1 Introduction
In this work we consider the following general nonconvex stochastic optimization problem:
(1) 
where
denotes the expectation with respect to the random variable
. We assume that is bounded from below, i.e., , and its gradient is Lipschitz continuous.Problem (1) arises from many statistical learning (e.g.
, logistic regression, AUC maximization) and deep learning models
goodfellow2016deep ; lecun2015deep. In general, one only has access to noisy estimates of
, as the expectation in problem (1) often can only be approximated as a finite sum. Hence, one of the most popular algorithms to solve problem (1) is Stochastic Gradient Decent (SGD) robbins1985stochastic ; bottou2018optimization :(2) 
where is the learning rate and is the noisy gradient estimate of in the th iteration. Its convergence rates for both convex and nonconvex settings have been established bottou1998online ; ghadimi2013stochastic .
However, vanilla SGD suffers from slow convergence, and its performance is sensitive to the learning rate—which is tricky to tune. Many techniques have been introduced to improve the convergence speed and robustness of SGD, such as variance reduction
defazio2014saga ; johnson2013accelerating ; nguyen2017sarah , adaptive learning rate duchi2011adaptive ; kingma2014adam , and momentum acceleration polyak1964some ; nesterov1983method ; levy2018online . Among them, adaptive learning rate and momentum acceleration techniques are most economic, since they merely require slightly more computation per iteration. SGD with adaptive learning rate was first proposed as AdaGrad mcmahan2010adaptive ; duchi2011adaptive and the learning rate is adjusted by cumulative gradient magnitudes:(3) 
where are fixed parameters. On the other hand, Heavy Ball (HB) polyak1964some ; ghadimi2015global and Nesterov Accelerated Gradient (NAG) nesterov1983method ; nesterov2013introductory are two most popular momentum acceleration techniques, which have been extensively studied for stochastic optimization problems ghadimi2016accelerated ; yan2018unified ; levy2018online :
(4) 
where , , and is the momentum factor.
Both the adaptive learning rate and momentum techniques have been individually investigated and have displayed to be effective in practice, so are independently and widely applied in tasks such as training deep networks krizhevsky2012imagenet ; sutskever2013importance ; kingma2014adam ; reddi2018convergence . It is natural to consider: Can we effectively incorporate both techniques at the same time so as to inherit their dual advantages and moreover develop convergence theory for this scenario, especially in the more difficult nonconvex stochastic setting? To the best of our knowledge, Levy et al. levy2018online firstly attempted to combine the adaptive learning rate with NAG momentum, which yields the AccAdaGrad algorithm. However, its convergence is limited to the stochastic convex setting. Yan et al. yan2018unified unified SHB and SNAG to a threestep iterate without considering the adaptive learning rate in Eq. (3).
In this work, we revisit the momentum acceleration technique polyak1964some ; nesterov1983method and adaptive learning rate duchi2011adaptive ; kingma2014adam , and propose weighted AdaGrad with unified stochastic momentum, dubbed as AdaUSM, to solve the general nonconvex stochastic optimization problem (1). Specifically, the proposed AdaUSM has two main features: it develops a novel Unified Stochastic Momentum (USM) scheme to cover SHB and SNAG, entirely different from the threestep scheme in yan2018unified , and it generalizes the adaptive learning rate in Eq. (3) to a more general weighted adaptive learning rate (see Sectiion 3) that can unify the adaptive learning rates of AdaGrad, AccAdaGrad, and Adam into a succinct framework. In contrast to those in AdaGrad duchi2011adaptive , the weighted adaptive learning rate in AdaUSM is estimated via a novel weighted gradient accumulation technique, which puts more weights on the most recent stochastic gradient estimates. Moreover, to make AdaUSM more practical for largescale problems, a coordinatewise weighted adaptive learning rate with a low computational cost is used.
We also characterize the convergence rate of AdaUSM in the nonconvex stochastic setting, when we take polynomially growing weights in AdaUSM. When momentum is NAG and weights are set as the same as those in levy2018online , AdaUSM reduces to AccAdaGrad levy2018online . In consequence, the convergence rate of AccAdaGrad in the nonconvex setting is derived directly as byproduct. Thus, our work generalizes AccAdaGrad levy2018online in three aspects: (i) more general weights in estimating the adaptive learning rate; (ii) new unified momentum including both NAG and HB; (iii) the convergence rate in the more difficult nonconvex stochastic setting. To the best of our knowledge, our work is the first to explore the convergence rates of adaptive stochastic algorithms with momentum acceleration in the nonconvex stochastic setting. Our contributions are threefold:

We develop a new weighted gradient accumulation technique to estimate the adaptive learning rate, and propose a novel unified stochastic momentum scheme to cover SHB and SNAG. We then integrate the weighted coordinatewise AdaGrad with a unified momentum mechanism, yielding a novel adaptive stochastic momentum algorithm, dubbed as AdaUSM.

We establish the nonasymptotic convergence rate of AdaUSM under the general nonconvex stochastic setting. Our assumptions are natural and mild.

We show that the adaptive learning rates of Adam and RMSProp correspond to taking exponentially growing weights in AdaUSM, which thereby provides a new perspective for understanding Adam and RMSProp.
Related Works. There exist several works to study the convergence rates of adaptive SGD in the nonconvex stochastic setting. For example, Li and Orabona li2018convergence first proved the global convergence of perturbed AdaGrad^{1}^{1}1In li2018convergence , the convergence rate of AdaGrad was established with perturbed adaptive learning rate: . The perturbation factor is unavoidable for its convergence argument.; Ward et al. ward2018adagrad established convergence rates for the original AdaGrad duchi2011adaptive and WNgrad wu2018wngrad ; the convergence rates of Adam/RMSProp zou2018sufficient and AMSGrad chen2018convergence were also established in the nonconvex stochastic setting, respectively.
2 Preliminaries
Notations.
denotes the maximum number of iterations. The noisy gradient of at the th iteration is denoted by for all . We use to denote expectation as usual, and as the conditional expectation with respect to conditioned on the random variables .
In this paper we allow differential learning rates across coordinates, so the learning rate
is a vector in
. Given a vector we denote its th coordinate by . The th coordinate of the gradient is denoted by . Given two vectors , the inner product between them is denoted by . We also heavily use the coordinatewise product between and , denoted as , with . Division by a vector is defined similarly. Given a vector , we define the weighted norm: Norm without any subscript is the Euclidean norm, and is defined as . Let and .Assumptions.
We assume that , are independent of each other. Moreover,

, i.e., stochastic gradient
is an unbiased estimator;
Notice that the condition (A2) is slightly weaker than that in chen2018convergence which assumes that stochastic gradient estimate is uniformly bounded, i.e., .
3 Weighted AdaGrad with Unified Momentum
We describe the two main ingredients of AdaUSM: the unified stochastic momentum formulation of SHB and SNAG (see Subsection 3.1), and the weighted adaptive learning rate (see Subsection 3.2).
3.1 Unified Stochastic Momentum (USM)
By introducing with , the iterate of SNAG can be equivalently written as
Comparing SHB and above SNAG, the difference lies in that SNAG has more weight on the current momentum . Hence, we can rewrite SHB and SNAG in the following unified form:
(5) 
where is a constant. When , it is SHB; when , it is SNAG. We call
the interpolation factor. For any
, can be chosen from the range .Remark 1.
Yan et al. yan2018unified unified SHB and SNAG as a threestep iterate scheme as follows:
(6) 
where . Its convergence rate has been established for . Notably, USM is slightly simpler than Eq. (6) and the learning rate in USM is adaptively determined.
3.2 Weighted Adaptive Learning Rate
We generalize the learning rate in Eq. (3) by assigning different weights to the past stochastic gradients accumulated. It is defined as follows:
(7) 
for , where and . Here, can be understood as the base learning rate. The classical AdaGrad corresponds to taking for all in Eq. (7), i.e., uniform weights. However, recent gradients tend to carry more information of local geometries than remote ones. Hence, it is natural to assign the recent gradients more weights. A typical choice for such weights is to choose for , which grows in a polynomial rate. For instance, in AccAdaGrad levy2018online weights are chosen to be for and for .
3.3 AdaUSM: Weighted AdaGrad with USM
In this subsection, we present the AdaUSM algorithm, which effectively integrates the weighted adaptive learning rate in Eq. (7) with the USM technique in Eq. (5), and establish its convergence rate.
Note that AdaUSM extends the AccAdaGrad in levy2018online by using more general weighted parameters and momentum accelerated mechanisms in the nonconvex stochastic setting. In addition, when , AdaUSM reduces to the weighted AdaGrad with a heavy ball momentum, which we denote by AdaHB for short. When , AdaUSM reduces to the weighted AdaGrad with Nesterov accelerated gradient momentum, which we denote by AdaNAG for short. The detailed iterate schemes of AdaHB and AdaNAG are placed in the supplementary material due to the space limit.
Theorem 1.
Let be a sequence generated by AdaUSM. Assume that the noisy gradient satisfies assumptions (A1) (A2), and the sequence of weights is nondecreasing in . Let be uniformly randomly drawn from . Then
(8) 
where , and constants and are defined as and , respectively.
Sketch of proof.
Our starting point is the following inequality which follows from the Lipschitz continuity of and the descent lemma in nesterov2013introductory :
The key point is to estimate the term , which involves the momentum. However, is far from an unbiased estimate of the true gradient. This difficulty is resolved in Lemma 3, where we establish an estimate for the term via iteration
Furthermore, we can derive an estimate on each in terms of (by Lemma 2). For the term , taking expectation of is tricky as is correlated with . We follow the idea of ward2018adagrad and consider the term , where is an approximation of independent of . Hence its expectation gives rise to the desired term . With a suitable choice of , in Lemma 4 we establish the following estimate
Summarizing the estimates above leads to the estimate in Lemma 7:
With the specific adaptive learning rate as Eq. (7), we can further show that the principal term is bounded by (by Lemma 5), while the term via Hölder inequality (by Lemma 6). This immediately gives rise to our theorem. ∎
Remark 2.
When we take for some constant power , then and conditions (i)(ii) are satisfied. Hence, AdaUSM with such weights has the convergence rate. In fact, AdaUSM is convergent as long as .
When interpolation factor , and for and for , AdaUSM reduces to coordinatewise AccAdaGrad levy2017online . In this case, . Thus, we have the following corollary for the convergence rate of AccAdaGrad in the nonconvex stochastic setting.
Corollary 1.
Assume the same setting as Theorem 1. Let be randomly selected from
with equal probability
. Then(9) 
where , and constants and are defined as and , respectively.
Remark 3.
The nonasymptotic convergence rate measured by the objective for AccAdaGrad has already been established in levy2018online in the convex stochastic setting. Corollary 1 provides the convergence rate of coordinatewise AccAdaGrad measured by gradient residual, which also supplements the results in levy2018online in the nonconvex stochastic setting.
4 Relationships with Adam and RMSProp
In this section, we show that the exponential moving average (EMA) technique in estimating adaptive learning rates in Adam kingma2014adam and RMSProp hinton2012lecture is a special case of the weighted adaptive learning rate in Eq. (7), i.e., their adaptive learning rates correspond to taking exponentially growing weights in AdaUSM, which thereby provides a new angle for understanding Adam and RMSProp.
4.1 Adam
For better comparison, we first represent the th iterate scheme of Adam kingma2014adam as follows:
for , where and are constants, and is a sufficiently small constant. Denoting , we can simplify the iterations of Adam as
Below, we show that AdaUSM and Adam differ in two aspects: momentum estimation and coordinatewise adaptive learning rate .
Momentum estimation. EMA technique is widely used in the momentum estimation step in Adam kingma2014adam and AMSGrad reddi2018convergence . Without loss of generality, we consider the simplified EMA step
(10) 
To show the difference clearly, we merely compare HB momentum with EMA momentum. Let . By the first equality in Eq. (10), we have
Comparing with HB, EMA has an extra error term which vanishes if for all , i.e., the stepsize is taken constant^{2}^{2}2Since the learning rates in AdaUSM and Adam are determined adaptively, we do not have .. In addition, EMA has a much smaller stepsize if the momentum factor is close to . More precisely, if we write the iterate in terms of stochastic gradients and eliminate , we obtain
One can see that AdaUSM uses the past stepsizes but EMA uses only the current one in exponential moving averaging. Moreover, when momentum factor is very close to , the update of via EMA could stagnate since . This dilemma will not appear in AdaUSM.
Adaptive learning rate. Note that . We have . Without loss of generality, we set . Hence, it holds that
Then, the adaptive learning rate in Adam can be rewritten as
(11) 
Let . Note that Hence, Eq. (11) can be further reformulated as
(12) 
For comparison, the adaptive learning rates of Adam and AdaUSM are summarized as follows:
Hence, the adaptive learning rate in Adam is actually equivalent to that in AdaUSM by specifying if is sufficiently small. For the parameter setting in Adam, it holds that . Thus, we gain an insight on understanding the convergence of Adam from the convergence results of AdaUSM in Theorem 1.
Remark 4.
Recently, Chen et al. chen2018convergence have also proposed AdaGrad with exponential moving average (AdaEMA) by setting as and removing the biascorrection steps in Adam. Its convergence rate in the nonconvex stochastic setting has been established under a slightly stronger assumption that the stochastic gradient estimate is required to be uniformly bounded. Compared with AdaEMA, AdaUSM not only adopts a general weighted sequence in estimating the adaptive learning rate, but also uses a different unified momentum that covers HB and NAG as special instances. The superiority of HB and NAG momentums over EMA has been pointed out in the above paragraph: Momentum estimation. In Section 5, we also experimentally demonstrate the effectiveness of AdaUSM against AdaEMA.
4.2 RMSProp
Coordinatewise RMSProp is another efficient solver for training DNNs hinton2012lecture ; mukkamala2017variants , which is defined as
Define . The adaptive learning rate of RMSProp denoted as can be rewritten as
When is a sufficiently small constant and , it is obvious that has a similar structure to after being sufficiently large. Based on the above analysis, AdaUSM can be interpreted as generalized RMSProp with HB and NAG momentums.
5 Experiments
In this section, we conduct experiments to validate the efficacy and theory of AdaHB (AdaUSM with ) and AdaNAG (AdaUSM with ) by applying them to train DNNs^{3}^{3}3https://github.com/kuangliu/pytorchcifar including LetNet lecun1998gradient , GoogLeNet szegedy2015going , ResNet he2016deep , and DenseNet huang2017densely on various datasets including MNIST lecun1998gradient , CIFAR10/100 krizhevsky2009learning
, and TinyImageNet
deng2009imagenet. The efficacies of AdaHB and AdaNAG are evaluated via the training loss, test accuracy, and test loss vs. epochs, respectively. In the experiments, we fix the batchsize as
and the weighted decay parameter as , respectively.Optimizers. We compare AdaHB/AdaNAG with five competing algorithms: SGD with momentum (SGDm) sutskever2013importance , AdaGrad duchi2011adaptive ; ward2018adagrad , AdaEMA chen2018convergence , AMSGrad reddi2018convergence ; chen2018convergence , and Adam kingma2014adam . The parameter settings of all compared optimizers are summarized in Table 1 (see Section D in the supplementary material) and are consistent with kingma2014adam ; reddi2018convergence ; chen2018convergence . To match the convergence theory, we take the diminishing base learning rate as uniformly across all the tested adaptive optimizers. Moreover, via the momentum estimation paragraph in Section 4.1, we know that the learning rates in AdaHB and AdaNAG will be times greater than those in AdaEMA, AMSGrad, and Adam if they share the same constant parameter . In addition, too large and small would lead to heavy oscillation and bad stagnation on the training loss, respectively, which would deteriorate the performances of the tested optimizers. Consequently, the base learning rate for each solver is chosen via grid search on the set . We report the base learning rate of each solver that can consistently contribute to the best performance.
Figures 4 4 illustrate the performance profiles of applying AdaHB, AdaNAG, SGDm, AdaGrad, AdaEMA, AMSGrad, and Adam to train LetNet on MNIST, GoogLeNet on CIFAR10, DenseNet on CIFAR100, and ResNet on TinyImageNet, respectively. More specifically, Figure 4 illustrates the performance of LeNet on MNIST which covers training examples and test examples. It can be seen that AdaHB decreases the training loss and test loss fastest among the seven tested optimizers, which simultaneously yields a higher test accuracy than the other tested optimizers. The performances of AdaEMA and AdaGrad are worse than AdaHB but better than SGDm, Adam, and AMSGrad. Figure 4 illustrates the performance of training GoogLeNet on CIFAR10 which covers training examples and test examples. It can be seen that SGDm decreases the training loss slightly faster than other optimizers, followed by AdaHB and AdaNAG, and that AMSGrad and Adam are the slowest optimizers. The test accuracies of AdaHB, AdaNAG, AdaGrad, and AdaEMA are comparable, which are all slightly better than SGDm and outperform Adam and AMSGrad. Figure 4 illustrates the performance of training DenseNet on CIFAR100 which covers training examples and test examples. It shows that SGDm has the worst training process and test accuracy, followed by Adam and AMSGrad. While AdaGrad, AdaEMA, AdaHB, and AdaNAG decrease the training loss and increase test accuracy at roughly the same speed. Figure 4 illustrates the performance of training ResNet on TinyImageNet which contains training examples and test examples. It can be seen that AdaHB and AdaNAG show the fastest speed to decrease the training loss and increase the test accuracy. SGDm is worse than AdaHB and AdaNAG, and better than AdaGrad, AdaEMA, and AMSGrad in terms of the training loss and test accuracy.
In summary, AdaHB and AdaNAG are more efficient and robust than SGDm, AdaGrad, AdaEMA, Adam, and AMSGrad in terms of both the training speed and generalization capacity. SGDm is also an efficient optimizer but it is highly sensitive to the handtuning learning rate. Moreover, the value of EMA is marginal and not as efficient as heavy ball and Nesterov accelerated gradient momentums, as revealed by the performance curves of AdaEMA, AdaGrad, AdaHB, and AdaNAG.
6 Conclusions
We integrated a novel weighted coordinatewise AdaGrad with unified momentum including heavy ball and Nesterov accelerated gradient momentums, yielding a new adaptive stochastic algorithm called AdaUSM. Its convergence rate was established in the nonconvex stochastic setting. Our work largely extends the convergence rate of accelerated AdaGrad in levy2018online to the general nonconvex stochastic setting. Moreover, we pointed out that the adaptive learning rates of Adam and RMSProp are essentially special cases of the weighted adaptive learning rate in AdaUSM, which provides a new angle to understand the convergences of Adam/RMSProp. We also experimentally verified the efficacy of AdaUSM in training deep learning models on several datasets. The promising results show that the proposed AdaUSM is more effective and robust than SGD with momentum, AdaGrad, AdaEMA, AMSGrad, and Adam in terms of training loss and test accuracy vs. epochs.
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Appendix A AdaHB and AdaNAG
Appendix B Preliminary Lemmas
In this section we provide preliminary lemmas that will be used to prove our main theorem. The readers may skip this part for the first time and come back whenever the lemmas are needed.
Lemma 1.
Let , where is a nonnegative sequence and . We have
Proof.
The finite sum can be interpreted as a Riemann sum as follows Since is decreasing on the interval , we have
The proof is finished. ∎
The following lemma is a direct result of the momentum updating rule.
Lemma 2.
Suppose with and . We have the following estimate
(13) 
Proof.
First, we have the following inequality due to convexity of :
(14)  
Taking sum of Eq. (14) from to and using , we have that
(15) 
Hence,
(16) 
The proof is finished. ∎
The following lemma is a result of the USM formulation for any general adaptive learning rate.
Lemma 3.
Let and be sequences generated by the following general SGD with USM momentum: starting from initial values and , and being updated through
where the momentum factor and the interpolation factor satisfy and , respectively. Suppose that the function is smooth. Then for any we have the following estimate
(17) 
In particular, the following estimate holds
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