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.Compare:

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., see the "Discussions" section, Dr. Smith's remark about the usage of the term "steep learning curve": "First, semantics. A steep learning curve is one where you gain proficiency over a short number of trials. That means the curve is steep. I think semantically we are really talking about a prolonged or long learning curve. I know it is a subtle distinction, but I can't miss the opportunity to make that point." 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. In 1936 Theodore Paul Wright described the effect of learning on production costs in the aircraft industry. This form, in which unit cost is plotted against total production, is sometimes called an experience curve, or Wright's law.

In psychology

Hermann Ebbinghaus' memory tests, published in 1885, 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.

The first known use of the term 'learning curve' is from 1903: "Bryan and Harter (6) found in their study of the acquisition of the telegraphic language a learning curve which had the rapid rise at the beginning followed by a period of slower learning, and was thus convex to the vertical axis."

Psychologist Arthur Bills gave a more detailed description of learning curves in 1934. He also discussed the properties of different types of learning curves, such as negative acceleration, positive acceleration, plateaus, and ogive curves.Bills, A.G. (1934). General experimental psychology. Longmans Psychology Series. pp. 192–215. New York: Longmans, Green and Co.

In economics

History

In 1936, Theodore Paul Wright described the effect of learning on production costs in the aircraft industry and proposed a mathematical model of the learning curve.

In 1952, the US Air Force published data on the learning curve in the airframe industry from 1940 to mid-1945.Air Materiel Command Wright-Patterson AFB OH. "Source Book of World War II Basic Data-Airframe Industry. Volume 1. Direct Man-Hours-Progress Curves." (1952): 0201. Specifically, they tabulated and plotted the direct man-hour cost of various products as a function of cumulative production. This formed the basis of many studies on learning curves in the 1950s.Asher, H. (1956). Cost-quantity relationships in the airframe industry (Doctoral dissertation, The Ohio State University).

In 1968 Bruce Henderson of the Boston Consulting Group (BCG) generalized the Unit Cost model pioneered by Wright, and specifically used a Power Law, which is sometimes called Henderson's Law. He named this particular version the experience curve.Henderson, Bruce (1968-01-01) The Experience Curve. Boston Consulting Group Research by BCG in the 1970s observed experience curve effects for various industries that ranged from 10 to 25 percent.

Models

The main statistical models for learning curves are as follows:

Wright's model ("log-linear"): , where is the cost of the -th unit, is the total number of units made, is the cost of the first unit made, is the exponent measuring the strength of learning. Plateau model: , where models the minimal cost achievable. In other words, the learning ceases after cost reaches a sufficiently low level. Stanford-B model: , where models worker's prior experience. DeJong's model: , where models the fraction of production done by machines (assumed to be unable to learn, unlike a human worker). S-curve model: , a combination of Stanford-B model and DeJong's model.

The key variable is the exponent measuring the strength of learning. It is usually expressed as , where is the "learning rate". In words, it means that the unit cost decreases by , for every doubling of total units made. Wright found that in aircraft manufacturing, meaning that the unit cost decreases by 20% for every doubling of total units made.

Applications The economic learning of productivity and efficiency generally follows the same kinds of experience curves and have interesting secondary effects. 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 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 in the 1880s and updated in the Khazzoom–Brookes Postulate in the 1980s.

A comprehensive understanding of the application of learning curve on managerial economics would provide plenty of benefits on strategic level. People could predict the appropriate timing of the introductions for new products and offering competitive pricing decisions, deciding investment levels by stimulate innovations on products and the selection of organizational design structures. Balachander and Srinivasan used to study a durable product and its pricing strategy on the principles of the learning curve. Based on the concepts that the growing experience in producing and selling a product would cause the decline of unit production cost, they found the potential best introductory price for this product. As for the problems of production management under the limitation of scarce resources, Liao observed that without including the effects of the learning curve on labor hours and machines hours, people might make incorrect managerial decisions. Demeester and Qi used the learning curve to study the transition between the old products' eliminating and new products' introduction. Their results indicated that the optimal switching time is determined by the characteristics of product and process, market factors, and the features of learning curve on this production. Konstantaras, Skouri, and Jaber applied the learning curve on demand forecasting and the economic order quantity. They found that the buyers obey to a learning curve, and this result is useful for decision-making on inventory management.

Learning curves have been used to model Moore's law in the semiconductor industry.

When wages are proportional to number of products made, workers may resist changing to a different post or having a new member on the team, since it would temporarily decrease productivity. Learning curves has been used to adjust for temporary dips so that workers are paid more for the same product while they are learning.

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

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:Newell, A. (1980) Mechanisms of skill acquisition and the law of practice. University of Southern CaliforniaRitter, F. E., & Schooler, L. J. (2002) "The learning curve". In International Encyclopedia of the Social and Behavioral Sciences, pp. 8602–8605. Amsterdam: Pergamon.

The S-Curve or Sigmoid function is the idealized general form of all learning curves, with slowly accumulating small steps at first followed by larger steps and then successively smaller ones later, as the learning activity reaches its limit. That idealizes the normal progression from discovery of something to learn about followed to the limit of learning about it. The other shapes of learning curves (4, 5 & 6) show segments of S-curves without their full extents. In this case the improvement of proficiency starts slowly, then increases rapidly, and finally levels off. Exponential growth; the proficiency can increase without limit, as in Exponential growth Exponential rise or fall to a Limit; proficiency can exponentially approach a limit in a manner similar to that in which a capacitor charges or discharges (exponential decay) through a resistor. The increase in skill or retention of information may increase rapidly to its maximum rate during the initial attempts, and then gradually levels out, meaning that the subject's skill does not improve much with each later repetition, with less new knowledge gained over time. Power law; similar in appearance to an exponential decay function, and is almost always used for a decreasing performance metric, such as cost. It also has the property that if plotted as the logarithm of proficiency against the logarithm of experience the result is a straight line, and it is often presented that way.

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. U.S. Department of Defense Manual Number 5000.2-M, mandates the use of learning curves for costing of defense programs (variable costs of production)

In machine learning

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.

Broader interpretations