| Algorithm | Oracle Complexity | MSS | Remarks |
| DSGD (Lian et al., 2017) | O\left(\max\left\{\frac{1}{n\epsilon^4}, \frac{\lambda^2n}{(1-\lambda)^2\epsilon^2}\right\}\right) | X | bounded heterogeneity |
| D2 (Tang et al., 2018) | O\left(\max\left\{\frac{1}{n\epsilon^4}, \frac{n}{(1-\lambda)^b\epsilon^2}\right\}\right) | X | b∈R+ is not explicitly shown in (Tang et al., 2018) |
| GT-DSGD (Xin et al., 2020e) | O\left(\max\left\{\frac{1}{n\epsilon^4}, \frac{\lambda^2n}{(1-\lambda)^3\epsilon^2}\right\}\right) | X | |
| D-PD-SGD (Yi et al., 2020) | O\left(\max\left\{\frac{1}{n\epsilon^4}, \frac{n}{(1-\lambda)^c\epsilon^2}\right\}\right) | X | c∈R+ is not explicitly shown in (Yi et al., 2020) |
| D-GET (Sun et al., 2020) | O\left(\frac{1}{(1-\lambda)^d\epsilon^3}\right) | ✓ | d∈R+ is not explicitly shown in (Sun et al., 2020) |
| D-SPIDER-SFO (Pan et al., 2020) | O\left(\frac{1}{(1-\lambda)^h\epsilon^3}\right) | ✓ | h∈R+ is not explicitly shown in (Pan et al., 2020) |
| GT-HSGD (this work) | O\left(\max\left\{\frac{1}{n\epsilon^3}, \frac{\lambda^4}{(1-\lambda)^3\epsilon^2}, \frac{\lambda^{1.5}n^{0.5}}{(1-\lambda)^{2.25}\epsilon^{1.5}}\right\}\right) | ✓ | |
+
+We adopt the following notations throughout the paper. We use lowercase bold letters to denote vectors and uppercase bold letters to denote matrices. The ceiling function is denoted as $\lceil \cdot \rceil$ . The matrix $\mathbf{I}_d$ represents the $d\times d$ identity; $\mathbf{1}_d$ and $\mathbf{0}_d$ are the $d$ -dimensional column vectors of all ones and zeros, respectively. We denote $[\mathbf{x}]_i$ as the $i$ -th entry of a vector $\mathbf{x}$ . The Kronecker product of two matrices $\mathbf{A}$ and $\mathbf{B}$ is denoted by $\mathbf{A}\otimes \mathbf{B}$ . We use $\| \cdot \|$ to denote the Euclidean norm of a vector or the spectral norm of a matrix. We use $\sigma (\cdot)$ to denote the $\sigma$ -algebra generated by the sets and/or random vectors in its argument.
+
+# 2. Problem Setup and GT-HSGD
+
+In this section, we introduce the mathematical model of the stochastic first-order oracle (SFO) at each node and the communication network. Based on these formulations, we develop the proposed GT-HSGD algorithm.
+
+# 2.1. Optimization and Network Model
+
+We work with a rich enough probability space $\{\Omega, \mathbb{P}, \mathcal{F}\}$ . We consider decentralized recursive algorithms of interest that generate a sequence of estimates $\{\mathbf{x}_t^i\}_{t \geq 0}$ of the first-order stationary points of $F$ at each node $i$ , where $\mathbf{x}_0^i$ is assumed constant. At each iteration $t$ , each node $i$ observes a random vector $\pmb{\xi}_t^i$ in $\mathbb{R}^q$ , which, for instance, may be considered as noise or as an online data sample. We then introduce the natural filtration (an increasing family of sub
+
+$\sigma$ -algebras of $\mathcal{F}$ ) induced by these random vectors observed sequentially by the networked nodes:
+
+$$
+\mathcal {F} _ {0} := \{\Omega , \phi \},
+$$
+
+$$
+\mathcal {F} _ {t} := \sigma \left(\left\{\boldsymbol {\xi} _ {0} ^ {i}, \boldsymbol {\xi} _ {1} ^ {i}, \dots , \boldsymbol {\xi} _ {t - 1} ^ {i}: i \in \mathcal {V} \right\}\right), \quad \forall t \geq 1, \tag {1}
+$$
+
+where $\phi$ is the empty set. We are now ready to define the SFO mechanism in the following. At each iteration $t$ , each node $i$ , given an input random vector $\mathbf{x} \in \mathbb{R}^p$ that is $\mathcal{F}_t$ -measurable, is able to query the local SFO to obtain a stochastic gradient of the form $\mathbf{g}_i(\mathbf{x}, \boldsymbol{\xi}_t^i)$ , where $\mathbf{g}_i: \mathbb{R}^p \times \mathbb{R}^q \to \mathbb{R}^p$ is a Borel measurable function. We assume that the SFO satisfies the following four properties.
+
+Assumption 1 (Oracle). For any $\mathcal{F}_t$ -measurable random vectors $\mathbf{x}, \mathbf{y} \in \mathbb{R}^p$ , we have the following: $\forall i \in \mathcal{V}, \forall t \geq 0$
+
+$\mathbb{E}\left[\mathbf{g}_i(\mathbf{x},\pmb {\xi}_t^i)|\mathcal{F}_t\right] = \nabla f_i(\mathbf{x});$
+$\mathbb{E}\left[\| \mathbf{g}_i(\mathbf{x},\pmb {\xi}_t^i) - \nabla f_i(\mathbf{x})\| ^2\right]\leq \nu_i^2,\overline{\nu}^2 := \frac{1}{n}\sum_{i = 1}^{n}\nu_i^2;$
+- the family $\{\pmb{\xi}_t^i : \forall t \geq 0, i \in \mathcal{V}\}$ of random vectors is independent;
+$\mathbb{E}\left[\| \mathbf{g}_i(\mathbf{x},\pmb {\xi}_t^i) - \mathbf{g}_i(\mathbf{y},\pmb {\xi}_t^i)\| ^2\right]\leq L^2\mathbb{E}\left[\| \mathbf{x} - \mathbf{y}\| ^2\right].$
+
+The first three properties above are standard and commonly used to establish the convergence of decentralized stochastic gradient methods. They however do not explicitly impose any structures on the stochastic gradient mapping $\mathbf{g}_i$ other
+
+than the measurability. On the other hand, the last property, the mean-squared smoothness, roughly speaking, requires that $\mathbf{g}_i$ is $L$ -smooth on average with respect to the input arguments $\mathbf{x}$ and $\mathbf{y}$ . As a simple example, Assumption 1 holds if $f_i(\mathbf{x}) = \frac{1}{2}\mathbf{x}^\top \mathbf{Q}_i\mathbf{x}$ and $\mathbf{g}_i(\mathbf{x},\boldsymbol {\xi}_i) = \mathbf{Q}_i\mathbf{x} + \boldsymbol {\xi}_i$ , where $\mathbf{Q}_i$ is a constant matrix and $\boldsymbol{\xi}_i$ has zero mean and finite second moment. We further note that the mean-squared smoothness of each $\mathbf{g}_i$ implies, by Jensen's inequality, that each $f_i$ is $L$ -smooth, i.e., $\| \nabla f_i(\mathbf{x}) - \nabla f_i(\mathbf{y})\| \leq L\| \mathbf{x} - \mathbf{y}\|$ , and consequently the global function $F$ is also $L$ -smooth.
+
+In addition, we make the following assumptions on $F$ and the communication network $\mathcal{G}$ .
+
+Assumption 2 (Global Function). $F$ is bounded below, i.e., $F^{*} \coloneqq \inf_{\mathbf{x} \in \mathbb{R}^{p}} F(\mathbf{x}) > -\infty$ .
+
+Assumption 3 (Communication Network). The directed network $\mathcal{G}$ admits a primitive and doubly-stochastic weight matrix $\underline{\mathbf{W}} = \{\underline{w}_{ij}\} \in \mathbb{R}^{n\times n}$ . Hence, $\underline{\mathbf{W}}\mathbf{1}_n = \underline{\mathbf{W}}^\top \mathbf{1}_n = \mathbf{1}_n$ and $\lambda := \|\underline{\mathbf{W}} - \frac{1}{n}\mathbf{1}_n\mathbf{1}_n^\top\| \in [0,1)$ .
+
+The weight matrix $\underline{\mathbf{W}}$ that satisfies Assumption 3 may be designed for strongly-connected weight-balanced directed graphs (and thus for arbitrary connected undirected graphs). For example, the family of directed exponential graphs is weight-balanced and plays a key role in decentralized training (Assran et al., 2019). We note that $\lambda$ is known as the second largest singular value of $\underline{\mathbf{W}}$ and measures the algebraic connectivity of the graph, i.e., a smaller value of $\lambda$ roughly means a better connectivity. We note that several existing approaches require strictly stronger assumptions on $\underline{\mathbf{W}}$ . For instance, D2 (Tang et al., 2018) and D-PD-SGD (Yi et al., 2020) require $\underline{\mathbf{W}}$ to be symmetric and hence are restricted to undirected networks.
+
+# 2.2. Algorithm Development and Description
+
+We now describe the proposed GT-HSGD algorithm and provide an intuitive construction. Recall that $\mathbf{x}_t^i$ is the estimate of an stationary point of the global cost $F$ at node $i$ and iteration $t$ . Let $\mathbf{g}_i(\mathbf{x}_t^i,\pmb {\xi}_t^i)$ and $\mathbf{g}_i(\mathbf{x}_{t - 1}^i,\pmb {\xi}_t^i)$ be the corresponding stochastic gradients returned by the local SFO queried at $\mathbf{x}_t^i$ and $\mathbf{x}_{t - 1}^i$ respectively. Motivated by the strong performance of recently introduced decentralized methods that combine gradient tracking and various variance reduction schemes for finite-sum problems (Xin et al., 2020b;c; Li et al., 2020a; Sun et al., 2020), we seek similar variance reduction for decentralized online problems with SFO. In particular, we focus on the following local hybrid variance reduced stochastic gradient estimator $\mathbf{v}_t^i$ introduced in (Liu et al., 2020; Tran-Dinh et al., 2020; Cutkosky & Orabona, 2019) for centralized online problems: $\forall t\geq 1$
+
+$$
+\mathbf {v} _ {t} ^ {i} = \mathbf {g} _ {i} (\mathbf {x} _ {t} ^ {i}, \pmb {\xi} _ {t} ^ {i}) + (1 - \beta) \big (\mathbf {v} _ {t - 1} ^ {i} - \mathbf {g} _ {i} (\mathbf {x} _ {t - 1} ^ {i}, \pmb {\xi} _ {t} ^ {i}) \big), \quad (2)
+$$
+
+for some applicable weight parameter $\beta \in [0,1]$ . This local gradient estimator $\mathbf{v}_t^i$ is fused, via a gradient tracking
+
+mechanism (Di Lorenzo & Scutari, 2016; Xu et al., 2015), over the network to update the global gradient tracker $\mathbf{y}_t^i$ , which is subsequently used as the descent direction in the $\mathbf{x}_t^i$ -update. The complete description of GT-HSGD is provided in Algorithm 1. We note that the update (2) of $\mathbf{v}_t^i$ may be equivalently written as
+
+$$
+\begin{array}{l} \mathbf {v} _ {t} ^ {i} = \boldsymbol {\beta} \cdot \underbrace {\mathbf {g} _ {i} \left(\mathbf {x} _ {t} ^ {i} , \boldsymbol {\xi} _ {t} ^ {i}\right)} \\ \text {S t o c h a s t i c g r a d i e n t} \\ + (1 - \beta) \cdot \underbrace {\left(\mathbf {g} _ {i} \left(\mathbf {x} _ {t} ^ {i} , \boldsymbol {\xi} _ {t} ^ {i}\right) - \mathbf {g} _ {i} \left(\mathbf {x} _ {t - 1} ^ {i} , \boldsymbol {\xi} _ {t} ^ {i}\right) + \mathbf {v} _ {t - 1} ^ {i}\right)} _ {\text {S A R A H}}, \\ \end{array}
+$$
+
+which is a convex combination of the local vanilla stochastic gradient returned by the SFO and a SARAH-type (Nguyen et al., 2017; Fang et al., 2018; Wang et al., 2019) gradient estimator. This discussion leads to the fact that GT-HSGD reduces to GT-DSGD (Pu & Nedich, 2020; Xin et al., 2020e; Lu et al., 2019) when $\beta = 1$ , and becomes the inner loop of GT-SARAH (Xin et al., 2020b) when $\beta = 0$ . However, our convergence analysis shows that GT-HSGD achieves its best oracle complexity and outperforms the existing decentralized online variance-reduced approaches (Sun et al., 2020; Pan et al., 2020) with a weight parameter $\beta \in (0,1)$ . It is then clear that neither GT-DSGD nor the inner loop of GT-SARAH, on their own, are able to outperform the proposed approach, making GT-HSGD a non-trivial algorithmic design for this problem class.
+
+Algorithm 1 GT-HSGD at each node $i$
+
+Require: $\mathbf{x}_0^i = \overline{\mathbf{x}}_0$ ; $\alpha$ ; $\beta$ ; $b_0$ ; $\mathbf{y}_0^i = \mathbf{0}_p$ ; $\mathbf{v}_{-1}^i = \mathbf{0}_p$ ; $T$ .
+
+1: Sample $\{\pmb{\xi}_{0,r}^{i}\}_{r = 1}^{b_{0}}$ and $\mathbf{v}_0^i = \frac{1}{b_0}\sum_{r = 1}^{b_0}\mathbf{g}_i(\mathbf{x}_0^i,\pmb{\xi}_{0,r}^i)$
+2: $\mathbf{y}_1^i = \sum_{j=1}^{n} \underline{w}_{ij} \left( \mathbf{y}_0^j + \mathbf{v}_0^j - \mathbf{v}_{-1}^j \right)$ ;
+3: $\mathbf{x}_1^i = \sum_{j=1}^{n} \underline{w}_{ij} \left( \mathbf{x}_0^j - \alpha \mathbf{y}_1^j \right)$ ;
+4: for $t = 1,2,\dots ,T - 1$ do
+5: Sample $\pmb{\xi}_t^i$ ;
+6: $\mathbf{v}_t^i = \mathbf{g}_i(\mathbf{x}_t^i,\pmb {\xi}_t^i) + (1 - \beta)\bigl (\mathbf{v}_{t - 1}^i -\mathbf{g}_i(\mathbf{x}_{t - 1}^i,\pmb {\xi}_t^i)\bigr).$
+7: $\mathbf{y}_{t + 1}^{i} = \sum_{j = 1}^{n}\underline{w}_{ij}\bigl (\mathbf{y}_{t}^{j} + \mathbf{v}_{t}^{j} - \mathbf{v}_{t - 1}^{j}\bigr);$
+8: $\mathbf{x}_{t + 1}^{i} = \sum_{j = 1}^{n}\underline{w}_{ij}\bigl (\mathbf{x}_{t}^{j} - \alpha \mathbf{y}_{t + 1}^{j}\bigr);$
+9: end for
+
+output $\widetilde{\mathbf{x}}_T$ selected uniformly at random from $\{\mathbf{x}_t^i\}_{0\leq t\leq T}^{i\in \mathcal{V}}$
+
+Remark 1. Clearly, each $\mathbf{v}_t^i$ is a conditionally biased estimator of $\nabla f_i(\mathbf{x}_t^i)$ , i.e., $\mathbb{E}[\mathbf{v}_t^i|\mathcal{F}_t]\neq \nabla f_i(\mathbf{x}_t^i)$ in general. However, it can be shown that $\mathbb{E}[\mathbf{v}_t^i] = \mathbb{E}[\nabla f_i(\mathbf{x}_t^i)]$ , meaning that $\mathbf{v}_t^i$ serves as a surrogate for the underlying exact gradient in the sense of total expectation.
+
+# 3. Main Results
+
+In this section, we present the main convergence results of GT-HSGD in this paper and discuss their salient features.
+
+The formal convergence analysis is deferred to Section 4.
+
+Theorem 1. If the weight parameter $\beta = \frac{48L^2\alpha^2}{n}$ and the step-size $\alpha$ is chosen as
+
+$$
+0 < \alpha < \min \left\{\frac {(1 - \lambda^ {2}) ^ {2}}{9 0 \lambda^ {2}}, \frac {\sqrt {n (1 - \lambda)}}{2 6 \lambda}, \frac {1}{4 \sqrt {3}} \right\} \frac {1}{L},
+$$
+
+then the output $\widetilde{\mathbf{x}}_T$ of $GT$ -HSGD satisfies: $\forall T \geq 2$ ,
+
+$$
+\begin{array}{l} \mathbb {E} \big [ \| \nabla F (\widetilde {\mathbf {x}} _ {T}) \| ^ {2} \big ] \leq \frac {4 (F (\overline {{\mathbf {x}}} _ {0}) - F ^ {*})}{\alpha T} + \frac {8 \beta \overline {{\nu}} ^ {2}}{n} + \frac {4 \overline {{\nu}} ^ {2}}{\beta b _ {0} n T} \\ + \frac {6 4 \lambda^ {4} \| \nabla \mathbf {f} (\mathbf {x} _ {0}) \| ^ {2}}{(1 - \lambda^ {2}) ^ {3} n T} + \frac {9 6 \lambda^ {2} \overline {{\nu}} ^ {2}}{(1 - \lambda^ {2}) ^ {3} b _ {0} T} + \frac {2 5 6 \lambda^ {2} \beta^ {2} \overline {{\nu}} ^ {2}}{(1 - \lambda^ {2}) ^ {3}}, \\ \end{array}
+$$
+
+where $\| \nabla \mathbf{f}(\mathbf{x}_0)\|^2 = \sum_{i=1}^n \| \nabla f_i(\overline{\mathbf{x}}_0)\|^2$
+
+Remark 2. Theorem 1 holds for $GT-HSGD$ with arbitrary initial minibatch size $b_0 \geq 1$ .
+
+Theorem 1 establishes a non-asymptotic bound, with no hidden constants, on the mean-squared stationary gap of GT-HSGD over any finite time horizon $T$ .
+
+Transient and steady-state performance over infinite time horizon. If $\alpha$ and $\beta$ are chosen according to Theorem 1, the mean-squared stationary gap $\mathbb{E}\left[||\nabla F(\widetilde{\mathbf{x}}_T)||^2\right]$ of GT-HSGD decays sublinearly at a rate of $O(1 / T)$ up to a steady-state error (SSE) such that
+
+$$
+\operatorname * {l i m s u p} _ {T \rightarrow \infty} \mathbb {E} \left[ \| \nabla F (\widetilde {\mathbf {x}} _ {T}) \| ^ {2} \right] \leq \frac {8 \beta \bar {\nu} ^ {2}}{n} + \frac {2 5 6 \lambda^ {2} \beta^ {2} \bar {\nu} ^ {2}}{\left(1 - \lambda^ {2}\right) ^ {3}}. \tag {3}
+$$
+
+In view of (3), the SSE of GT-HSGD is bounded by the sum of two terms: (i) the first term is in the order of $O(\beta)$ and the division by $n$ demonstrates the benefit of increasing the network size $^2$ ; (ii) the second term is in the order of $O(\beta^2)$ and reveals the impact of the spectral gap $(1 - \lambda)$ of the network topology. Clearly, the SSE can be made arbitrarily small by choosing small enough $\beta$ and $\alpha$ . Moreover, since the spectral gap $(1 - \lambda)$ only appears in a higher order term of $\beta$ in (3), its impact reduces as $\beta$ becomes smaller, i.e., as we require a smaller SSE.
+
+The following corollary is concerned with the finite-time convergence rate of GT-HSGD with specific choices of the algorithmic parameters $\alpha, \beta$ , and $b_0$ .
+
+Corollary 1. Setting $\alpha = \frac{n^{2/3}}{8LT^{1/3}}$ , $\beta = \frac{3n^{1/3}}{4T^{2/3}}$ , and $b_0 = \left\lceil \frac{T^{1/3}}{n^{2/3}} \right\rceil$ in Theorem 1, we have:
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \| \nabla F (\widetilde {\mathbf {x}} _ {T}) \| ^ {2} \right] \leq \frac {3 2 L (F (\overline {{\mathbf {x}}} _ {0}) - F ^ {*}) + 1 2 \overline {{\nu}} ^ {2}}{(n T) ^ {2 / 3}} \\ + \frac {6 4 \lambda^ {4} \| \nabla \mathbf {f} (\mathbf {x} _ {0}) \| ^ {2}}{(1 - \lambda^ {2}) ^ {3} n T} + \frac {2 4 0 \lambda^ {2} n ^ {2 / 3} \overline {{\nu}} ^ {2}}{(1 - \lambda^ {2}) ^ {3} T ^ {4 / 3}}, \\ \end{array}
+$$
+
+for all $T > \max \left\{\frac{1424\lambda^6n^2}{(1 - \lambda^2)^6},\frac{35\lambda^3n^{0.5}}{(1 - \lambda)^{1.5}}\right\}$ . As a consequence, $GT-HSGD$ achieves an $\epsilon$ -accurate stationary point $\mathbf{x}^*$ of the global cost $F$ such that $\mathbb{E}[\|\nabla F(\mathbf{x}^*)\|] \leq \epsilon$ with
+
+$$
+\mathcal {H} = O \left(\max \left\{\mathcal {H} _ {\text {o p t}}, \mathcal {H} _ {\text {n e t}} \right\}\right)
+$$
+
+iterations $^3$ , where $\mathcal{H}_{opt}$ and $\mathcal{H}_{net}$ are given respectively by
+
+$$
+\begin{array}{l} \mathcal {H} _ {o p t} = \frac {(L (F (\overline {{\mathbf {x}}} _ {0}) - F ^ {*}) + \overline {{\nu}} ^ {2}) ^ {1 . 5}}{n \epsilon^ {3}}, \\ \mathcal {H} _ {n e t} = \max \left\{\frac {\lambda^ {4} \| \nabla \mathbf {f} (\mathbf {x} _ {0}) \| ^ {2}}{(1 - \lambda^ {2}) ^ {3} n \epsilon^ {2}}, \frac {\lambda^ {1 . 5} n ^ {0 . 5} \overline {{\nu}} ^ {1 . 5}}{(1 - \lambda^ {2}) ^ {2 . 2 5} \epsilon^ {1 . 5}} \right\}. \\ \end{array}
+$$
+
+The resulting total number of oracle queries at each node is thus $\lceil \mathcal{H} + \mathcal{H}^{1 / 3}n^{-2 / 3}\rceil$
+
+Remark 3. Since $\mathcal{H}^{1/3}n^{-2/3}$ is much smaller than $\mathcal{H}$ , we treat the oracle complexity of $GT-HSGD$ as $\mathcal{H}$ for the ease of exposition in Table 1 and the following discussion.
+
+An important implication of Corollary 1 is given in the following.
+
+A regime for network topology-independent oracle complexity and linear speedup. According to Corollary 1, the oracle complexity of GT-HSGD at each node is bounded by the maximum of two terms: (i) the first term $\mathcal{H}_{opt}$ is independent of the network topology and, more importantly, is $n$ times smaller than the oracle complexity of the optimal centralized online variance-reduced methods that execute on a single node for this problem class (Fang et al., 2018; Wang et al., 2019; Pham et al., 2020; Tran-Dinh et al., 2020; Liu et al., 2020); (ii) the second term $\mathcal{H}_{net}$ depends on the network spectral gap $1 - \lambda$ and is in the lower order of $1 / \epsilon$ . These two observations lead to the interesting fact that the oracle complexity of GT-HSGD becomes independent of the network topology, i.e., $\mathcal{H}_{opt}$ dominates $\mathcal{H}_{net}$ , if the required error tolerance $\epsilon$ is small enough such that $4\epsilon \lesssim \min \left\{\lambda^{-4}(1 - \lambda)^3 n^{-1}, \lambda^{-1}(1 - \lambda)^{1.5} n^{-1}\right\}$ . In this regime, GT-HSGD thus achieves a network topology-independent oracle complexity $\mathcal{H}_{opt} = O(n^{-1}\epsilon^{-3})$ , exhibiting a linear speed up compared with the aforementioned centralized optimal algorithms (Fang et al., 2018; Wang et al., 2019; Pham et al., 2020; Tran-Dinh et al., 2020; Liu et al., 2020; Zhou et al., 2020), in the sense that the total number of oracle queries required to achieve an $\epsilon$ -accurate stationary point at each node is reduced by a factor of $1 / n$ .
+
+Remark 4. The small error tolerance regime in the above discussion corresponds to a large number of oracle queries, which translates to the scenario where the required total number of iterations $T$ is large. Note that a large $T$ further implies that the step-size $\alpha$ and the weight parameter $\beta$ are small; see the expression of $\alpha$ and $\beta$ in Corollary 1.
+
+# 4. Outline of the Convergence Analysis
+
+In this section, we outlines the proof of Theorem 1, while the detailed proofs are provided in the Appendix. We let Assumptions 1-3 hold throughout the rest of the paper without explicitly stating them. For the ease of exposition, we write the $\mathbf{x}_{t^{-}}$ and $\mathbf{y}_t$ -update of GT-HSGD in the following equivalent matrix form: $\forall t\geq 0$
+
+$$
+\mathbf {y} _ {t + 1} = \mathbf {W} \left(\mathbf {y} _ {t} + \mathbf {v} _ {t} - \mathbf {v} _ {t - 1}\right), \tag {4a}
+$$
+
+$$
+\mathbf {x} _ {t + 1} = \mathbf {W} \left(\mathbf {x} _ {t} - \alpha \mathbf {y} _ {t + 1}\right), \tag {4b}
+$$
+
+where $\mathbf{W} := \underline{\mathbf{W}} \otimes \mathbf{I}_p$ and $\mathbf{x}_t, \mathbf{y}_t, \mathbf{v}_t$ are square-integrable random vectors in $\mathbb{R}^{np}$ that respectively concatenate the local estimates $\{\mathbf{x}_t^i\}_{i=1}^n$ of a stationary point of $F$ , gradient trackers $\{\mathbf{y}_t^i\}_{i=1}^n$ , stochastic gradient estimators $\{\mathbf{v}_t^i\}_{i=1}^n$ . It is straightforward to verify that $\mathbf{x}_t$ and $\mathbf{y}_t$ are $\mathcal{F}_t$ -measurable while $\mathbf{v}_t$ is $\mathcal{F}_{t+1}$ -measurable for all $t \geq 0$ . For convenience, we also denote
+
+$$
+\nabla \mathbf {f} (\mathbf {x} _ {t}) := \left[ \nabla f _ {1} \left(\mathbf {x} _ {t} ^ {1}\right) ^ {\top}, \dots , \nabla f _ {n} \left(\mathbf {x} _ {t} ^ {n}\right) ^ {\top} \right] ^ {\top}
+$$
+
+and introduce the following quantities:
+
+$$
+\mathbf {J} := \left(\frac {1}{n} \mathbf {1} _ {n} \mathbf {1} _ {n} ^ {\top}\right) \otimes \mathbf {I} _ {p}
+$$
+
+$$
+\overline {{\mathbf {x}}} _ {t} := \frac {1}{n} (\mathbf {1} _ {n} ^ {\top} \otimes \mathbf {I} _ {p}) \mathbf {x} _ {t}, \quad \overline {{\mathbf {y}}} _ {t} := \frac {1}{n} (\mathbf {1} _ {n} ^ {\top} \otimes \mathbf {I} _ {p}) \mathbf {y} _ {t},
+$$
+
+$$
+\overline {{\mathbf {v}}} _ {t} := \frac {1}{n} (\mathbf {1} _ {n} ^ {\top} \otimes \mathbf {I} _ {p}) \mathbf {v} _ {t}, \quad \overline {{\nabla \mathbf {f}}} (\mathbf {x} _ {t}) := \frac {1}{n} (\mathbf {1} _ {n} ^ {\top} \otimes \mathbf {I} _ {p}) \nabla \mathbf {f} (\mathbf {x} _ {t}).
+$$
+
+In the following lemma, we enlist several well-known results in the context of gradient tracking-based algorithms for decentralized stochastic optimization, whose proofs may be found in (Di Lorenzo & Scutari, 2016; Qu & Li, 2017; Xin et al., 2020d; Pu & Nedich, 2020).
+
+Lemma 1. The following relationships hold.
+
+(a) $\| \mathbf{W}\mathbf{x} - \mathbf{J}\mathbf{x}\| \leq \lambda \| \mathbf{x} - \mathbf{J}\mathbf{x}\|$ ,VxE np.
+(b) $\overline{\mathbf{y}}_{t + 1} = \overline{\mathbf{v}}_t,\forall t\geq 0.$
+(c) $\left\| \overline{\nabla}\mathbf{f}(\mathbf{x}_t) - \nabla F(\overline{\mathbf{x}}_t)\right\|^2 \leq \frac{L^2}{n}\left\| \mathbf{x}_t - \mathbf{J}\mathbf{x}_t\right\|^2, \forall t \geq 0.$
+
+We note that Lemma 1(a) holds since $\underline{\mathbf{W}}$ is primitive and doubly-stochastic, Lemma 1(b) is a direct consequence of the gradient tracking update (4a) and Lemma 1(c) is due to the $L$ -smoothness of each $f_{i}$ . By the estimate update of GT-HSGD described in (4b) and Lemma 1(b), it is straightforward to obtain:
+
+$$
+\overline {{\mathbf {x}}} _ {t + 1} = \overline {{\mathbf {x}}} _ {t} - \alpha \overline {{\mathbf {y}}} _ {t + 1} = \overline {{\mathbf {x}}} _ {t} - \alpha \overline {{\mathbf {v}}} _ {t}, \quad \forall t \geq 0. \tag {5}
+$$
+
+Hence, the mean state $\overline{\mathbf{x}}_t$ proceeds in the direction of the average of local stochastic gradient estimators $\overline{\mathbf{v}}_t$ . With the help of (5) and the $L$ -smoothness of $F$ and each $f_i$ , we establish the following descent inequality which sheds light on the overall convergence analysis.
+
+Lemma 2. If $0 < \alpha \leq \frac{1}{2L}$ , then we have: $\forall T \geq 0$ ,
+
+$$
+\begin{array}{l} \sum_ {t = 0} ^ {T} \left\| \nabla F (\overline {{\mathbf {x}}} _ {t}) \right\| ^ {2} \leq \frac {2 (F (\overline {{\mathbf {x}}} _ {0}) - F ^ {*})}{\alpha} - \frac {1}{2} \sum_ {t = 0} ^ {T} \| \overline {{\mathbf {v}}} _ {t} \| ^ {2} \\ + 2 \sum_ {t = 0} ^ {T} \left| \left| \overline {{\mathbf {v}}} _ {t} - \overline {{\nabla \mathbf {f}}} (\mathbf {x} _ {t}) \right| \right| ^ {2} + \frac {2 L ^ {2}}{n} \sum_ {t = 0} ^ {T} \left\| \mathbf {x} _ {t} - \mathbf {J} \mathbf {x} _ {t} \right\| ^ {2}. \\ \end{array}
+$$
+
+In light of Lemma 2, our approach to establishing the convergence of GT-HSGD is to seek the conditions on the algorithmic parameters of GT-HSGD, i.e., the step-size $\alpha$ and the weight parameter $\beta$ , such that
+
+$$
+\begin{array}{l} - \frac {1}{2 T} \sum_ {t = 0} ^ {T} \mathbb {E} \left[ \| \overline {{\mathbf {v}}} _ {t} \| ^ {2} \right] + \frac {2}{T} \sum_ {t = 0} ^ {T} \mathbb {E} \left[ \| \overline {{\mathbf {v}}} _ {t} - \overline {{\nabla \mathbf {f}}} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ + \frac {2 L ^ {2}}{n T} \sum_ {t = 0} ^ {T} \mathbb {E} \left[ \| \mathbf {x} _ {t} - \mathbf {J} \mathbf {x} _ {t} \| ^ {2} \right] = O \left(\alpha , \beta , \frac {1}{b _ {0}}, \frac {1}{T}\right), \tag {6} \\ \end{array}
+$$
+
+where $O(\alpha, \beta, 1/b_0, 1/T)$ represents a nonnegative quantity which may be made arbitrarily small by choosing small enough $\alpha$ and $\beta$ along with large enough $T$ and $b_0$ . If (6) holds, then Lemma 2 reduces to
+
+$$
+\begin{array}{l} \frac {1}{T + 1} \sum_ {t = 0} ^ {T} \mathbb {E} \left[ \| \nabla F (\overline {{\mathbf {x}}} _ {t}) \| ^ {2} \right] \\ \leq \frac {2 (F (\overline {{\mathbf {x}}} _ {0}) - F ^ {*})}{\alpha T} + O \left(\alpha , \beta , \frac {1}{b _ {0}}, \frac {1}{T}\right), \\ \end{array}
+$$
+
+which leads to the convergence arguments of GT-HSGD. For these purposes, we quantify $\sum_{t=0}^{T} \mathbb{E}\left[\|\overline{\mathbf{v}}_t - \overline{\nabla\mathbf{f}}(\mathbf{x}_t)\|^2\right]$ and $\sum_{t=0}^{T} \mathbb{E}\left[\|\mathbf{x}_t - \mathbf{J}\mathbf{x}_t\|^2\right]$ next.
+
+# 4.1. Contraction Relationships
+
+First of all, we establish upper bounds on the gradient variances $\mathbb{E}\left[\|\overline{\mathbf{v}}_t - \overline{\nabla\mathbf{f}}(\mathbf{x}_t)\|^2\right]$ and $\mathbb{E}\left[\|\mathbf{v}_t - \nabla \mathbf{f}(\mathbf{x}_t)\|^2\right]$ by exploiting the hybrid and recursive update of $\mathbf{v}_t$ .
+
+Lemma 3. The following inequalities hold: $\forall t\geq 1$
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \left\| \overline {{\mathbf {v}}} _ {t} - \overline {{\nabla \mathbf {f}}} (\mathbf {x} _ {t}) \right\| ^ {2} \right] \\ \leq (1 - \beta) ^ {2} \mathbb {E} \left[ \| \overline {{\mathbf {v}}} _ {t - 1} - \overline {{\nabla \mathbf {f}}} (\mathbf {x} _ {t - 1}) \| ^ {2} \right] \\ + \frac {6 L ^ {2} \alpha^ {2}}{n} (1 - \beta) ^ {2} \mathbb {E} \left[ \| \overline {{\mathbf {v}}} _ {t - 1} \| ^ {2} \right] + \frac {2 \beta^ {2} \overline {{\nu}} ^ {2}}{n} \\ + \frac {6 L ^ {2}}{n ^ {2}} (1 - \beta) ^ {2} \mathbb {E} \left[ \| \mathbf {x} _ {t} - \mathbf {J} \mathbf {x} _ {t} \| ^ {2} + \| \mathbf {x} _ {t - 1} - \mathbf {J} \mathbf {x} _ {t - 1} \| ^ {2} \right], \tag {7} \\ \end{array}
+$$
+
+and, $\forall t\geq 1$
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \left\| \mathbf {v} _ {t} - \nabla \mathbf {f} (\mathbf {x} _ {t}) \right\| ^ {2} \right] \\ \leq (1 - \beta) ^ {2} \mathbb {E} \left[ \| \mathbf {v} _ {t - 1} - \nabla \mathbf {f} (\mathbf {x} _ {t - 1}) \| ^ {2} \right] \\ + 6 n L ^ {2} \alpha^ {2} (1 - \beta) ^ {2} \mathbb {E} \left[ \| \overline {{\mathbf {v}}} _ {t - 1} \| ^ {2} \right] + 2 n \beta^ {2} \overline {{\nu}} ^ {2} \\ + 6 L ^ {2} (1 - \beta) ^ {2} \mathbb {E} \left[ \| \mathbf {x} _ {t} - \mathbf {J} \mathbf {x} _ {t} \| ^ {2} + \| \mathbf {x} _ {t - 1} - \mathbf {J} \mathbf {x} _ {t - 1} \| ^ {2} \right]. \tag {8} \\ \end{array}
+$$
+
+Remark 5. Since $\mathbf{v}_t$ is a conditionally biased estimator of $\nabla \mathbf{f}(\mathbf{x}_t)$ , (7) and (8) do not directly imply each other and need to be established separately.
+
+We emphasize that the contraction structure of the gradient variances shown in Lemma 3 plays a crucial role in the convergence analysis. The following contraction bounds on the consensus errors $\mathbb{E}\left[\| \mathbf{x}_t - \mathbf{J}\mathbf{x}_t\|^2\right]$ are standard in decentralized algorithms based on gradient tracking, e.g., (Pu & Nedich, 2020; Xin et al., 2020b); in particular, it follows directly from the $\mathbf{x}_t$ -update (4b) and Young's inequality.
+
+Lemma 4. The following inequalities hold: $\forall t\geq 0$
+
+$$
+\begin{array}{l} \left\| \mathbf {x} _ {t + 1} - \mathbf {J} \mathbf {x} _ {t + 1} \right\| ^ {2} \leq \frac {1 + \lambda^ {2}}{2} \left\| \mathbf {x} _ {t} - \mathbf {J} \mathbf {x} _ {t} \right\| ^ {2} \\ + \frac {2 \alpha^ {2} \lambda^ {2}}{1 - \lambda^ {2}} \left\| \mathbf {y} _ {t + 1} - \mathbf {J} \mathbf {y} _ {t + 1} \right\| ^ {2}. \tag {9} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \left\| \mathbf {x} _ {t + 1} - \mathbf {J} \mathbf {x} _ {t + 1} \right\| ^ {2} \leq 2 \lambda^ {2} \left\| \mathbf {x} _ {t} - \mathbf {J} \mathbf {x} _ {t} \right\| ^ {2} \\ + 2 \alpha^ {2} \lambda^ {2} \left\| \mathbf {y} _ {t + 1} - \mathbf {J} \mathbf {y} _ {t + 1} \right\| ^ {2}. \tag {10} \\ \end{array}
+$$
+
+It is then clear from Lemma 4 that we need to further quantify the gradient tracking errors $\mathbb{E}\left[\| \mathbf{y}_t - \mathbf{J}\mathbf{y}_t\|^2\right]$ in order to bound the consensus errors. These error bounds are shown in the following lemma.
+
+Lemma 5. We have the following.
+
+(a) $\mathbb{E}\left[\| \mathbf{y}_1 - \mathbf{J}\mathbf{y}_1\|^2\right] \leq \lambda^2\left\|\nabla \mathbf{f}(\mathbf{x}_0)\right\|^2 + \lambda^2 n\nu^2 / b_0.$
+(b) If $0 < \alpha \leq \frac{1 - \lambda^2}{2\sqrt{42}\lambda^2L}$ , then $\forall t \geq 1$ ,
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \left\| \mathbf {y} _ {t + 1} - \mathbf {J} \mathbf {y} _ {t + 1} \right\| ^ {2} \right] \\ \leq \frac {3 + \lambda^ {2}}{4} \mathbb {E} \left[ \| \mathbf {y} _ {t} - \mathbf {J} \mathbf {y} _ {t} \| ^ {2} \right] + \frac {2 1 \lambda^ {2} n L ^ {2} \alpha^ {2}}{1 - \lambda^ {2}} \mathbb {E} [ \| \overline {{\mathbf {v}}} _ {t - 1} \| ^ {2} ] \\ + \frac {6 3 \lambda^ {2} L ^ {2}}{1 - \lambda^ {2}} \mathbb {E} \left[ \| \mathbf {x} _ {t - 1} - \mathbf {J} \mathbf {x} _ {t - 1} \| ^ {2} \right] \\ + \frac {7 \lambda^ {2} \beta^ {2}}{1 - \lambda^ {2}} \mathbb {E} \left[ \| \mathbf {v} _ {t - 1} - \nabla \mathbf {f} (\mathbf {x} _ {t - 1}) \| ^ {2} \right] + 3 \lambda^ {2} n \beta^ {2} \bar {\nu} ^ {2}. \\ \end{array}
+$$
+
+We note that establishing the contraction argument of gradient tracking errors in Lemma 5 requires a careful examination of the structure of the $\mathbf{v}_t$ -update.
+
+# 4.2. Error Accumulations
+
+To proceed, we observe, from Lemma 3, 4, and 5, that the recursions of the gradient variances, consensus, and gradient tracking errors admit similar forms. Therefore, we abstract out formulas for the accumulation of the error recursions of this type in the following lemma.
+
+Lemma 6. Let $\{V_t\}_{t \geq 0}$ , $\{R_t\}_{t \geq 0}$ and $\{Q_t\}_{t \geq 0}$ be nonnegative sequences and $C \geq 0$ be some constant such
+
+that $V_{t} \leq qV_{t - 1} + qR_{t - 1} + Q_{t} + C, \forall t \geq 1$ , where $q \in (0,1)$ . Then the following inequality holds: $\forall T \geq 1$
+
+$$
+\sum_ {t = 0} ^ {T} V _ {t} \leq \frac {V _ {0}}{1 - q} + \frac {1}{1 - q} \sum_ {t = 0} ^ {T - 1} R _ {t} + \frac {1}{1 - q} \sum_ {t = 1} ^ {T} Q _ {t} + \frac {C T}{1 - q}. \tag {11}
+$$
+
+Similarly, if $V_{t + 1} \leq qV_t + R_{t - 1} + C, \forall t \geq 1$ , then we have: $\forall T \geq 2$
+
+$$
+\sum_ {t = 1} ^ {T} V _ {t} \leq \frac {V _ {1}}{1 - q} + \frac {1}{1 - q} \sum_ {t = 0} ^ {T - 2} R _ {t} + \frac {C T}{1 - q}. \tag {12}
+$$
+
+Applying Lemma 6 to Lemma 3 leads to the following upper bounds on the accumulated variances.
+
+Lemma 7. For any $\beta \in (0,1)$ , the following inequalities hold: $\forall T\geq 1$
+
+$$
+\begin{array}{l} \sum_ {t = 0} ^ {T} \mathbb {E} \left[ \| \overline {{\mathbf {v}}} _ {t} - \overline {{\nabla \mathbf {f}}} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ \leq \frac {\overline {{\nu}} ^ {2}}{\beta b _ {0} n} + \frac {6 L ^ {2} \alpha^ {2}}{n \beta} \sum_ {t = 0} ^ {T - 1} \mathbb {E} \left[ \| \overline {{\mathbf {v}}} _ {t} \| ^ {2} \right] \\ + \frac {1 2 L ^ {2}}{n ^ {2} \beta} \sum_ {t = 0} ^ {T} \mathbb {E} \left[ \left\| \mathbf {x} _ {t} - \mathbf {J} \mathbf {x} _ {t} \right\| ^ {2} \right] + \frac {2 \beta \bar {\nu} ^ {2} T}{n}, \tag {13} \\ \end{array}
+$$
+
+and, $\forall T\geq 1$
+
+$$
+\begin{array}{l} \sum_ {t = 0} ^ {T} \mathbb {E} \left[ \| \mathbf {v} _ {t} - \nabla \mathbf {f} (\mathbf {x} _ {t}) \| ^ {2} \right] \\ \leq \frac {n \bar {\nu} ^ {2}}{\beta b _ {0}} + \frac {6 n L ^ {2} \alpha^ {2}}{\beta} \sum_ {t = 0} ^ {T - 1} \mathbb {E} \left[ \| \overline {{\mathbf {v}}} _ {t} \| ^ {2} \right] \\ + \frac {1 2 L ^ {2}}{\beta} \sum_ {t = 0} ^ {T} \mathbb {E} \left[ \left\| \mathbf {x} _ {t} - \mathbf {J} \mathbf {x} _ {t} \right\| ^ {2} \right] + 2 n \beta \bar {\nu} ^ {2} T. \tag {14} \\ \end{array}
+$$
+
+It can be observed that (13) in Lemma 7 may be used to refine the left hand side of (6). The remaining step, naturally, is to bound $\sum_{t}\mathbb{E}\left[\| \mathbf{x}_t - \mathbf{J}\mathbf{x}_t\| ^2\right]$ in terms of $\sum_{t}\mathbb{E}\left[\| \overline{\mathbf{v}}_t\| ^2\right]$ . This result is provided in the following lemma that is obtained with the help of Lemma 4, 5, 6, and 7.
+
+Lemma 8. If $0 < \alpha \leq \frac{(1 - \lambda^2)^2}{70\lambda^2L}$ and $\beta \in (0,1)$ , then the following inequality holds: $\forall T \geq 2$ ,
+
+$$
+\begin{array}{l} \sum_ {t = 0} ^ {T} \frac {\mathbb {E} \left[ \| \mathbf {x} _ {t} - \mathbf {J} \mathbf {x} _ {t} \| ^ {2} \right]}{n} \leq \frac {2 0 1 6 \lambda^ {4} L ^ {2} \alpha^ {4}}{(1 - \lambda^ {2}) ^ {4}} \sum_ {t = 0} ^ {T - 2} \mathbb {E} \left[ \| \overline {{\mathbf {v}}} _ {t} \| ^ {2} \right] \\ + \frac {3 2 \lambda^ {4} \alpha^ {2}}{(1 - \lambda^ {2}) ^ {3}} \frac {\| \nabla \mathbf {f} (\mathbf {x} _ {0}) \| ^ {2}}{n} + \left(\frac {7 \beta}{1 - \lambda^ {2}} + 1\right) \frac {3 2 \lambda^ {4} \overline {{\nu}} ^ {2} \alpha^ {2}}{(1 - \lambda^ {2}) ^ {3} b _ {0}} \\ + \left(\frac {1 4 \beta}{1 - \lambda^ {2}} + 3\right) \frac {3 2 \lambda^ {4} \beta^ {2} \bar {\nu} ^ {2} \alpha^ {2} T}{(1 - \lambda^ {2}) ^ {3}}. \\ \end{array}
+$$
+
+Finally, we note that Lemma 7 and 8 suffice to establish (6) and hence lead to Theorem 1; see the Appendix for details.
+
+
+
+
+
+
+Figure 1. A comparison of GT-HSGD with other decentralized online stochastic gradient algorithms over the undirected exponential graph of 20 nodes on the a9a, covertype, KDD98, and MiniBooNE datasets.
+
+
+
+# 5. Numerical Experiments
+
+In this section, we illustrate our theoretical results on the convergence of the proposed GT-HSGD algorithm with the help of numerical experiments.
+
+Model. We consider a non-convex logistic regression model (Antoniadis et al., 2011) for decentralized binary classification. In particular, the decentralized non-convex optimization problem of interest takes the form $\min_{\mathbf{x} \in \mathbb{R}^p} F(\mathbf{x}) \coloneqq \frac{1}{n} \sum_{i=1}^{n} f_i(\mathbf{x}) + r(\mathbf{x})$ , such that
+
+$$
+f _ {i} (\mathbf {x}) = \frac {1}{m} \sum_ {j = 1} ^ {m} \log \left[ 1 + e ^ {- \langle \mathbf {x}, \pmb {\theta} _ {i j} \rangle l _ {i j}} \right]
+$$
+
+and
+
+$$
+r (\mathbf {x}) = R \sum_ {k = 1} ^ {p} \frac {[ \mathbf {x} ] _ {k} ^ {2}}{1 + [ \mathbf {x} ] _ {k} ^ {2}},
+$$
+
+where $\theta_{i,j}$ is the feature vector, $l_{i,j} \in \{-1, +1\}$ is the corresponding binary label, and $r(\mathbf{x})$ is a non-convex regularizer. To simulate the online SFO setting described in Section 2, each node $i$ is only able to sample with replacement from its local data $\{\theta_{i,j}, l_{i,j}\}_{j=1}^{m}$ and compute the corresponding (minibatch) stochastic gradient. Throughout all experiments, we set the number of the nodes to $n = 20$ and the regularization parameter to $R = 10^{-4}$ .
+
+Data. To test the performance of the applicable decentralized algorithms, we distribute the a9a, covertype, KDD98, MiniBooNE datasets uniformly over the nodes and normalize the feature vectors such that $\| \pmb{\theta}_{i,j}\| = 1,\forall i,j$ . The statistics of these datasets are provided in Table 2.
+
+Table 2. Datasets used in numerical experiments, all available at https://www.openml.org/.
+
+