Fast and Faster Convergence of SGD for Over-Parameterized Models and an Accelerated Perceptron

Modern machine learning focuses on highly expressive models that are able to fit or interpolate the data completely, resulting in zero training loss. For such models, we show that the stochastic gradients of common loss functions satisfy a strong growth condition. Under this condition, we prove that constant step-size stochastic gradient descent (SGD) with Nesterov acceleration matches the convergence rate of the deterministic setting for both convex and strongly-convex functions. In the non-convex setting, this condition implies that SGD can find a first-order stationary point as efficiently as full gradient descent. Under interpolation, we also show that all smooth loss functions with a finite-sum structure satisfy a weaker growth condition. Given this weaker condition, we prove that SGD with a constant step-size attains the deterministic convergence rate in both the strongly-convex and convex settings. Under additional assumptions, the above results enable us to prove an O(1/k^2) mistake bound for k iterations of a stochastic perceptron algorithm using the squared-hinge loss. Finally, we validate our theoretical findings with experiments on synthetic and real datasets.

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1 Introduction

Modern machine learning models are typically trained with iterative stochastic first-order methods [7, 32, 12, 24, 11, 6]. Stochastic gradient descent (SGD) and related methods such as Adagrad [7] or Adam [12] compute the gradient with respect to one or a mini-batch of training examples in each iteration and take a descent step using this gradient. Since these methods use only a small part of the data in each iteration, they are the preferred way for training models on large datasets. However, in order to converge to the solution, these methods require the step-size to decay to zero in terms of the number of iterations. This implies that the gradient descent procedure takes smaller steps as the training progresses. Consequently, these methods result in slow sub-linear rates of convergence. Specifically, if is the number of iterations, then SGD-like methods achieve a convergence rate of and for strongly-convex and convex functions respectively [16]. In practice, these methods are augmented with some form of momentum or acceleration [20, 18] that results in faster empirical convergence [28]. Recently, there has been some theoretical analysis for the use of such acceleration in the stochastic setting [5]. Other related work includes algorithms specifically designed to achieve an accelerated rate of convergence in the stochastic setting [1, 13, 8].

Another recent trend in the literature has been to use variance-reduction techniques 

[24, 11, 6] that exploit the finite-sum structure of the loss function in machine-learning applications. These methods do not require the step-size to decay to zero and are able to achieve the optimal rate of convergence. However, they require additional bookkeeping [24, 6] or need to compute the full gradient periodically [11], both of which are difficult in the context of training complex models on large datasets.

In this paper, we take further advantage of the optimization properties specific to modern machine learning models. In particular, we make use of the fact that models such as non-parametric regression or over-parameterized deep neural networks are expressive enough to fit or

interpolate the training dataset completely [33, 15]. For an SGD-like algorithm, this implies that the gradient with respect to each training example converges to zero at the optimal solution. This property of interpolation is also true for boosting [23]

and for simple linear classifiers on separable data. For example, the perceptron algorithm 

[22] was first shown to converge to the optimal solution under a linear separability assumption on the data [19]. This assumption implies that the linear perceptron is able to fit the complete dataset.

There has been some related work that takes advantage of the interpolation property in order to obtain faster rates of convergence for SGD [25, 15, 4]. Specifically, Schmidt and Le Roux [25] assume a strong growth condition on the stochastic gradients. This condition relates the norms of the stochastic gradients to that of the full gradient. Under this assumption, they prove that constant step-size SGD can attain the same convergence rates as full gradient descent in both the strongly-convex and convex cases. Other related work has used the strong growth condition to prove convergence rates for incremental gradient methods [27, 29]. Ma et al. [15] show that under weaker conditions, SGD with constant step-size results in linear convergence for strongly-convex functions. They also investigate the effect of batch-size on the convergence and theoretically justify the linear-scaling rule

used for training deep learning models in practice 

[10]. Recently, Cevher and Vũ showed the linear convergence of proximal stochastic gradient descent under a weaker growth condition for restricted strongly convex functions [4]. They also analyse the effect of an additive error term on the convergence rate.

In contrast to the above mentioned work, we first show that the strong growth condition (SGC) [25]

implies that SGD with a constant step-size and Nesterov momentum 

[18] achieves the accelerated convergence rate of the deterministic setting for both strongly-convex and convex functions (Section 3). Our result gives some theoretical justification behind the empirical success of using Nesterov acceleration with SGD [28]. In Section 4, we prove that under the SGC, constant step-size SGD is able to find a first-order stationary point as efficiently as deterministic gradient descent. To the best of our knowledge, this is the first work to study accelerated and non-convex rates under the SGC. Next, we relax the strong growth condition to a more practical weak growth condition (WGC). In Section 5, we prove that the weak growth condition is sufficient to obtain the optimal convergence of constant step-size SGD for smooth strongly-convex and convex functions.

To demonstrate the applicability of our growth conditions in practice, we first show that for models interpolating the data, the WGC is satisfied for all smooth loss functions with a finite-sum structure (Section 6.1). Furthermore, we prove that functions satisfying the WGC and the Polyak- Lojasiewicz inequality [21] also satisfy the SGC. Under additional assumptions, we show that it is also satisfied for the squared-hinge loss. This result enables us to prove an mistake bound for iterations of the stochastic perceptron algorithm using the squared-hinge loss (Section 7). Finally, in Section 8, we evaluate our claims with experiments on synthetic and real datasets.

2 Background

In this section, we give the required background and set up the necessary notation. Our aim is to minimize a differentiable function . Depending on the context, this function can be strongly-convex, convex or non-convex. We assume that we have access to noisy gradients for the function and use stochastic gradient descent (SGD) for iterations in order to minimize it. The SGD update rule in iteration can be written as: . Here, and are the SGD iterates, is the gradient noise and is the step-size at iteration . We assume that the gradients are unbiased, implying that for all and that .

While most of our results apply for general SGD methods, a subset of our results rely on the function having a finite-sum structure meaning that . In the context of supervised machine learning, given a training dataset of points, the term corresponds to the loss function for the point when the model parameters are equal to . Here and

refer to the feature vector and label for point

respectively. Common choices of the loss function include the squared loss where , the hinge loss where or the squared-hinge loss where

. The finite sum setting includes both simple models such as logistic regression or least squares and more complex models like non-parametric regression and deep neural networks.

In the finite-sum setting, SGD consists of choosing a point and its corresponding loss function (typically uniformly) at random and evaluating the gradient with respect to that function. It then performs a gradient descent step: where is the random loss function selected at iteration . The unbiasedness property is automatically satisfied in this case, i.e. for all . Note that in this case, the random selection of points for computing the gradient is the source of the noise . In order to converge to the optimum, SGD requires the step-size to decrease with ; specifically at a rate of for convex functions and at a rate for strongly-convex functions. Decreasing the step-size with results in sub-linear rates of convergence for SGD.

In order to derive convergence rates, we need to make additional assumptions about the function  [16]. Beyond differentiability, our results assume that the function satisfies some or all of the following common assumptions. For all points , and for constants , , and ;

(Bounded below)
(Convexity)
( Strong-convexity)
( Smoothness)

Note that some of our results in Section 6 rely on the finite-sum structure and we explicitly state when we need this additional assumption.

In this paper, we consider the case where the model is able to interpolate or fit the labelled training data completely. This is true for expressive models such as non-parametric regression and over-parametrized deep neural networks. For common loss functions that are lower-bounded by zero, interpolating the data results in zero training loss. Interpolation also implies that the gradient with respect to each point converges to zero at the optimum. Formally, in the finite-sum setting, if the function is minimized at , i.e., if , then for all functions , .

The strong growth condition (SGC) used connects the rates at which the stochastic gradients shrink relative to the full gradient. Formally, for any point

and the noise random variable

, the function satisfies the strong growth condition with constant if,

(1)
Equivalently, in the finite-sum setting,
(2)

For this inequality to hold, if , then for all . Thus, functions satisfying the SGC necessarily satisfy the above interpolation property. Schmidt and Le Roux’s work [25] derives optimal convergence rates for constant step-size SGD under the above condition for both convex and strongly-convex functions. In the next section, we show that the SGC implies the accelerated rate of convergence for constant step-size SGD with Nesterov momentum.

3 SGD with Nesterov acceleration under the strong growth condition

We first describe constant step-size SGD with Nesterov acceleration. The algorithm consists of three sequences () updated in each iteration [17]. Specifically, it consists of the following update rules:

(3)
(4)
(5)

Here, is the constant step-size for the SGD step and , , are tunable parameters to be set according to the properties of .

In order to derive a convergence rate for the above algorithm under the SGC, we first observe that a form of the SGC is satisfied in the case of coordinate descent [30]. In this case, we choose a coordinate (typically at random) and perform a gradient descent step with respect to that coordinate. The notion of a coordinate in this case is analogous to that of an individual loss function in the finite sum case. For coordinate descent, a zero gradient at the optimal solution implies that the partial derivative with respect to each coordinate is also equal to zero. This is analogous to the SGC in the finite-sum case, although we note the results in this section do not require the finite-sum assumption.

We use this analogy formally in order to extend the proof of Nesterov’s accelerated coordinate descent [17] to derive convergence rates for the above algorithm when using the SGC. This enables us to prove the following theorems (with proofs in Appendices B.1.1 and B.1.3) in both the strongly-convex and convex settings.

Theorem 1 (Strongly convex).

Under -smoothness and strong-convexity, if satisfies the SGC with constant , then SGD with Nesterov acceleration with the following choice of parameters,

results in the following convergence rate:

Theorem 2 (Convex).

Under -smoothness and convexity, if satisfies the SGC with constant , then SGD with Nesterov acceleration with the following choice of parameters,

results in the following convergence rate:

The above theorems show that constant step-size SGD with Nesterov momentum achieves the accelerated rate of convergence up to a factor for both strongly-convex and convex functions.

In Appendix A, we consider the SGC with an extra additive error term, resulting in the following condition: . We analyse the rate of convergence of the above algorithm under this modified condition and obtain a similar dependence on as in Cohen et al. [5].

4 SGD under the strong growth condition

In this section, we show that the SGC results in an improvement over the rate for SGD in the non-convex setting [9]. In particular, we show that under the strong growth condition, constant step-size SGD is able to find a first-order stationary point as efficiently as deterministic gradient descent. We prove the following theorem (with the proof in Appendix B.2),

Theorem 3 (Non-Convex).

Under -smoothness, if satisfies SGC with constant , then SGD with a constant step-size attains the following convergence rate:

The above theorem shows that under the SGC, SGD with a constant step-size can attain the optimal rate for non-convex functions. To the best of our knowledge, this is the first result for non-convex functions under interpolation-like conditions. Under these conditions, constant step-size SGD has a better convergence rate than algorithms which have recently been proposed to improve on SGD [2, 3]. Our results also provide some theoretical justification for the effectiveness of SGD for non-convex over-parameterized models like deep neural networks.

5 Weak growth condition

In this section, we relax the strong growth condition to a more practical condition which we refer to as the weak growth condition (WGC). Formally, if the function is -smooth and has a minima at , then it satisfies the WGC with constant , if for all points and noise random variable ,

(6)
Equivalently, in the finite-sum setting,
(7)

In the above condition, notice that if , then for all points . Thus, the WGC implies the interpolation property explained in Section 2.

5.1 Relation between WGC and SGC

In this section, we relate the two growth conditions. We first prove that SGC implies WGC with the same without any additional assumptions, formally showing that the WGC is indeed weaker than the corresponding SGC. For the converse, a function satisfying the WGC satisfies the SGC with a worse constant if it also satisfies the Polyak- Lojasiewicz (PL) inequality [21]. The above relations are captured by the following proposition, proved in Appendix B.5

Proposition 1.

If is -smooth, satisfies the WGC with constant and the PL inequality with constant , then it satisfies the SGC with constant .

Conversely, if is -smooth and satisfies the SGC with constant , then it also satisfies the WGC with the same constant .

5.2 SGD under the weak growth condition

Using the WGC, we obtain the following convergence rates for SGD with a constant step-size.

Theorem 4 (Strongly-convex).

Under -smoothness and strong-convexity, if satisfies the WGC with constant , then SGD with a constant step-size achieves the following rate:

Theorem 5 (Convex).

Under -smoothness and convexity, if satisfies the WGC with constant , then SGD with a constant step-size and iterate averaging achieves the following rate:

Here, is the averaged iterate after iterations.

The proofs for Theorems 4 and 5 are deferred to Appendices B.3 and B.4 respectively.

In these cases, the WGC is sufficient to show that constant step-size SGD can attain the deterministic rates up to a factor of . Since this condition is weaker than the corresponding strong growth condition, our results subsume the SGC results [25]. In the next section, we characterize the functions satisfying the growth conditions in practice.

6 Growth conditions in practice

In this section, we give examples of functions that satisfy the weak and strong growth conditions. In Section 6.1, we first show that for models interpolating the data, the WGC is satisfied by all smooth functions with a finite-sum structure. In section 6.2, we show that the SGC is satisfied by the squared-hinge loss under additional assumptions.

6.1 Functions satisfying WGC

To characterize the functions satisfying the WGC, we first prove the following proposition (with the proof in Appendix B.6):

Proposition 2.

If the function has a finite-sum structure for a model that interpolates the data and is the maximum smoothness constant amongst the functions , then for all ,

(8)

Comparing the above equation to Equation 7, we see that any smooth finite-sum problem under interpolation satisfies the WGC with . The WGC is thus satisfied by common loss functions such as the squared and squared-hinge loss. For these loss functions, if for all , then Theorem 4 implies that SGD with results in linear convergence for strongly-convex functions. This matches the recently proved result of Ma et al. [15], whereas Theorem 5 allows us to generalize their result beyond strongly-convex functions.

6.2 Functions satisfying SGC

We now show that under additional assumptions on the data, the squared-hinge loss also satisfies the SGC. We first assume that the data is linearly separable with a margin equal to , implying that for all , . Here, is the support of the distribution of the features . Note that the above assumption implies the existence of a classifier such that . In addition to this, we assume that the features have a finite support, meaning that the set is finite and has a cardinality equal to . Under these assumptions, we prove the following lemma in Appendix B.7,

Lemma 1.

For linearly separable data with margin and a finite support of size , the squared-hinge loss satisfies the SGC with the constant .

In the next section, we use the above lemma to prove a mistake bound for the perceptron algorithm using the squared-hinge loss.

7 Implication for Faster Perceptron

In this section, we use the strong growth property of the squared-hinge function in order to prove a bound on the number of mistakes made by the perceptron algorithm [22] using a squared-hinge loss. The perceptron algorithm is used for training a linear classifier for binary classification and is guaranteed to converge for linearly separable data [19]. It can be considered as stochastic gradient descent on the loss .

The common way to characterize the performance of a perceptron is by bounding the number of mistakes (in the binary classification setting) after iterations of the algorithm. In other words, we care about the quantity . Assuming linear separability of the data and that for all points , the perceptron achieves a mistake bound of  [19].

In this paper, we consider a modified perceptron algorithm using the squared-hinge function as the loss. Note that since we assume the data to be linearly separable, a linear classifier is able to fit all the training data. Since the squared-hinge loss function is smooth, the conditions of Proposition 2 are satisfied, which implies that it satisfies the WGC with . Also observe that since we assume that , . Using these facts with Theorem 5 and assuming that we start the optimization with , we obtain the following convergence rate using SGD with ,

To see this, recall that and the loss is equal to zero at the optima, implying that .

The above result gives us a bound on the training loss. We use the following lemma (proved using the Markov inequality in Appendix B.8) to relate the mistake bound to the training loss.

Lemma 2.

If represents the loss on the point , then

Combining the above results, we obtain a mistake bound of when using the squared-hinge loss on linearly separable data. We thus recover the standard results for the stochastic perceptron.

Note that for a finite amount of data (when the expectation is with respect to a discrete distribution), if we use batch accelerated gradient descent (which is not one of the stochastic gradient algorithms studied in this paper, and for which no growth condition is needed), we obtain a mistake bound that decreases as . This improves on existing mistake bounds that scale as  [26, 31]. Note that both sets of algorithms have the same dependence on the margin , but this deterministic accelerated method would require evaluating gradients on each iteration.

From Lemma B.8, we know that the squared-hinge loss satisfies the SGC with . Under the same conditions as above, this lemma along with the result of Theorem 2 gives us the following bound:

Using the result from Lemma 2, this results in a mistake bound of the order while only requiring one gradient per iteration. Hence, the use of acceleration leads to an improved novel dependence of , but requires the additional assumptions of Lemma B.8 and has a worse dependence on the margin .

8 Experiments

(a)
(b)
(c)
(d)
Figure 1: Comparison of SGD and variants of accelerated SGD on a synthetic linearly separable dataset with margin . Accelerated SGD with leads to faster convergence as compared to SGD with

. Accelerated SGD with a line-search to estimate

is stable and results in good performance.
(a) CovType
(b) Protein
Figure 2: Comparison of SGD and accelerated SGD for learning a linear classifier with RBF features on the (a) CovType and (b) Protein datasets. Accelerated SGD with line-search leads to better performance as compared to SGD with .

In this section, we empirically validate our theoretical results. For the first set of experiments (Figures 1(a)-1(d)), we generate a synthetic binary classification dataset with and the dimension . We ensure that the data is linearly separable with a margin , thus satisfying the interpolation property for training a linear classifier. We seek to minimize the finite-sum squared-hinge loss, . In Figure 1, we vary the margin and plot the logarithm of the loss with the number of effective passes (one pass is equal to iterations of SGD) over the data. In all of our experiments, we estimate the value of the smoothness parameter

as the maximum eigenvalue of the Gram matrix

.

We evaluate the performance of constant step-size SGD with and without acceleration. Since the squared-hinge loss satisfies the WGC with (Proposition 2), we use SGD with a constant step-size 111Note that using lead to consistently better results as compared to using as suggested by Theorem 5. (denoted as SGD(T) in the plots). For using Nesterov acceleration, we experimented with the dependence of the margin on the constant in the SGC. We found that setting results in consistently stable but fast convergence across different choices of . We thus use a step-size and set the tunable parameters in the update Equations 3-5 as specified by Theorem 2. We denote this variant of accelerated SGD as Acc-SGD(T) in the subsequent plots.

In addition, we propose a line-search heuristic to dynamically estimate the value of

. Our heuristic is inspired from the line-search used in SAG [24] and can be described as follows: we start with an initial estimate and in each iteration, we halve the estimate when the condition is not satisfied. We denote this variant as Acc-SGD(LS) in the plots.

In each of the Figures 1(a)-1(d), we make the following observations: (i) SGD(T) results in reasonably slow convergence. This observation is in line with other SGD methods using as the step-size [24]. (ii) Acc-SGD(T) with is consistently stable and as suggested by the theory, it results in faster convergence as compared to using SGD. (iii) Acc-SGD(LS) either matches or outperforms Acc-SGD(T). We plan to investigate better line-search methods for both SGD [24] and Acc-SGD [14] in the future. (iv) For larger values of (Figures 1(a)1(b)), the training loss becomes equal to zero, verifying the interpolation property.

The next set of experiments (Figure 2) considers binary classification on the CovType222http://osmot.cs.cornell.edu/kddcup and Protein333http://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets datasets. For this, we train a linear classifer using the radial basis (non-parametric) features. Non-parametric regression models of this form are capable of interpolating the data [15] and thus satisfies our assumptions. We subsample random points from the datasets and use the squared-hinge loss as above. Note that we do not have a good estimate of in this case and only compare the performance of SGD(T) and Acc-SGD(LS).

From Figures 2(a) and 2(b), we make the following observations: (i) In Figure 2(a), both variants have similar performance. (ii) In Figure 2(b), the Acc-SGD(LS) leads to considerably faster convergence as compared to SGD(T). (iii) Accelerated SGD in conjunction with our line-search heuristic is stable across datasets. These experiments show that in cases where the interpolation property is satisfied, both SGD and accelerated SGD with a constant step-size can result in good empirical performance.

9 Conclusion

In this paper, we showed that under interpolation, the stochastic gradients of common loss functions satisfy specific growth conditions. Under these conditions, we proved that it is possible for constant step-size SGD (with and without Nesterov acceleration) to achieve the convergence rates of the corresponding deterministic settings. These are the first results achieving optimal rates in the accelerated and non-convex settings under interpolation-like conditions. We used these results to demonstrate the fast convergence of the stochastic perceptron algorithm employing the squared-hinge loss. We showed that both SGD and accelerated SGD with a constant step-size can lead to good empirical performance when the interpolation property is satisfied. As opposed to determining the step-size and the schedule for annealing it for current SGD-like methods, our results imply that under interpolation, we only need to automatically determine the constant step-size for SGD. In the future, we hope to develop line-search techniques for automatically determining this step-size for both the accelerated and non-accelerated variants.

10 Acknowledgements

We acknowledge support from the European Research Council (grant SEQUOIA 724063) and the CIFAR program on Learning with Machines and Brains. We also thank Nicolas Flammarion for discussions related to this work.

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Appendix A Incorporating additive error for Nesterov acceleration

For this section, we assume an additive error in the the strong growth condition implying that the following equation is satisfied for all , .

In this case, we have the counterparts of Theorems 1 and 2 as follows:

Theorem 6 (Strongly convex).

Under -smoothness and strongly-convexity, if satisfies SGC with constant and an additive error , then SGD with Nesterov acceleration with the following choice of parameters,

results in the following convergence rate:

Theorem 7 (Convex).

Under -smoothness and convexity, if satisfies SGC with constant and an additive error , then SGD with Nesterov acceleration with the following choice of parameters,

results in the following convergence rate:

The above theorems are proved in appendices B.1.1 and B.1.3

Appendix B Proofs

b.1 Proofs for SGD with Nesterov Acceleration

Recall the update equations for SGD with Nesterov acceleration as follows:

Since the stochastic gradients are unbiased, we obtain the following equation,
(10)
For the proof, we consider the more general strong-growth condition with an additive error .
(11)
We choose the parameters , , , , such that the following equations are satisfied:
(12)
(13)
(14)
(15)
(16)

We now prove the following lemma assuming that the function is -smooth and strongly-convex.

Lemma 3.

Assume that the function is -smooth and strongly-convex and satisfies the strong-growth condition in Equation 11. Then, using the updates in Equation 3-5 and setting the parameters according to Equations 1216, if , then the following relation holds:

Proof.
Let , then using equation 5
Taking expecation wrt to ,
(By convexity of )
(From equation 4)
(By convexity)
By strong convexity,
(18)
By Lipschitz continuity,
Taking expectation wrt and using equations 1011
If ,
(19)
From equations 18 and 19,
(Since )
Since and ,
Multiplying by ,
Since , ,