  Learning is a fundamental characteristic of living systems, enabling them to
comprehend their environments and make informed decisions. These
decision-making processes are inherently influenced by available information
about their surroundings and specific objectives. An intriguing perspective is
that each process could be optimal under a given set of conditions. Here, a
geometric method is introduced for analytically exploring such a perspective.
The probability distribution describing the state of the composite system of
the environment, termed the information bath, and a decision-maker is discussed
on the basis of the entropic concept. This enables one to study the system in
analogy with thermodynamics. Learning processes are expressed as the changes of
parameters contained in the distribution. For a geometric interpretation of the
processes, the manifold endowed with the Fisher-Rao metric as the Riemannian
metric is considered. This framework allows one to conceptualize the optimality
of each process as a state change along a geodesic curve on the manifold, which
is referred to here as geodesic learning. Then, the bivariate Gaussian model is
presented for illustrating this approach, and its rigorous solution is
obtained. There, the correspondence between learning and cooling is made
manifest. Evolution of the entropy is evaluated for geodesic learning. The
findings are then interpreted in the context of adaptation. Geodesic learning
plays a role of the case of reference and may give novel insights into the
mechanics of learning in living systems.
