The interplay between star formation (SF) activity and active galactic nuclei (AGN) governs the co-evolution of supermassive black holes (SMBHs) and their host galaxies. AGN feedback has been hailed as the de facto process to suppress, or even shut down SF within the framework of hierarchical galaxy merger based on the current ΛCDM paradigm. However, it is unclear what physical processes regulate the growth of SMBHs and how SMBHs and their evolution are interconnected with their host galaxies when SMBHs and host galaxies are of hugely different physical scales. In fact, there has been no observational evidence to show that AGN feedback works, but rather some evidence to speculate that the more powerful AGNs reside in the more actively star-forming host galaxies. While it is difficult to measure the amount of SF from AGN host galaxies, polycyclic aromatic hydrocarbon (PAH) emission features emerged as good proxies for this purpose. Although having several caveats as SFR indicators, such as metallicity dependency, and non-SF contribution from evolved stellar populations, or AGNs, PAH emissions have been utilized to investigate SF activity of AGN host galaxies with varying results. Utilizing the slitless spectroscopic apability of the AKARI Infrared Camera, we obtained the spectra in the wavelength range of 2∼5 μm from extended regions of 79 type 1 AGN host galaxies to detect and measure the 3.3 μm (PAH) emission feature as star formation rate proxy. Based on 18 sample galaxies, we found that the luminosity of the 3.3 μm PAH emission feature is strongly correlated with AGN luminosity, except for ultra-luminous infrared galaxies (ULIRGs). Therefore, we suggest that host galaxies with stronger AGN activities have stronger star formation activities. However, it is still unclear why ULIRGs deviate from the correlation, not to mention why the detection rate of the 3.3 μm emission feature is so low. High spatial resolution imaging not only for the circumnuclear region of AGN host galaxies, but also for entire galaxies should help the cause. We present the prospective studies to diagnose SF regulation for AGN host galaxies with various space telescope facilities, such as JWST, and SPHEREx.