Proximal-like incremental aggregated gradient method with Bregman distance in weakly convex optimization problems

Abstract

We focus on a special nonconvex and nonsmooth composite function, which is the sum of the smooth weakly convex component functions and a proper lower semi-continuous weakly convex function. An algorithm called the proximal-like incremental aggregated gradient (PLIAG) method proposed in Zhang et al. (Math Oper Res 46(1): 61–81, 2021) is proved to be convergent and highly efficient to solve convex minimization problems. This algorithm can not only avoid evaluating the exact full gradient which can be expensive in big data models but also weaken the stringent global Lipschitz gradient continuity assumption on the smooth part of the problem. However, under the nonconvex case, there is few analysis on the convergence of the PLIAG method. In this paper, we prove that the limit point of the sequence generated by the PLIAG method is the critical point of the weakly convex problems. Under further assumption that the objective function satisfies the Kurdyka–Łojasiewicz (KL) property, we prove that the generated sequence converges globally to a critical point of the problem. Additionally, we give the convergence rate when the Łojasiewicz exponent is known.

Appendix

Proof of Lemma 3.1

Considering the $L_j$-smooth of the pair $( f_j,\omega )$ according to Assumption 2(a), we can obtain that

$$\begin{aligned} f_{j}(x_{k+1})&\le f_{j}(x_{k-\tau _{k}^{j}})+\left\langle \nabla f_{j}(x_{k-\tau _{k}^{j}}),x_{k+1}-x_{k-\tau _{k}^{j}} \right\rangle +L_j\cdot D_{\omega }(x_{k+1},x_{k-\tau _k^j})\nonumber \\&\le f_j(x)+\frac{\beta _j}{2}\left\| x-x_{k-\tau _k^j}\right\| ^2+ \left\langle \nabla f_{j}(x_{k-\tau _{k}^{j}}),x_{k+1}-x \right\rangle +L_j\cdot D_{\omega }(x_{k+1},x_{k-\tau _k^j}), \end{aligned}$$

(31)

where the second inequality follows from the convexity of $f_j(\cdot )+\frac{\beta _j}{2}\Vert \cdot \Vert ^2$ according to Assumption 2(b). Summing (31) over all $j\in {\mathcal {J}}_k$, we can get

$$\begin{aligned} \sum \limits _{j\in {\mathcal {J}}_{k}}f_{j}(x_{k+1})\le&\sum \limits _{j\in {\mathcal {J}}_{k}}f_{j}(x)+\frac{\gamma _1}{2}\sum \limits _{j\in {\mathcal {J}}_{k}}\left\| x-x_{k-\tau _{k}^{j}}\right\| ^{2}+ \left\langle \sum \limits _{j\in {\mathcal {J}}_{k}}\nabla f_{j}(x_{k-\tau _{k}^{j}}),x_{k+1}-x\right\rangle \nonumber \\&+\sum \limits _{j\in {\mathcal {J}}_{k}}L_{j} \cdot D_{\omega }(x_{k+1},x_{k-\tau _{k}^{j}}). \end{aligned}$$

(32)

where $\gamma _1$ is defined in Assumption 2 as $\gamma _1{:}{=}\sum _{n=1}^N\beta _n$. By the optimality of $x_{k+1}$, we have

$$\begin{aligned} -\sum \limits _{j\in {\mathcal {J}}_{k}}\nabla f_{j}(x_{k-\tau _k^j})-\frac{1}{\alpha } \left( \nabla \omega (x_{k+1})-\nabla \omega (x_k)\right) \in \partial h(x_{k+1})+\sum \limits _{i\in {\mathcal {I}}_{k}}\nabla f_i (x_{k+1}). \end{aligned}$$

(33)

Using the subgradient inequality for the convex function $h(x)+\sum \limits _{i\in {\mathcal {I}}_{k}}f_i (x)+\frac{\gamma _1+\gamma _2}{2}\Vert x\Vert ^2$ at $x_{k+1}$, we have

$$\begin{aligned}&h(x_{k+1})+\sum \limits _{i\in {\mathcal {I}}_{k}}f_i (x_{k+1}) \nonumber \\ \le&h(x)+\sum \limits _{i\in {\mathcal {I}}_{k}}f_{i}(x)+ \left\langle \sum \limits _{j\in {\mathcal {J}}_{k}}\nabla f_{j}(x_{k-\tau _{k}^{j}})+\frac{1}{\alpha }\left( \nabla \omega (x_{k+1}) -\nabla \omega (x_k)\right) ,x-x_{k+1}\right\rangle \nonumber \\&+ \frac{\gamma _{1}+\gamma _{2}}{2}\Vert x-x_{k+1}\Vert ^2 \nonumber \\ =&h(x)+\sum \limits _{i\in {\mathcal {I}}_{k}}f_{i}(x)+ \left\langle \sum \limits _{j\in {\mathcal {J}}_{k}}\nabla f_{j}(x_{k-\tau _{k}^{j}}),x-x_{k+1}\right\rangle + \frac{\gamma _{1}+\gamma _{2}}{2}\Vert x-x_{k+1}\Vert ^2 \nonumber \\&+\frac{1}{\alpha }\langle \nabla \omega (x_{k+1})- \nabla \omega (x_k),x-x_{k+1}\rangle \nonumber \\ =&h(x)+\sum \limits _{i\in {\mathcal {I}}_{k}}f_{i}(x)+ \left\langle \sum \limits _{j\in {\mathcal {J}}_{k}}\nabla f_{j}(x_{k-\tau _{k}^{j}}),x-x_{k+1}\right\rangle + \frac{\gamma _{1}+\gamma _{2}}{2}\Vert x-x_{k+1}\Vert ^{2} \nonumber \\&+\frac{1}{\alpha }D_{\omega }(x,x_{k})- \frac{1}{\alpha }D_{\omega }(x,x_{k+1})- \frac{1}{\alpha }D_{\omega }(x_{k+1},x_{k}), \end{aligned}$$

(34)

where the last equality follows from the three-point identity of the Bregman distance. Adding (34) to (32), we get

$$\begin{aligned} \Phi (x_{k+1})&\le \Phi (x)+\frac{\gamma _{1}+\gamma _{2}}{2}\Vert x-x_{k+1}\Vert ^{2} +\frac{\gamma _{1}}{2}\sum \limits _{j\in {\mathcal {J}}_{k}}\Vert x-x_{k-\tau _{k}^{j}}\Vert ^{2}\nonumber \\&\quad +\frac{1}{\alpha }D_{\omega }(x,x_{k})- \frac{1}{\alpha }D_{\omega }(x,x_{k+1})- \frac{1}{\alpha }D_{\omega }(x_{k+1},x_{k})+ \sum \limits _{j\in {\mathcal {J}}_{k}}L_{j}\cdot D_{\omega }(x_{k+1},x_{k-\tau _{k}^{j}}). \end{aligned}$$

(35)

According to Assumption 1(d), we deduce

$$\begin{aligned} \sum _{i\in {\mathcal {J}}_{k}}L_{i}\cdot D_{\omega }(x_{k+1},x_{k-\tau _{k}^{i}})&\le \sum _{i\in {\mathcal {J}}_{k}}L_{i}\cdot \ell (\tau _k^i+1)\cdot \sum _{j=k-\tau _k^i}^k D_{\omega }(x_{j+1},x_j)\\&\le \sum _{i\in {\mathcal {J}}_{k}}L_{i}\cdot \ell (\tau +1)\cdot \sum _{j=k -\tau }^k D_{\omega }(x_{j+1},x_j)\\&=\ell (\tau +1)\cdot \sum _{i\in {\mathcal {J}}_k}L_i\sum _{j=k-\tau }^{k}D_{\omega }(x_{j+1},x_{j})\\&\le L\cdot \ell (\tau +1)\cdot \sum _{j=k-\tau }^{k}D_{\omega }(x_{j+1},x_{j}), \end{aligned}$$

where the second inequality is due to the fact that $\tau _k^i$ is bounded above by $\tau$ and the last inequality follows from $L{:}{=}\sum _{j=1}^N L_j$. Together with the above inequality, (35) can be rewritten as

$$\begin{aligned} \Phi (x_{k+1})\le&\Phi (x)+\frac{\gamma _{1}+\gamma _{2}}{2}\Vert x-x_{k+1}\Vert ^{2} +\frac{\gamma _{1}}{2}\sum \limits _{j\in {\mathcal {J}}_{k}}\Vert x-x_{k-\tau _{k}^{j}}\Vert ^{2}\nonumber \\&+\frac{1}{\alpha }D_{\omega }(x,x_{k})- \frac{1}{\alpha }D_{\omega }(x,x_{k+1})- \frac{1}{\alpha }D_{\omega }(x_{k+1},x_{k})+ L\cdot \ell (\tau +1)\cdot \sum \limits _{j=k-\tau }^{k}D_{\omega }(x_{j+1},x_{j}). \end{aligned}$$

(36)

Invoking (36) with $x=x_k$ implies

$$\begin{aligned} \begin{aligned} \Phi (x_{k+1}) \le&\Phi (x_k)+\frac{\gamma _{1}+\gamma _{2}}{2}\Vert x_k-x_{k+1}\Vert ^{2} +\frac{\gamma _{1}}{2}\sum \limits _{j\in {\mathcal {J}}_{k}}\Vert x_k-x_{k-\tau _{k}^{j}}\Vert ^{2}\\&-\frac{1}{\alpha }D_{\omega }(x_k,x_{k+1})- \frac{1}{\alpha }D_{\omega }(x_{k+1},x_{k})+ L\cdot \ell (\tau +1)\cdot \sum \limits _{j=k-\tau }^{k}D_{\omega }(x_{j+1},x_{j}). \end{aligned} \end{aligned}$$

(37)

The component $\sum _{j=k-\tau }^{k}D_{\omega }(x_{j+1},x_{j})$ in (37) can be written as

$$\begin{aligned} \sum \limits _{j=k-\tau }^{k}D_{\omega }(x_{j+1},x_{j})=&(\tau +1)D_\omega (x_{k+1},x_k)+ \sum _{i=1}^{\tau }i D_{\omega }(x_{k-\tau +i},x_{k-\tau +i-1}) -\sum \limits _{i=1}^{\tau }i D_{\omega }(x_{k+1-\tau +i},x_{k-\tau +i}), \end{aligned}$$

(38)

and the component $\Vert x_k-x_{k-\tau _{k}^{j}}\Vert ^{2}$ in (37) has the following inequality

$$\begin{aligned}&\left\| x_{k}-x_{k-\tau _{k}^{j}}\right\| ^{2}\le \tau _k^j\sum \limits _{j=k-\tau _k^j}^{k-1}\Vert x_{j+1}-x_{j}\Vert ^{2}\le \tau \sum \limits _{j=k-\tau }^{k}\Vert x_{j+1}-x_{j}\Vert ^{2}\nonumber \\ =&\tau (\tau +1)\Vert x_{k+1}-x_{k}\Vert ^{2} +\tau \sum \limits _{i=1}^{\tau }i\Vert x_{k-\tau +i}-x_{k-\tau +i-1}\Vert ^{2} -\tau \sum \limits _{i=1}^{\tau }i\Vert x_{k+1-\tau +i}-x_{k-\tau +i}\Vert ^{2}. \end{aligned}$$

Using the above inequaity, we have

$$\begin{aligned}&\sum _{j\in {\mathcal {J}}_{k}}\left\| x_{k}-x_{k-\tau _{k}^{j}}\right\| ^{2}\nonumber \\ \le&J_k\left[ \tau (\tau +1)\Vert x_{k+1}-x_{k}\Vert ^{2} +\tau \sum \limits _{i=1}^{\tau }i\Vert x_{k-\tau +i}-x_{k-\tau +i-1}\Vert ^{2} -\tau \sum \limits _{i=1}^{\tau }i\Vert x_{k+1-\tau +i}-x_{k-\tau +i}\Vert ^{2}\right] \nonumber \\ \le&N\left[ \tau (\tau +1)\Vert x_{k+1}-x_{k}\Vert ^{2} +\tau \sum \limits _{i=1}^{\tau }i\Vert x_{k-\tau +i}-x_{k-\tau +i-1}\Vert ^{2} -\tau \sum \limits _{i=1}^{\tau }i\Vert x_{k+1-\tau +i}-x_{k-\tau +i}\Vert ^{2}\right] . \end{aligned}$$

(39)

Plugging (38) and (39) into (37) and minusing $\upsilon ({\mathcal {P}})$ on both sides of the inequality, we can get

$$\begin{aligned}&\Phi (x_{k+1})-\upsilon ({\mathcal {P}})+L\cdot \ell (\tau +1)\cdot \sum \limits _{i=1}^{\tau }i D_{\omega }(x_{k+1-\tau +i},x_{k-\tau +i}) +\frac{\gamma _{1}\tau N}{2}\sum \limits _{i=1}^{\tau }i\Vert x_{k+1-\tau +i}-x_{k-\tau +i}\Vert ^{2}\nonumber \\ \le&\Phi (x_{k})-\upsilon ({\mathcal {P}}) +L\cdot \ell (\tau +1)\cdot \sum \limits _{i=1}^{\tau }i D_{\omega }(x_{k-\tau +i},x_{k-\tau +i-1}) +\frac{\gamma _{1}\tau N}{2}\sum \limits _{i=1}^{\tau }i\Vert x_{k-\tau +i}-x_{k-\tau +i-1}\Vert ^{2}\nonumber \\&-\frac{1}{\alpha }D_{\omega }(x_{k},x_{k+1}) -\left[ \frac{1}{\alpha }-L\cdot \ell (\tau +1)\cdot (\tau +1)\right] D_{\omega }(x_{k+1},x_{k})\nonumber \\&+\frac{\gamma _1+\gamma _2}{2}\Vert x_{k}-x_{k+1}\Vert ^{2}+{\frac{\gamma _{1}\tau N}{2}(\tau +1)}\Vert x_{k+1}-x_{k}\Vert ^{2}. \end{aligned}$$

(40)

Recalling the definition of $T_k$, $C_1$ and $C_2$, (40) can be rewritten as

$$\begin{aligned} T_{k+1}\le T_{k}-\frac{1}{\alpha }D_{\omega }(x_{k},x_{k+1}) -C_1D_{\omega }(x_{k+1},x_{k})+C_2\Vert x_{k+1}-x_{k}\Vert ^{2}, \end{aligned}$$

thus complete the proof. $\square$