the estimators that achieve the optimal sample complexity to estimate the entropy rate of the English language in the Penn Treebank and the Google One Billion Words corpora, which provides a natural benchmark for language modeling and relates it directly to the widely used perplexity measure.

Societies often rely on human experts to take a wide variety of decisions affecting their members, from jail-or-release decisions taken by judges and stop-and-frisk decisions taken by police officers to accept-or-reject decisions taken by academics. In this context, each decision is taken by an expert who is typically chosen uniformly at random from a pool of experts. However, these decisions may be imperfect due to limited experience, implicit biases, or faulty probabilistic reasoning. Can we improve the accuracy and fairness of the overall decision making process by optimizing the assignment between experts and decisions?

Machine learning models are changing the paradigm of molecular modeling, which is a fundamental tool for material science, chemistry, and computational biology. Of particular interest is the inter-atomic potential energy surface (PES). Here we develop Deep Potential - Smooth Edition (DeepPot-SE), an end-to-end machine learning-based PES model, which is able to efficiently represent the PES for a wide variety of systems with the accuracy of ab initio quantum mechanics models. By construction, DeepPot-SE is extensive and continuously differentiable, scales linearly with system size, and preserves all the natural symmetries of the system. Further, we show that DeepPot-SE describes finite and extended systems including organic molecules, metals, semiconductors, and insulators with high fidelity.

We present a differentiable physics engine that can be integrated as a module in deep neural networks for end-to-end learning. As a result, structured physics knowledge can be embedded into larger systems, allowing them, for example, to match observations by performing precise simulations, while achieves high sample efficiency. Specifically, in this paper we demonstrate how to perform backpropagation analytically through a physical simulator defined via a linear complementarity problem. Unlike traditional finite difference methods, such gradients can be computed analytically, which allows for greater flexibility of the engine. Through experiments in diverse domains, we highlight the system's ability to learn physical parameters from data, efficiently match and simulate observed visual behavior, and readily enable control via gradient-based planning methods. Code for the engine and experiments is included with the paper.

In this paper, we revisit the Empirical Risk Minimization problem in the non-interactive local model of differential privacy. In the case of constant or low dimensions ($p\ll n$), we first show that if the loss function is $(\infty, T)$-smooth, we can avoid a dependence of the sample complexity, to achieve error $\alpha$, on the exponential of the dimensionality $p$ with base $1/\alpha$ ({\em i.e.,} $\alpha^{-p}$), which answers a question in \cite{smith2017interaction}. Our approach is based on polynomial approximation. Then, we propose player-efficient algorithms with $1$-bit communication complexity and $O(1)$ computation cost for each player. The error bound is asymptotically the same as the original one. With some additional assumptions, we also give an efficient algorithm for the server. In the case of high dimensions ($n\ll p$), we show that if the loss function is a convex generalized linear function, the error can be bounded by using the Gaussian width of the constrained set, instead of $p$, which improves the one in \cite{smith2017interaction}.

We address the problem of algorithmic fairness: ensuring that sensitive information does not unfairly influence the outcome of a classifier. We present an approach based on empirical risk minimization, which incorporates a fairness constraint into the learning problem. It encourages the conditional risk of the learned classifier to be approximately constant with respect to the sensitive variable. We derive both risk and fairness bounds that support the statistical consistency of our methodology. We specify our approach to kernel methods and observe that the fairness requirement implies an orthogonality constraint which can be easily added to these methods. We further observe that for linear models the constraint translates into a simple data preprocessing step. Experiments indicate that the method is empirically effective and performs favorably against state-of-the-art approaches.

Learning low-dimensional embeddings of knowledge graphs is a powerful approach used to predict unobserved or missing edges between entities. However, an open challenge in this area is developing techniques that can go beyond simple edge prediction and handle more complex logical queries, which might involve multiple unobserved edges, entities, and variables. For instance, given an incomplete biological knowledge graph, we might want to predict "em what drugs are likely to target proteins involved with both diseases X and Y?" -- a query that requires reasoning about all possible proteins that might interact with diseases X and Y. Here we introduce a framework to efficiently make predictions about conjunctive logical queries -- a flexible but tractable subset of first-order logic -- on incomplete knowledge graphs. In our approach, we embed graph nodes in a low-dimensional space and represent logical operators as learned geometric operations (e.g., translation, rotation) in this embedding space. By performing logical operations within a low-dimensional embedding space, our approach achieves a time complexity that is linear in the number of query variables, compared to the exponential complexity required by a naive enumeration-based approach. We demonstrate the utility of this framework in two application studies on real-world datasets with millions of relations: predicting logical relationships in a network of drug-gene-disease interactions and in a graph-based representation of social interactions derived from a popular web forum.