General Convergence Analysis of Stochastic First-Order Methods for Composite Optimization

Abstract

In this paper, we consider stochastic composite convex optimization problems with the objective function satisfying a stochastic bounded gradient condition, with or without a quadratic functional growth property. These models include the most well-known classes of objective functions analyzed in the literature: nonsmooth Lipschitz functions and composition of a (potentially) nonsmooth function and a smooth function, with or without strong convexity. Based on the flexibility offered by our optimization model, we consider several variants of stochastic first-order methods, such as the stochastic proximal gradient and the stochastic proximal point algorithms. Usually, the convergence theory for these methods has been derived for simple stochastic optimization models satisfying restrictive assumptions, and the rates are in general sublinear and hold only for specific decreasing stepsizes. Hence, we analyze the convergence rates of stochastic first-order methods with constant or variable stepsize under general assumptions covering a large class of objective functions. For constant stepsize, we show that these methods can achieve linear convergence rate up to a constant proportional to the stepsize and under some strong stochastic bounded gradient condition even pure linear convergence. Moreover, when a variable stepsize is chosen we derive sublinear convergence rates for these stochastic first-order methods. Finally, the stochastic gradient mapping and the Moreau smoothing mapping introduced in the present paper lead to simple and intuitive proofs.

Appendix

In this appendix, we present several objective functions satisfying Assumptions 2.1 and 2.2 . First, we provide two examples of objective functions satisfying the stochastic bounded gradient condition (2).

Example 1

(Nonsmooth (Lipschitz) functions satisfy Assumption 2.1) Assume that the functions $f(\cdot ,\xi )$ and $g(\cdot ,\xi )$ have bounded (sub)gradients:

$$\begin{aligned} \Vert \nabla f(x,\xi )\Vert \le B_f \quad \text {and} \quad \Vert \nabla g(x,\xi ) \Vert \le B_g \quad \forall x \in \text {dom} \ F. \end{aligned}$$

Then, obviously Assumption 2.1 holds with $L=0 \quad \text {and} \quad B^2 = 2 B_f^2 + 2 B_g^2.$

Example 2

(Smooth (Lipschitz gradient) functions satisfy Assumption 2.1) Condition (2) contains the class of functions formed as a sum of two terms: one having Lipschitz continuous gradient and the other having bounded subgradients. Indeed, let us assume that $f(\cdot ,\xi )$ has Lipschitz continuous gradient, i.e., there exists $L(\xi )>0$ such that:

$$\begin{aligned} \Vert \nabla f(x, \xi ) - \nabla f(\bar{x},\xi )\Vert \le L(\xi ) \Vert x - \bar{x}\Vert \quad \forall x \in \text {dom} \ F. \end{aligned}$$

Then, using standard arguments we have [19]:

$$\begin{aligned} f(x, \xi ) - f(\bar{x},\xi ) \ge \langle \nabla f(\bar{x}, \xi ), x -\bar{x} \rangle + \frac{1}{2L(\xi )} \Vert \nabla f(x, \xi ) - \nabla f(\bar{x}, \xi ) \Vert ^2. \end{aligned}$$

Assuming that $L(\xi ) \le L_f$ for all $\xi \in \varOmega$ and $g(\cdot ,\xi )$ convex, then adding $g(x,\xi ) - g(\bar{x},\xi ) \ge \langle \nabla g(\bar{x},\xi ), x -\bar{x} \rangle$ in the previous inequality, where $\nabla g(\bar{x},\xi ) \in \partial g(\bar{x},\xi )$, and then taking expectation w.r.t. $\xi$, we get:

$$\begin{aligned} F(x) - F(\bar{x})&\ge \langle \nabla F(\bar{x}), x -\bar{x} \rangle + \frac{1}{2L_f} \mathbf {E}\left[ \Vert \nabla f(x, \xi ) - \nabla f(\bar{x}, \xi ) \Vert ^2\right] , \end{aligned}$$

where we used that $\nabla f(\bar{x},\xi )$ and $\nabla g(\bar{x},\xi )$ are unbiased stochastic estimates of the (sub)gradients of f and g and thus $\nabla F(\bar{x}) = \mathbf {E}\left[ \nabla g(\bar{x},\xi ) + \nabla g(\bar{x},\xi )\right] \in \partial F(\bar{x})$. Using the optimality conditions for (1) in $\bar{x}$, i.e., $0 \in \partial F(\bar{x})$, we get:

$$\begin{aligned} F(x) - F^*&\ge \frac{1}{2L_f} \mathbf {E}\left[ \Vert \nabla f(x, \xi ) - \nabla f(\bar{x}, \xi ) \Vert ^2\right] . \end{aligned}$$

Therefore, for any $\nabla g(x,\xi ) \in \partial g(x,\xi )$ we have:

$$\begin{aligned} \mathbf {E}\left[ \Vert \nabla F(x, \xi ) \Vert ^2\right]&= \mathbf {E}\left[ \Vert \nabla f(x, \xi ) - \nabla f(\bar{x}, \xi ) +\nabla g(x,\xi ) + \nabla f(\bar{x}, \xi ) \Vert ^2\right] \nonumber \\&\le 2 \mathbf {E}[\Vert \nabla f(x, \xi ) - \nabla f(\bar{x}, \xi )\Vert ^2 ] +2 \mathbf {E}[\Vert \nabla g(x,\xi ) + \nabla f(\bar{x}, \xi )\Vert ^2 ]\\&\le 4L_f (F(x) - F^*) +2 \mathbf {E}[\Vert \nabla g(x,\xi ) + \nabla f(\bar{x}, \xi )\Vert ^2 ]. \end{aligned}$$

Assuming now that the regularization function $g(x,\xi )$ has bounded subgradients, i.e., $\Vert \nabla g(x,\xi )\Vert \le B_g$, then we get that the stochastic bounded gradient condition (2) holds with:

$$\begin{aligned} L = 4 L_f \quad \text {and} \quad B^2 = 4 (B_g^2 +\min _{\bar{x} \in X^*} \mathbf {E}[\Vert \nabla f(\bar{x}, \xi )\Vert ^2]). \end{aligned}$$

Further, many practical problems satisfy the quadratic functional growth condition (3); the most relevant one is given next.

Example 3

(Composition between a strongly convex function and a linear map satisfy Assumption 2.2): Assume $F(x)= {{\hat{f}}}(A^Tx) + g(x)$, where ${{\hat{f}}}$ is a strongly convex function with constant $\sigma _f>0$, A is a matrix of appropriate dimension and g is a polyhedral function. Since g has a polyhedral epigraph, then the optimization problem (1) can be equivalently written as:

$$\begin{aligned} \min _{x,\zeta } {{\hat{f}}}(A^Tx) + \zeta \qquad \text {s.t.}: \quad Cx + c \zeta \le d, \end{aligned}$$

for some appropriate matrix C and vectors c and d of appropriate dimensions. In conclusion, this reformulation leads to the following extended problem:

$$\begin{aligned} \hat{F^*} = \min _{{\hat{x}}=[x^T \; \zeta ]^T} {{\hat{F}}}({\hat{x}}) \quad \left( = {{\hat{f}}}({\hat{A}} {\hat{x}}) + {\hat{c}}^T {\hat{x}} \right) \qquad \text {s.t.}: \quad {\hat{C}} {\hat{x}} \le {\hat{d}}, \end{aligned}$$

where ${\hat{A}}=[A \; 0], \; {\hat{c}} =[0 \; 1]^T, \;{\hat{C}} = [C \; c]$ and ${\hat{d}} =d$. It can be easily seen that ${\hat{x}}^* = [(x^*)^T \; \zeta ^*]^T$ is an optimal point of this extended optimization problem if $x^*$ is optimal for the original problem and $g(x^*) = \zeta ^*$. Moreover, we have $\hat{F^*}= F^*$. Following a standard argument, as, e.g., in [12], there exist $b^*$ and $s^*$ such that the optimal set of the extended optimization problem is given by ${\hat{X}}^* =\{{\hat{x}}: \; {\hat{A}} {\hat{x}} = b^*, \; {\hat{c}}^T {\hat{x}}= s^*, \; {\hat{C}} {\hat{x}} \le {\hat{d}} \}$. Further, since ${{\hat{f}}}$ is strongly convex function with constant $\sigma _f>0$, it follows from [12][Theorem 10] that for any $M>0$ on any sublevel set defined in terms of M the function ${\hat{F}}$ satisfies a quadratic functional growth condition of the form:

$$\begin{aligned} {\hat{F}}({\hat{x}}) - \hat{F^*} \ge \frac{\mu (M)}{2} \Vert {\hat{x}} - {\hat{x}}^* \Vert ^2 \quad \forall {\hat{x}}: \; {\hat{F}}({\hat{x}}) - \hat{F^*} \le M, \end{aligned}$$

where $\mu (M) = \frac{\sigma _f}{\theta ^2 (1 + M \sigma _f + 2 \Vert \nabla {{\hat{f}}}({\hat{A}} {\hat{x}}^*)\Vert ^2)}$, with ${\hat{x}}^* \in {\hat{X}}^*$ and $\theta$ is the Hoffman bound for the optimal polyhedral set ${\hat{X}}^*$. Now, setting $\zeta = g(x)$ in the previous inequality, we get $F(x) - F({\overline{x}}) \!\ge \! \frac{\mu (M)}{2} ( \Vert x - {\overline{x}}\Vert ^2 + (\zeta - \zeta ^*)^2) \!\ge \! \frac{\mu (M)}{2} \Vert x - {\overline{x}}\Vert ^2$ for all $x: F(x) - F({\overline{x}}) \!\le \! M$. In conclusion, the objective function F satisfies the quadratic functional growth condition (3) on any sublevel set, that is, for any $M>0$ there exists $\mu (M)>0$ defined above such that:

$$\begin{aligned} F(x) - F({\overline{x}}) \ge \frac{\mu (M)}{2} \Vert x - {\overline{x}}\Vert ^2 \quad \forall x: \; F(x) - F({\overline{x}}) \le M. \end{aligned}$$

The quadratic functional growth condition (3) is a relaxation of strong convexity notion, see [12] for a more detailed discussion. Clearly, any strongly convex function F satisfies (3) (see, e.g., [19]).