 # Asymptotic Network Independence in Distributed Optimization for Machine Learning

We provide a discussion of several recent results which have overcome a key barrier in distributed optimization for machine learning. Our focus is the so-called network independence property, which is achieved whenever a distributed method executed over a network of n nodes achieves comparable performance to a centralized method with the same computational power as the entire network. We explain this property through an example involving of training ML models and sketch a short mathematical analysis.

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## 1 Introduction: Distributed Optimization and Its Limitations

First-order optimization methods, ranging from vanilla gradient descent to Nesterov acceleration and its many variants, have emerged over the past decade as the principal way to train Machine Learning (ML) models. There is a great need for techniques to train such models quickly and reliably in a distributed fashion over networks where the individual processors or GPUs may be scattered across the globe and communicate over an unreliable network which may suffer from message losses, delays, and asynchrony (see [1, 2, 3, 4, 5]).

Unfortunately, what often happens is that the gains from having many different processors running an optimization algorithm are squandered by the cost of coordination, shared memory, message losses and latency. This effect is especially pronounced when there are many processors and they are spread across geographically distributed data centers. As is widely recognized by the distributed systems community, “throwing” more processors at a problem will not, after a certain point, result in better performance.

This is typically reflected in the convergence time bounds obtained for distributed optimization in the literature. The problem formulation is that one must solve

 z∗=argminz∈Rdn∑i=1fi(z), (1)

over a network of nodes (see Figure 1 for an example). Figure 1: Example of a network. Two nodes are connected if there is an edge between them.

Only node has knowledge of the function , and the standard assumption is that, at every step when it is awake, node can compute the gradient of its own local function . These functions are assumed to be convex. The problem is to compute this minimum in a distributed manner over the network based on peer-to-peer communication, possible message losses, delays, and asynchrony.

This relatively simple formulation captures a large variety of learning problems. Suppose each agent stores training data points , where

are vectors of features and

are the associated responses (either discrete or continuous). We are interested to learn a predictive model , parameterized by parameters , so that for all . In other words, we are looking for a model that fits all the data throughout the network. This can be accomplished by empirical risk minimization

 θ∗=argminθ∈Rdn∑i=1ci(θ,Xi), (2)

where

 ci(θ,Xi)=∑(xj,yj)∈Xiℓ(h(xj;θ),yj)

measures how well the parameter fits the data at node , with

being a loss function measuring the difference between

and . Much of modern machine learning is built around such a formulation, including regression, classification, and regularized variants .

It is also possible that each agent does not have a static dataset, but instead collects streaming data points repetitively over time, where represents an unknown distribution of . In this case we can find through expected risk minimization

 θ∗=argminθ∈Rdn∑i=1fi(θ), (3)

where

 fi(θ)=E(xi,yi)∼Piℓ(g(xi;θ),yi).

This paper is concerned with the current limitations of distributed optimization and how to get past them. To illustrate our main concern, let us consider the distributed subgradient method in the simplest possible setting, namely the problem of computing the median of a collection of numbers in a distributed manner over a fixed graph. Each agent in the network holds value , and the global objective is to find the median of . This can be incorporated in the framework of (1) by choosing

 fi(z)=|z−mi|,∀i.

The distributed subgradient method (see [7, 8]) uses the subgradients of at any point , to have agent update as

 zi(k+1)=n∑j=1wij(zj(k)−αksj(zj(k))), (4)

where denotes the stepsize at iteration , and are the weights agent assigns to agent ’s solutions: two agents and are able to exchange information if and only if ( otherwise). The weights are assumed to be symmetric. For comparison, the centralized subgradient method updates the solution at iteration according to

 z(k+1)=z(k)−αk1nn∑j=1sj(z(k)). (5)

In Figure 2, we show the performance of Algorithm (4) as a function of the network size assuming the agents communicate over a ring network. As can be clearly seen, when the network size grows it takes a longer time for the algorithm to reach a certain performance threshold. Figure 2: Performance of Algorithm (4) as a function of the network size n. The agents communicate over a ring network (see Figure 3(b)) and choose the Metropolis weights (see Section 2.1 for the definition). Stepsizes αk=1√k, and mi are evenly distributed in [−10,10]. The time k to reach 1n∑ni=1|yi(k)|<ϵ is plotted, where yi(k)=1k∑k−1ℓ=0zi(ℓ) and ϵ=0.1.

Clearly, this is an undesirable property. Glancing at the figure, we see that distributing computation over nodes can result in a convergence time on the order of . Few practitioners will be enthusiastic about distributed optimization if the final effect is vastly increased convergence time.

One might hope that this phenomenon, demonstrated for the problem of median computation – considered here because it is arguably the simplest problem to which one can apply the subgradient method – will not hold for the more sophisticated optimization problems in the ML literature. Unfortunately, most work in distributed optimization replicates this undesirable phenomenon. We next give an extremely brief discussion of known convergence times in the distributed setting (for a much more extended discussion, we refer the reader to the recent survey ).

We would like to confine our discussion to the following point: most known convergence times in the distributed optimization literature imply bounds of the form

 Timen,ϵ(decentralized)≤p(n)Timen,ϵ(centralized), (6)

where denotes the time for the decentralized algorithm on nodes to reach accuracy (error ), and is the time for the centralized algorithm which can query gradients per time step to reach the same level of accuracy. The function can usually be bounded in terms of some polynomial in the number of nodes .

For instance, in the subgradient methods, Corollary 9 of  gives that

 Timen,ϵ(decentralized)=O(max{∥1n∑ni=1zi(0)−z∗∥2,G4h(n)}ϵ2), Timen,ϵ(centralized)=O(max{∥z(0)−z∗∥2,G4}ϵ2),

where

are initial estimates,

denotes the optimal solution and bounds the -norm of the subgradients. The function is the inverse of the spectral gap corresponding to the graph, and will typically grow with ; hence when is large, . In particular if the communication graphs are 1) path graphs, then ; 2) star graphs, then ; 3) geometric random graphs, then . in , but typically is at least .

By comparing and , we are keeping the computational power the same in both cases. Naturally, the centralized is always better: anything that can be done in a decentralized way could be done in a centralized way. The question, though, is how much better.

Framed in this way, the polynomial scaling in the quantity is extremely disconcerting. It is hard, for example, to argue that an algorithm should be run in a distributed manner with, say, if the quantity in Eq. (6) satisfies ; that would imply the distributed variant would be times slower than the centralized one with the same computational power.

Sometimes is written as the inverse spectral gap

in terms of the second-eigenvalue of some matrix. Because the second-smallest eigenvalue of an undirected graph Laplacian is approximately

away from zero, such bounds will translate into at least quadratic scalings with in the worst-case. Over time-varying -connected graphs, the best-known bounds on will be cubic in using the results of .

There are a number of caveats to the pessimistic argument outlined above. For example, in a multi-agent scenario where data sharing is not desirable or feasible, decentralized computation might be the only available option. Generally speaking, however, fast-growing will preclude the widespread applicability of distributed optimization. Indeed, returning to the back-of-the-envelope calculation above, if a user has to pay a multiplicative factor of 10,000 in convergence speed to use an algorithm, the most likely scenario is that the algorithm will not be used.

There is one scenario which avoids the pessimistic discussion above: when the underlying graph is an expander, the associated spectral gap is constant (see Chapter 6 of  for a definition of these terms as well as an explanation). In particular, on a random Erdos-Renyi random graph, the quantity

is constant with high probability (Corollary 9, part 9 in

). Unfortunately, this is a very special case which will not occur in geographically distributed systems. By way of comparison, a random graph where nodes are associated with random locations, with links between nodes close together, will not have constant spectral gap and will thus have that grows with (Corollary 9, part 10 of ). The Erdos-Renyi graph escapes this because, if we again associate nodes with locations, the average link in the E-R graph is a “long range” one connecting nodes that are geographically far apart. By contrast, graphs built on geographic nearest-neighbor communications will not have constant spectral gaps.

## 2 Asymptotic Network Independence in Distributed Stochastic Optimization

In this paper, we provide a discussion of several recent papers which have obtained that, for a number of settings, , as long as is large enough. In other words, asymptotically, the distributed algorithm performs as well as a centralized algorithm with the same computational power.

We call this property asymptotic network independence: it is as if the network is not even there. Asymptotic network independence provides an answer to the concerns raised in the previous section.

We begin by illustrating these results with a simulation from , shown in Figure 3

. Here the problem to be solved is classification with a smooth support vector machine between overlapping clusters of points. The performance of the centralized algorithm is shown in orange, and the performance of the decentralized algorithm is shown in dark blue. The graph is a ring of 50 nodes, and the problem being solved is the search for a support vector classifier. The light blue line shows the disagreement among nodes.

The graph illustrates the main result, which is that a network of 50 nodes performs as well in the limit as a centralized method with 50x the computational power of one node. Indeed, after iterations the orange and dark blue lines are almost indistinguishable. (a) A total number of 1000 data points and their labels for SVM classification. The data points are randomly generated around 50 cluster centers.

We mention that similar simulations are available for other machine learning methods (training neural networks, logistic regression, elastic net regression, etc.). The asymptotic network independence property enables us to efficiently distribute the training process for a variety of existing learning methods.

The name “asymptotic network independence” is a slight misnomer, as we actually do not care if the asymptotic performance depends in some complicated way on the network. All we want is that the decentralized performance can be bounded by times the performance of the centralized method.

These results were developed in the papers [12, 13, 14, 11, 15] in the setting of distributed optimization of strongly convex functions in the presence of noise. The very first paper  contained the basic idea. By assuming sufficiently small constant stepsize, it approximated the distributed stochastic gradient method by a stochastic differential equation in continuous time. The work showed that the distributed method outperforms a centralized scheme with synchronization overhead. However, it did not lead to straightforward algorithmic bounds. The paper 

gave the first crisp statement of the relationship between centralized and distributed methods by means of a central limit theorem. It considered a general stochastic approximation setting which is not limited to strongly convex optimization; the proof proceeded based on certain technical properties of stochastic approximation methods. In our recent work

, we generalized the results to graphs which are time-varying, with delays, message losses, and asynchrony. In a parallel recent work , a similar result was demonstrated with a further compression technique which allowed nodes to save on communication.

When the objective functions are not assumed to be convex, several recent works have obtained asymptotic network independence for distributed stochastic gradient descent. The work in  was the first to show that distributed algorithms could achieve a speedup like a centralized method when the number of computing steps is large enough. Such a result was generalized to the setting of directed communication networks in  for training deep neural networks, where the push-sum technique was combined with the standard distributed stochastic gradient scheme.

In the rest of this section, we will give a simple and readable explanation of the asymptotic network independence phenomenon in the context of distributed stochastic optimization over smooth and strongly convex objective functions. 111For more references on the topic of distributed stochastic optimization, the readers may refer to [17, 18, 19, 20, 21, 22, 23, 24, 25].

### 2.1 Setup

We are interested in minimizing Eq. (1) over a network of communicating agents. Regarding the objective functions we make the following standing assumption.

###### Assumption 1

Each is -strongly convex with -Lipschitz continuous gradients, i.e., for any ,

 ⟨∇fi(z)−∇fi(z′),z−z′⟩≥μ∥z−z′∥2,∥∇fi(z)−∇fi(z′)∥≤L∥z−z′∥. (7)

Under Assumption 1, Problem (1) has a unique optimal solution , and the function defined as

 f(z)=1nn∑i=1fi(z)

has the following contraction property (see  Lemma 10).

###### Lemma 1

For any and , we have

 ∥z−α∇f(z)−z∗∥≤(1−αμ)∥z−z∗∥.

In other words, gradient descent with a small stepsize reduces the distance between the current solution and .

In the stochastic optimization setting, we assume each agent is able to obtain noisy gradient estimates that satisfy the following condition.

###### Assumption 2

For all and , each random vector is independent, and

 Eξi[gi(z,ξi)∣z]=∇fi(z),Eξi[∥gi(z,ξi)−∇fi(z)∥2∣z]≤σ2,\ for some σ>0. (8)

This assumption is satisfied for many distributed learning problems. For instance, in empirical risk minimization (2), the gradient estimation of can introduce noise from various sources, such as approximation and discretization errors. For another example, when minimizing the expected risk in (3), where independent data points are gathered over time,

is an unbiased estimator of

satisfying Assumption 2.

The algorithm we discuss is the standard Distributed Stochastic Gradient Descent (DSGD) method. We let each agent in the network hold a local copy of the decision vector denoted by , and its value at iteration/time is written as . Denote for short. At each step , every agent performs the following update:

 zi(k+1)=n∑j=1wij(zj(k)−αkgj(k)), (9)

where is a sequence of nonnegative non-increasing stepsizes. The initial vectors are arbitrary for all , and is a mixing matrix.

DSGD belongs to the class of so-called consensus-based distributed optimization methods, where different agents mix their estimates at each iteration to reach a consensus of the solutions, i.e., for all and in the long run. To achieve consensus, the following condition is assumed on the mixing matrix and the communication topology among agents.

###### Assumption 3

The graph of agents is undirected and connected (there exists a path between any two agents). The mixing matrix is nonnegative, symmetric and doubly stochastic, i.e., and , where is the all one vector. In addition, for some .

Some examples of undirected connected graphs are presented in Figure 4 below.

Because of Assumption 3, the mixing matrix has an important contraction property.

###### Lemma 2

Let Assumption 3 hold, and let denote the eigenvalues of the matrix . Then, and

 ∥Wω−1¯¯¯ω∥≤λ∥ω−1¯¯¯ω∥

for all , where .

As a result, when running a consensus algorithm (which is just (9) without gradient descent)

 zi(k+1)=n∑j=1wijzj(k), (10)

the speed of reaching consensus is determined by . In particular, if we adopt the so-called lazy Metropolis rule for defining the weights, the dependency of on the network size is upper bounded by for some constant .

Lazy Metropolis rule for constructing :

Notation: denotes the degree (number of “neighbors”) of node . Correspondingly, is the set of “neighbors” for agent .

Despite the fact that may be very close to with large , the consensus algorithm (10) enjoys geometric convergence speed, i.e.,

 n∑i=1∥∥ ∥∥zi(k)−1nn∑j=1zj(k)∥∥ ∥∥2≤λkn∑i=1∥∥ ∥∥zi(0)−1nn∑j=1zj(0)∥∥ ∥∥2.

By contrast, the optimal rate of convergence for any stochastic gradient methods is sublinear, asymptotically (see ). This difference suggests that a consensus-based distributed algorithm for stochastic optimization may match the centralized methods in the long term: any errors due to consensus will decay at a fast-enough rate so that they ultimately do not matter.

In what follows, we discuss and compare the performance of the centralized stochastic gradient descent (SGD) method and DSGD. We will show that both methods asymptotically converge at the rate . Furthermore, the time needed for DSGD to approach the asymptotic convergence rate turns out to scale as .

### 2.2 Centralized Stochastic Gradient Descent (SGD)

The benchmark for evaluating the performance of DSGD is the centralized stochastic gradient descent (SGD) method, which we now describe. At each iteration , the following update is executed:

 z(k+1)=z(k)−αk¯g(k), (11)

where stepsizes satisfy

 αk=1μk,

and

 ¯g(k)=1nn∑i=1gi(z(k),ξi(k)),

i.e., is the average of noisy gradients evaluated at (by utilizing gradients at each iteration, we are keeping the computational power the same for SGD and DSGD). As a result, the gradient estimation is more accurate than using just one gradient. Indeed, from Assumption 2 we have

 E[∥¯g(k)−∇f(z(k))∥2]=1n2n∑i=1E[∥gi(z(k),ξi(k))−∇fi(z(k))∥2]≤σ2n. (12)

We measure the performance of SGD by , the expected squared distance between the solution at time and the optimal solution. Theorem 1 characterizes the convergence rate of , which is optimal for such stochastic gradient methods (see [27, 28]).

###### Theorem 1

Under SGD (11), supposing Assumptions 1-3 hold, we have

 R(k)≤σ2nμ2k+Ok(1k2). (13)

To compare with the analysis for DSGD later, we briefly describe how to obtain (13). Note that

 R(k+1)=E[∥z(k)−αk¯g(k)−z∗∥2]=E[∥z(k)−αk∇f(z(k))−z∗∥2]+α2kE[∥∇f(z(k))−¯g(k)∥2].

For large , in light of Lemma 1 and relation (12), we have the following inequality that relates to .

 R(k+1)≤(1−αkμ)2R(k)+α2kσ2n=(1−1k)2R(k)+σ2nμ21k2. (14)

A simple induction then gives Eq. (13).

### 2.3 Distributed Stochastic Gradient Descent (DSGD)

We assume the same stepsize policy for DSGD and SGD. To analyze DSGD starting from Eq. (9), define

 ¯¯¯z(k)=1nn∑i=1zi(k) (15)

as the average of all the iterates in the network. Differently from the analysis for SGD, we will be concerned with two error terms. The first term , called the expected optimization error, defines the expected squared distance between and , and the second term , called the expected consensus error, measures the dissimilarities of individual estimates among all the agents. Given any individual iterate , its squared distance to the optimum is bounded by . Hence exploring the two terms will provide us with insights into the performance of DSGD. To simplify notation, denote

 U(k)=E[∥¯¯¯z(k)−z∗∥2],V(k)=n∑i=1E[∥zi(k)−¯¯¯z(k)∥2],∀k. (16)

Inspired by the analysis for SGD, we first look for an inequality that bounds , which is analogous to in SGD. One such relation turns out to be :

 U(k+1)≤(1−1k)2U(k)+2L√nμ√U(k)V(k)k+L2nμ2V(k)k2+σ2nμ21k2. (17)

Comparing (17) to (14), we find two additional terms on the right-hand side of the inequality. Both terms involve the expected consensus error , thus reflecting the additional disturbances caused by the dissimilarities of solutions. Relation (17) also suggests that the convergence rate of can not be better than for SGD, which is expected. Nevertheless, if decays fast enough compared to , it is likely that the two additional terms are negligible in the long run, and we would guess that the convergence rate of is comparable to for SGD.

This indeed turns out to be the case, as it is shown in  that when , we have that

 U(k)≤σ2nμ2kO(1).

In other words, we have the network independence phenomenon: after a transient, DSGD performs comparably to a centralized stochastic gradient descent method with the same computational power (e.g., which can query the same number of gradients per step as the entire network).

### 2.4 Numerical Illustration

We provide a numerical example to illustrate the asymptotic network independence property of DSGD. Consider the on-lineRidge regression problem

 z∗=argminz∈Rdn∑i=1fi(z)(=Eui,vi[(u⊺iz−vi)2+ρ∥z∥2]), (18)

where is a penalty parameter. Each agent collects data points in the form of continuously over time with representing the features and being the observed outputs. Suppose each

is uniformly distributed, and

is drawn according to , where are predefined parameters uniformly situated in , and

are independent Gaussian random variables with mean

and variance

. Given a pair , agent can compute an estimated gradient of : , which is unbiased. Problem (18) has a unique solution given by

 z∗=(n∑i=1Eui[uiu⊺i]+nρI)−1n∑i=1Eui[uiu⊺i]~zi.

In the experiments, we consider two instances. In the first instance, we assume agents constitute a random network for DSGD, where every two agents are linked with probability . In the second instance, we let agents form a grid network. We use Metropolis weights in both instances. The problem dimension is set to and , the zero vector, for all . The penalty parameter is set to and the stepsizes . For both SGD and DSGD, we run the simulations times and average the results to approximate the expected errors.

The performance of SGD and DSGD is shown in Figure 5. We notice that in both instances the expected consensus error for DSGD converges to faster than the expected optimization error as predicted from our previous discussion. Regarding the expected optimization error, DSGD is slower than SGD in the first (resp., ) iterations for random network (resp., grid network). But after that, their performance is almost indistinguishable. The difference in the transient times is due to the stronger connectivity (or smaller ) of the random network compared to the grid network.

## 3 Conclusions

In this paper, we provided a discussion of recent results which have overcome a key barrier in distributed optimization methods for machine learning. These results established an asymptotic network independence property, that is, asymptotically, the distributed algorithm performs comparable to a centralized algorithm with the same computational power. We explain the property by examples of training ML models and provide a short mathematical analysis.

Along the line of achieving asymptotic network independence in distributed optimization, there are various future research directions, including considering nonconvex objective functions, reducing communication costs and transient time, and using exact gradient information. We next briefly discuss these.

First, distributed training of deep neural networks - the state-of-the-art machine learning approach in many application areas - involves minimizing nonconvex objective functions which are different from the main objectives considered in this paper. This area is largely unexplored with a few recent works in [16, 3, 4].

In distributed algorithms, the costs associated with communication among the agents are often non-negligible and may become the main burden for large networks. It is therefore important to explore communication reduction techniques that do not sacrifice the asymptotic network independence property. The recent papers [4, 14] have touched upon this point.

When considering asymptotic network independence for distributed optimization, an important factor is the transient time to reach the asymptotic convergence rate, as it may take a long time before the distributed implementation catches up with the corresponding centralized method. In fact, as we have shown in Section 2.1, this transient time can be a function of the network topology and grows with the network size. Reducing the transient time is thus a key future objective.

Finally, while several recent works have established the asymptotic network independence property in distributed optimization, they are mainly constrained to using stochastic gradient information. If the exact gradient is available, will distributed methods still be able to compete with the centralized ones? As we know, centralized algorithms typically enjoy a faster convergence speed with exact gradients. For example, plain gradient descent achieves linear convergence for strongly convex and smooth objective functions. With the exception of , which considered a restricted range of smoothness/strong convexity parameters, results on asymptotic network independence in this setting are currently lacking.

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