Learning Curve

street fighter by David Soames

A learning curve graphically represents the amount of experience it takes to learn a given task. Skills with a steep learning curve are difficult to learn quickly, but progress comes rapidly once past the initial hurdle. Activities with a shallow learning curve, by contrast, are said to be ‘easy to learn, hard to master’ (Bushnell’s Law of video game design).

The term can refer to individual tasks repeated in a series of trials or a body of knowledge is learned over time. It was first described by German psychologist Hermann Ebbinghaus in 1885. His tests involved memorizing series of nonsense syllables, and recording the success over a number of trials. The translation does not use the term learning curve—but he presents diagrams of learning against trial number. He also notes that the score can decrease, or even oscillate.

In 1936, aeronautical engineer Theodore Paul Wright examined the effect of learning on production costs in the US aircraft industry. This form, in which unit cost is plotted against total production, is sometimes called an ‘experience curve,’ which shows if efficiency gains are worth the investment in the effort. Efficiency and productivity improvement can be considered as whole organization or industry or economy learning processes, as well as for individuals. The general pattern is of first speeding up and then slowing down, as the practically achievable level of methodology improvement is reached. The effect of reducing local effort and resource use by learning improved methods paradoxically often has the opposite latent effect on the next larger scale system, by facilitating its expansion, or economic growth, as discussed in the Jevons paradox (increasing the efficiency of resource consumption leads to more, not less, of the resource being consumed) and updated in the Khazzoom-Brookes Postulate (increased energy efficiency paradoxically tends to lead to increased energy consumption).

Plots relating performance to experience are widely used in machine learning. Performance is the error rate or accuracy of the learning system, while experience may be the number of training examples used for learning or the number of iterations used in optimizing the system model parameters. The machine learning curve is useful for many purposes including comparing different algorithms, choosing model parameters during design, adjusting optimization to improve convergence, and determining the amount of data used for training.

Initially introduced in educational and behavioral psychology, the term has acquired a broader interpretation over time, and expressions such as ‘experience curve,’ ‘improvement curve,’ ‘progress function,’ ‘startup curve,’ and ‘efficiency curve’ are often used interchangeably. In economics the subject is rates of ‘development,’ as development refers to a whole system learning process with varying rates of progression. Generally speaking all learning displays incremental change over time, but describes an ‘S’ curve which has different appearances depending on the time scale of observation. It has now also become associated with the evolutionary theory of punctuated equilibrium (the theory that evolutionary change happens in brief stops and starts, with organisms spending most of their geological time in relative stasis) and other kinds of revolutionary change in complex systems generally, relating to innovation, organizational behavior and the management of group learning, among other fields. These processes of rapidly emerging new form appear to take place by complex learning within the systems themselves, which when observable, display curves of changing rates that accelerate and decelerate.

Learning curves, also called experience curves, relate to the much broader subject of natural limits for resources and technologies in general. Such limits generally present themselves as increasing complications that slow the learning of how to do things more efficiently, like the well-known limits of perfecting any process or product or to perfecting measurements. These practical experiences match the predictions of the second law of thermodynamics for the limits of waste reduction generally. Approaching limits of perfecting things to eliminate waste meets geometrically increasing effort to make progress, and provides an environmental measure of all factors seen and unseen changing the learning experience. Perfecting things becomes ever more difficult despite increasing effort despite continuing positive, if ever diminishing, results. The same kind of slowing progress due to complications in learning also appears in the limits of useful technologies and of profitable markets applying to product life cycle management and software development cycles). Remaining market segments or remaining potential efficiencies or efficiencies are found in successively less convenient forms. Efficiency and development curves typically follow a two-phase process of first bigger steps corresponding to finding things easier, followed by smaller steps of finding things more difficult. It reflects bursts of learning following breakthroughs that make learning easier followed by meeting constraints that make learning ever harder, perhaps toward a point of cessation.

‘Natural limits’ are one of the key studies in the area concerns diminishing returns on investments generally, either physical or financial, pointing to whole system limits for resource development or other efforts. The most studied of these may be ‘Energy Return on Energy Invested’ (EROEI), referred to as ‘Hubert curves.’ The energy needed to produce energy is a measure of our difficulty in learning how to make remaining energy resources useful in relation to the effort expended. Energy returns on energy invested have been in continual decline for some time, caused by natural resource limits and increasing investment. Energy is both nature’s and our own principal resource for making things happen. The ‘point of diminishing returns’ is when increasing investment makes the resource more expensive. As natural limits are approached, easily used sources are exhausted and ones with more complications need to be used instead.



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