Second-order Properties of Noisy Distributed Gradient Descent

We consider the optimization problem

where each $f_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is smooth but possibly non-convex, and $\mathbf{x}\in\mathbb{R}^{n}$ is the decision vector. The aim is to employ $m$ agents to iteratively solve the optimization problem in (1), over an undirected and connected network graph $\mathcal{G}(\mathcal{V},\mathcal{E})$. Each agent $i\in\mathcal{V}:=\{1,\ldots,m\}$ only knows the corresponding function $f_{i}$ and its gradient. The pair of agents $(i,j)\in\mathcal{V}\times\mathcal{V}$ is able to directly exchange information if and only if $(i,j)\in\mathcal{E}$. Collaborative distributed optimization over a network is of significant interest in the contexts of control, learning and estimation, particularly in large-scale system scenarios, such as unmanned vehicle systems [1], electric power systems [2], transit systems [3], and wireless sensor networks [4].

In the field of optimization, two primary classes of distributed methods can be identified: dual decomposition methods and consensus-based methods. Dual decomposition methods involve minimizing an augmented Lagrangian formulated on the basis of constraints that enforce agreement between agents, via iterative updates of the corresponding primal and dual variables [5]. The distributed dual decomposition algorithm in [6] involves agents alternating between updating their primal and dual variables and communicating with their neighbors. In [7], it is established that distributed alternating direction method of multipliers (ADMM) exhibits linear convergence rates in strongly convex settings. Consensus-based methods can be traced back to the distributed computation models proposed in [8], which seek to eliminate agent disagreements through local iterate exchange and weighted averaging to achieve consensus. This idea underlies the distributed (sub)gradient methods proposed in [9] and [10] to solve problem (1) with all $f_{i}$ convex. In the case of diminishing step-size, each agent converges to an optimizer [10]; with constant step-size, convergence is typically faster, but only to the vicinity of an optimizer [9].

The focus in this paper is on consensus-based distributed (sub)gradient methods, for the simplicity as first-order methods, enabling easy adaptation to diverse situations. In particular, a fixed step-size Distributed Gradient Descent (DGD) algorithm is considered, as an instance of the distributed (sub)gradient methods. In unperturbed form, the update for each agent $i\in\mathcal{V}$ at iteration $k$ is given by

where $\alpha>0$ is the constant step-size, $\nabla f_{i}$ is the gradient of $f_{i}$, $\hat{\mathbf{x}}_{i}\in\mathbb{R}^{n}$ is the local copy of the decision vector $\mathbf{x}$ at agent $i\in\mathcal{V}$, and $\mathbf{W}_{ij}$ is the scalar entry in the $i$-th row and $j$-th column of a given mixing matrix $\mathbf{W}\in\mathbb{R}^{m\times m}$. The mixing matrix is consistent with the graph $\mathcal{G}(\mathcal{V},\mathcal{E})$, in the sense that $\mathbf{W}_{ii}>0$ for all $i\in\mathcal{V}$, $\mathbf{W}_{ij}>0$ if $(i,j)\in\mathcal{E}$, and $\mathbf{W}_{ij}=0$ otherwise. The convergence rates of fixed step-size and diminishing step-size DGD algorithms in strongly convex settings are examined in [11] and [12], respectively. Several variants of DGD have been proposed in (strongly) convex settings. Nesterov momentum is used in the distributed gradient descent update with diminishing step-sizes in [13] to improve convergence rates. Inexact proximal-gradient method is considered in [14] for problems involving non-smooth functions. To reach exact consensus with a constant step-size, the gradient tracking method [15, 16] and [17] is used to neutralize $-\alpha\nabla f_{i}$ in (2), as the gradient descent part cannot vanish itself.

In non-convex settings, gradient descent methods may face issues due to saddle points, as converging to a first-order stationary point does not guarantee the local minimality. While Hessian-based methods can avoid saddle points, their computational cost can be prohibitive for large-scale problems. In [18], the fixed step-size DGD algorithm is shown to retain the property of convergence to a neighborhood of a consensus stationary solution under some regularity assumptions in non-convex settings. It is also shown in [19] that DGD with a constant step-size converges almost surely to a neighborhood of second-order stationary solutions. However, this requires random initialization to avoid the zero measure manifold of saddle point attraction, and moreover, the underlying analysis does not support techniques for actively escaping saddle points.

Recently, it has been shown that standard (centralized) gradient descent methods can take exponential time to escape saddle points [20], while noise (random perturbations) have been proven effective for escaping saddle points in non-convex optimization. The Noisy Gradient Descent algorithm in [21] and [22] is proven to be able to escape saddle points efficiently while converging to a neighborhood of a local minimizer with high probability. However, all of these works are limited to centralized methods. In [23] it is shown that distributed stochastic gradient descent converges to local minima almost surely when diminishing step sizes are used. Consider constant step-size, the diffusion strategy with stochastic gradient in [24] and [25] only returns approximately second-order stationary points rather than an outcome that lies in a neighborhood of a local minimizer with controllable size.

In this paper, the main contribution is that we analyze a fixed step-size noisy distributed gradient descent (NDGD) algorithm for solving the optimization problem in (1) by expanding upon and combining ideas from [21] and [22] on centralized stochastic gradient descent, and from [11, 18, 19] on distributed unperturbed gradient descent. In this combination, random perturbations are added to the gradient descent directions at each step to actively evade saddle points. It is established that under certain regularity conditions, and with a suitable step-size, each agent converges to a neighborhood of a local minimizer. In particular, we determine a probabilistic upper bound for the distance between the iterate at each agent and the set of local minimizers after a sufficient number of iterations. A numerical example is presented to illustrate the effectiveness of the algorithm in terms of escaping from the vicinity of a saddle point in fewer iterations than the standard (i.e., unperturbed) fixed step-size DGD method.

In this section, we present a reformulation of the optimization problem defined in (1) and provide a list of assumptions used in subsequent analysis. Then, we briefly recall aspects of the fixed step-size DGD algorithm (2) and present some existing results for non-convex optimization problems. Next, some intermediate results are derived to establish certain properties of the local minimizers of $f$ defined in (1) (see Theorem 2.1), on the basis of a collection of supporting lemmas (see Lemma 2.3.1-2.3.4).

In this section, a Noisy Distributed Gradient Descent algorithm (see Algorithm 1), a variant of the fixed step-size DGD algorithm, is formulated. The main analysis result, also formulated in this section, establishes the second-order properties of the NDGD algorithm. The key idea is to add random noise to the distributed gradient descent directions at each iteration. The required properties of the noise $\xi_{i}^{k}$ in Algorithm 1 is presented in Theorem 3.1.

Recall that $\mathcal{X}^{*}_{f}$ and $\hat{\mathcal{X}}^{*}_{Q_{\alpha}}$ denote the set of local minimizers of $f$ and $Q_{\alpha}$ respectively, as per (9). Given $\epsilon>0$, $\gamma>0$, $\mu>0$, $\delta>0$, and $\alpha>0$, define

where $\hat{\mathcal{X}}^{\prime}:=\{\hat{\mathbf{x}}:\|\nabla F(\hat{\mathbf{x}})+\alpha^{-1}(\mathbf{I}_{mn}-\hat{\mathbf{W}})\hat{\mathbf{x}}\|=0,~{}\lambda_{\min}(\nabla^{2}F(\hat{\mathbf{x}}))>0\}$. The next theorem, which establishes second-order properties of the NDGD algorithm, is the main result of the paper; a proof is given in Section IV. Note that this result focuses on the dependency on the given step-size $\alpha$ and confidence parameter $\zeta$, hiding the factors that have polynomial dependence on all other parameters (including $\Delta_{1}$, $\epsilon$, $\gamma$, $\mu$, $\delta$ and $\sigma$ ).

Second-order guarantees of DGD have been studied in [19] and [23] based on the almost sure non-convergence to saddle points with random initialization. In this paper, we propose to use random perturbations to actively evade saddle points. The second-order guarantees of NDGD stated in Theorem 3.1 do not require any additional initialization conditions. Second-order guarantees of the stochastic variant of DGD have been studied in [24] and [25], although they only show the convergence to an approximate second-order stationary point. Here, an upper bound is given for the distance between the iterate at each agent and the set of local minimizers after a sufficient number of iterations.

A proof of Theorem 3.1 is provided in Section IV. First, we consider three different cases to study the behavior of NDGD for three different cases, in line with the development of the related result in [21] for centralized gradient descent. i) large in norm $\nabla Q_{\alpha}(\hat{\mathbf{x}}^{k})$ (see Lemma 4.1.1); ii) sufficiently negative $\lambda_{\min}(\nabla^{2}Q_{\alpha}(\hat{\mathbf{x}}^{k}))$ (see Lemma 4.1.2); and iii) $\hat{\mathbf{x}}^{k}$ in a neighborhood of the local minimizers of $Q_{\alpha}$ with local strong convexity (see Lemma 4.1.3). Combining the outcome of this with Theorem 2.1, we then prove that with probability at least $1-\zeta$, after $K(\alpha,\zeta)$ iterations, NDGD yields a point of which the component at each agent is $\Delta_{1}+\Delta_{2}(\alpha,\zeta)$-close to some local minimizer of $f$.