  We revisit $M$-ary classification of Gutman (TIT 1989), where one is tasked
to determine whether a testing sequence is generated with the same distribution
as one of the $M$ training sequences or not. Our main result is a two-phase
test, its theoretical analysis and its optimality guarantee. Specifically, our
two-phase test is a special case of a sequential test with only two decision
time points: the first phase of our test is a fixed-length test with a reject
option, the second-phase of our test proceeds only if a reject option is
decided in the first phase and the second phase of our test does \emph{not}
allow a reject option. To provide theoretical guarantee for our test, we derive
achievable error exponents using the method of types and derive a converse
result for the optimal sequential test using the techniques recently proposed
by Hsu, Li and Wang (ITW, 2022) for binary classification. Analytically and
numerically, we show that our two phase test achieves the performance of an
optimal sequential test with proper choice of test parameters. In particular,
similarly as the optimal sequential test, our test does not need a final reject
option to achieve the optimal error exponent region while an optimal
fixed-length test needs a reject option to achieve the same region. Finally, we
specialize our results to binary classification when $M=2$ and to $M$-ary
hypothesis testing when the ratio of the lengths of training sequences and
testing sequences tends to infinity so that generating distributions can be
estimated perfectly.
