Convergence and error estimates for time-discrete consensus-based optimization algorithms

Abstract

We present convergence and error estimates of modified versions of the time-discrete consensus-based optimization (CBO) algorithm proposed in Carrillo et al. (ESAIM: Control Optim Calc Var, 2020) for general non-convex functions. In authors’ recent work (Ha et al. in Math Models Meth Appl Sci 30:2417–2444, 2020), rigorous error analysis of a modified version of the first-order consensus-based optimization algorithm proposed in Carrillo et al. (2020) was studied at the particle level without resorting to the kinetic equation via a mean-field limit. However, the error analysis for the corresponding time- discrete algorithm was not done mainly due to lack of discrete analogue of Itô’s stochastic calculus. In this paper, we provide a simple and elementary convergence and error analysis for a general time-discrete consensus-based optimization algorithm, which includes modifications of the three discrete algorithms in Carrillo et al. (2020), two of which are present in Ha et al. (2020). Our analysis provides numerical stability and convergence conditions for the three algorithms, as well as error estimates to the global minimum.

Error estimate under an alternative framework

In this subsection, we provide an alternative result for an error estimate for time-discrete CBO scheme (1.1) without using Laplace’s principle under a slightly different framework $({{\mathcal {B}}}1) - ({{\mathcal {B}}}2)$. Below, we present a framework for the objective function L, global minimum point $X_*$ and reference random variable ${X_\text {in}}$ as follows:

• $({{\mathcal {B}}}1)$: Let $L = L(x)$ be a $C^2$-objective function satisfying the following relations:

  $$\begin{aligned} L_m{:}{=}\min _{x \in {\mathbb {R}}^d} L(x) >0 \quad \text{ and } \quad C_L{:}{=}\sup _{x\in {\mathbb {R}}^d}\Vert \nabla ^2L(x)\Vert _2 <\infty , \end{aligned}$$

  where $\Vert \cdot \Vert _2$ denotes the spectral norm.

• $({{\mathcal {B}}}2)$: Let ${X_\text {in}}$ be a reference random variable associated with a law whose support ${{\tilde{D}}}$ is compact and contains $X_*$.

Note that the condition $({{\mathcal {B}}}1)$ is exactly the same as $({{\mathcal {A}}}1)$, whereas the condition $({{\mathcal {B}}}2)$ is different from $({{\mathcal {A}}}3)$ in which the probability measures associated with ${X_\text {in}}$ is absolutely continuous with respect to Lebesgue measure. Moreover, notice that this new framework does not need any condition on ${\text {det}}\left( \nabla ^2 L(X_*)\right)$ as in $({{\mathcal {A}}}2)$.

Next, we study how the common consensus state $X_\infty$ is close to the global minimum $X_*$ in suitable sense. Now, we are ready to provide an error estimate for the discrete CBO algorithm, which is analogous to Theorem 4.1 in [14] for the continuous case.

Theorem 5.1

Suppose that the framework $({{\mathcal {B}}}1) - ({{\mathcal {B}}}2)$ holds, and parameters $\beta , \gamma , \zeta , \delta$ and the initial data $\{X_0^i \}$ satisfy

$$\begin{aligned} \begin{aligned}&\beta> 0, \quad \delta > 0, \quad (\gamma -1)^2 + \zeta ^2< 1, \quad X_0^i: i,i.d, \quad X_0^i \sim {X_\text {in}}, \quad \sup _{x \in {\tilde{{\mathcal {D}}}}} L(x) - L_m < \delta , \\&(1-\varepsilon ){\mathbb {E}} \Big [ e^{-\beta L({X_\text {in}})} \Big ] \\&\ge \frac{2C_L \sqrt{ \big (1+ (1-\gamma )^2+\zeta ^2\big ) \big ( \gamma ^2+\zeta ^2\big )} \beta e^{-\beta L_m}}{1-e^{-[ 1-(\gamma - 1)^2 - \zeta ^2]}} \sum _{l=1}^d \left( {\mathbb {E}}\max _{1\le i\le N} (x_0^{i,l} -{{\bar{x}}}^l_0)^2\right) , \end{aligned} \end{aligned}$$

(5.1)

for some $0<\varepsilon <1$. Then, for a solution $\{X_n^i\}_{1\le i\le N}$ to (1.1),

$$\begin{aligned} \Big | \mathop {\mathrm{essinf}}\limits _{\omega \in \Omega } L(X_\infty ) - L_m \Big | \le \delta + \Big | \frac{\log \varepsilon }{\beta } \Big |. \end{aligned}$$

(5.2)

Before we provide a proof, we give several comments on the result of this theorem. 1. In the proof of Theorem 5.1, we will first derive the estimate:

$$\begin{aligned} \mathop {\mathrm{essinf}}\limits _{\omega \in \Omega } L(X_\infty )\le \sup \limits _{x\in {{\tilde{D}}}} L(x)-\frac{1}{\beta }\log \varepsilon . \end{aligned}$$

(5.3)

2. For any given $\delta> 0,~\varepsilon > 0$ and $\beta > 0$, the conditions (5.1) can be attained with suitable ${X_\text {in}}$. To see this, we choose the law of ${X_\text {in}}$ such that ${{\tilde{D}}}$ is a small neighborhood of a global minimum $X_*$ satisfying the following two relations:

$$\begin{aligned} \sup \limits _{x\in {{\tilde{D}}}} L(x)-L_m<\delta , \end{aligned}$$

(5.4)

and

$$\begin{aligned} 1-\varepsilon \ge \frac{2C_L \sqrt{ \big (1+ (1-\gamma )^2+\zeta ^2\big ) \big ( \gamma ^2+\zeta ^2\big )} \beta e^{\beta (\sup \limits _{x\in D}L(x)- L_m)}}{1-e^{-[ 1-(\gamma - 1)^2 - \zeta ^2]}} \big ( {\text {diam}}({{\mathcal {R}}}_{in})\big )^2,\nonumber \\ \end{aligned}$$

(5.5)

where ${{\mathcal {R}}}_{in} = [a_1, b_1] \times \cdots \times [a_d, b_d]$ is the smallest closed d-dimensional rectangle containing ${{\tilde{D}}}$ so that

$$\begin{aligned} {\mathbb {E}} \Big [ \max _{1\le i\le N} (x_0^{i,l} -{{\bar{x}}}^l_0)^2 \Big ] \le |b_\ell - a_\ell |^2, \end{aligned}$$

(5.6)

for each $\ell = 1, \ldots , d$. Then, due to (5.6), the relation (5.5) implies the condition (5.1)$_2$. Hence, we can apply the estimate (5.3) and (5.4) to get the desired error estimate:

$$\begin{aligned} \mathop {\mathrm{essinf}}\limits _{\omega \in \Omega } L(X_\infty )\le L_m+\left( \sup \limits _{x\in {{\tilde{D}}}} L(x)-L_m\right) -\frac{1}{\beta }\log \varepsilon <L_m+\delta -\frac{\log \varepsilon }{\beta }. \end{aligned}$$

Now we are ready to provide a proof of Theorem 5.1. Similar to the proof of Theorem 3.2, we obtain

$$\begin{aligned} \mathbb Ee^{-\beta L(X_\infty )}&\ge \frac{1}{N}\sum _{i=1}^N {\mathbb {E}} e^{-\beta L(X_0^i)} \\&\quad -\frac{2C_L \sqrt{ \big (1+ (1-\gamma )^2+\zeta ^2\big ) \big ( \gamma ^2+\zeta ^2\big )} \beta e^{-\beta L_m}}{1-e^{-(2\gamma - \gamma ^2 -\zeta ^2)}} \\&\quad \sum _{l=1}^d \left( {\mathbb {E}}\max _{1\le i\le N} (x_0^{i,l} -{{\bar{x}}}^l_0)^2\right) \\&\ge \varepsilon {\mathbb {E}} e^{-\beta L({X_\text {in}})}. \end{aligned}$$

Hence one has

$$\begin{aligned} e^{-\beta \mathop {\mathrm{essinf}}\limits _{\omega \in \Omega } L(X_\infty )} =\mathbb Ee^{-\beta \mathop {\mathrm{essinf}}\limits _{\omega \in \Omega }L(X_\infty )}\ge \mathbb Ee^{-\beta L(X_\infty )}\ge \varepsilon {\mathbb {E}} e^{-\beta L({X_\text {in}})}\ge \varepsilon e^{-\beta \sup \limits _{x\in {{\tilde{D}}}} L(x)}. \end{aligned}$$

Finally, we take logarithm to the both sides of the above relation to get the desired estimate (5.3).