Categorical variables are a natural choice for representing discrete structure in the world. However, stochastic neural networks rarely use categorical latent variables due to the inability to backpropagate through samples. In this work, we present an efficient gradient estimator that replaces the non-differentiable sample from a categorical distribution with a differentiable sample from a novel Gumbel-Softmax distribution. This distribution has the essential property that it can be smoothly annealed into a categorical distribution. We show that our Gumbel-Softmax estimator outperforms state-of-the-art gradient estimators on structured output prediction and unsupervised generative modeling tasks with categorical latent variables, and enables large speedups on semi-supervised classification.
For unlabeled data, difficulties arise because the categorical distribution is not reparameterizable. @cite_0 approach this by marginalizing out @math over all classes, so that for unlabeled data, inference is still on @math for each @math . The lower bound on unlabeled data is:
@cite_0
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We study the potential benefits of unlabeled data to classification prediction to the learner. We compare learning in the semi-supervised model to the standard, supervised PAC (distribution free) model, considering both the realizable and the unrealizable (agnostic) settings. Roughly speaking, our conclusion is that access to unlabeled samples cannot provide sample size guarantees that are better than those obtainable without access to unlabeled data, unless one postulates very strong assumptions about the distribution of the labels. In particular, we prove that for basic hypothesis classes over the real line, if the distribution of unlabeled data is ‘smooth’, knowledge of that distribution cannot improve the labeled sample complexity by more than a constant factor (e.g., 2). We conjecture that a similar phenomena holds for any hypothesis class and any unlabeled data distribution. We also discuss the utility of semi-supervised learning under the common cluster assumption concerning the distribution of labels, and show that even in the most accommodating cases, where data is generated by two uni-modal label-homogeneous distributions, common SSL paradigms may be misleading and inflict poor prediction performance. Multiclass classification problems such as image annotation can involve a large number of classes. In this context, confusion between classes can occur, and single label classification may be misleading. We provide in the present paper a general device that, given an unlabeled dataset and a score function defined as the minimizer of some empirical and convex risk, outputs a set of class labels, instead of a single one. Interestingly, this procedure does not require that the unlabeled dataset explores the whole classes. Even more, the method is calibrated to control the expected size of the output set while minimizing the classification risk. We show the statistical optimality of the procedure and establish rates of convergence under the Tsybakov margin condition. It turns out that these rates are linear on the number of labels. We apply our methodology to convex aggregation of confidence sets based on the V-fold cross validation principle also known as the superlearning principle. We illustrate the numerical performance of the procedure on real data and demonstrate in particular that with moderate expected size, w.r.t. the number of labels, the procedure provides significant improvement of the classification risk. We introduce a new approach to characterizing the unobserved portion of a distribution, which provides sublinear--sample estimators achieving arbitrarily small additive constant error for a class of properties that includes entropy and distribution support size. Additionally, we show new matching lower bounds. Together, this settles the longstanding question of the sample complexities of these estimation problems, up to constant factors. Our algorithm estimates these properties up to an arbitrarily small additive constant, using O(n log n) samples, where n is a bound on the support size, or in the case of estimating the support size, 1 n is a lower bound on the probability of any element of the domain. Previously, no explicit sublinear--sample algorithms for either of these problems were known. Our algorithm is also computationally extremely efficient, running in time linear in the number of samples used. In the second half of the paper, we provide a matching lower bound of Ω(n log n) samples for estimating entropy or distribution support size to within an additive constant. The previous lower-bounds on these sample complexities were n 2O(√log n). To show our lower bound, we prove two new and natural multivariate central limit theorems (CLTs); the first uses Stein's method to relate the sum of independent distributions to the multivariate Gaussian of corresponding mean and covariance, under the earthmover distance metric (also known as the Wasserstein metric). We leverage this central limit theorem to prove a stronger but more specific central limit theorem for "generalized multinomial" distributions---a large class of discrete distributions, parameterized by matrices, that represents sums of independent binomial or multinomial distributions, and describes many distributions encountered in computer science. Convergence here is in the strong sense of statistical distance, which immediately implies that any algorithm with input drawn from a generalized multinomial distribution behaves essentially as if the input were drawn from a discretized Gaussian with the same mean and covariance. Such tools in the multivariate setting are rare, and we hope this new tool will be of use to the community. We consider in this paper the problem of sampling a high-dimensional probability distribution @math having a density with respect to the Lebesgue measure on @math , known up to a normalization constant @math . Such problem naturally occurs for example in Bayesian inference and machine learning. Under the assumption that @math is continuously differentiable, @math is globally Lipschitz and @math is strongly convex, we obtain non-asymptotic bounds for the convergence to stationarity in Wasserstein distance of order @math and total variation distance of the sampling method based on the Euler discretization of the Langevin stochastic differential equation, for both constant and decreasing step sizes. The dependence on the dimension of the state space of these bounds is explicit. The convergence of an appropriately weighted empirical measure is also investigated and bounds for the mean square error and exponential deviation inequality are reported for functions which are measurable and bounded. An illustration to Bayesian inference for binary regression is presented to support our claims. For statistical learning, categorical variables in a table are usually considered as discrete entities and encoded separately to feature vectors, e.g., with one-hot encoding. “Dirty” non-curated data give rise to categorical variables with a very high cardinality but redundancy: several categories reflect the same entity. In databases, this issue is typically solved with a deduplication step. We show that a simple approach that exposes the redundancy to the learning algorithm brings significant gains. We study a generalization of one-hot encoding, similarity encoding, that builds feature vectors from similarities across categories. We perform a thorough empirical validation on non-curated tables, a problem seldom studied in machine learning. Results on seven real-world datasets show that similarity encoding brings significant gains in predictive performance in comparison with known encoding methods for categories or strings, notably one-hot encoding and bag of character n-grams. We draw practical recommendations for encoding dirty categories: 3-gram similarity appears to be a good choice to capture morphological resemblance. For very high-cardinalities, dimensionality reduction significantly reduces the computational cost with little loss in performance: random projections or choosing a subset of prototype categories still outperform classic encoding approaches. We develop a Bayesian nonparametric approach to a general family of latent class problems in which individuals can belong simultaneously to multiple classes and where each class can be exhibited multiple times by an individual. We introduce a combinatorial stochastic process known as the negative binomial process ( @math ) as an infinite-dimensional prior appropriate for such problems. We show that the @math is conjugate to the beta process, and we characterize the posterior distribution under the beta-negative binomial process ( @math ) and hierarchical models based on the @math (the @math ). We study the asymptotic properties of the @math and develop a three-parameter extension of the @math that exhibits power-law behavior. We derive MCMC algorithms for posterior inference under the @math , and we present experiments using these algorithms in the domains of image segmentation, object recognition, and document analysis. Categorical variables are a natural choice for representing discrete structure in the world. However, stochastic neural networks rarely use categorical latent variables due to the inability to backpropagate through samples. In this work, we present an efficient gradient estimator that replaces the non-differentiable sample from a categorical distribution with a differentiable sample from a novel Gumbel-Softmax distribution. This distribution has the essential property that it can be smoothly annealed into a categorical distribution. We show that our Gumbel-Softmax estimator outperforms state-of-the-art gradient estimators on structured output prediction and unsupervised generative modeling tasks with categorical latent variables, and enables large speedups on semi-supervised classification. Categorical variables are a natural choice for representing discrete structure in the world. However, stochastic neural networks rarely use categorical latent variables due to the inability to backpropagate through samples. In this work, we present an efficient gradient estimator that replaces the non-differentiable sample from a categorical distribution with a differentiable sample from a novel Gumbel-Softmax distribution. This distribution has the essential property that it can be smoothly annealed into a categorical distribution. We show that our Gumbel-Softmax estimator outperforms state-of-the-art gradient estimators on structured output prediction and unsupervised generative modeling tasks with categorical latent variables, and enables large speedups on semi-supervised classification. We investigate the class of computable probability distributions and explore the fundamental limitations of using this class to describe and compute conditional distributions. In addition to proving the existence of noncomputable conditional distributions, and thus ruling out the possibility of generic probabilistic inference algorithms (even inefficient ones), we highlight some positive results showing that posterior inference is possible in the presence of additional structure like exchangeability and noise, both of which are common in Bayesian hierarchical modeling. This theoretical work bears on the development of probabilistic programming languages (which enable the specification of complex probabilistic models) and their implementations (which can be used to perform Bayesian reasoning). The probabilistic programming approach is particularly well suited for defining infinite-dimensional, recursively-defined stochastic processes of the sort used in nonparametric Bayesian statistics. We present a new construction of the Mondrian process as a partition-valued Markov process in continuous time, which can be viewed as placing a distribution on an infinite kd-tree data structure. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.) Let @math be a nontrivial @math -ary predicate. Consider a random instance of the constraint satisfaction problem @math on @math variables with @math constraints, each being @math applied to @math randomly chosen literals. Provided the constraint density satisfies @math , such an instance is unsatisfiable with high probability. The problem is to efficiently find a proof of unsatisfiability. We show that whenever the predicate @math supports a @math - probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree @math (which runs in time @math ) refute a random instance of @math . In particular, the polynomial-time SOS algorithm requires @math constraints to refute random instances of CSP @math when @math supports a @math -wise uniform distribution on its satisfying assignments. Together with recent work of [LRS15], our result also implies that polynomial-size semidefinite programming relaxation for refutation requires at least @math constraints. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random CSPs. For every constraint predicate @math , they give a three-way hardness tradeoff between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent algorithmic results of [AOW15] and [RRS16], this full three-way tradeoff is , up to lower-order factors. We consider the problem of using a large unlabeled sample to boost performance of a learning algorit,hrn when only a small set of labeled examples is available. In particular, we consider a problem setting motivated by the task of learning to classify web pages, in which the description of each example can be partitioned into two distinct views. For example, the description of a web page can be partitioned into the words occurring on that page, and the words occurring in hyperlinks t,hat point to that page. We assume that either view of the example would be sufficient for learning if we had enough labeled data, but our goal is to use both views together to allow inexpensive unlabeled data to augment, a much smaller set of labeled examples. Specifically, the presence of two distinct views of each example suggests strategies in which two learning algorithms are trained separately on each view, and then each algorithm’s predictions on new unlabeled examples are used to enlarge the training set of the other. Our goal in this paper is to provide a PAC-style analysis for this setting, and, more broadly, a PAC-style framework for the general problem of learning from both labeled and unlabeled data. We also provide empirical results on real web-page data indicating that this use of unlabeled examples can lead to significant improvement of hypotheses in practice. *This research was supported in part by the DARPA HPKB program under contract F30602-97-1-0215 and by NSF National Young investigator grant CCR-9357793. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. TO copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and or a fee. COLT 98 Madison WI USA Copyright ACM 1998 l-58113-057--0 98 7... 5.00 92 Tom Mitchell School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213-3891 mitchell+@cs.cmu.edu Let @math be a @math -ary predicate over a finite alphabet. Consider a random CSP @math instance @math over @math variables with @math constraints. When @math the instance @math will be unsatisfiable with high probability, and we want to find a refutation - i.e., a certificate of unsatisfiability. When @math is the @math -ary OR predicate, this is the well studied problem of refuting random @math -SAT formulas, and an efficient algorithm is known only when @math . Understanding the density required for refutation of other predicates is important in cryptography, proof complexity, and learning theory. Previously, it was known that for a @math -ary predicate, having @math constraints suffices for refutation. We give a criterion for predicates that often yields efficient refutation algorithms at much lower densities. Specifically, if @math fails to support a @math -wise uniform distribution, then there is an efficient algorithm that refutes random CSP @math instances @math whp when @math . Indeed, our algorithm will "somewhat strongly" refute @math , certifying @math , if @math then we get the strongest possible refutation, certifying @math . This last result is new even in the context of random @math -SAT. Regarding the optimality of our @math requirement, prior work on SDP hierarchies has given some evidence that efficient refutation of random CSP @math may be impossible when @math . Thus there is an indication our algorithm's dependence on @math is optimal for every @math , at least in the context of SDP hierarchies. Along these lines, we show that our refutation algorithm can be carried out by the @math -round SOS SDP hierarchy. Finally, as an application of our result, we falsify assumptions used to show hardness-of-learning results in recent work of Daniely, Linial, and Shalev-Shwartz. The development of accurate models and ecient algorithms for the analysis of multivariate categorical data are important and longstanding problems in machine learning and computational statistics. In this paper, we focus on modeling categorical data using Latent Gaussian Models (LGMs). We propose a novel stick-breaking likelihood function for categorical LGMs that exploits accurate linear and quadratic bounds on the logistic log-partition function, leading to an eective variational inference and learning framework. We thoroughly compare our approach to existing algorithms for multinomial logit probit likelihoods on several problems, including inference in multinomial Gaussian process classification and learning in latent factor models. Our extensive comparisons demonstrate that our stick-breaking model eectively captures correlation in discrete data and is well suited for the analysis of categorical data. We investigate a local reparameterizaton technique for greatly reducing the variance of stochastic gradients for variational Bayesian inference (SGVB) of a posterior over model parameters, while retaining parallelizability. This local reparameterization translates uncertainty about global parameters into local noise that is independent across datapoints in the minibatch. Such parameterizations can be trivially parallelized and have variance that is inversely proportional to the mini-batch size, generally leading to much faster convergence. Additionally, we explore a connection with dropout: Gaussian dropout objectives correspond to SGVB with local reparameterization, a scale-invariant prior and proportionally fixed posterior variance. Our method allows inference of more flexibly parameterized posteriors; specifically, we propose variational dropout, a generalization of Gaussian dropout where the dropout rates are learned, often leading to better models. The method is demonstrated through several experiments. Unlabeled samples can be intelligently selected for labeling to minimize classification error. In many real-world applications, a large number of unlabeled samples arrive in a streaming manner, making it impossible to maintain all the data in a candidate pool. In this work, we focus on binary classification problems and study selective labeling in data streams where a decision is required on each sample sequentially. We consider the unbiasedness property in the sampling process, and design optimal instrumental distributions to minimize the variance in the stochastic process. Meanwhile, Bayesian linear classifiers with weighted maximum likelihood are optimized online to estimate parameters. In empirical evaluation, we collect a data stream of user-generated comments on a commercial news portal in 30 consecutive days, and carry out offline evaluation to compare various sampling strategies, including unbiased active learning, biased variants, and random sampling. Experimental results verify the usefulness of online active learning, especially in the non-stationary situation with concept drift. We present a new solution to the ecological inference'' problem, of learning individual-level associations from aggregate data. This problem has a long history and has attracted much attention, debate, claims that it is unsolvable, and purported solutions. Unlike other ecological inference techniques, our method makes use of unlabeled individual-level data by embedding the distribution over these predictors into a vector in Hilbert space. Our approach relies on recent learning theory results for distribution regression, using kernel embeddings of distributions. Our novel approach to distribution regression exploits the connection between Gaussian process regression and kernel ridge regression, giving us a coherent, Bayesian approach to learning and inference and a convenient way to include prior information in the form of a spatial covariance function. Our approach is highly scalable as it relies on FastFood, a randomized explicit feature representation for kernel embeddings. We apply our approach to the challenging political science problem of modeling the voting behavior of demographic groups based on aggregate voting data. We consider the 2012 US Presidential election, and ask: what was the probability that members of various demographic groups supported Barack Obama, and how did this vary spatially across the country? Our results match standard survey-based exit polling data for the small number of states for which it is available, and serve to fill in the large gaps in this data, at a much higher degree of granularity. Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with @math variables and @math clauses, there is a value of @math beyond which the CSP will be unsatisfiable with high probability. Strong refutation is the problem of certifying that no variable assignment satisfies more than a constant fraction of clauses; this is the natural algorithmic problem in the unsatisfiable regime (when @math ). Intuitively, strong refutation should become easier as the clause density @math grows, because the contradictions introduced by the random clauses become more locally apparent. For CSPs such as @math -SAT and @math -XOR, there is a long-standing gap between the clause density at which efficient strong refutation algorithms are known, @math , and the clause density at which instances become unsatisfiable with high probability, @math . In this paper, we give spectral and sum-of-squares algorithms for strongly refuting random @math -XOR instances with clause density @math in time @math or in @math rounds of the sum-of-squares hierarchy, for any @math and any integer @math . Our algorithms provide a smooth transition between the clause density at which polynomial-time algorithms are known at @math , and brute-force refutation at the satisfiability threshold when @math . We also leverage our @math -XOR results to obtain strong refutation algorithms for SAT (or any other Boolean CSP) at similar clause densities. Our algorithms match the known sum-of-squares lower bounds due to Grigoriev and Schonebeck, up to logarithmic factors. Additionally, we extend our techniques to give new results for certifying upper bounds on the injective tensor norm of random tensors. This paper considers the problem of knowledge inference on large-scale imperfect repositories with incomplete coverage by means of embedding entities and relations at the first attempt. We propose IIKE (Imperfect and Incomplete Knowledge Embedding), a probabilistic model which measures the probability of each belief, i.e. @math , in large-scale knowledge bases such as NELL and Freebase, and our objective is to learn a better low-dimensional vector representation for each entity ( @math and @math ) and relation ( @math ) in the process of minimizing the loss of fitting the corresponding confidence given by machine learning (NELL) or crowdsouring (Freebase), so that we can use @math to assess the plausibility of a belief when conducting inference. We use subsets of those inexact knowledge bases to train our model and test the performances of link prediction and triplet classification on ground truth beliefs, respectively. The results of extensive experiments show that IIKE achieves significant improvement compared with the baseline and state-of-the-art approaches. Motivated by an application to unsupervised part-of-speech tagging, we present an algorithm for the Euclidean embedding of large sets of categorical data based on co-occurrence statistics. We use the CODE model of but constrain the embedding to lie on a high-dimensional unit sphere. This constraint allows for efficient optimization, even in the case of large datasets and high embedding dimensionality. Using k-means clustering of the embedded data, our approach efficiently produces state-of-the-art results. We analyze the reasons why the sphere constraint is beneficial in this application, and conjecture that these reasons might apply quite generally to other large-scale tasks. We propose an efficient method to estimate the accuracy of classifiers using only unlabeled data. We consider a setting with multiple classification problems where the target classes may be tied together through logical constraints. For example, a set of classes may be mutually exclusive, meaning that a data instance can belong to at most one of them. The proposed method is based on the intuition that: (i) when classifiers agree, they are more likely to be correct, and (ii) when the classifiers make a prediction that violates the constraints, at least one classifier must be making an error. Experiments on four real-world data sets produce accuracy estimates within a few percent of the true accuracy, using solely unlabeled data. Our models also outperform existing state-of-the-art solutions in both estimating accuracies, and combining multiple classifier outputs. The results emphasize the utility of logical constraints in estimating accuracy, thus validating our intuition.