We present 29 likely members of the young $rho$ Oph or Upper Sco regions of recent star formation that exhibit accretion burst type light curves in $K2$ time series photometry. The bursters were identified by visual examination of their ~80 day light curves, though all satisfy the $M < -0.25$ flux asymmetry criterion for burst behavior defined by Cody et al. (2014). The burst sources represent $approx$9% of cluster members with strong infrared excess indicative of circumstellar material. Higher amplitude burster behavior is correlated with larger inner disk infrared excesses, as inferred from $WISE$ $W1-W2$ color. The burst sources are also outliers in their large H$alpha$ emission equivalent widths. No distinction between bursters and non-bursters is seen in stellar properties such as multiplicity or spectral type. The frequency of bursters is similar between the younger, more compact $rho$ Oph region, and the older, more dispersed Upper Sco region. The bursts exhibit a range of shapes, amplitudes (~10-700%), durations (~1-10 days), repeat time scales (~3-80 days), and duty cycles (~10-100%). Our results provide important input to models of magnetospheric accretion, in particular by elucidating the properties of accretion-related variability in the low state between major longer duration events such as EX Lup and FU Ori type accretion outbursts. We demonstrate the broad continuum of accretion burst behavior in young stars -- extending the phenomenon to lower amplitudes and shorter timescales than traditionally considered in the theory of pre-main sequence accretion history.
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