  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.
