Recent advances in neural forecasting have produced major improvements in accuracy for probabilistic demand prediction. In this work, we propose novel improvements to the current state of the art by incorporating changes inspired by recent advances in Transformer architectures for Natural Language Processing. We develop a novel decoder-encoder attention for contextalignment, improving forecasting accuracy by allowing the network to study its own history based on the context for which it is producing a forecast. We also present a novel positional encoding that allows the neural network to learn context-dependent seasonality functions as well as arbitrary holiday distances. Finally we show that the current state of the art MQ-Forecaster (Wen et al., 2017) models display excess variability by failing to leverage previous errors in the forecast to improve accuracy. We propose a novel decoder-self attention scheme for forecasting that produces significant improvements in the excess variation of the forecast.
Mental stress results in a marked heart response consistent with a marked neurohormonal effect. This response is effectively blunted by a 5-week moderately intensive exercise programme. These results should encourage endorsement of a regular exercise programme as an important lifestyle modification for improving maladaptive responses to stress.
Multi-horizon probabilistic time series forecasting has wide applicability to real-world tasks such as demand forecasting. Recent work in neural time-series forecasting mainly focus on the use of Seq2Seq architectures Sutskever et al. (2014). For example, MQTransformer (Eisenach et al., 2020) -an improvement of MQCNN (Wen et al., 2017) -has shown the state-of-the-art performance in probabilistic demand forecasting. In this paper, we consider incorporating crossentity information to enhance model performance by adding a cross-entity attention mechanism along with a retrieval mechanism to select which entities to attend over. We demonstrate how our new neural architecture, MQRetNN, leverages the encoded contexts from a pretrained baseline model on the entire population to improve forecasting accuracy. Using MQCNN as the baseline model (due to computational constraints, we do not use MQTransformer), we first show on a small demand forecasting dataset that it is possible to achieve ∼3% improvement in test loss by adding a cross-entity attention mechanism where each entity attends to all others in the population. We then evaluate the model with our proposed retrieval methods -as a means of approximating an attention over a large population -on a large-scale demand forecasting application with over 2 million products and observe ∼1% performance gain over the MQCNN baseline.
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Motivated by the task of clustering either d variables or d points into K groups, we investigate efficient algorithms to solve the Peng-Wei (P-W) K-means semi-definite programming (SDP) relaxation. The P-W SDP has been shown in the literature to have good statistical properties in a variety of settings, but remains intractable to solve in practice. To this end we propose FORCE, a new algorithm to solve this SDP relaxation. Compared to the naive interior point method, our method reduces the computational complexity of solving the SDP from O(d 7 log −1 ) to O(d 6 K −2 −1 ) arithmetic operations for an -optimal solution. Our method combines a primal first-order method with a dual optimality certificate search, which when successful, allows for early termination of the primal method. We show for certain variable clustering problems that, with high probability, FORCE is guaranteed to find the optimal solution to the SDP relaxation and provide a certificate of exact optimality. As verified by our numerical experiments, this allows FORCE to solve the P-W SDP with dimensions in the hundreds in only tens of seconds. For a variation of the P-W SDP where K is not known a priori a slight modification of FORCE reduces the computational complexity of solving this problem as well: from O(d 7 log −1 ) using a standard SDP solver to O(d 4 −1 ).
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