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  1. Variational methods, such as mean-field (MF) and tree-reweighted (TRW), provide computationally efficient approximations of the log-partition function for generic graphical models but their approximation ratio is generally not quantified. As the primary contribution of this work, we provide an approach to quantify their approximation ratio for any discrete pairwise graphical model with non-negative potentials through a property of the underlying graph structure G. Specifically, we argue that (a variant of) TRW produces an estimate within factor K(G) which captures how far G is from tree structure. As a consequence, the approximation ratio is 1 for trees. The quantity K(G) is the solution of a min-max problem associated with the spanning tree polytope of G that can be evaluated in polynomial time for any graph. We provide a near linear-time variant that achieves an approximation ratio depending on the minimal (across edges) effective resistance of the graph. We connect our results to the graph partition approximation method and thus provide a unified perspective.
  2. A major bottleneck of the current Machine Learning (ML) workflow is the time consuming, error prone engineering required to get data from a datastore or a database (DB) to the point an ML algorithm can be applied to it. This is further exacerbated since ML algorithms are now trained on large volumes of data, yet we need predictions in real-time, especially in a variety of time-series applications such as finance and real-time control systems. Hence, we explore the feasibility of directly integrating prediction functionality on top of a data store or DB. Such a system ideally: (i) provides an intuitive prediction query interface which alleviates the unwieldy data engineering; (ii) provides state-of-the-art statistical accuracy while ensuring incremental model update, low model training time and low latency for making predictions. As the main contribution we explicitly instantiate a proof-of-concept, tspDB which directly integrates with PostgreSQL. We rigorously test tspDB’s statistical and computational performance against the state-of-the-art time series algorithms, including a Long-Short-Term-Memory (LSTM) neural network and DeepAR (industry standard deep learning library by Amazon). Statistically, on standard time series benchmarks, tspDB outperforms LSTM and DeepAR with 1.1-1.3x higher relative accuracy. Computationally, tspDB is 59-62x and 94-95x faster compared to LSTM andmore »DeepAR in terms of median ML model training time and prediction query latency, respectively. Further, compared to PostgreSQL’s bulk insert time and its SELECT query latency, tspDB is slower only by 1.3x and 2.6x respectively. That is, tspDB is a real-time prediction system in that its model training / prediction query time is similar to just inserting, reading data from a DB. As an algorithmic contribution, we introduce an incremental multivariate matrix factorization based time series method, which tspDB is built off. We show this method also allows one to produce reliable prediction intervals by accurately estimating the time-varying variance of a time series, thereby addressing an important problem in time series analysis.« less
  3. We discuss the question of learning distribution over permutations of a given set of choices, options or items based on partial observations. This is central to capturing the so called ``choice'' in a variety of contexts: understanding preferences of consumers over a collection of products based on purchasing and browsing data in the setting of retail and e-commerce, learning public opinion amongst a collection of socio-economic issues based on sparse polling data, deciding a ranking of teams or players based on outcomes of games, electing leaders based on votes, and more generally collaborative decision making based on collective judgement such as accepting paper(s) in a competitive academic conference. The question of learning distribution over permutations arises beyond capturing ``choice'' as well. For example, tracking a collection of objects using noisy cameras, or aggregating ranking of web-pages using outcomes of multiple search engines. It is only natural that such a topic has been extensively studied in Economics, Political Science and Psychology for more than a century, and more so recently in Computer Science, Electrical Engineering, Statistics and Operations Research. Here we shall focus on the task of learning distribution over permutations from its marginal distributions of two types: first-order marginals andmore »pair-wise comparisons. There has been a lot of progress made on this topic in the last decade. The ideal goal is to provide a comprehensive overview of the state-of-art on this topic. We shall provide detailed overview of selective aspects, biased by author's perspective of the topic. And provide sufficient pointers to aspects not covered here. We shall emphasize on ability to identify the entire distribution over permutation as well as ``best ranking''.« less
  4. We propose an algorithm to impute and forecast a time series by transforming the observed time series into a matrix, utilizing matrix estimation to recover missing values and de-noise observed entries, and performing linear regression to make predictions. At the core of our analysis is a representation result, which states that for a large class of models, the transformed time series matrix is (approximately) low-rank. In effect, this generalizes the widely used Singular Spectrum Analysis (SSA) in the time series literature, and allows us to establish a rigorous link between time series analysis and matrix estimation. The key to establishing this link is constructing a Page matrix with non-overlapping entries rather than a Hankel matrix as is commonly done in the literature (e.g., SSA). This particular matrix structure allows us to provide finite sample analysis for imputation and prediction, and prove the asymptotic consistency of our method. Another salient feature of our algorithm is that it is model agnostic with respect to both the underlying time dynamics and the noise distribution in the observations. The noise agnostic property of our approach allows us to recover the latent states when only given access to noisy and partial observations a la amore »Hidden Markov Model; e.g., recovering the time-varying parameter of a Poisson process without knowing that the underlying process is Poisson. Furthermore, since our forecasting algorithm requires regression with noisy features, our approach suggests a matrix estimation based method-coupled with a novel, non-standard matrix estimation error metric-to solve the error-in-variable regression problem, which could be of interest in its own right. Through synthetic and real-world datasets, we demonstrate that our algorithm outperforms standard software packages (including R libraries) in the presence of missing data as well as high levels of noise.« less
  5. We propose an algorithm to impute and forecast a time series by transforming the observed time series into a matrix, utilizing matrix estimation to recover missing values and de-noise observed entries, and performing linear regression to make predictions. At the core of our analysis is a representation result, which states that for a large class of models, the transformed time series matrix is (approximately) low-rank. In effect, this generalizes the widely used Singular Spectrum Analysis (SSA) in the time series literature, and allows us to establish a rigorous link between time series analysis and matrix estimation. The key to establishing this link is constructing a Page matrix with non-overlapping entries rather than a Hankel matrix as is commonly done in the literature (e.g., SSA). This particular matrix structure allows us to provide finite sample analysis for imputation and prediction, and prove the asymptotic consistency of our method. Another salient feature of our algorithm is that it is model agnostic with respect to both the underlying time dynamics and the noise distribution in the observations. The noise agnostic property of our approach allows us to recover the latent states when only given access to noisy and partial observations a la amore »Hidden Markov Model; e.g., recovering the time-varying parameter of a Poisson process without knowing that the underlying process is Poisson. Furthermore, since our forecasting algorithm requires regression with noisy features, our approach suggests a matrix estimation based method—coupled with a novel, non-standard matrix estimation error metric—to solve the error-in-variable regression problem, which could be of interest in its own right. Through synthetic and real-world datasets, we demonstrate that our algorithm outperforms standard software packages (including R libraries) in the presence of missing data as well as high levels of noise.« less
  6. Appeared in AISTAT 2018, published as part of Proceedings of Journal of Machine Learning Research.