1 Introduction
With rapidly growing data volumes and variety, the need to scale up machine learning has sparked broad interests in developing efficient parallel optimization algorithms. A typical parallel optimization algorithm usually decomposes the original problem into multiple subproblems, each handled by a worker node. Each worker iteratively downloads the global model parameters and computes its local gradients to be sent to the master node or servers for model updates. Recently, asynchronous parallel optimization algorithms (Niu et al., 2011; Li et al., 2014b; Lian et al., 2015), exemplified by the Parameter Server architecture (Li et al., 2014a), have been widely deployed in industry to solve practical largescale machine learning problems. Asynchronous algorithms can largely reduce overhead and speedup training, since each worker may individually perform model updates in the system without synchronization. Another trend to deal with large volumes of data is the use of stochastic algorithms. As the number of training samples increases, the cost of updating the model taking into account all error gradients becomes prohibitive. To tackle this issue, stochastic algorithms make it possible to update using only a small subset of all training samples at a time.
Stochastic gradient descent (SGD) is one of the first algorithms widely implemented in an asynchronous parallel fashion; its convergence rates and speedup properties have been analyzed for both convex (Agarwal and Duchi, 2011; Mania et al., 2017) and nonconvex (Lian et al., 2015) optimization problems. Nevertheless, SGD is mainly applicable to the case of smooth optimization, and yet is not suitable for problems with a nonsmooth term in the objective function, e.g., an norm regularizer. In fact, such nonsmooth regularizers are commonplace in many practical machine learning problems or constrained optimization problems. In these cases, SGD becomes ineffective, as it is hard to obtain gradients for a nonsmooth objective function.
We consider the following nonconvex regularized optimization problem:
(1) 
where takes a finitesum form of , and each is a smooth (but not necessarily convex) function. The second term is a convex (but not necessarily smooth
) function. This type of problems is prevalent in machine learning, as exemplified by deep learning with regularization
(Dean et al., 2012; Chen et al., 2015; Zhang et al., 2015), LASSO (Tibshirani et al., 2005), sparse logistic regression
(Liu et al., 2009), robust matrix completion (Xu et al., 2010; Sun and Luo, 2015), and sparse support vector machine (SVM)
(Friedman et al., 2001). In these problems,is a loss function of model parameters
, possibly in a nonconvex form (e.g., in neural networks), while
is a convex regularization term, which is, however, possibly nonsmooth, e.g., the norm regularizer.Many classical deterministic (nonstochastic) algorithms are available to solve problem (1), including the proximal gradient (ProxGD) method (Parikh et al., 2014) and its accelerated variants (Li and Lin, 2015) as well as the alternating direction method of multipliers (ADMM) (Hong et al., 2016). These methods leverage the socalled proximal operators (Parikh et al., 2014) to handle the nonsmoothness in the problem. Although implementing these deterministic algorithms in a synchronous parallel fashion is straightforward, extending them to asynchronous parallel algorithms is much more complicated than it appears. In fact, existing theory on the convergence of asynchronous proximal gradient (PG) methods for nonconvex problem (1) is quite limited. An asynchronous parallel proximal gradient method has been presented in (Li et al., 2014b) and has been shown to converge to stationary points for nonconvex problems. However, (Li et al., 2014b) has essentially proposed a nonstochastic algorithm and has not provided its convergence rate.
In this paper, we propose and analyze an asynchronous parallel proximal stochastic gradient descent (ProxSGD) method for solving the nonconvex and nonsmooth problem (1), with provable convergence and speedup guarantees. The analysis of ProxSGD has attracted much attention in the community recently. Under the assumption of an increasing minibatch size used in the stochastic algorithm, the nonasymptotic convergence of ProxSGD to stationary points has been shown in (Ghadimi et al., 2016) for problem (1) with a convergence rate of , being the times the model is updated. Moreover, additional variance reduction techniques have been introduced (Reddi et al., 2016) to guarantee the convergence of ProxSGD, which is different from the stochastic method we discuss here. The stochastic algorithm considered in this paper assumes that each worker selects a minibatch of randomly chosen training samples to calculate the gradients at a time, which is a scheme widely used in practice. To the best of our knowledge, the convergence behavior of ProxSGD—under a constant minibatch size without variance reduction—is still unknown (even for the synchronous or sequential version).
Our main contributions are summarized as follows:

We propose asynchronous parallel ProxSGD (a.k.a. AsynProxSGD) and prove that it can converge to stationary points of nonconvex and nonsmooth problem (1) with an ergodic convergence rate of , where is the number of times that the model is updated. This rate matches the convergence rate known for asynchronous SGD. The latter, however, is suitable only for smooth problems. To our knowledge, this is the first work that offers convergence rate guarantees for any stochastic proximal methods in an asynchronous parallel setting.

Our result also suggests that the sequential (or synchronous parallel) ProxSGD can converge to stationary points of problem (1), with a convergence rate of . To the best of our knowledge, this is also the first work that provides convergence rates of any stochastic algorithm for nonsmooth problem (1) under a constant batch size, while prior literature on such stochastic proximal methods assumes an increasing batch size or relies on variance reduction techniques.

We provide a linear speedup guarantee as the number of workers increases, provided that the number of workers is bounded by . This result has laid down a theoretical ground for the scalability and performance of our AsynProxSGD algorithm in practice.
2 Preliminaries
In this paper, we use as the one defined in (1), and
as a function whose stochastic nature comes from the random variable
representing a random index selected from the training set . We use to denote the norm of the vector , and to denote the inner product of two vectors and . We use to denote the “true” gradient and use to denote the stochastic gradient for a function . For a random variable or vector , let be the conditional expectation of w.r.t. a sigma algebra . We denote as the subdifferential of . A point is a critical point of , iff .2.1 Stochastic Optimization Problems
In this paper, we consider the following stochastic optimization problem instead of the original deterministic version (1):
(2) 
where the stochastic nature comes from the random variable , which in our problem settings, represents a random index selected from the training set . Therefore, (2) attempts to minimize the expected loss of a random training sample plus a regularizer . In this work, we assume the function is proper, closed and convex, yet not necessarily smooth.
2.2 Proximal Gradient Descent
The proximal operator is fundamental to many algorithms to solve problem (1) as well as its stochastic variant (2).
Definition 1 (Proximal operator).
The proximal operator of a point under a proper and closed function with parameter is defined as:
(3) 
In its vanilla version, proximal gradient descent performs the following iterative updates:
for , where is the step size at iteration .
To solve stochastic optimization problem (2), we need a variant called proximal stochastic gradient descent (ProxSGD), with its update rule at each (synchronized) iteration given by
(4) 
where is the minibatch size. In ProxSGD, the aggregate gradient over all the samples is replaced by the gradients from a random subset of training samples, denoted by at iteration . Since is a random variable indicating a random index in , is a random loss function for the random sample , such that .
2.3 Parallel Stochastic Optimization
Recent years have witnessed rapid development of parallel and distributed computation frameworks for largescale machine learning problems. One popular architecture is called parameter server (Dean et al., 2012; Li et al., 2014a), which consists of some worker nodes and server nodes. In this architecture, one or multiple master machines play the role of parameter servers, which maintain the model . Since these machines serve the same purpose, we can simply treat them as one server node for brevity. All other machines are worker nodes that communicate with the server for training machine learning models. In particular, each worker has two types of requests: pull the current model from the server, and push the computed gradients to the server.
Before proposing an asynchronous Proximal SGD algorithm in the next section, let us first introduce its synchronous version. Let us use an example to illustrate the idea. Suppose we execute ProxSGD with a minibatch of 128 random samples on 8 workers. We can let each worker randomly take 16 samples, and compute a summed gradient on these 16 samples, and push it to the server. In the synchronous case, the server will finally receive 8 summed gradients (containing information of all 128 samples) in each iteration. The server then updates the model by performing the proximal gradient descent step. In general, if we have workers, each worker will be assigned random samples in an iteration.
Note that in this scenario, all workers contribute to the computation of the sum of gradients on random samples in parallel, which corresponds to data parallelism in the literature (e.g., (Agarwal and Duchi, 2011; Ho et al., 2013)). Another type of parallelism is called model parallelism, in which each worker uses all random samples in the batch to compute a partial gradient on a specific block of (e.g., (Niu et al., 2011; Pan et al., 2016)). Typically, data parallelism is more suitable when , i.e., large dataset with moderate model size, and model parallelism is more suitable when . We focus on data parallelism.
3 Asynchronous Proximal Gradient Descent
We now present our asynchronous proximal gradient descent (AsynProxSGD) algorithm, which is the main contribution in this paper. In the asynchronous algorithm, different workers may be in different local iterations due to random delays in computation and communication.
For ease of presentation, let us first assume each worker uses only one random sample at a time to compute its stochastic gradient, which naturally generalizes to using a minibatch of random samples to compute a stochastic gradient. In this case, each worker will independently and asynchronously repeat the following steps:

Pull the latest model from the server;

Calculate a gradient based on a random sample locally;

Push the gradient to the server.
Here we use to emphasize that the gradient computed on workers may be delayed. For example, all workers but worker have completed their tasks of iteration , while worker still works on iteration . In this case, the gradient is not computed based on the current model but from a delayed one .
In our algorithm, the server will perform an averaging over the received sample gradients as long as gradients are received and perform an proximal gradient descent update on the model , no matter where these gradients come from; as long as gradients are received, the averaging is performed. This means that it is possible that the server may have received multiple gradients from one worker while not receiving any from another worker.
In general, when each minibatch has samples, and each worker processes random samples to calculate a stochastic gradient to be pushed to the server, the proposed AsynProxSGD algorithm is described in Algorithm 1 leveraging a parameter server architecture. The server maintains a counter . Once reaches , the server has received gradients that contain information about random samples (no matter where they come from) and will perform a proximal model update.
4 Convergence Analysis
To facilitate the analysis of Algorithm 1, we rewrite it in an equivalent global view (from the server’s perspective), as described in Algorithm 2. In this algorithm, we use an iteration counter to keep track of how many times the model has been updated on the server; increments every time a push request (model update request) is completed. Note that such a counter is not required by workers to compute gradients and is different from the counter in Algorithm 1— is maintained by each worker to count how many sample gradients have been computed locally.
In particular, for every stochastic sample gradients received, the server simply aggregates them by averaging:
(5) 
where indicates that the stochastic gradient received at iteration could have been computed based on an older model due to communication delay and asynchrony among workers. Then, the server updates to using proximal gradient descent.
4.1 Assumptions and Metrics
We make the following assumptions for convergence analysis. We assume that is a smooth function with the following properties:
Assumption 1 (Lipschitz Gradient).
For function there are Lipschitz constants such that
(6) 
As discussed above, assume that is a proper, closed and convex function, which is yet not necessarily smooth. If the algorithm has been executed for iterations, we let denote the set that consists of all the samples used up to iteration . Since for all , the collection of all such forms a filtration
. Under such settings, we can restrict our attention to those stochastic gradients with an unbiased estimate and bounded variance, which are common in the analysis of
stochastic gradient descent or stochastic proximal gradient algorithms, e.g., (Lian et al., 2015; Ghadimi et al., 2016).Assumption 2 (Unbiased gradient).
For any , we have .
Assumption 3 (Bounded variance).
The variance of the stochastic gradient is bounded by .
We make the following assumptions on the delay and independence:
Assumption 4 (Bounded delay).
All delay variables are bounded by : .
Assumption 5 (Independence).
All random variables for all and in Algorithm 2 are mutually independent.
The assumption of bounded delay is to guarantee that gradients from workers should not be too old. Note that the maximum delay is roughly proportional to the number of workers in practice. This is also known as stale synchronous parallel (Ho et al., 2013) in the literature. Another assumption on independence can be met by selecting samples with replacement, which can be implemented using some distributed file systems like HDFS (Borthakur et al., 2008). These two assumptions are common in convergence analysis for asynchronous parallel algorithms, e.g., (Lian et al., 2015; Davis et al., 2016).
4.2 Theoretical Results
We present our main convergence theorem as follows:
Theorem 1.
Taking a closer look at Theorem 1, we can properly choose the learning rate as a constant value and derive the following convergence rate:
Corollary 1.
Let the step length be a constant, i.e.,
(9) 
If the delay bound satisfies
(10) 
then the output of Algorithm 1 satisfies the following ergodic convergence rate:
(11) 
Remark 1 (Consistency with ProxSGD).
Remark 2 (Linear speedup w.r.t. the staleness).
From (11) we can see that linear speedup is achievable, as long as the delay is bounded by (if other parameters are constants). The reason is that by (10) and (11), as long as is no more than , the iteration complexity (from a global perspective) to achieve optimality is , which is independent from .
Remark 3 (Linear speedup w.r.t. number of workers).
As the iteration complexity is to achieve optimality, it is also independent from the number of workers if assuming other parameters are constants. It is worth noting that the delay bound is roughly proportional to the number of workers. As the iteration complexity is independent from , we can conclude that the total iterations will be shortened to of a single worker’s iterations if workers work in parallel, achieving nearly linear speedup.
Remark 4 (Comparison with AsynSGD).
Compared with asynchronous SGD (Lian et al., 2015), in which or the number of workers should be bounded by to achieve linear speedup, here AsynProxSGD is more sensitive to delays and more suitable for a smaller cluster.
5 Experiments
We now present experimental results to confirm the capability and efficiency of our proposed algorithm to solve challenging nonconvex nonsmooth machine learning problems. We implemented our algorithm on TensorFlow
(Abadi et al., 2016), a flexible and efficient deep learning library. We execute our algorithm on Ray (Moritz et al., 2017), a generalpurpose framework that enables parallel and distributed execution of Python as well as TensorFlow functions. A key feature of Ray is that it provides a unified taskparallel abstraction, which can serve as workers, and actor abstraction, which stores some states and acts like parameter servers.We use a cluster of 9 instances on Google Cloud. Each instance has one CPU core with 3.75 GB RAM, running 64bit Ubuntu 16.04 LTS. Each server or worker uses only one core, with 9 CPU cores and 60 GB RAM used in total. Only one instance is the server node, while the other nodes are workers.
Setup: In our experiments, we consider the problem of nonnegative principle component analysis (NNPCA) (Reddi et al., 2016). Given a set of samples , NNPCA solves the following optimization problem
(12) 
This NNPCA problem is NPhard in general. To apply our algorithm, we can rewrite it with for all samples . Since the feasible set is convex, we can replace the optimization constraint by a regularizer in the form of an indicator function , such that if and otherwise.
datasets  dimension  sample size 

a9a  123  32,561 
mnist  780  60,000 
The hyperparameters are set as follows. The step size is set using the popular inverse step size choice , which is the same as the one used in (Reddi et al., 2016). Here determine how learning rates change, and controls for how many steps the learning rate would change.
We conduct experiments on two datasets ^{1}^{1}1Available at http://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/, with their information summarized in Table 1. All samples have been normalized, i.e., for all . In our experiments, we use a batch size of in order to evaluate the performance and speedup behavior of the algorithm under constant batches.
We consider the function suboptimality value as our performance metric. In particular, we run proximal gradient descent (ProxGD) for a large number of iterations with multiple random initializations, and obtain a solution . For all experiments, we evaluate function suboptimality, which is the gap , against the number of sample gradients processed by the server (divided by the total number of samples ), and then against time.
Results: Empirically, Assumption 4 (bounded delays) is observed to hold for this cluster. For our proposed AsynProxSGD algorithm, we are particularly interested in the speedup in terms of iterations and running time. In particular, if we need iterations (with sample gradients processed by the server) to achieve a certain suboptimality level using one worker, and iterations (with sample gradients processed by the server) to achieve the same suboptimality with workers, the iteration speedup is defined as (Lian et al., 2015). Note that all iterations are counted on the server side, i.e., how many sample gradients are processed by the server. On the other hand, the running time speedup is defined as the ratio between the running time of using one worker and that of using workers to achieve the same suboptimality.
The iteration and running time speedups on both datasets are shown in Fig. 1 and Fig. 2, respectively. Such speedups achieved at the suboptimality level of are presented in Table 2 and 3. We observe that nearly linear speedup can be achieved, although there is a loss of efficiency due to communication as the number workers increases.
Workers  1  2  4  8 

Iteration Speedup  1.000  1.982  3.584  5.973 
Time Speedup  1.000  2.219  3.857  5.876 
Workers  1  2  4  8 

Iteration Speedup  1.000  2.031  3.783  7.352 
Time Speedup  1.000  2.285  4.103  5.714 
6 Related Work
Stochastic optimization problems have been studied since the seminal work in 1951 (Robbins and Monro, 1951), in which a classical stochastic approximation algorithm is proposed for solving a class of strongly convex problems. Since then, a series of studies on stochastic programming have focused on convex problems using SGD (Bottou, 1991; Nemirovskii and Yudin, 1983; Moulines and Bach, 2011). The convergence rates of SGD for convex and strongly convex problems are known to be and , respectively. For nonconvex optimization problems using SGD, Ghadimi and Lan (Ghadimi and Lan, 2013) proved an ergodic convergence rate of , which is consistent with the convergence rate of SGD for convex problems.
When in (1) is not necessarily smooth, there are other methods to handle the nonsmoothness. One approach is closely related to mirror descent stochastic approximation, e.g., (Nemirovski et al., 2009; Lan, 2012). Another approach is based on proximal operators (Parikh et al., 2014), and is often referred to as the proximal stochastic gradient descent (ProxSGD) method. Duchi et al. (Duchi and Singer, 2009) prove that under a diminishing learning rate for strongly convex objective functions, ProxSGD can achieve a convergence rate of . For a nonconvex problem like (1), rather limited studies on ProxSGD exist so far. The closest approach to the one we consider here is (Ghadimi et al., 2016), in which the convergence analysis is based on the assumption of an increasing minibatch size. Furthermore, Reddi et al. (Reddi et al., 2016) prove convergence for nonconvex problems under a constant minibatch size, yet relying on additional mechanisms for variance reduction. We fill the gap in the literature by providing convergence rates for ProxSGD under constant batch sizes without variance reduction.
To deal with big data, asynchronous parallel optimization algorithms have been heavily studied. Recent work on asynchronous parallelism is mainly limited to the following categories: stochastic gradient descent for smooth optimization, e.g., (Niu et al., 2011; Agarwal and Duchi, 2011; Lian et al., 2015; Pan et al., 2016; Mania et al., 2017) and deterministic ADMM, e.g. (Zhang and Kwok, 2014; Hong, 2017). A nonstochastic asynchronous ProxSGD algorithm is presented by (Li et al., 2014b), which however did not provide convergence rates for nonconvex problems.
7 Concluding Remarks
In this paper, we study asynchronous parallel implementations of stochastic proximal gradient methods for solving nonconvex optimization problems, with convex yet possibly nonsmooth regularization. However, compared to asynchronous parallel stochastic gradient descent (AsynSGD), which is targeting smooth optimization, the understanding of the convergence and speedup behavior of stochastic algorithms for the nonsmooth regularized optimization problems is quite limited, especially when the objective function is nonconvex. To fill this gap, we propose an asynchronous proximal stochastic gradient descent (AsynProxSGD) algorithm with convergence rates provided for nonconvex problems. Our theoretical analysis suggests that the same order of convergence rate can be achieved for asynchronous ProxSGD for nonsmooth problems as for the asynchronous SGD, under constant minibatch sizes, without making additional assumptions on variance reduction. And a linear speedup is proven to be achievable for both asynchronous ProxSGD when the number of workers is bounded by .
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Appendix A Auxiliary Lemmas
Lemma 1 ([Ghadimi et al., 2016]).
For all , we have:
(13) 
Due to slightly different notations and definitions in [Ghadimi et al., 2016], we provide a proof here for completeness. We refer readers to [Ghadimi et al., 2016] for more details.
Proof.
By the definition of proximal function, there exists a such that:
which proves the lemma.
Lemma 2 ([Ghadimi et al., 2016]).
For all , if is a convex function, we have
(14) 
Proof.
Lemma 3 ([Ghadimi et al., 2016]).
For any and , we have
(21) 
Proof.
It can be obtained by directly applying Lemma 2 and the definition of gradient mapping.
Lemma 4 ([Reddi et al., 2016]).
Suppose we define for some . Then for , the following inequality holds:
(22) 
for all .
We recall and define some notations for convergence analysis in the subsequent. We denote as the average of delayed stochastic gradients and as the average of delayed true gradients, respectively:
Moreover, we denote as the difference between these two differences.
Appendix B Convergence analysis for AsynProxSGD
b.1 Milestone lemmas
We put some key results of convergence analysis as milestone lemmas listed below, and the detailed proof is listed in B.4.
Lemma 5 (Decent Lemma).
(23) 
Lemma 6.
Lemma 7.
Suppose we have a sequence by Algorithm 2, , then we have:
(25) 
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