Covariance Matrix Adaptation Evolution Strategy for Low Effective Dimensionality

TL;DR

Countermeasures for LED are incorporated into the CMA-ES and the experimental results show the CMA-ES-LED outperforms the CMA-ES on benchmark functions with LED.

Abstract

Despite the state-of-the-art performance of the covariance matrix adaptation evolution strategy (CMA-ES), high-dimensional black-box optimization problems are challenging tasks. Such problems often involve a property called low effective dimensionality (LED), in which the objective function is formulated with redundant dimensions relative to the intrinsic objective function and a rotation transformation of the search space. The CMA-ES suffers from LED for two reasons: the default hyperparameter setting is determined by the total number of dimensions, and the norm calculations in step-size adaptations are performed including elements on the redundant dimensions. In this paper, we incorporate countermeasures for LED into the CMA-ES and propose CMA-ES-LED. We tackle with the rotation transformation using the eigenvectors of the covariance matrix. We estimate the effectiveness of each dimension in the rotated search space using the element-wise signal-to-noise ratios of the mean vector update and the rank-$μ$ update, both of which updates can be explained as the natural gradient ascent. Then, we adapt the hyperparameter using the estimated number of effective dimensions. In addition, we refine the cumulative step-size adaptation and the two-point step-size adaptation to measure the norms only on the effective dimensions. The experimental results show the CMA-ES-LED outperforms the CMA-ES on benchmark functions with LED.

Figures (10)

• Figure 1: The conceptual image of function with LED. We simply consider the case where the objective function contains a rotation matrix $\boldsymbol{R} \in \mathbb{R}^{N \times N}$ and an intrinsic objective function $\tilde{f}: \mathbb{R}^{N_{\mathrm{eff}}} \to \mathbb{R}$. The objective function value at $\boldsymbol{x}$ is given by $f(\boldsymbol{x}) = \tilde{f}(\psi(\boldsymbol{R} \boldsymbol{x}))$, where $\psi(\boldsymbol{y}) = (\boldsymbol{y}_{1}, \cdots, \boldsymbol{y}_{N_{\mathrm{eff}}})^\mathrm{T} \in \mathbb{R}^{N_{\mathrm{eff}}}$. This figure shows an example with $N=2$ and $N_{\mathrm{eff}} = 1$. See Section \ref{['sec:proposed']} for detail.
• Figure 2: The transitions of the norms of rotated eigenvectors of the covariance matrix on the effective dimensions $\| \bar{\boldsymbol{b}}_i \|_\mathrm{eff}$. We also plot the number of lines above and below 0.5. This is a typical result of CMA-ES with the CSA on the Sphere function. We set $N = 16$ and $N_{\mathrm{eff}}=8$. The rotation matrix $\boldsymbol{R}$ was randomly given.
• Figure 3: The transitions of the elements of $\boldsymbol{v}_{\mathrm{snr}}$ on the sphere function (green) and the random function (gray). The solid lines and shaded areas show the median and ranges between the minimum and maximum, respectively. Note that these lines are obtained by a single trial of the CMA-ES with the CSA.
• Figure 4: The maximum element of $\boldsymbol{v}_{\mathrm{snr}}$ from 1,000-th iteration to 2,000-th iteration in the CMA-ES on the random function. The average values over ten runs are displayed.
• Figure 5: Comparison of the number of function evaluations divided by the success rate and the number of effective dimensions on the benchmark functions with redundant dimensions. The median values and the interquartile ranges over 20 trials are displayed for each $N$. The ratio of successful trials is shown when less than 15 trials were successful.