  Entropy maximization and free energy minimization are general physical
principles for modeling the dynamics of various physical systems. Notable
examples include modeling decision-making within the brain using the
free-energy principle, optimizing the accuracy-complexity trade-off when
accessing hidden variables with the information bottleneck principle (Tishby et
al., 2000), and navigation in random environments using information
maximization (Vergassola et al., 2007). Built on this principle, we propose a
new class of bandit algorithms that maximize an approximation to the
information of a key variable within the system. To this end, we develop an
approximated analytical physics-based representation of an entropy to forecast
the information gain of each action and greedily choose the one with the
largest information gain. This method yields strong performances in classical
bandit settings. Motivated by its empirical success, we prove its asymptotic
optimality for the two-armed bandit problem with Gaussian rewards. Owing to its
ability to encompass the system's properties in a global physical functional,
this approach can be efficiently adapted to more complex bandit settings,
calling for further investigation of information maximization approaches for
multi-armed bandit problems.
