  Feedback from active galactic nuclei (AGN) and supernovae can affect
measurements of integrated SZ flux of halos ($Y_\mathrm{SZ}$) from CMB surveys,
and cause its relation with the halo mass ($Y_\mathrm{SZ}-M$) to deviate from
the self-similar power-law prediction of the virial theorem. We perform a
comprehensive study of such deviations using CAMELS, a suite of hydrodynamic
simulations with extensive variations in feedback prescriptions. We use a
combination of two machine learning tools (random forest and symbolic
regression) to search for analogues of the $Y-M$ relation which are more robust
to feedback processes for low masses ($M\lesssim 10^{14}\, h^{-1} \, M_\odot$);
we find that simply replacing $Y\rightarrow Y(1+M_*/M_\mathrm{gas})$ in the
relation makes it remarkably self-similar. This could serve as a robust
multiwavelength mass proxy for low-mass clusters and galaxy groups. Our
methodology can also be generally useful to improve the domain of validity of
other astrophysical scaling relations.
  We also forecast that measurements of the $Y-M$ relation could provide
percent-level constraints on certain combinations of feedback parameters and/or
rule out a major part of the parameter space of supernova and AGN feedback
models used in current state-of-the-art hydrodynamic simulations. Our results
can be useful for using upcoming SZ surveys (e.g., SO, CMB-S4) and galaxy
surveys (e.g., DESI and Rubin) to constrain the nature of baryonic feedback.
Finally, we find that the an alternative relation, $Y-M_*$, provides
complementary information on feedback than $Y-M$
