Bi-fidelity stochastic gradient descent for structural optimization under uncertainty

Abstract

The presence of uncertainty in material properties and geometry of a structure is ubiquitous. The design of robust engineering structures, therefore, needs to incorporate uncertainty in the optimization process. Stochastic gradient descent (SGD) method can alleviate the cost of optimization under uncertainty, which includes statistical moments of quantities of interest in the objective and constraints. However, the design may change considerably during the initial iterations of the optimization process which impedes the convergence of the traditional SGD method and its variants. In this paper, we present two SGD based algorithms, where the computational cost is reduced by employing a low-fidelity model in the optimization process. In the first algorithm, most of the stochastic gradient calculations are performed on the low-fidelity model and only a handful of gradients from the high-fidelity model is used per iteration, resulting in an improved convergence. In the second algorithm, we use gradients from the low-fidelity models to be used as control variate, a variance reduction technique, to reduce the variance in the search direction. These two bi-fidelity algorithms are illustrated first with a conceptual example. Then, the convergence of the proposed bi-fidelity algorithms is studied with two numerical examples of shape and topology optimization and compared to popular variants of the SGD method that do not use low-fidelity models. The results show that the proposed use of a bi-fidelity approach for the SGD method can improve the convergence. Two analytical proofs are also provided that show linear convergence of these two algorithms under appropriate assumptions.

Appendices

Proof of Theorem 1

Assume ${{\varvec{\theta }}}_{k}$ is the vector of optimization parameters after k iterations of Algorithm 4. ${\mathbf {h}}_\mathrm {low}({{\varvec{\theta }}}_k)$ and ${\mathbf {h}}_\mathrm {high}({{\varvec{\theta }}}_k)$ are gradients of the objective with respect to ${{\varvec{\theta }}}_k$ using the low- and high-fidelity models, respectively. Under the assumption of strong convexity¹ of low-fidelity and high-fidelity objectives,

$$\begin{aligned} \begin{aligned} ({{\varvec{\theta }}}_k - {{\varvec{\theta }}}^*)^T{\mathbf {h}}_\mathrm {low}({{\varvec{\theta }}}_k)&\ge \mu _\mathrm {low} \Vert {{\varvec{\theta }}}_k - {{\varvec{\theta }}}^* \Vert ^2,\\ ({{\varvec{\theta }}}_k - {{\varvec{\theta }}}^*)^T{\mathbf {h}}_\mathrm {high}({{\varvec{\theta }}}_k)&\ge \mu _\mathrm {high} \Vert {{\varvec{\theta }}}_k - {{\varvec{\theta }}}^* \Vert ^2,\\ \end{aligned} \end{aligned}$$

(42)

where $\mu _\mathrm {low}$ and $\mu _\mathrm {high}$ are constants. Similarly, if the low- and high-fidelity gradients are Lipschitz continuous,

$$\begin{aligned} \begin{aligned} \Vert {\mathbf {h}}_\mathrm {low}({{\varvec{\theta }}}_k) \Vert ^2&\le L^2_\mathrm {low} \Vert {{\varvec{\theta }}}_k - {{\varvec{\theta }}}^* \Vert ^2,\\ \Vert {\mathbf {h}}_\mathrm {high}({{\varvec{\theta }}}_k) \Vert ^2&\le L^2_\mathrm {high} \Vert {{\varvec{\theta }}}_k - {{\varvec{\theta }}}^* \Vert ^2,\\ \end{aligned} \end{aligned}$$

(43)

where $L_\mathrm {low},L_\mathrm {high}$ are the Lipschitz constants for low- and high-fidelity gradients, respectively. The parameters are updated in Algorithm 4 using

$$\begin{aligned} {{\varvec{\theta }}}_{k+1} = {{\varvec{\theta }}}_{k} - \eta {\widehat{{\mathbf {h}}}}_k. \end{aligned}$$

(44)

The expected value of the search direction ${\widehat{{\mathbf {h}}}}_k$ at iteration k is

$$\begin{aligned} {\mathbb {E}}[{\widehat{{\mathbf {h}}}}_k|{{\varvec{\theta }}}_{k}] = p_l {\mathbf {h}}_\mathrm {low}({{\varvec{\theta }}}_{k}) + p_h {\mathbf {h}}_\mathrm {high}({{\varvec{\theta }}}_{k}) + (1- p_l -p_h) {\mathbf {d}}_{k-1}, \end{aligned}$$

(45)

where $p_l = N_l/N$ and $p_h = N_h/N$.

Next, we evaluate the following expectation

$$\begin{aligned} \begin{aligned} {\mathbb {E}}[\Vert {{\varvec{\theta }}}_{k+1} - {{\varvec{\theta }}}^*\Vert ^2|{{\varvec{\theta }}}_{k}]&= {\mathbb {E}}[\Vert {{\varvec{\theta }}}_{k} - {{\varvec{\theta }}}^* -\eta {\widehat{{\mathbf {h}}}}_k\Vert ^2|{{\varvec{\theta }}}_{k}]\\&= \Vert {{\varvec{\theta }}}_{k} - {{\varvec{\theta }}}^* \Vert ^2 - 2 \eta ({{\varvec{\theta }}}_{k} - {{\varvec{\theta }}}^*)^T{\mathbb {E}}[{\widehat{{\mathbf {h}}}}_k| {{\varvec{\theta }}}_{k}] \\&\quad + \eta ^2 {\mathbb {E}}[\Vert {\widehat{{\mathbf {h}}}}_k\Vert ^2| {{\varvec{\theta }}}_{k} ]\\&\le \Vert {{\varvec{\theta }}}_{k} - {{\varvec{\theta }}}^* \Vert ^2 - 2 \eta ({{\varvec{\theta }}}_{k} - {{\varvec{\theta }}}^*)^T{\mathbb {E}}[{\widehat{{\mathbf {h}}}}_k| {{\varvec{\theta }}}_{k}] \\&\quad + \eta ^2 L^2 \Vert {{\varvec{\theta }}}_{k} - {{\varvec{\theta }}}^* \Vert ^2, \end{aligned} \end{aligned}$$

(46)

where $L^2 = \max \left\{ p_l(1-p_l-p_h)^{(k-j)}\frac{L_\mathrm {low}^2\Vert {{\varvec{\theta }}}_j - {{\varvec{\theta }}}^* \Vert ^2}{\Vert {{\varvec{\theta }}}_k - {{\varvec{\theta }}}^* \Vert ^2},p_h(1-p_l-p_h)^{(k-j)}\frac{L_\mathrm {high}^2\Vert {{\varvec{\theta }}}_j - {{\varvec{\theta }}}^* \Vert ^2}{\Vert {{\varvec{\theta }}}_k - {{\varvec{\theta }}}^* \Vert ^2}\right\}$ for $j=1,\ldots ,k$. Using the strong convexity property of $J({{\varvec{\theta }}})$,

$$\begin{aligned} \begin{aligned} {\mathbb {E}}[\Vert {{\varvec{\theta }}}_{k+1} - {{\varvec{\theta }}}^*\Vert ^2|{{\varvec{\theta }}}_{k}]&\le (1 - 2 \eta \mu + \eta ^2 L^2) \Vert {{\varvec{\theta }}}_{k} - {{\varvec{\theta }}}^* \Vert ^2,\\ {\mathbb {E}}[\Vert {{\varvec{\theta }}}_{k+1} - {{\varvec{\theta }}}^*\Vert ^2|{{\varvec{\theta }}}_{0}]&\le (1 - \mu ^2/L^2)^k \Vert {{\varvec{\theta }}}_{0} - {{\varvec{\theta }}}^* \Vert ^2, \end{aligned} \end{aligned}$$

(47)

where $\mu = \min \left\{ p_l(1-p_l-p_h)^{(k-j)}\frac{\mu _\mathrm {low}\Vert {{\varvec{\theta }}}_j - {{\varvec{\theta }}}^* \Vert ^2}{\Vert {{\varvec{\theta }}}_k - {{\varvec{\theta }}}^* \Vert ^2}\right.$, $\left. p_h(1-p_l-p_h)^{(k-j)}\frac{\mu _\mathrm {high}\Vert {{\varvec{\theta }}}_j - {{\varvec{\theta }}}^* \Vert ^2}{\Vert {{\varvec{\theta }}}_k - {{\varvec{\theta }}}^* \Vert ^2}\right\}$ for $j=1,\ldots ,k$; and learning rate is chosen as $\eta = \mu /L^2$ subject to $\mu ^2/L^2\le 1$. This completes the proof of Theorem 1.

The constants $\mu$ and $L^2$ in (47) are affected by the parameter update history as mentioned in Sect. 3.1. To see this, let us define

$$\begin{aligned} \begin{aligned} c^k_{\min }&= \underset{j}{\min } \left\{ (1-p_l-p_h)^{(k-j)} \Vert {\varvec{\theta }}_j-{\varvec{\theta }}^* \Vert ^2 \right\} ;\\ c^k_{\max }&= \underset{j}{\max } \left\{ (1-p_l-p_h)^{(k-j)} \Vert {\varvec{\theta }}_j-{\varvec{\theta }}^* \Vert ^2 \right\} ; \\&\quad \text {for } j=1,\ldots ,k-1.\\ \end{aligned} \end{aligned}$$

(48)

Hence, the constants $\mu$ and $L^2$ can be written as

$$\begin{aligned} \begin{aligned} \mu&= \frac{c^k_{\min }}{\Vert {\varvec{\theta }}_k-{\varvec{\theta }}^*\Vert ^2} \min \left\{ p_l\mu _\mathrm {low},p_h\mu _\mathrm {high} \right\} ; \\ L^2&= \frac{c^k_{\max }}{\Vert {\varvec{\theta }}_k-{\varvec{\theta }}^*\Vert ^2} \max \left\{ p_lL^2_\mathrm {low},p_hL^2_\mathrm {high} \right\} . \\ \end{aligned} \end{aligned}$$

(49)

Note that, if $p_l$ and $p_h$ are fixed $\mu$ depends on $c^k_{\min }$, i.e., on $\min \left\{ (1-p_l-p_h)^{(k-j)} \Vert {\varvec{\theta }}_j-{\varvec{\theta }}^* \Vert ^2 \right\}$ for $j=1,\ldots ,k$. Further, $(1-p_l-p_h)^{(k-j)}$ increases with j since $1-p_l-p_h<1$ but $\Vert {\varvec{\theta }}_j-{\varvec{\theta }}^* \Vert ^2$ depends on the parameter updates $\left\{ {{\varvec{\theta }}}_j\right\} _{j=1}^k$. Similarly, $L^2$ depends on $c^k_{\max }$ and in turn on $\left\{ {{\varvec{\theta }}}_j\right\} _{j=1}^k$. Hence, the parameter update history affects $\mu$ and $L^2$.

Proof of Theorem 2

Using the assumption of strong convexity of objective obtained from the high-fidelity models,

$$\begin{aligned} \begin{aligned} ({{\varvec{\theta }}}_k - {{\varvec{\theta }}}^*)^T{\mathbf {h}}_\mathrm {high}({{\varvec{\theta }}}_k)&\ge \mu _\mathrm {high} \Vert {{\varvec{\theta }}}_k - {{\varvec{\theta }}}^* \Vert ^2,\\ \end{aligned} \end{aligned}$$

(50)

where $\mu _\mathrm {high}$ is a constant. Similarly, if the high-fidelity gradients are Lipschitz continuous

$$\begin{aligned} \begin{aligned} \Vert {\mathbf {h}}_\mathrm {high}({{\varvec{\theta }}}_k) \Vert ^2&\le L^2_\mathrm {high} \Vert {{\varvec{\theta }}}_k - {{\varvec{\theta }}}^* \Vert ^2,\\ \end{aligned} \end{aligned}$$

(51)

where $L_\mathrm {high}$ is the Lipschitz constant. For the inner iterations, we can evaluate the following expectation

$$\begin{aligned} \begin{aligned} {\mathbb {E}}[\Vert&{{\varvec{\theta }}}_{k+1} - {{\varvec{\theta }}}^*\Vert ^2|{{\varvec{\theta }}}_{k},{{\varvec{\theta }}}_\mathrm {prev}] \\&= {\mathbb {E}}\left[ \Bigg \Vert {{\varvec{\theta }}}_{k} - {{\varvec{\theta }}}^* -\eta \left[ {{\widehat{{\mathbf {h}}}}}_\mathrm {high}-\frac{{\varvec{\alpha }}}{N_h}\sum _{i=1}^{N_h}\left( {\mathbf {h}}_\mathrm {low}({\varvec{\theta }}_\mathrm {prev};{\varvec{\xi }}_{i})\right. \right. \right. \\&\quad \left. \left. \left. -{{\widehat{{\mathbf {h}}}}}_\mathrm {low}\right) \right] \Bigg \Vert ^2\Bigg |{{\varvec{\theta }}}_{k},{{\varvec{\theta }}}_\mathrm {prev}\right] \\&= \Vert {{\varvec{\theta }}}_{k} - {{\varvec{\theta }}}^* \Vert ^2 - 2 \eta ({{\varvec{\theta }}}_k-{{\varvec{\theta }}}^*)^T{\mathbf {h}}_\mathrm {high}({{\varvec{\theta }}}_k)+\eta ^2\Vert {\mathbf {h}}_\mathrm {high}({{\varvec{\theta }}}_k)\Vert ^2 \\&\quad + \eta ^2 \sum _{q=1}^{n_{{\varvec{\theta }}}}\mathrm {Var}\left( {{\widehat{h}}}_{\mathrm {high},q}-\frac{{\alpha }_{qq}}{N_h}\sum _{i=1}^{N_h}\left( h_{\mathrm {low},q}({\varvec{\theta }}_\mathrm {prev};{\varvec{\xi }}_{i})\right. \right. \\&\quad \left. \left. -\,{{\widehat{h}}}_{\mathrm {low},q}\right) \Bigg | {{\varvec{\theta }}}_k,{{\varvec{\theta }}}_\mathrm {prev} \right) , \end{aligned} \end{aligned}$$

(52)

where ${h}_q$ is the gradient with respect to $\theta _q$ and $\mathrm {Var}(\cdot )$ denotes variance of its argument. Note that, if ${\widehat{{\mathbf {h}}}}_\mathrm {low}={\mathbb {E}}[{\mathbf {h}}_\mathrm {low}({{\varvec{\theta }}};{\varvec{\xi }})]$ exactly,

$$\begin{aligned} \begin{aligned}&\mathrm {Var}\Bigg ( {{\widehat{h}}}_{\mathrm {high},q}-\frac{\alpha _{qq}}{N_h} \sum _{i=1}^{N_h}\big (h_{\mathrm {low},q}({\varvec{\theta }}_\mathrm {prev}; {\varvec{\xi }}_{i})-{{\widehat{h}}}_{\mathrm {low},q}\big ) \Bigg |{{\varvec{\theta }}}_k, {{\varvec{\theta }}}_\mathrm {prev}\Bigg )\\&\quad = \frac{1}{N_h} (1-\rho _{hl,q}^2)\mathrm {Var}(h_{\mathrm {high},q}( {{\varvec{\theta }}}_k;{\varvec{\xi }})), \\ \end{aligned} \end{aligned}$$

(53)

where $\alpha _{qq}=\mathrm {Cov}(h_{\mathrm {low},q}({{\varvec{\theta }}}_\mathrm {prev};{\varvec{\xi }}),h_{\mathrm {high},q}({{\varvec{\theta }}}_k;{\varvec{\xi }}))/\mathrm {Var}(h_{\mathrm {low},q}({{\varvec{\theta }}}_\mathrm {prev};{\varvec{\xi }}))$ and the correlation coefficient $\rho _{hl,q}=\mathrm {Cov}(h_{\mathrm {low},q}({{\varvec{\theta }}}_\mathrm {prev};{\varvec{\xi }}),h_{\mathrm {high},q}({{\varvec{\theta }}}_k;{\varvec{\xi }}))/\sqrt{\mathrm {Var}(h_{\mathrm {low},q}({{\varvec{\theta }}}_\mathrm {prev};{\varvec{\xi }}))\mathrm {Var}(h_{\mathrm {high},q}({{\varvec{\theta }}}_{k};{\varvec{\xi }}))}$. On the other hand, if we use $N_l$ samples to estimate ${\widehat{{\mathbf {h}}}}_\mathrm {low}$, i.e., ${\widehat{{\mathbf {h}}}}_\mathrm {low}=\frac{1}{N_l}\sum _{i=1}^{N_l}{\mathbf {h}}_\mathrm {low}({\varvec{\theta }}_\mathrm {prev};{\varvec{\xi }}_i)$ then we can write

$$\begin{aligned} \begin{aligned}&\mathrm {Var}\Bigg ( {{\widehat{h}}}_{\mathrm {high},q}-\frac{\alpha _{qq}}{N_h} \sum _{i=1}^{N_h}\big (h_{\mathrm {low},q} ({\varvec{\theta }}_\mathrm {prev};{\varvec{\xi }}_{i})-{{\widehat{h}}}_{\mathrm {low},q}\big ) \Bigg |{{\varvec{\theta }}}_k, {{\varvec{\theta }}}_\mathrm {prev}\Bigg )\\&\quad = \frac{1}{N_h} \left( 1-\frac{\rho _{hl,q}^2}{1+N_h/N_l}\right) \mathrm {Var}(h_{\mathrm {high},q}( {{\varvec{\theta }}}_k;{\varvec{\xi }})), \\ \end{aligned} \end{aligned}$$

(54)

where the coefficient $\alpha _{qq}$ is obtained by minimizing the mean-squared error in ${\widehat{h}}_{\mathrm {high},q}$ [33, 76], i.e.,

$$\begin{aligned} \alpha _{qq}=\frac{\mathrm {Cov}(h_{\mathrm {low},q}({{\varvec{\theta }}}_\mathrm {prev};{\varvec{\xi }}),h_{\mathrm {high},q}({{\varvec{\theta }}}_k;{\varvec{\xi }}))}{\mathrm {Var}(h_{\mathrm {low},q}({{\varvec{\theta }}}_\mathrm {prev};{\varvec{\xi }}))}\left( \frac{1}{1+N_h/N_l} \right) , \end{aligned}$$

(55)

and the correlation coefficient $\rho _{hl,q}$ is same as before. Next, let us assume

$$\begin{aligned}&\frac{1}{N_h} \left( 1-\frac{\rho _{hl,q}^2}{1+N_h/N_l}\right) \mathrm {Var}(h_{\mathrm {high},q}({{\varvec{\theta }}}_k;{\varvec{\xi }}))\nonumber \\&\quad \le L^2_\mathrm {high} \delta _{k,q} \Vert {{\varvec{\theta }}}_{k} - {{\varvec{\theta }}}^* \Vert ^2 \end{aligned}$$

(56)

for some constants $\delta _{k,q}$. Further, assume $\delta _k = \max \{1,\delta _{k,q} \}$ for $q=1,\ldots ,n_{{\varvec{\theta }}}$. Hence,

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}[\Vert {{\varvec{\theta }}}_{k+1} - {{\varvec{\theta }}}^*\Vert ^2|{{\varvec{\theta }}}_{k},{{\varvec{\theta }}}_\mathrm {prev}]\\&\quad \le (1-2\eta \mu _\mathrm {high}+ \eta ^2 L^2_\mathrm {high}) \Vert {{\varvec{\theta }}}_{k} - {{\varvec{\theta }}}^* \Vert ^2 \\&\qquad + \eta ^2 L^2_\mathrm {high}\delta _k \Vert {{\varvec{\theta }}}_{k} - {{\varvec{\theta }}}^* \Vert ^2\\&\quad \le (1-2\eta \mu _\mathrm {high}+ 2 \eta ^2 L^2_\mathrm {high}\delta _k) \Vert {{\varvec{\theta }}}_{k} - {{\varvec{\theta }}}^* \Vert ^2 \end{aligned} \end{aligned}$$

(57)

At kth inner iteration let us use the learning rate $\eta = \frac{\mu _\mathrm {high}}{2L^2_\mathrm {high}\delta _k}$. This leads to

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}[\Vert {{\varvec{\theta }}}_{k+1} - {{\varvec{\theta }}}^*\Vert ^2|{{\varvec{\theta }}}_{k},{{\varvec{\theta }}}_\mathrm {prev}] \le \left( 1-\frac{\mu ^2_\mathrm {high}}{2L^2_\mathrm {high}\delta _k}\right) \Vert {{\varvec{\theta }}}_{k} - {{\varvec{\theta }}}^* \Vert ^2,\\&{\mathbb {E}}[\Vert {{\varvec{\theta }}}_{k+1} - {{\varvec{\theta }}}^*\Vert ^2|{{\varvec{\theta }}}_\mathrm {prev}] \le \left( 1-\frac{\mu ^2_\mathrm {high}}{2L^2_\mathrm {high}{\underline{\delta }}}\right) ^k \Vert {{\varvec{\theta }}}_\mathrm {prev} - {{\varvec{\theta }}}^* \Vert ^2, \end{aligned} \end{aligned}$$

(58)

where ${\underline{\delta }}=\min \{\delta _i\}_{i=1}^k$; and ${{\varvec{\theta }}}_1={{\varvec{\theta }}}_\mathrm {prev}$. Similarly, for jth outer iteration,

$$\begin{aligned} \begin{aligned} {\mathbb {E}}[\Vert {{\widetilde{{{\varvec{\theta }}}}}}_{j} - {{\varvec{\theta }}}^*\Vert ^2|{{\varvec{\theta }}}_0]&\le \left( 1-\frac{\mu ^2_\mathrm {high}}{2L^2_\mathrm {high}\delta }\right) ^{jm} \Vert {{\varvec{\theta }}}_{0} - {{\varvec{\theta }}}^* \Vert ^2, \end{aligned} \end{aligned}$$

(59)

where $\delta = \min \{{{\underline{\delta }}}_i\}_{i=1}^j$ subjected to $\frac{\mu ^2_\mathrm {high}}{2L^2_\mathrm {high}\delta }\le 1$ and this proves Theorem 2.

Note that, if $\rho _{hl}$ is close to 1, i.e., the low- and the high-fidelity models are highly correlated, then $\delta _k$ can be assumed small. This implies that $\delta$ will be close to 1 and, thus, we can use a larger learning rate $\eta$. This, in turn, leads to a smaller right hand in (59) and a tighter bound on ${\mathbb {E}}[\Vert {{\widetilde{{{\varvec{\theta }}}}}}_{j} - {{\varvec{\theta }}}^*\Vert ^2|{{\varvec{\theta }}}_0]$.