  We introduce and study Swap Agnostic Learning. The problem can be phrased as
a game between a predictor and an adversary: first, the predictor selects a
hypothesis $h$; then, the adversary plays in response, and for each level set
of the predictor $\{x \in \mathcal{X} : h(x) = v\}$ selects a (different)
loss-minimizing hypothesis $c_v \in \mathcal{C}$; the predictor wins if $h$
competes with the adaptive adversary's loss. Despite the strength of the
adversary, we demonstrate the feasibility Swap Agnostic Learning for any convex
loss.
  Somewhat surprisingly, the result follows through an investigation into the
connections between Omniprediction and Multicalibration. Omniprediction is a
new notion of optimality for predictors that strengthtens classical notions
such as agnostic learning. It asks for loss minimization guarantees (relative
to a hypothesis class) that apply not just for a specific loss function, but
for any loss belonging to a rich family of losses. A recent line of work shows
that omniprediction is implied by multicalibration and related multi-group
fairness notions. This unexpected connection raises the question: is
multi-group fairness necessary for omniprediction?
  Our work gives the first affirmative answer to this question. We establish an
equivalence between swap variants of omniprediction and multicalibration and
swap agnostic learning. Further, swap multicalibration is essentially
equivalent to the standard notion of multicalibration, so existing learning
algorithms can be used to achieve any of the three notions. Building on this
characterization, we paint a complete picture of the relationship between
different variants of multi-group fairness, omniprediction, and Outcome
Indistinguishability. This inquiry reveals a unified notion of OI that captures
all existing notions of omniprediction and multicalibration.
