$\beta$–$M_{\mathrm{UV}}$ relation at $M_{\mathrm{UV}} \sim -19.0$, and the dependence of $\beta$ on $M_{\mathrm{UV}}$ becomes weaker at $M_{\mathrm{UV}} \gtrsim -19.0$ than at $M_{\mathrm{UV}} < -19.0$ [@Bouw14]. The change of the dependence is interpreted as the change of the dependence of the dust extinction on the UV luminosity or the stellar mass (e.g., [@Pann09]; [@Redd10]). The semi-analytic model predicts the sudden change of the dust-to-gas mass ratio at the critical metallicity and the dust mass rapidly increase at the metallicity level larger than the critical metallicity (e.g., [@Hira11]). The change of the dependence of $\beta$ on $M_{\mathrm{UV}}$ perhaps indicates the existence of the critical metallicity. The redshift dependence of $\beta$ is interpreted as the dust attenuation history or the dust production history in star-forming galaxies. Interestingly, the dust attenuation history at $z \gtrsim 3$ estimated from the redshift dependence of $\beta$ is smoothly connected with the dust attenuation history at $z \lesssim 3$ estimated from the direct measurements of both IR and UV luminosity [@Burg13]. The dust attenuation history is also used for revealing the history of the true (dust-corrected) cosmic Star Formation Rate (SFR) density in the high-z universe because it is still difficult to obtain the IR luminosity for high-z 