Learning curve

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An example of a subject becoming more proficient at a task as they spend more time doing it. In this example, proficiency increases rapidly at first but at later stages there are diminishing returns.

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An example of what the common (yet confusing) expression "steep learning curve" is referring to. The subject spends a great amount of time but does not see an increase in proficiency at first.

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]

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".^{[citation needed]}

The learning curve may refer to a specific task or a

In psychology

Hermann Ebbinghaus' memory tests, published in 1885, involved memorizing series of

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."[5]^[3]

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.^[8]

In economics

History

In 1936,

In 1952, the US Air Force published data on the learning curve in the airframe industry from 1940 to mid-1945.^[9] 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.^[10]

In 1968

Models

The main statistical models for learning curves are as follows:^[15][16]

• Wright's model ("log-linear"): ${\displaystyle y=Kx^{n}}$, where
  • ${\displaystyle y}$ is the cost of the ${\displaystyle x}$-th unit,
  • ${\displaystyle x}$ is the total number of units made,
  • ${\displaystyle K}$ is the cost of the first unit made,
  • ${\displaystyle n}$ is the exponent measuring the strength of learning.
• Plateau model: ${\displaystyle y=\max(Kx^{n},K_{0})}$, where ${\displaystyle K_{0}}$ models the minimal cost achievable. In other words, the learning ceases after cost reaches a sufficiently low level.
• Stanford-B model: ${\displaystyle y=K(x+B)^{n}}$, where ${\displaystyle B}$ models worker's prior experience.
• DeJong's model: ${\displaystyle y=K(M+(1-M)x^{n})}$, where ${\displaystyle M}$ models the fraction of production done by machines (assumed to be unable to learn, unlike a human worker).
• S-curve model: ${\displaystyle y=K(M+(1-M)(x+B)^{n})}$, a combination of Stanford-B model and DeJong's model.

The key variable is the exponent ${\displaystyle n}$ measuring the strength of learning. It is usually expressed as ${\displaystyle n=\log(2)/\log(\phi )}$, where ${\displaystyle \phi }$ is the "learning rate". In words, it means that the unit cost decreases by ${\displaystyle 1-\phi }$, for every doubling of total units made. Wright found that ${\displaystyle \phi \approx 80\%}$ 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

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.^[17] 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.^[18] As for the problems of production management under the limitation of scarce resources, Liao ^[19] 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 ^[20] 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 ^[21] 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.^[22]

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.^[15]

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.

• 
  S-curve or sigmoid function
• 
  Exponential growth
• 
  Exponential rise or fall to a limit
• 
  Power law

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

• 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

Plots relating performance to experience are widely used in

• 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
  Energy Return on Energy Invested 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.
• 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.

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.)

• 
  Short and long learning curves
• 
  Product A has lower functionality and a short learning curve. Product B has greater functionality but takes longer to learn.

The term 'learning curve' with meanings of 'easy' and 'difficult' can be described with adjectives like 'short' and 'long' rather than steep and 'shallow'.^[2] If two products have similar functionality then the one with a "steep" curve is probably better, because it can be learned in a shorter time. On the other hand, if two products have different functionality, then one with a short curve (a short time to learn) and limited functionality may not be as good as one with a long curve (a long time to learn) and greater functionality.

For example, the Windows program

Ben Zimmer discusses the use of the term "on a steep learning curve" in Downton Abbey, a television series set in the early 20th century, concentrating mainly on whether use of the term is an anachronism. "Matthew Crawley, the presumptive heir of Downton Abbey and now the co-owner of the estate, says, 'I've been on a steep learning curve since arriving at Downton.' By this he means that he has had a difficult time learning the ways of Downton, but people did not start talking that way until the 1970s."^[3][34]

Zimmer also comments that the popular use of steep as difficult is a reversal of the technical meaning. He identifies the first use of steep learning curve as 1973, and the arduous interpretation as 1978.

Difficulty curves in video games

The idea of learning curves is often translated into

• Forgetting curve
• Learning speed
• Labor productivity
• Learning-by-doing (economics)
• Population growth
• Trial and error

References