Learning curve

A learning curve is a graphical representation of the relationship between how proficient people are at a task and the amount of experience they have. Proficiency (measured on the vertical axis) usually increases with increased experience (the horizontal axis), that is to say, the more someone, groups, companies or industries perform a task, the better their performance at the task.^[1]

For other uses, see Learning curve (disambiguation).

The common expression "a steep learning curve" is a misnomer suggesting that an activity is difficult to learn and that expending much effort does not increase proficiency by much, although a learning curve with a steep start actually represents rapid progress.^[2]^[3] In fact, the gradient of the curve has nothing to do with the overall difficulty of an activity, but expresses the expected rate of change of learning speed over time. An activity that it is easy to learn the basics of, but difficult to gain proficiency in, may be described as having "a steep learning curve".

The learning curve may refer to a specific task or a body of knowledge. Hermann Ebbinghaus first described the learning curve in 1885 in the field of the psychology of learning, although the name did not come into use until 1903.^[4]^[5] In 1936 Theodore Paul Wright described the effect of learning on production costs in the aircraft industry.^[6] This form, in which unit cost is plotted against total production, is sometimes called an experience curve, or Wright's law.

Plateau model: $y=\max(Kx^{n},K_{0})$ , where $K_{0}$ models the minimal cost achievable. In other words, the learning ceases after cost reaches a sufficiently low level.

Stanford-B model: $y=K(x+B)^{n}$ , where $B$ models worker's prior experience.

DeJong's model: $y=K(M+(1-M)x^{n})$ , where $M$ models the fraction of production done by machines (assumed to be unable to learn, unlike a human worker).

S-curve model: $y=K(M+(1-M)(x+B)^{n})$ , a combination of Stanford-B model and DeJong's model.

The horizontal axis represents experience either directly as time (clock time, or the time spent on the activity), or can be related to time (a number of trials, or the total number of units produced).

The vertical axis is a measure representing 'learning' or 'proficiency' or other proxy for "efficiency" or "productivity". It can either be increasing (for example, the score in a test), or decreasing (the time to complete a test).

A learning curve is a plot of proxy measures for implied learning (proficiency or progression toward a limit) with experience.

For the performance of one person in a series of trials the curve can be erratic, with proficiency increasing, decreasing or leveling out in a plateau.

When the results of a large number of individual trials are averaged then a smooth curve results, which can often be described with a mathematical function.

Several main functions have been used:^[23]^[24]^[25]

The specific case of a plot of Unit Cost versus Total Production with a power law was named the experience curve: the mathematical function is sometimes called Henderson's Law. This form of learning curve is used extensively in industry for cost projections.^[26]

Broader interpretations[edit]

Initially introduced in educational and behavioral psychology, the term has acquired a broader interpretation over time, and expressions such as "experience curve", "improvement curve", "cost improvement curve", "progress 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 and other kinds of revolutionary change in complex systems generally, relating to innovation, organizational behavior and the management of group learning, among other fields.^[31] 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.

Natural Limits 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 or EROEI, discussed at length in an Encyclopedia of the Earth article and in an OilDrum article and series also 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. As an environmental signal persistently diminishing EROI indicates an approach of whole system limits in our ability to make things happen.

Energy Return on Energy Invested

Useful Natural Limits EROEI measures the return on invested effort as a ratio of R/I or learning progress. The inverse I/R measures learning difficulty. The simple difference is that if R approaches zero R/I will too, but I/R will approach infinity. When complications emerge to limit learning progress the limit of useful returns, uR, is approached and R-uR approaches zero. The difficulty of useful learning I/(R-uR) approaches infinity as increasingly difficult tasks make the effort unproductive. That point is approached as a vertical asymptote, at a particular point in time, that can be delayed only by unsustainable effort. It defines a point at which enough investment has been made and the task is done, usually planned to be the same as when the task is complete. For unplanned tasks it may be either foreseen or discovered by surprise. The usefulness measure, uR, is affected by the complexity of environmental responses that can only be measured when they occur unless they are foreseen.

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.^[32] 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.

In culture[edit]

"Steep learning curve"[edit]

The expression "steep learning curve" is used with opposite meanings. Most sources, including the Oxford Dictionary of English, the American Heritage Dictionary of the English Language, and Merriam-Webster's Collegiate Dictionary, define a learning curve as the rate at which skill is acquired, so a steep increase would mean a quick increment of skill.^[2]^[33] However, the term is often used in common English with the meaning of a difficult initial learning process.^[3]^[33]

The common English usage aligns with a metaphorical interpretation of the learning curve as a hill to climb. (A steeper hill is initially hard, while a gentle slope is less strainful, though sometimes rather tedious. Accordingly, the shape of the curve (hill) may not indicate the total amount of work required. Instead, it can be understood as a matter of preference related to ambition, personality and learning style.)

Learning curve

Broader interpretations[edit]

Energy Return on Energy Invested

In culture[edit]

"Steep learning curve"[edit]

Forgetting curve

Learning speed

Labor productivity

Learning-by-doing (economics)

Population growth

Trial and error

Learning curve