Revisit last-iterate convergence of mSGD under milder requirement on step size
Authors: ruinan Jin
Xingkang He
Lang Chen
Difei Cheng
Vijay Gupta
Venue: NeurIPS 2022 Accept (2022)
Keywords: stochastic gradient descent
momentum
convergence

Abstract:  Understanding convergence of SGD-based optimization algorithms can help deal with enormous machine learning problems. To ensure last-iterate convergence of   SGD and momentum-based SGD (mSGD),
  the existing studies  usually constrain the step size $\epsilon_{n}$ to decay as $\sum_{n=1}^{+\infty}\epsilon_{n}^{2}<+\infty$, which however is rather conservative and may lead to slow convergence in the early stage of the iteration. In this paper, we relax this requirement by studying an alternate step size for the mSGD. First, we relax the requirement of the decay on step size to $\sum_{n=1}^{+\infty}\epsilon_{n}^{2+\eta_{0}}<+\infty\ (0\le\eta_{0}<1/2)$. This implies that a larger step size, such as $\epsilon_{n}=\frac{1}{\sqrt{n}}$  can  be utilized for accelerating the mSGD in the early stage.  Under this new step size and some common conditions, we prove that the  gradient norm of mSGD for non-convex loss functions   asymptotically decays to zero. In addition, we show that this step size can indeed help make the  convergence into a neighborhood of the stationary points quicker in the early stage. In addition, we establish the convergence of   mSGD  under a constant step size $\epsilon_n\equiv\epsilon>0$ by removing the common requirement in the literature on the strong convexity of the loss function. 
  Some experiments are given to illustrate the developed results.

Revisit last-iterate convergence of mSGD under milder requirement on step size
Ruinan Jin Xingkang He²* Lang Chen Difei Cheng Vijay Gupta 2,5
The Chinese University of Hong Kong, Shenzhen, China DDpparmentt of Electrical Engineering, University of Notre Dame, IN, USA CColleee of Information Engineering, Shenyang University of Chemical Technology, Shenyang, China *Institute of Automation, Chinese Academy of Sciences, Beijing, China $ Elmore Family School of Electrical and Computer Engineering, Purdue University, IN, USA
Abstract
Understanding convergence of stochastic gradient descent (SGD) based optimiza- tion algorithms can help deal with enormous machine learning problems. To ensure last-iterate convergence of SGD and momentum--asee SGD (mSGD), the existing studies usually constrain the step size en to decay as Σ+s e < +o0, which however rather conservative and may lead to slow convergence the early stage of the iteration. In this paper, we relax this requirement by studying an alternate step size for the mSGD. First, we relax the requirement of the decay on step size Σts1 24 +00 (0 S no 1/2). This implies that larger step size, such as En - can utilized for accelerating the mSGD in the early stage. Under this
new step size and some common conditions, we prove that the gradient norm mSGD for class of non-convex loss functions asymptotically decays zero. In addition, we show that this step size can indeed help make the iterates of mSGD converge into neighborhood of the stationary points quicker the early stage. Fi- nally, we establish the convergence of mSGD under a constant step size ex2 = e > 0 by removing common requirement the literature on strong convexity of the loss functions. Some experiments are given to illustrate the developed results.
Introduction
The booming development of machine learning over the past decade relies on the employment of effective optimization algorithms for training parameterized machine learning models (c.g., neural networks). A large number of such optimization algorithms are based on gradient descent (GD). The optimization problem in machine learning can be cast as minimizing a loss function g(0) e E over the choice of an N-dimensional real-valued parameter vector 6, i.e., by solving the problem
arg min g(0). BERN
This problem can be solved with a typical GD algorithm through an iteration of the form
e 8, en Vo..(On).
(2)
where O, is the estimate of 0* at step n, en is a positive step size (learning rate) to be designed, and Vo, (0,)) stands for the gradient of g(O,,, at step Under certain technical conditions, 0,, in (2) can asymptotically (as n + oo) converge to the optimal solution 8" However, (2) is not efficient in machine leaning applications with enormous training data. To accelerate (2), one attempt is stochastic
"Corespooning author: xhe9@nd.ech edu
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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ments on step size. Specifically, seek to relax the condition = & < +oo [1, 7, 15, 16, 21] that has been required in the literature for proving last-iterate convergence of mSGD. We consider two possible step sizes: step size that decay as e + and constant step size e, = e > 0. Using such larger step size is expected to lead to the last iterate of mSGD (i.c., 6x2) converging faster. We gradient descent (SGD) originating from [1]. Instead of calculating Vo,,(9n)) over all the training data, SGD computes gradient estimate Vo.g(On: ( Sn) by sampling a subset of data (following the distribution of sampling noise (fn) and then updates the iterate as follows
0x+1 = p, - Cn Vo,, (6n, En)-
(3)
In addition to sampling noise, &n can be used to model any external noises arising from the gradient computation.
However the price to pay for this data efficiency is that the employment of stochastic gradients (3) displays a relatively slow convergence rate during the learning process. In order to improve the convergence rate of SGD, momentum- based stochastic gradient descend (mSGD), which reduces the update variance by averaging the past gradients ([2)), has been proposed. A typical iteration form of mSGD is as follows (13, 4)):
Un = 00h-1 + éa Von. 0n+1 0, Un:
(4)
where a E [0, 1) and en > 0 are momentum coefficient and step size (learning rate), respectively. An alternative formulation of mSGD, named stochastic heavy ball (SHB) [5, 6]. has also been proposed as follows
Un = Bn + (1 V,, 9,, En), 0u+1 = On In Un:
(5)
where 3n E (0, 1) and 7n are momentum coefficient and step size. It has been shown that mSGD and SHB are essentially equivalent [7]. In recent years, mSGD has been widely employed in many applications of deep learning such as image classification [8], fault diagnosis [9], statistical image reconstruction [10], among others. Moreover, number of variants on momentum have been emerging, see, e.g., synthesized Nesterov variants [11], robust momentum [12], and PID-control based methods [13]. The importance of momentum in deep learning has been illustrated in [4] through experiments.
There has been long line of literature analyzing the convergence of SGD and mSGD algorithms. Regarding SGD, authors in [14] studied the last-iterate convergence when the step size is chosen to decay as 1/n. Authors in [15, 16] established the last-iterate convergence of SGD for strongly convex loss functions without the bounded gradients assumption. For the normalized mSGD (SHB), Polyak [2, 17] and Kaniovski [18] established the convergence (subsequence convergence and convergence of time averages) properties for convex loss functions. Igor Gitman [5] provided convergence results of mSGD (SHB) for non-convex loss functions under somewhat restrictive requirement on uniform boundedness of noise term [E[[ 9. (0,, En V e.. g g(,))l) S. 8. It has been pointed out [4, 19] that the designs of momentum coefficients in [2, 5, 17, 18] may not be consistent with the requirements of some practical applications. The last- convergence of mSGD for non-convex loss functions
n= 2 < +oc, which has been widely utilized in stochastic optimization [1, 15, 16, 21]. However, this condition requires the step size to decay relatively fast, so it may lead to slow convergence especially in the early stage of the iteration. With the requirement that the loss function is strongly convex, [22] showed that the last-iterate of mSGD can converge to neighbor of the stationary point when the step size is constant.
has recently been established in [7, 20] with the step size condition Σ +:00
Problem of interest and contributions
In this paper, we consider the problem of last-iterate convergence of mSGD under relaxed require-
vn
note that since the SGD (3) is special case of mSGD (4), the developed results in this paper for mSGD also work for SGD.
The contributions of this paper are summarized as follows:
1) Under the setting of step size decaying as 6, - we prove that the gradient norm of mSGD for a class of non-convex loss functions asymptotically decays to zero. This result more general than mean-square gradient convergence established in [20]. In fact, this result holds for any step sizes of

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the form Σts1 20 < too (0 S 70 < <1/2), which is a superset of the widely required condition Σta & < +oo [1, 7, 15, 16, 21]. Given this relaxed step size condition, one run mSGD by employing larger step size like en. V
for last-iterate convergence.
2) Under some mild conditions, provide estimate of the convergence rate of mSGD. Further- more, under a probability- convergence speed in the early stage of the iteration.
metric, we show that the new step size can help to improve the
3) Under the setting of constant step size, first we prove that given any small neighborhood of stationary points, it is feasible to design step size, such that there is convergent iterate subsequence of mSGD staying within the neighborhood almost surely. In addition, we prove that the mean-square gradient can be arbitrary small by tunning the step size. Comparing with [22]. remove the requirement of strong convexity on the loss functions.
Regarding the definitions on sequence convergence, the following ones from literature are typical. For a stochastic variable sequence { E R with as the unique limit point, the sequence is said to satisfy last-iterate convergence if lim ||E..|| 0, a.s.; and time-average (mean-square convergence) if -im : Eh=1 E(I|&x||)) O. can be proved that last-iterate convergence implies time-average convergence under the general setting of the considered problem.
Paper outline. The rest of the paper is organized as follows. Section 3, we provide the main results of the paper on last-iterate convergence of mSGD under decaying step size and constant step size, respectively. In Section 4, two simulation experiments are given. Conclusion is made in Section The main proofs are given in Appendix.
3
Main results
In this section, we provide the main results of this paper on the last-iterate convergence for mSGD (4) under two possible step sizes: step size that decays as vπ and constant step size 6, = 0.
3.1 Convergence of mSGD under decaying step size
The following assumptions are needed in this paper.
Assumption 3.1. Loss function g(0) in (1) satisfies the following conditions:
1. Noise sequence { are mutually independent and independent of 02 and vo, such that g(0) = EE,, (g(0, En)) for any 0 E N
2. g(0) is non-negative and continuously differentiable function. The set of its stationary points [0]l] [0)]| 0} is a bounded set which has only finite connected components JI,..., Jn. In addition. there is ε₁ >> O, such that for any i and 0 < d(0, < ε1, it holds that [9(0) 9.| # O. where = [0 E J.Y is a constant.
3. There are two constants M' > and af > O such that for any O E R and n E N+,
Ee,. (I) (oog( (.)) s M'|[oo[(0)] +a.
4. The gradient Veg(0) satisfies the Lipschitz condition, i.e., there is a constant >> such that for any x, y E IR,,
V V,,((v) S cl|x vll-
Conditions I and 4 in Assumption 3.1 are common in the literature [1, 15, 16, 21J. [7] required
a condition on strong growth, i.e., Eg., V/ ( 2 < M Veg(0)) which however makes close to the deterministic case. In contrast, condition 3 allows more randomness of data sampling. Condition is a mild condition for the reasons as follows. In some works, the nen-negative condition may be replaced by a lower bound condition g > lo > -0. These two conditions are essentially g lo under the lower bound condition,
equivalent, since one can construct new loss function such that the new loss function is nog-negative. The rest of condition 2 is quite general, since allows the loss function to have multiple stationary points and to be non-convex.
3

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Assumption 3.2. In the mSGD (4), momentim coefficient a E (0,1) and the sequence step size cn is positive and monotonically decreasing zero, such that Enst En to0 and Σ nn. where 0 S To < 1/2 is a constant.
2+
< +00,
In Assumption 3.2, the momentum coefficient [0, 1) is a constant. Comparing with the setting in [5. 6, 17. 18J. where (x) tends to 1 or O, constant momentum coefficient is more common in practice [4, 7, 19]. Regarding the step size condition, it is more general than the one in many existing works on last-iterate convergence of SGD and mSGD [1, 7, 15, 13, 16, 21J, i... Σn nn=i E. < +00, which obtained from Assumption 3.2 if 7oo O. Under the step size condition in Assumption 3.2, one can choose larger step size like c, = which however is not feasible in the commonly required
condition Σtei 2 < +oc. Since the new step size can decay more slowly than the existing one, provides more space for step size fine tuning, such that the algorithm can quickly converge to neighborhood of the stationary point [23, 24, 22].
Then attain the first main result in this paper on last-iterate convergence of mSGD under the step size condition in Assumption 3.2.
Theorem 3.1. Consider the mSGD in (4) with any to E R- and 0 E RRN. Under Assump- tions 3.1 and 3.2, the gradient norm tends O almost surely, i.e.
lim R-++000
(o,)((n)) 0 a.s..
Due to page constraint, the complete proof is given Appendix. Here, we provide a proof outline to present the main proof ideas.
Proof Sketch of Theorem 3.1: We aim prove that Vo..((n,) 0 a.s. via the following key steps.
Step I: In this step we aim to prove E g(n)) is uniformly upper bounded. We present this result as Lemma B.8 in Appendix.
Step 2: We prove there exists a subsequence of ((,) which is convergent to 0 a.s.. It is attained by proving E En 2,22 [ ((,), < too a.s.. We present this result as Lemma B.10 Appendix.
Step 3: We aim to extend the subsequence convergence in Step 2 to asymptotic convergence. The basic idea is to prove that the adjacent terms g(0,) and g(0n+1) are "close" enough, such that
g(0n+1) - g(0)) S ke, 1+ + Qu:
where i: > 0 is a constant and {Qn] is a sequence, such that Σ On converges a.s.. This result is presented in Lemma B.11.
Step 4: Finally, we use some techniques to attain the result of asymptotic convergence
g e[Vo,, (0.)) + a.s..
In the last-iterate convergence analysis of mSGD, there are several technical challenges when try to extend the condition Σ Σm=1 e < +00 to Σ Sim 2+ < too with 0 No 1/2. One of the challenges is: When we try to prove Σ Sn Σt= G, Vo, g(..) (((..)(((,) (V... (... (0,,((0)) is convergent a.s., which is the vital step to get the last-iterate convergence, we usually use the Martingale convergence theorem, through which we just need to prove D ((()) is convergent. the condition Σ= 2 < +oo holds, we can get Σte E(G)) < E Ke t00 (K is constant) directly. However, under the new condition > too 2+60, we can not use this approach. Instead, we introduce more technical approach handle this problem.
Comparing with the corresponding result in [7] (Theorem in [7]), we relax the condition on the decay of step size. Under this new condition, one utilize step size with more slow decaying speed in the iteration of mSGD. Let a = in mSGD, then we can get similar result for SGD
Corollary 3.1. Consider the SGD in (3) with any 01 E RN Under Assumptions 3.1 and 3.2 with 0, the gradient norm tends to 0 almost surely, i.e.
lim
VV.. = 0 a.s..

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In the following, we aim to estimate the last-iterate convergence rate of Ve,.((n)) Instead. the is quite meaningful. In general, if we need to estimate the convergence rate of the last iterate, we need probability - retention assumption. i... Vko > Ba > O, making P(]O,,]] || > a) ~ P[]]0 > Koe ||0 > koc ), and we usually need some extra assumptions to establish quantitative relationship between g and Vg. Existing works were established usually under the stroglly-onvve assumption [25, 26, 13], local P-L condition [7], or convex loss function satisfying some stability conditions, e.g.. E V v.v.ge ((,)))l < 8 < +00. However, these assumptions are rather relatively strong, especially the stability condition which is very difficult to verify. In our paper, we need a milder assumption than the above conditions as follow.
convergence rate of E (I
Assumption 3.3. Regarding the loss function g(t) defined in (I), there exists dg > 0. To > 0, such that for any 0- E inleERs g(0).
- g²) < do}, it holds that g(0) < To. where g" =
This assumption requires the value of the loss function g(0) to be bounded in the set S -
(o]]i -))) < 6o}. The motivation of this assumption is: For a stochastic algo- rithm, there is always a probability at which the iterates are far away from the true value. Since second-derivative information has more positive effect on convergence (such as strongly convex loss functions), this condition restricts iterates which are far away due to extreme noise, so that it does not have a significant effect on global updates. This requirement is milder than the strong convexity condition, which is commonly used in existing works [25, 26, 13] for the analysis of the last-iterate convergence. In addition, this assumption allows the loss function to be non-convex, and does not need the local P-L condition used in [7] on the second derivative information near stationary points. Note that Assumption 3.3 can be satisfied by many loss functions commonly encountered in machine learning. First of all, many common loss functions are bounded [27-30], and thus satisfy Assumption 3.3. For unbounded loss functions, there are number of instances satisfying Assumption 3.3. For linear regression functions, since they are strongly convex, it is easy to verify the assumption. In the Appendix, explain (not in a strict but an illustrating manner) this assumption can also be satisfied by a two layer neural network with Relu active function under a square loss function.
In the following theorem, we provide an estimate of the convergence rate for the mSGD in (4). Theorem 3.2. Consider the mSGD in (4) with a unique stationary point 0", Then for any uo E R under Assumptions 3.1-3.3 and probability- retention assumption. Then exists a > 0, for any [[Vo,g(0)) (0))) there is
= (1Vv)((.11)==0(( (0 22 (
(IIVog(0
(
(Σe 1=1
n
Gea ofz (l
(7)
where è = minfa/2To. }
Comparing with [7], remove the strong growth condition, i.e., Eg, Il Vgg(O, E,)l1 S
M[]Veg(0)11 and relax the step size condition to be Σm= 2++6 -2+6 (0 < 8n S 1/2). In the comparison with [20], we do not require the convexity of loss function or E (. 12 < G. In addition, we relax the step size condition in [20]. There are quite a few results on the average iterates in the literature, such as [13, 31, 32]. In [13, 31, 32], the convergence rate 0 1 of mSGD
vn
is established with the step size -/-. According our result, let en -/ it holds that
E(IV p. ((w))) - 0 Σ
k O
It can be proved that the convergence rate of average iterates is also of
From (7), we see that larger step size may not lead to larger convergence rate. This is reasonable. since the convergence rate reflects how fast the iterate O, converges when n + too and a larger step size usually makes the iterate converges to a neighborhood of stationary point quicker. We have the following theorem on this point.
Theorem 3.3. Consider the mSGD in (4) with any Up E R.N. Under Assumptions 3.1-3.3, given any à >>>00 O, if VV, Vo.0(0:)122 > à, then it holds that
(()(0) 2 n) = 0(e (1 28
5

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where 7()) Ve. < â} and é min 260
Theorem 3.3 provides probability description of how fast the iterates of mSGD converge into preset neighborhood of stationary points. It can be illustrated as follows. It follows from Theorem 3.3 that
P(7(a) < <n) > kie (1-0000 (1-a)² Ei.
where k > O is a constant. From the inequality, as n increases, the event [Vo,, < a occurs with higher and higher probability tending probability 1. The influence of momentum coefficient a and step size en is also quantified. We note that with the measure in Theorem 3.3, the convergence to a neighborhood of the stationary point is faster than under the traditional condition Σn e < too by setting the step size with relatively slow decay speed as Σ nn 2+00 (0 < 60 S 1/2), such as En vi and ?/u 1/4. Similar results (the large step size make the iteration quickly converge
neighborhood of stationary point, but may converge slowly in the late stage of the algorithm) appear in [23, 24, 22] through experiments.
3.2 Convergence of mSGD under constant step size
In the previous subsection, we establish the last-iterate convergence of mSGD under relaxed decay condition on step size. In this subsection, we study the convergence of mSGD under a constant step size 6n > We start with the following result on subsequence convergence.
Theorem 3.4. Consider the mSGD in (4) with any vo R.N and 01 RRN Under Assumption 3.1,
(v)) for any > 0, there exists μo (4) > O, such that for any 0 < en = c < Po it holds that
S p, a.s.
where (B( is subsequence {.,}.
Proof. Due to page constraint, the complete proof is given Appendix. Here, we provide a proof outline present the main proof ideas.
To study the mSGD with constant step size, we notice that the impact of the noise cannot be totally eliminated through the step size. But when ||V (0,)) is relatively large (meaning the distance between On and the stationary point 0" is relatively large). it holds that E,, (On. < M'I e.. e..g((, (02 a' < (M' + V)][Ve.. (0,)) This implies that with a proper step size e, the noise can be controlled. As a result, the algorithm can be stabilized around stationary point. In the following, several key steps in the proof are provided.
Step I: We define an event by A(r) = (l ((11))l 9, [[Vo, > p...., Vo.. } and characteristic function of this event by r((1)_
Step 2: We prove that the probability of (1)) is tending to as co. First, attain recursion formula:
E g1441g(0 (0 E
( g(0.)
S <-a 5-a- (0) E Vo. (0,)
122
By taking the sum of multiple inequalities as above, we have
E (I < t00.
1=2
This leads to
E (1 (0) V (o))())))) + 0.
From the Chebyshev inequality, attain P(A (1)) + 0.

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0,03
0.05
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344
0,03
0.00
0,02
0.12
0.00
0.00
200
400
600
800
1001
200
400
600
100)
Epochs
Epochs
(a) Three-layer neural network
(b) Four-layer neural network
Figure I: Training loss-iteration chart on the Boston house-price data.
Step 3: Step 2 implics that with probability there exists at least one instant n at which (Vo.g((,)) < sp. We note that for the time instant at which ((.)) > 40, use a new start point. By repeating the process Step 2, we attain the conclusion of this theorem.
Theorem 3.4 shows that under the common conditions in the literature, the constant step size can guarantee the subsequence convergence of mSGD (4) in any small neighborhood of stationary point. In addition, we have a similar result on o (0.))ll)) as follows.
Proposition 3.1. Suppose {0,} is sequence generated by (4) with any 01 E RN and E R. Under Assumptions 3.1 and 3.3, if (1) has a unique solution and inf inf [Vgg(0) >
(sp) > and no > such that for any (46 > 0), then for any 40 > O, there exist scalars po 6n < 140
(sp)) and n. > no, it holds that
(IV (0.))1 S 40-
This result is substantial extension of the existing results on SGD with constant step size mSGD. Comparing with the results in [22], we remove the requirement of f--strong convexity on loss function.
4 Experiments
Since this paper focuses on convergence analysis on the well-known mSGD, two relatively simple experiments on regression and classification tasks are given respectively to show the effectiveness of the results.
4.1 Regression task
In this subsection, we study a house price prediction problem with neural networks trained by mSGD, Network Architectures. We respectively employ 3-layer and 4-layer fully-connected neural network with ReLu active function and squared loss function.
Implementation. We implement two neural networks to train on this dataset using Keras. The first neural network consists of three fully connected layers with 13, 8, and neurons. The second neural network consists of four fully connected layers with 13, 32, 16, neurons, respectively. We initialize weights use glorot uniform algorithm. We use mSGD with a mini-batch size of 16, and use mean square error loss function to train the model. The momentum term coefficient is 0.9 and the model is trained for up to 1000 epochs. The training takes about two hours time in 3080GPU. We do use dropout.
7

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0.4
699
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1.4
033
200
100
(00
800
1000
200
400
600
1000
Egochs
Epochs
(a) The training loss
(b) Accuracy of ResNer20 CIFAR-10
Figure 2: Training and prediction performance on CIFAR-10.
Dataset. The training dataset we use is from Boston House Price Dataset This dataset a regression prediction dataset, which consists of 506 sets of data with dimension 13. During the training process, we randomly divide 30% as the test set and normalize the data before training.
पo.. and En प... The experiment results are given in Figure 1. The figure shows that 1) the loss decays to zero under the three step size settings; 2) 6n noअ neighborhood of zero fastest among the three step size settings. This conforms to the theoretical analysis in Theorems 3.1-3.3.
Results. We use the mSGD in (4) under three different step sizes respectively: 6,
+ can make the loss tend small
4.2 Classification task
In this subsection, we study an image classification problem with neural networks trained by mSGD,
Network Architectures. We employ 20-layer ResNet network. The convolutional layers mostly have 3x3 filters and follow two simple design rules: (i) for the same output feature map size, the layers have the same number of filters; (ii) if the feature map size is halved, the number of filters doubled so as to preserve the time complexity per layer. We perform downsampling directly by convolutional layers that have stride of 2 and padding with valid. The network ends with average pooling layer and 10-way fully-connected layer with softmax.
Implementation. We implement the ResNer20 network using Keras. We initialize the weights use glorot uniform algorithm. We use mSGD with a mini-batch size of 64, momentum coefficient 0.9 and use categorical crossentropy loss function to train the model. The models are trained for up to 1000 epochs, taking about two hours time 3080GPU. We do not use dropout.
Dataset. We use two different datasets CIFAR-I0 and CIFAR-100'. CIFAR-IO consists of 50k training images and 10k testing images in 10 classes. CIFAR-I0 a color image dataset close to ubiquitous objects, and the image size is 32x32x3. CIFAR- 100 consists of 50k training images and 10k testing images in 100 classes. Each category contains 500 training images and 100 testing images. CIFAR- 100 is a color image dataset closer to ubiquitous objects, and the image size is 32x32x3. Normalize the dataset between 0-1 before training.
Results. We use the mSGD in (4) under three different step sizes: 6, = 1 6n = T..,, and 6, w.3 The experiment results are given in Figure and Figure 3. The figures show that 1) the loss decays to zero and the accuracy goes up to under the three step size settings; 2) e, 200. can make the loss tend to small neighborhood of zero and make the accuracy tend small neighborhood of fastest among the three step size settings, followed by the setting of e, प... These results conform to the theoretical analysis in Theorems 3.1-3.3.
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0
200
100
600
800
1030
200
400
600
500
1009
Epochs
Epochs
(a) The training loss
(b) Accuracy of ResNet20 CIFAR-100
Figure 3: Training and prediction performance on CIFAR-100.
5
Conclusion
In this paper, we studied the last-iterate convergence of momentum-based stochastic gradient descent (mSGD) under relaxed conditions on step sizes. First, under the relaxed condition Σ 2 < too (0 7/0 1/2), proved the last-iterate convergence mSGD for a class of non-convex loss functions. In addition, we showed that this step size can indeed help to improve the convergence speed in the early stage of the algorithm by quantifying the influence of the step size and momentum coefficient. This implies that a larger step size, such as
1 v can be utilized with guaranteed 0) can
convergence. We also proved that the algorithm with constant step size (i.e., c,, ensure the last-iterate convergence of mSGD) without requiring the strong convexity assumption on loss functions.
6 Acknowledgments and Disclosure of Funding
Xingkang He was supported in part by ARO Grant W911NF191048 and AFOSR Grant FA9550-21- 1-0231. Vijay Gupta were supported in part by NSF Grants 2020246, 2225978, and 2208794. We thank the anonymous reviewers for helpful comments and suggestions.

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References
[1] H. Robbins and S. Monro, "A stochastic approximation method," Annals of Mathematical Statistics, vol. 22, no, 3. pp. 400-407, 1951.
[2] B. T. Polyak, "Some methods of speeding up the convergence iteration methods," USSR Computational Mathematics & Mathematical Physics, vol. 4, no. 5, pp. 1-17, 1964.
[3] S. Zareha, A. Gonezarek, J. M. Tomczak, and Swiatek, "Accelerated learning for restricted Boltzmann machine with momentum term." in Progress in Sysiems Engineering, pp. 187-192, 2015.
[4] I. Sutskever, J. Martens, G. Dahl, and G. E. Hinton, "On the importance of initialization and momentum in deep learning," in International Conference Machine Learning, pp. 1139-1147, 2013.
[5] I. Gitman, H. Lang, Zhang, and Xiao, "Understanding the role of momentum in stochastic gradient methods," " Advances in Neural Information Processing Systems, vol. 32, 2019.
[6] A. M. Gupal and L. G. Bazhenov, "Stochastic analog of the conjugate- method," Cybernetics. vol. 8, no. I, pp. 138-140, 1972.
[7] R. Jin, Y. Xing, and X. He, "On the convergence of mSGD and AdaGrad for stochastic optimization," in International Conference on Learning Representations, 2022.
[8] A. Krizhevsky. I. Sutskever, and G. B. Hinton, "Imagenct classification with deep convolutional neural networks," Advances in Neural Information Processing Systems, vol. 25, pp. 1097-1105, 2012.
[9] S. Tang, C. Shen, D. Wang. S. Li, W. Huang, and Z. Zhu, "Adaptive deep feature learning network with Nesterov momentum and its application to rotating machinery fault diagnosis," Neurocomputingg vol. 305, pp. 1-14, 2018.
[10] D. Kim, S. Ramani, and J. A. Fessler, "Combining ordered subsets and momentum for accelerated X-ray CT image reconstruction," IEEE Transactions on Medical Imaging, vol. 34, no. 1. pp. 167-178, 2014.
[11] M. O. Williams and T. F. Lovett, "Numerical diagnostics for systems of differential algebraic equations." arXiv preprint arxiv:/602 002000550 2016.
[12] S. Cyrus, B. Hu, B. Van Scoy, and L.. Lessard, "A robust accelerated optimization algorithm for strongly convex functions," in American Control Conference, pp. 1376-1381, 2018
[13] An, H. Wang, Q. Sun, J. Xu, Q. Dai, and Zhang, "A PID controller approach for stochastic optimization of deep networks," in Proceedings of the IEEE Conference on Computer Vision and Pottern Recognition, pp. 8522-8531, 2018.
[14] O. Shamir, "Making gradient descent optimal for strongly convex stochastic optimization," Omnipress, 2011.
[15] L. M. Nguyen, P. H. Nguyen, P. Richtárik, K. Scheinberg, M. Taká, and M. V. Dijk, "New convergence aspects of stochastic gradient algorithms," 2018.
[16] L. Nguyen, H. Nguyen, M. Dijk, P. Richtárik, K. Scheinberg, and M. Takáe, "SGD and Hogwild! convergence without the bounded gradients assumption," in International Conference on Machine Learning, pp. 3750-3758, 2018
[17] T. Polyak, "Comparison of convergence rate of one-step and multistep optimization algorithms in the presence of noise." Izv. Akad. Nauk SSSR, Tekh. Kihern., vol. 1, pp. 9-12, 1977.
[18] Y. M. Kaniovskii, "Behaviour the limit of iterations of the stochastic two-step method," Zuurnal Vychislitel noi Matematiki Matematicheskoi Fiziki, vol. 23, no. 1, pp. 13-20, 1983.
[19] S. L. Smith, P.J. Kindermans, Ying, and Q Le, "Don't 't decay the learning rate, increase the batch size," 2018.
[20] X. Li, M. Liu, and F. Orahona, "On the last iterate convergence of momentum methods," 2021.
[21] H.-F Chen, Stochastic Approximation and Its Applications. Springer Science & Business Media, 2006.
[22] Y. Liu, Y. Gao, and W. Yin, "An improved analysis of stochastic gradient descent with momentum," arXiv preprint arXiv:2007.07998 2020.
10

---

[23] N. Iyer. V. Thejas. N. Kwatra, R. Ramjee, and M. Sivathanu. "Lrtuner: A learning rate tuner for deep neural networks," 2021.
[24] K. N. S. S. Sastry, "Step size matters in deep learning," 2018.
[25] T. Zhang, "Solving large scale linear prediction problems using stochastic gradient descent algorithms," in P Proceedings of the twenty-first international conference Muchine learning, 116, 2004.
[26] O. Shamir and T. Zhang, "Stochastic gradient descent for non-smooth optimization: Convergence results and optimal averaging schemes," JMI.R.or. 2012.
[27] L. Yang, Z. Hui, and S. Yuan, "Bayes estimation of parameter of exponential distribution under a bounded loss function," Research Journal of Mathematics Statistics, vol. 5, no. 4, 28-31, 2013.
[28] R. Leon and C. Wu, "A theory of performance measures in parameter design," Statistica Sinica, pp. 335-358, 1992.
[29] Berger and O. James, "Statistical decision theory and bayesian analysis," vol. 10..111007777911114777--42222233 2004.
[30] M. Towhidi and J. Behboodian, "Minimax estimation of a bounded parameter under some bounded loss functions," Far East Journal of Theoretical Statistics, vol. 6, no. 1, pp. 39 48, 2002.
[31] V. Mai and M. Johansson, "Convergence of stochastic gradient method with momentum for nonsmooth nonconvex optimization," in International Conference Machine Learning, pp. 6630-6639, 2020.
[32] T. Yang. Q Lin, and Z. Li, "Unified convergence analysis of stochastic momentum methods for convex and non-convex optimization," 2016.
[33] Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course. Introductory Lectures on Convex Optimization: Basic Course, 2004.
[34] Z.zz. Wang, Y. Dong, and F. Ding, "On almost sure convergence for sums of stochastic sequence," Communications in Satistics-Theon and Methods, vol. 48, no. 14, pp. 3609-3621 2019.

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12


Reviews:
[{"review_id": "yYluByktZ6O", "reviewer": "Reviewer_Kgxh", "summary": "This work considers mSGD and SGD under more weaker conditions on the step size as considered before. This can be important since in practice, picking a large step size initially is shown to improve convergence. Therefore, a theory of convergence with large step sizes can be useful. It is standard to consider step sizes such that $\\sum_t \\epsilon_t = \\infty$ and $\\sum_t \\epsilon_t^2 < \\infty$ whereas this work improves this to $\\sum_t \\epsilon_t^{2+\\eta_0} < \\infty$. Under these conditions, along with standard assumptions like smoothness, asymptotic convergence to a stationary point is established (Theorem 3.1).\n\nUnder additional growth assumptions of the function, the Theorem 3.2 establishes rates of convergence as a function of the step size sequence, which shows the dependence of the momentum parameter alpha. However, since this does not establish how picking larger step sizes can be effective, the authors show Theorem 3.3 where the upper bounds are such that picking a larger step size can help the algorithm reach a point better than the initial point more reliably. \n\nThe proof techniques and assumptions mostly seem to be modifications of [7]. \n\n[7] R. Jin, Y. Xing, and X. He, “On the convergence of mSGD and AdaGrad for stochastic optimization,” in\n312 International Conference on Learning Representations, 2022", "questions": "1. Given the observations above, how do we justify using larger step sizes based on the theoretical results provided?\n\n2. The authors should to carefully compare the techniques used in [7] to the current work since a lot of the main techniques and assumptions seem very similar. They currently mainly compare the results obtained in [7]. \n\n3. Below Theorem 3.2, the authors discuss that [7] uses the condition $\\mathbb{E}\\|\\nabla g(\\theta,\\zeta)\\|^2 \\leq M \\|\\nabla g(\\theta)\\|^2$. However, when I checked the ICLR 2022 paper, this work did not contain this assumption in Theorem 2 (See (4) in Assumption 2). However, I do agree Theorem 2 in [7] cannot be established without such an assumption and the original work [7] could have a typo. The authors should elaborate more on this. ", "limitations": "The authors have been forthright about several of the weaknesses in the analysis in their discussions (as pointed above), however several of the limitations need to be discussed more as pointed out in the rest of the review. ", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "*Strengths*:\nThe work establishes convergence for widely used stochastic gradient methods in regimes which have not been considered in the literature but nevertheless seem very important. \n\n*Weaknesses*:\nThe work does not justify why picking larger step sizes is useful. Theorem 3.2, like the authors discuss, does not show that picking a step size sequence such that $\\sum_t \\epsilon_t^2 = \\infty$ is helpful in terms of convergence rates.\n\nTheorem 3.3, on the other hand, considers a non-standard stopping time $\\tau^{(a)}$. This seems to be quite a weak metric to compare algorithms since this is the first time when the algorithm's output is better than the initial point. Also, arguing about how certain step sizes are better based on upper bounds is fallacious since we do not know if the upper bounds are sharp. \n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "wBkQ3Hym6wA", "reviewer": "Reviewer_947T", "summary": "The authors improve the step size requirement for mSGD and prove the convergence result. Specifically, they have extra assumption condition 2 in Assumption 3.1 then they allow the step size to decay more slowly. Additionally, they prove the subsequence convergence with constant step size. They implement the experiments to verify their theoretical results. ", "questions": "1) I am not totally sure if Condition 2 in Assumption 3.1is a reasonable one. Is the set of stationary points bounded? For overparameterized models, I believe it's not. Is it critical in the proof? Can you clarify it a bit more?\n\n2) The equation in Assumption 3.3 is very close to PL condition. In my understanding, you want the loss to decrease exponentially out of this set? I suggest the authors have more discussion about the relation to PL condition.\n\n3) Can you clarify why we care about the convergence result for subsequence i.e., Theorem 3.4? \n", "limitations": "yes", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\nThe problem studied in the paper is very interesting and important for the community. The authors improve the results which are verified by the experiments. The neural network example to support Assumption 3.3 is nice.\n\nWeaknesses:\n\nI don't see any major weakness. Maybe the authors can point out the intuition of the extra assumption (Condition 2 in Assumption 3.1) and how to use it to prove the results. \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "uh4tqnJq3z8", "reviewer": "Reviewer_hkDq", "summary": "This paper establishes the convergence result of the last iterate of momentum algorithms. In contrast to the existing results that require the step size to be squared summable, this paper allows a larger spectrum of step size rules and even constant step size. The proof technique is different from the standard martingale-based argument.", "questions": "1. I didn't check all proofs, but I think it would be helpful to sketch the proof of Theorem 3.1, in addition to mentioning the main challenges. In particular, how does the proof technique differ from those in [7]?\n\n2.  I am wondering if the rate of $2+\\eta_0$, $\\eta\\in[0,1/2)$, is the best achievable in the setting of Section 3.1. This can be answered either theoretically, or numerically.\n\n3. For Proposition 3.1, the result improves the previous one [22] that imposes strong convexity. But is there any substantial difference in terms of the proof compared with [22]?\n\n4. Minor: I cannot find the definition of $\\alpha^{(n)}$ in line 121.", "limitations": "This is a theoretical work so I think there's not much potential negative societal impact. The authors do not discuss much about the limitations of their results.", "rating": 7, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n\n1. The presentation is clear, and the authors make efforts to explain the meaning and use of each assumption, condition in their results, and compare their results to relevant literature.\n\n2. From the practical point of view, it establishes provable convergence for larger step sizes, so the results look nice to me, although I am not an expert in this field. \n\n3. The additional numerical results certify the theoretical analysis.\n\nWeakness:\n\n1. I am not sure how strong Assumption 3.3 is, even though it is weaker than the P-L condition. The paper illustrates that this assumption holds for a two layer ReLU network, but I doubt it works for more layers.\n\n2. Some illustration of the proof seems important. See my questions below.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "EyNWcy8C2Gr", "reviewer": "Reviewer_61dG", "summary": "The paper proves last-iterate convergence for SGD with and without momentum for larger step sizes and weaker assumptions. In particular, the paper provides asymptotic convergence rates for mSGD when the learning rate decay is slower than what was needed by previous works, hence showing theoretically why mSGD can work with larger learning rates than shown by previous theory. The paper also provides non-asymptotic rates under a more general assumption than strong convexity. The paper has theorems 3.3 and 3.2 which show that even though larger step size may lead to one iterate being very close to a stationary point, the expected error at a fixed iteration number might not necessarily get smaller with larger step size.", "questions": "1. Although Theorem 3.2 hints that larger step sizes might not necessarily lead to faster convergence, to actually prove that it is indeed the case, a lower bound needs to be shown. Is there any lower bound existing in literature that matches or closely follows your upper bound from Theorem 3.2? If not, then a discussion should be added in the paper about this.\n\n2. It will be nice to have some experiments that show that large step sizes that satisfy assumption 3.2 can still be slower for convergence than smaller step sizes, to support the discussion for Theorem 3.2 (lines 190-192).", "limitations": "Most of the results in the paper are asymptotic in nature and only prove convergence in the limit (that is, they do not have asymptotic convergence rates). Hence they do not provide much information about the actual rates of convergence. ", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The paper proves asymptotic convergence for mSGD for learning rates that decay slower than what was proved by previous theory.\n2. The paper proves non-asymptotic convergence rates for mSGD in a milder condition than strong convexity.\n3. The paper proves that for larger step sizes, indeed convergence is faster in the sense that there will be one iterate that is closer to the optimizer (Theorem 3.3)\n\n\n\nWeaknesses:\n1. Most of the results in the paper are asymptotic in nature and only prove convergence in the limit (that is, they do not have asymptotic convergence rates). Hence they do not provide much information about the actual rates of convergence. \n\n\nSuggestions:\n1. The paper should include proof sketches or ideas for the main theorems in the main paper. Since the part where Assumption 3.3 is demonstrated for shallow neural networks is not rigorous, maybe that can be moved to the appendix to make space for proof sketches.\n\n\nNOTE: I have not verified the proofs in the appendix.\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "AJlMt_RJG6N", "reviewer": "Reviewer_aWD4", "summary": "The paper considers SGD with momentum (and classical SGD if momentum coefficient is equal to zero) and provides last-iterate convergence rates for such an algorithm with (i) allowing for the bigger stepsizes, and (ii) by relaxing the PL-condition. \n", "questions": "7. What would be the optimal stepsize to give the fastest convergence? \n\n8. Can you show some advantages in the theory of using momentum ? \n\n9. how “a” is defined in theorem 3.2. Should it be a’ ? if not, what is the dependance on a’ in the convergence rate ? \n", "limitations": "-", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "1. Overall, understanding the theoretical convergence of SGD and momentum SGD is an important problem in machine learning. I think it is an interesting and solid result that both SGD and momentum SGD allow for the larger learning rates (even without improving the convergence rate). \n\n2. The paper misses the discussion on the tightness of the obtained result: is the obtained convergence rate tighter than existing ones (also the ones that use PL-condition or strong-convexity)? Also, does it recover known convergence rates of momentum SGD for the average iterates? Are obtained results recover best-known last iterate convergence rates of SGD when momentum is set to zero? \n\n3. I found Theorem 3.1 to be confusing. I quickly checked the proof and it seems that you basically show there that ||\\nabla g(theta_n) || -> 0 as n -> \\infty. I think it would be less confusing to state the theorem in terms of the norm of the gradient. I also did not find that you prove that \\theta_n stays bounded, thus it can happen that ||\\theta_n|| -> \\infty, e.g. if the function behaves similarly to e^{-theta} for \\theta -> + \\infty. Thus the algorithm might not converge to any connected component of stationary points. \n\n4. Regarding presentation, assumptions 3.1, 2) and 3.3. are not common in the literature. It would be useful for the reader to spend more time to explain the connections to the common assumptions in the literature: e.g. for assumption 3.3. explaining how does delta_0 relates to the strong convexity parameter, or \\mu in PL condition (I think delta_0 <= \\mu). It would be also helpful to provide a table with existing last-iterate convergence results under \n\n5. Overall the proofs are hard to read. \n\n6. Less important, I found the notation in the paper to be confusing, usually the stochastic noise is denoted as \\sigma, and lipschitz smoothness as L. \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept