Time-Smoothed Gradients for Online Forecasting

TL;DR

This paper introduces a time-smoothed gradient method for online forecasting with SGD, reducing sensitivity to learning rate tuning and providing more stable, computationally efficient results.

Contribution

It proposes a novel time-smoothed gradient update rule for SGD in online forecasting, addressing learning rate sensitivity and stability issues.

Findings

More stable forecasting results on GEFCom2014 dataset

Approach is computationally efficient

Outperforms existing methods in stability

Abstract

Here, we study different update rules in stochastic gradient descent (SGD) for online forecasting problems. The selection of the learning rate parameter is critical in SGD. However, it may not be feasible to tune this parameter in online learning. Therefore, it is necessary to have an update rule that is not sensitive to the selection of the learning parameter. Inspired by the local regret metric that we introduced previously, we propose to use time-smoothed gradients within SGD update. Using the public data set-- GEFCom2014, we validate that our approach yields more stable results than the other existing approaches. Furthermore, we show that such a simple approach is computationally efficient compared to the alternatives.

Figures

Equations

$$H{R}_w(T)\triangleq \sum _{t=1}^T\Vert \nabla F_{t,w}(x_t){\Vert }^2$$

$$P{R}_w(T)\triangleq \sum _{t=1}^T{\Vert \nabla S_{t,w}(x_t)\Vert }^2$$

$$L_q(y,\hat{y})=q\max (y-\hat{y},0)+(1-q)\max (\hat{y}-y,0)$$

Full text

Machine Learning, ICML, forecasting, online learning

1 Introduction

Our goal is to design efficient stochastic gradient descent (SGD) algorithms for online time-series forecasting problems. Imagine training a complex machine learning (ML) model such as recurrent neural networks (RNN). As we observe more data sets, we may need to update our model since the relationship between the inputs and the targets might change over time. In large scale ML, re-training such complex models using the entire data set will be time consuming. Ideally, we should update our model using only the new data set and automate this process.

Hazan et al. (2017) introduced a notion of local regret for online non-convex problems. They also proposed efficient algorithms that have non-linear convergence according to their proposed regret. The main idea is averaging the gradients of the most recent loss functions within a window that are evaluated at the current forecast. However, such regret definition of local regret is not suitable for forecasting problems. In forecasting, we would like to evaluate our performance on the recent loss functions that are evaluated at their corresponding forecasts instead of the current forecast.

Recently, we introduced another definition of local regret that is more interpretable for forecasting problems (Aydore et al., 2018). Under certain theoretical conditions, this regret is equivalent to the average of gradients at their corresponding forecasts over a sliding window. Inspired by this regret, we suggest using time-smoothed gradients in SGD where each gradient is computed at the corresponding forecast.

We study the stability of our approach against learning rate which is an important parameter in SGD . During online learning, tuning this parameter will not be practical. Therefore, it is important to use an algorithm which is not very sensitive to the learning rate. Moreover, an update rule in SGD should not introduce a computational bottleneck.

In this work, using a real-world time-series data set, we show that smoothing the gradients at their corresponding forecast values is an effective way for online forecasting. The advantages are: (i) it is inspired by a local regret that fits forecasting problems better, (ii) it is less sensitive to the changes in learning rate, (iii) it is faster than other alternatives.

2 Setting

In online forecasting, our goal is to update $x_{t}$ at each $t$ in order to incorporate the most recently available information. Assume that $t\in\mathcal{T}=\left\{1,\cdots,T\right\}$ represents a collection of $T$ consecutive points where $T$ is an integer and $t=1$ represents an initial forecast point. $f_{1},\cdots,f_{T}:\mathcal{K}\rightarrow\mathbb{R}$ are loss functions on some convex subset $\mathcal{K}\subseteq\mathbb{R}^{d}$. To put in another way, $x_{t}$ represents the parameters of an ML model at time $t$, $f_{t}(x_{t})$ represents the loss function computed using the available data at time $t$ given the model parameters $x_{t}$. In order to update $x_{t}$ at each $t$ using SGD, we consider the following two regret definitions.

Definition 2.1**.**

(Hazan’s local regret) The $w$-local regret of an online algorithm is defined as:

[TABLE]

when $\mathcal{K}=\mathbb{R}^{d}$ and $F_{t,w}(x_{t})\triangleq\frac{1}{w}\sum_{i=0}^{w-1}f_{t-i}(x_{t})$. Hazan et al. (2017) proposed various gradient descent algorithms where the regret $HR$ is sublinear.

Definition 2.2**.**

(Proposed Regret) We propose a $w$-local regret as:

[TABLE]

where $S_{t,w}(x_{t})\triangleq\frac{1}{W}\sum_{i=0}^{w-1}\alpha^{i}f_{t-i}(x_{t-i})$, $\alpha\rightarrow 1_{-}$, $W\triangleq\sum_{i=0}^{w-1}\alpha^{i}$ and $f_{t}(x_{t})=0$ for $t\leq 0$.

Using our definition of regret, we effectively evaluate an online learning algorithm by computing the exponential average of gradients at the corresponding forecast values over a sliding window. This way, we assign larger weights to the most recent gradients. Hazan et al. (2017)’s local regret, on the other hand, computes average of previous gradients computed on the current forecast. We believe that our definition of regret is more applicable to forecasting problems as evaluating today’s forecast on previous loss functions might be misleading. Algorithm 1 represents Hazan’s time-smoothed SGD (HTS-SGD) algorithm which is sub-linear according to the the regret in Definition 1. Inspired by HTS-SGD, we propose time-smoothed SGD (PTS-SGD) as represented in Algorithm 2 where gradients of loss functions are calculated at their corresponding forecasts.

In the following sections, we study the performance of these two algorithms and standard SGD for online forecasting as well as standard SGD for offline learning. The details of these four models are described in Section 4.2.

3 Forecasting Overview

Standard mean squared error as a loss function summarizes the average relationship between a set of features (regressors) and targets. The resulting forecast will be a point forecast which is the conditional mean of the value to be predicted given the input features, i.e. the most likely outcome. However, point forecasts provide only partial information about the conditional distribution of outcomes. Many business applications such as inventory planning require richer information than just the point forecasts.

Quantile loss, on the other hand, minimizes a sum that gives asymmetric penalties for overprediction and underprediction. For example, in demand forecasting, the penalty for overprediction and underprediction could be formulated as overage cost and opportunity cost, respectively. Hence, the loss for the ML model can be designed so that the profit is maximized. Therefore, using quantile loss as an objective function is often more desirable in forecasting applications.

The quantile loss for a given quantile $q$ between true value $y$ and the predicted value $\hat{y}$ is defined as:

[TABLE]

where $q\in(0,1)$.

Typically, forecasting systems produce outputs for multiple quantiles and horizons. The total quantile loss function to be minimized in such situations can be written as: $\sum_{t}\sum_{k}\sum_{q}L_{q}(y_{t+k},\hat{y}^{q}_{t+k})$ where $\hat{y}_{t+k}^{q}$ is the output of the machine learning model, e.g. RNN, to forecast the q-th quantile of horizon k at forecast creation time t. This way, the model learns several quantiles of the conditional distribution such that $\mathbb{P}\left(y_{t+k}\leq{y}_{t+k}^{q}\mid y_{:t}\right)=q$.

We use quantile loss as our cost function in the following section to forecast electric demand values from a time series data set.

4 Experimental Results

We conduct experiments on a real-world time series data set to evaluate the performance of our approach and compare with other SGD algorithms. We use the LSTM model since it has been shown that LSTMs are very efficient in modeling sequential data. A brief description of the data and model can be found below.

4.1 Time Series Data set

We use the data from GEFCom2014 (Barta et al., 2017) for our experiments. It is a public data set for competition in 2014. The data contains 4 sub-data sets among which we use the data that has electrical loads. The electrical load directory contains 16 sub-directories: Task1-Task15 and Solution of Task 15. Each Task1-Task15 directory contains two CSV files: benchmark.csv and train.csv. Each train.csv file contains electrical load values per hour and temperature values measured by 25 stations. The train.csv file in Task 1 contains data from January 2005 to September 2010. The other folders have one month of data from October 2010 to December 2012. Each benchmark.csv file has forecasts of the electrical load values for the next month. These forecasts are generated from the benchmark method.

4.2 Implementation Details

The general flow chart of our experiments is illustrated in Figure 1. We use the data from January 2005 to September 2010 for training and we set the forecast time between October 2010 and December 2012. We assume that 5-year data arrives in monthly intervals. Therefore, we update the LSTM model every time new monthly data is observed.

LSTM Model: LSTMs are special kind of RNNs that are developed to deal with exploding and vanishing gradient problems by introducing input, output and forget gates (Hochreiter & Schmidhuber, 1997). The model contains two LSTM layers and three fully connected linear layers where each represents one of the three quantiles. The architecture of our LSTM model is illustrated in Figure 2. We use multi-step LSTM to forecast multiple horizons. We use electrial load value, hours of the day, days of the week and months of the year as features so that the total number of features is 44. The input to our LSTM model is 48 $\times$ 44 where 48 is hours in two days. The output is the prediction of three quantiles of next day’s values.

Training: During the update, we make only one pass to the data, which means that the epoch number is set to $1$. In order to make learning curves smoother, we adjust the learning rate at each update $t$ so that $\eta_{t}\leftarrow\eta/\sqrt{t}$ where $\eta$ is the initial value for the learning rate. In our experiments, we use 1, 3, 5, 9 for the value of $\eta$.

Metrics: After updating the model once, we evaluate the performance on the 15 months of test data (October 2010 - December 2012). We compute quantile loss for each month and report the average of these which we call $QL_{grand}$. Low $QL_{grand}$ indicates better performance.

Methods: We use one offline and three online methods for training. Offline model uses standard SGD algorithm and is re-trained from scratch to incorporate all the available data each time new data arrives. We see this strategy as the best strategy to be achieved, but as the most expensive in terms of computation. We call this SGD offline in our experiments. The online models are updated only once each time new data is observed. We use standard SGD (called SGD online), Hazan’s time smoothed SGD (called HTS-SGD) and our proposed time smoothed SGD (called PTS-SGD) for online models.

Computational Details: We use Python 3.7 for implementation (Oliphant, 2007) using open source library PyTorch (Paszke et al., 2017). We use 2 NVIDIA GeForce RTX 2080 Ti GPUs with 512 GB Memory to run our experiments.

5 Results

We compare the performance of three online models in terms of their (i) stability against the selection of learning rate, and (ii) computational efficiency.

Stability Against Learning Rate: Figure 3 shows the comparison of models in terms of $QL_{grand}$ for different learning rates. When the learning rate is $1$ (Figure 3(a)), the performance of all four approaches are similar. However, we would expect SGD offline model to perform the best. The results in Figure 3(a) indicate that SGD offline has not converged yet and we need to use a different learning rate. We plot the performances for larger learning rates in Figures 3(b), 3(c) and 3(d). The results show that larger learning rate is needed for SGD offline and it is the best performing model as expected. However, the results for SGD online and HTS-SGD oscillate a lot indicating that they are very sensitive to the changes in learning rate. Our proposed approach PTS-SGD, on the other hand, stays robust as we increase the learning rate. Note that, for $\eta=9$, the values for HTS-SGD became nan (not a number) due to very large losses after some number of iterations, hence are not shown in the Figure.

Computation Time: We further investigate the computation time of each model. Figure 4 shows the amount of time spent in terms of GPU seconds at each update for $\eta=9$ and varying $w$ for HTS-SGD and PTS-SGD. Note that, these results will not be different for other learning rates since computation time does not depend on the learning rate. The figure shows that the elapsed time increases for HTS-SGD and PTS-SGD as $w$ increases as expected. When $w=200$, HTS-SGD takes the most time. However, it can be seen that the time elapsed curve for SGD offline looks more exponential than linear. This means that at some point in future, HTS-SGD will be more efficient than SGD offline. The computation time for our PTS-SGD is already more efficient than SGD offline after 350-th observation even when $w=200$. The reason why HTS-SGD is not as efficient as PTS-SGD is because it needs to store previous losses and compute the gradients using the current parameters. Unsurprisingly, SGD online is the most efficient but its accuracy results in Figure 3 were not as stable as that of PTS-SGD.

6 Conclusion

In this work, we propose exponentially time-smoothed gradient descent for online forecasting. Our approach is inspired by the regret that is more applicable for forecasting problems. The main idea is to smooth the gradients in time when an update is performed using the new data set. We evaluate the performance of this approach compared to the existing approaches as well as the offline model. We use a real-world data set to compare all models in terms of computation time and stability against learning rate. Our results show that our proposed algorithm (PTS-SGD) has the following benefits: (i) it is not sensitive to the learning rate, and (ii) it is computationally efficient compared to the alternatives. We believe that our contribution can have a significant impact on applications for online forecasting problems.