What we learn from the learning rate

Abstract

The learning rate is an information-theoretical quantity for bipartite Markov chains describing two coupled subsystems. It is defined as the rate at which transitions in the downstream subsystem tend to increase the mutual information between the two subsystems, and is bounded by the dissipation arising from these transitions. Its physical interpretation, however, is unclear, although it has been used as a metric for the sensing performance of the downstream subsystem. In this paper, we explore the behaviour of the learning rate for a number of simple model systems, establishing when and how its behaviour is distinct from the instantaneous mutual information between subsystems. In the simplest case, the two are almost equivalent. In more complex steady-state systems, the mutual information and the learning rate behave qualitatively distinctly, with the learning rate clearly now reflecting the rate at which the downstream system must update its information in response to changes in the upstream system. It is not clear whether this quantity is the most natural measure for sensor performance, and, indeed, we provide an example in which optimising the learning rate over a region of parameter space of the downstream system yields an apparently sub-optimal sensor. arXiv:1702.06041v2 [cond-mat.stat-mech]

Energy Dissipation and Information Flow in Coupled Markovian Systems

“…is a learning rate which quantifies the information gained by the system about the current environmental state [17]. The denominator follows from the facts that at steady state − Q = W (due to energy conservation) and W = W diss [7].…”

Theory of Nonequilibrium Free Energy Transduction by Molecular Machines

“…287,288 At steady state this nostalgia (for unit-time steps) equals the learning rate, the rate at which a system (due to its own dynamics) increases its mutual information with the environment. [289][290][291][292][293] Intuitively, this says that thermodynamic efficiency is accomplished by forgetting (i.e., rapidly randomizing and hence relaxing to equilibrium with respect to) degrees of freedom in one's environment that are not predictive of future fluctuations. Possible implications for efficient transduction of environmental fluctuations by molecular machines are that they must ignore the myriad aspects of their environment (e.g., sundry collisions from water molecules) that have no bearing on mechanically meaningful future environmental fluctuations.…”

Role of sufficient statistics in stochastic thermodynamics and its implication to sensory adaptation

“…To characterize the directional information flow from one system to the other, we now consider the two informational quantities: the learning rate [11,16,17,19,26] and the transfer entropy [5-7, 10, 16, 17]. The learning rate has recently been introduced in Ref.…”

“…In particular, thermodynamics of autonomous measurement and feedback has been developed [4][5][6][7][8][9][10][11][12][13][14][15][16][17] and applied to biological systems [16][17][18][19][20][21][22][23][24], where the concept of continuous information flow has played a significant role. Specifically, the transfer entropy [4-7, 10, 16, 17, 25] and the learning rate [4,11,16,17,19,26] have been shown related to the second law of thermodynamics. The ratio of these two informational quantities is referred to as the sensory capacity [16,17], which is argued to be a measure of the effectiveness of stochastic sensors.…”