Abstract. Making use of predictions is a crucial, but under-explored, area of online algorithms. This paper studies a class of online optimization problems where we have external noisy predictions available. We propose a stochastic prediction error model that generalizes prior models in the learning and stochastic control communities, incorporates correlation among prediction errors, and captures the fact that predictions improve as time passes. We prove that achieving sublinear regret and constant competitive ratio for online algorithms requires the use of an unbounded prediction window in adversarial settings, but that under more realistic stochastic prediction error models it is possible to use Averaging Fixed Horizon Control (AFHC) to simultaneously achieve sublinear regret and constant competitive ratio in expectation using only a constant-sized prediction window. Furthermore, we show that the performance of AFHC is tightly concentrated around its mean.
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 a 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.
Making use of predictions is a crucial, but under-explored, area of online algorithms. This paper studies a class of online optimization problems where we have external noisy predictions available. We propose a stochastic prediction error model that generalizes prior models in the learning and stochastic control communities, incorporates correlation among prediction errors, and captures the fact that predictions improve as time passes. We prove that achieving sublinear regret and constant competitive ratio for online algorithms requires the use of an unbounded prediction window in adversarial settings, but that under more realistic stochastic prediction error models it is possible to use Averaging Fixed Horizon Control (AFHC) to simultaneously achieve sublinear regret and constant competitive ratio in expectation using only a constant-sized prediction window. Furthermore, we show that the performance of AFHC is tightly concentrated around its mean.
In this work, we aim to design a data marketplace; a robust real-time matching mechanism to efficiently buy and sell training data for Machine Learning tasks. While the monetization of data and pre-trained models is an essential focus of industry today, there does not exist a market mechanism to price training data and match buyers to sellers while still addressing the associated (computational and other) complexity. The challenge in creating such a market stems from the very nature of data as an asset: (i) it is freely replicable; (ii) its value is inherently combinatorial due to correlation with signal in other data; (iii) prediction tasks and the value of accuracy vary widely; (iv) usefulness of training data is difficult to verify a priori without first applying it to a prediction task. As our main contributions we: (i) propose a mathematical model for a two-sided data market and formally define the key associated challenges; (ii) construct algorithms for such a market to function and analyze how they meet the challenges defined. We highlight two technical contributions: (i) a new notion of "fairness" required for cooperative games with freely replicable goods; (ii) a truthful, zero regret mechanism to auction a class of combinatorial goods based on utilizing Myerson's payment function and the Multiplicative Weights algorithm. These might be of independent interest.
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