How Machine (Deep) Learning Helps Us Understand Human Learning: the Value of Big Ideas

TL;DR

This paper uses neural network simulations to explore how big ideas and regularization improve human learning, aligning with psychological research and highlighting the role of expert teachers and diverse learning conditions.

Contribution

It demonstrates how regularization in neural networks models the importance of big ideas in human learning and compares different teaching scenarios.

Findings

Regularization enhances the transmission of big ideas.

Learning from a teacher outperforms data-only learning.

Simulation results align with psychological literature.

Abstract

I use simulation of two multilayer neural networks to gain intuition into the determinants of human learning. The first network, the teacher, is trained to achieve a high accuracy in handwritten digit recognition. The second network, the student, learns to reproduce the output of the first network. I show that learning from the teacher is more effective than learning from the data under the appropriate degree of regularization. Regularization allows the teacher to distinguish the trends and to deliver "big ideas" to the student. I also model other learning situations such as expert and novice teachers, high- and low-ability students and biased learning experience due to, e.g., poverty and trauma. The results from computer simulation accord remarkably well with finding of the modern psychological literature. The code is written in MATLAB and will be publicly available from the author's web page.

Equations

$$\sigma \left(x\right)=\frac{1}{1+e^{-x}},$$

$$J\left(\theta \right)=\frac{1}{m}\sum _{i=1}^m\sum _{k=1}^{10}\left[-y_k^i\log (h_k\left(\theta ,X^i\right))-(1-y_k^i)\log (1-h_k\left(\theta ,X^i\right))\right]+\frac{\lambda }{2m}\sum _{j=1}^{10,285}{\theta }_j^2,$$

$$\frac{\partial J}{\partial {\theta }_n}=\sum _{i=1}^m{\delta }_{n+1}^i{a}_n^i+\frac{\lambda }{m}{\theta }_n,$$

$${\delta }_3^i=h\left(\theta ,X^i\right)-y^i,$$

$${\delta }_n^i=\left({\delta }_{n+1}^i\times {\theta }_n\right).\times \left[\sigma (z_n^i).\times (\sigma (z_n^i)-1)\right],$$

$$\theta ⟵\theta -\alpha \times \frac{\partial J}{\partial \theta },$$

Full text

$Key$ $Words:$ deep learning, neural networks, machine learning, human learning

1 Introduction

Nowadays, it is hard to find an area of human life where artificial intelligence would not have its applications, starting from composing music and writing novels to self-driving cars and anti-spam filters. As is evident from the term machine learning, the main goal of the research in artificial intelligence is to teach machines. ”Artificial neurons” were designed to perform logical operations in the same way humans do. But the learning process can go in the opposite direction as well. In particular, brain scientists believe that we can get a better understanding of how humans learn by studying how machines learn.111In spite of similarities, researchers found some important differences between humans and machines. Specifically, our bodies help our memories; metaphors are powerful learning tools; there is no substitute for learning-by-doing; good teaching draws on shared experience; see Briggs (2017) for a discussion. Researchers, educators and students can benefit from using machine learning to explain the learning process undertaken by humans.

Human learning can be studied in many different ways (e.g., by analyzing the existing empirical data, by running lab experiments, by looking at the highlighted areas of a human brain). Recent developments in deep learning have opened new possibilities for studying human learning by means of computer simulation; see Domingos (2015). This is the approach I follow in the present paper.

I use deep-learning neural networks to analyze how the design of human learning process can influence its outcome. One important question is under what conditions do humans learn better from teachers than from their own experience (represented by raw data from the real world). I address these questions in the context of an image recognition problem, namely, classification of handwritten numbers. Handwritten character recognition is one of the most challenging problems in the field of pattern recognition; see Purohit and Chauhan (2016) for a survey of the literature on handwritten character recognition. The techniques used in this field rely on standard deep learning; see Goodfellow et al. (2016) and Hastie et al. (2008). Industry developments in software, e.g., TensorFlow software, make it feasible to handle problems of very large dimensionality; see Hope et al. (2017).

The main novelty of the present paper is the way of modeling the interaction between the teacher and student, which I propose in the context of classification problem. Learning from a teacher means that a previously trained neural network (teacher) observes an image and produces probabilities that the given image corresponds to digits 0, 1, …, 9. These probabilitities are provided as input to train the student network. On opposite, a student who learns from its own experience observes the handwritten digit images and has no other information. I design the learning environment to mimic several real world learning scenarios in which I vary the number of training samples (expert versus novice teacher), regularization level (the teacher’s ability to categorize information into big ideas) and the selection of the training samples (constructing ”bad” teacher with a biased perception of the learning process). I also vary the student’s learning rate (representing the ability of students) and the selection of training samples (modeling the students with atypical learning experience due to, e.g., poverty and trauma). To measure the success of learning, I report two statistics: the predictive accuracy of image recognition and the cost on a test set. The code is written in MATLAB and will be publicly available on my web site.

My findings are as follows: (i) If the teacher is sufficiently trained, then a student with a high learning rate learns faster from the teacher than from the data and attains a higher accuracy of digit recognition; in contrast, a student with a low learning rate does not significantly benefit from the teacher. (ii) The success of learning depends on the qualification of the teacher relative to the qualification of the student; overqualification of the teacher does not improve the learning outcomes but underqualification worsens such outcomes; (iii) Learning from a ”bad” teacher **(**i.e., with a biased perception of the learning process) is less effective than learning from the data, and regularization only increases the bias. (iv) Low-ability students benefit less from a teacher than high-ability students. (v) Learning from an unbiased teacher can correct biased views of the students derived from their atypical experience (e.g., poverty and trauma); and moderate regularization increases the effectiveness of the teacher.

The main novelty and methodological contribution of the present paper is to show how to model the learning of one artificial intelligence from another. My analysis highlights the importance of ”big ideas” for successful learning: a student network progresses most in my experiments when the teacher’s input is sufficiently regularized. The degree of regularization determines how well the teacher is able to distinguish and generalize the regularities in the data. This finding accords with the conclusion of modern psychological literature about the importance of learning experiences which specifically enhance the students’ abilities to recognize meaningful patterns of information; see Section 2 for a survey of the related literature. The knowledge of experts is not simply a list of facts and formulas that are relevant to their domain; instead, their knowledge is organized around core concepts or “big ideas” that guide their thinking about their domains.

The paper is organized as follows: Section 2 describes the evidence on the key determinants of successful learning, Section 3 outlines the methodology of my computer simulation. Section 4 investigates how the teacher’s characteristics affect the learning outcomes. Section 5 discusses how the learning outcome depends on the student’s characteristics. Finally, Section 7 concludes.

2 Evidence on human learning

Learning plays a critical role in both personal development of each individual and the evolution of the society as a whole. Some relevant evidence on human learning is summarized below.

i) The critical role of the teacher in a student’s learning

outcomes.

There is a large body of modern psychological literature that emphasizes the critical role of a teacher in successful learning outcomes. For example, Hattie (2003) found that excellence in teaching is the single most powerful influence on achievement and found that teachers account for 30% of the variance in the students’ learning outcomes. Ulug et al. (2011) concluded that teachers’ positive attitudes have positive effects on students’ performance and personality developments and are the second-highest determining factor in the development of individuals, after parents. Finland is often cited to have the best educational system in the world and the key factor for its success seems to be the training of teachers: Finland’s teachers are required to have Master’s degrees, and teaching careers are the most competitive in the country.

ii) Expert teachers and the importance of big ideas.

But what does it mean to be an expert teacher? Bransford, Brown, and Cocking, (2000) survey the literature investigating how teachers-experts differ from teachers-novices in such areas as chess, physics, mathematics, electronics, and history. Their conclusion is that experts’ knowledge is organized around important ”big” ideas that lead to deeper conceptual understanding of their domain. As an illustration, consider an experiment from DeGroot (1965). A chess master and a novice player were given 5 seconds to memorize a chess board position. The master was able to reconstruct the position far more accurately than the novice but only when the chess pieces were arranged in configurations that conformed to meaningful games of chess; when the pieces were randomized both the master and novice had similar recall. Hatano and Inagaki (1986) argue that the key to effective learning is adaptive expertise: The experts not only use their current knowledge but attempt to continually improve their expertise. The main challenge for theory of learning is therefore to understand how the ”virtuosos” organize their knowledge into big ideas.

iii) Bad teachers are insufficiently trained and have a biased

perception of the learning process.

Another piece of evidence is provided by Foote et al. (2000) who conducted a questionnaire about secondary school teachers. One common response amongst students, teachers, administrators and parents was that bad secondary teachers do not posses the knowledge of their discipline in the degree that is sufficient for effective teaching. The other common response was that bad teachers do not have adequate perception of the learning process, making their lessons too fast, slow, easy, or difficult. In other words, the teacher’s perception (experience) is biased in the sense that it does not accord well with the learning needs of the students.

iv) High-ability students benefit from enriched teaching but not

low-ability students.

There is also evidence on learning outcomes by students groups. Kulik and Kulik (1982) summarize the results from 52 studies of grouping students by their abilities in secondary schools. One robust finding was that studies in which high-ability students received enriched instructions in honors classes produced significant learning improvements, while studies of average and below-average students produced near-zero effects. In other words, only high ability students benefited from more advanced teaching, in contrast to low ability students.

iv) Students can learn from the educator what they missed in their

own experience.

Finally, there is abundant evidence about learning of students from low socioeconomic backgrounds. The handbook by Izard (2016) edited by the US National Educational Association states: ”Because students impacted by poverty and trauma have not learned appropriate emotional responses, when the educator models appropriate social behaviors, the student’s mirror neurons that were neglected or harmed earlier can still pick up on clues from the educator to learn now what they missed earlier”. That is, the educator can become the missing person and fill in the socioemotional gap for those students whose own perception is affected by the negativity of poverty and trauma.

3 Modeling the learning process

In this section, I present the methodology of our analysis. I first describe the topology of the neural network studied, then explain how the teacher and students are trained, and finally, show how our computer simulation is connected to the evidence on human learning.

3.1 Topology of neural network

All neural networks studied in this paper have the same topology. They have three layers, $n$ = 1, 2, and 3: the input layer ($400\times 25$), one hidden layer ($25\times 10$), and the output layer ($10\times 1$). Thus, the number of nodes in the three layers are $400$, $25$ and $10$, respectively. The output layer returns a $\left(10\times 1\right)$ vector; ten elements in this vector represent the probabilities that the image is a digit $0,1,...,9$, respectively.

The neural network’s activation function is a sigmoid function given by

[TABLE]

for any $x\in\mathbb{R}$. The cost function is given by

[TABLE]

where $m$ is the number of training samples, $X^{i}\in\mathbb{R}^{400\times 1}$ is the input image and $y^{i}\in\mathbb{R}^{10\times 1}$ is the actual output vector for a sample $i$, $\theta\in\mathbb{R}^{10,285\times 1}$ is the vector of weights of the neural network, $h:\mathbb{R}^{400\times 10,285}\rightarrow\mathbb{R}^{10\times 1}$ is the predicted output vector, and $\lambda\in\left[0,\infty\right)$ is the regularization parameter; see Goodfellow et al. (2016) for details.

We use back propagation to calculate the gradient of every node. These values are used to train the networks. The partial derivative of the $n$th layer for weight matrix $\theta_{n\text{ }}$is

[TABLE]

where $a_{n}^{i}$ is the activated output, and $\delta_{n}^{i}$ is the error for the current layer $n$ and sample $i$. For the last layer $n$ = 3, $\delta_{n}^{i}\in\mathbb{R}^{10\times 1}$ is

[TABLE]

where $h\left(\theta,X^{i}\right)$ is the output of the neural network, and $y^{i}$ is the real digit vector, both for sample $i$.

For all other layers, i.e., $n=1,2$, $\delta_{n}^{i}$ is

[TABLE]

where $\theta_{n}$ is the matrix containing the weights for layer $n$, $\sigma$ is the sigmoid function, and $z_{n}^{i}$ is the inactivated output of layer $n$ ($z_{n}^{i}=\theta_{n}\times a_{n-1}^{i}$) for sample $i$. The operator $.\times$ represents an element-wise matrix multiplication, and the operator $\times$ is a matrix multiplication.

3.2 The learning process

In this section, I describe the data, as well as the training procedures for the teacher and students.

3.2.1 Data

We train the neural networks with a handwritten digit classification problem $20\times 20$ pixel images—these images come from the database provided by Ng (2017). We have 5,000 total training samples composed of $\left(X^{i},y^{i}\right)$, where $X^{i}\in\mathbb{R}^{400\times 1}$, $y^{i}\in\mathbb{R}^{10\times 1}$, for sample $i$ from $1$ to $m$. We begin by randomizing the order of the samples. Then, we set apart 1,000 samples as the test set. These samples will be used to evaluate the performance of neural networks later. No neural network will have access to these samples. Then, we take 500 samples for training the students, and we call this the student training set $(X_{student},y_{student})$. To train the teacher, we use (up to) 3,500 of the remaining samples $(X_{teacher},y_{teacher})$. The teacher training set and the student training set will never overlap.

3.2.2 The criteria of learning success

As a criteria of the learning success, we consider two measurements: accuracy (number of samples predicted correctly on a test set) and cost (the value of cost function (1) on the test set, which is a real number showing how far off the predictions are from the correct answers). To reduce sampling errors, I report the average accuracy and cost which are averaged over ten simulations.

3.2.3 Training the teacher network

We train the teacher on the teacher’s training set $(X_{teacher},y_{teacher})$ by minimizing the cost function (1). We use the MATLAB optimization routine fmincg that guarantees convergence: we don’t want to study how quickly the teacher learns but we want the teacher to learn as much as possible from the data available (in contrast, for students, we will use stochastic gradient which allows us to investigate the learning speed). In Figure 1, I show the accuracy and cost of the trained teacher for 4 training sets consisting of 500, 1000, 2000 and 4000 handwritten samples (I connect the 3 points on the graph with a line for expositional convenience); we vary the learning rate $\lambda\in\left\{0,5,10\right\}$.

First, we see that even 500 training samples produce sufficiently good training in the sense that the teacher correctly classifies more than 80% of the test samples, while adding the other 3500 training samples increases the accuracy by another 10–15%.

Second, the performance of the teacher is visibly affected by the regularization constant $\lambda$ that controls how a neural network fits to the data. We consider three regularization levels: $\lambda=0,5,10$. A large value of $\lambda$ penalizes large weight values; it results in a more general model that possibly underfits the data (if $\lambda$ is big enough), leading to a high bias. A small value of $\lambda$ allows large weight values; it results in a more specific model that possibly overfits the data (if $\lambda$ is small enough), leading to a high variance. When $\lambda$ is 0, the training algorithm does not penalize large weight numbers. An overfit model will do badly on unseen samples, while an underfit model will do badly overall. We see that $\lambda=5$ delivers better results than $\lambda=0$ (underfit) and $\lambda=10$ (overfit) in terms of accuracy and cost.

3.2.4 Training the student network

The student’s network differs from the teacher’s network in two respects: First, the student can learn either from the data or from the trained teacher, and second, its network is trained by stochastic gradient descent, which allows us to assess the learning speed. In all experiments, we set the student’s regularization $\lambda$ equal to 0 because the student is not trying to generalize anything; rather, it is simply trying to learn.

Learning from a teacher versus learning from data.

We consider two alternative learning procedures for the student: one is from the data and the other is from the teacher. In the former case, the student is an independent learner trained on the student training samples $X_{student}$ and the corresponding actual output $y_{student}$. In the latter case, the student is trained on the same samples $X_{student}$ but uses the teacher’s predicted output $\hat{y}_{student}$.

What is the difference between $y_{student}$ and $\hat{y}_{student}$? In the former case, we assert that $y_{student}$ represents a certain digit and does not represent all of the other digits. As a result, each $y_{student}$ is a $10\times 1$ vector that has 9 zero values and 1 unit value. For example, the $y_{student}$ vector is (1,0,0,0,0,0,0,0,0,0) if the digit is a 0. Another possible interpretation is that the image is the given digit with probability 1 and any other digit with probability 0.

In turn, the teacher’s predicted output $\hat{y}_{student}$ is a $10\times 1$ vector where each element represents the probability that the image is a certain digit as perceived by the trained teacher. Thus, $\hat{y}_{student}$ has same structure as $y_{student}$ but has all nonzero (positive) elements $(p_{1},p_{2},...,p_{10})$ normalized to one by $\sum_{i=1}^{10}p_{i}=1$. We will not round $\hat{y}_{student}$ to the most probable output because we imagine that the teacher considers various alternatives rather than just one answer and informs the student about the corresponding probability distribution.

Gradient descent and the learning rate.

We will train a student neural network using stochastic gradient descent. The student will train on one sample at a time: for each sample, we find the gradient (partial derivatives of weights) based on the current sample. Then, we update the neural network weights based on this gradient. The update is performed according to the gradient descent rule:

[TABLE]

where $\alpha$ is a learning rate. This parameter describes how fast the student learns from the current sample. I use two values of $\alpha$, namely, 0.1 and 0.5. In my model, a high learning rate $\alpha=0.5$ represents a high-ability student, and a low learning rate $\alpha=0.1$ corresponds to a low-ability student. Models with a high learning rate may converge quicker but are more prone to fluctuations and instability.

4 The role of a teacher in the learning process

In Section 2, we reviewed evidence from modern psychological literature about the determinant of successful human learning. In Section 3, we designed a computer simulation in the way that allows to capture some of these determinants: For the teacher, we account for the ”expert-novice” effect by varying the number of handwritten samples that are used for training from 500 to 3,500; we model the teacher’s ability to categorize information into big ideas by varying the level of regularization $\lambda=0,5,10$; and we construct examples of ”bad” teachers with inadequate (biased) perception of the learning process by training it on biased data sets, such as too hard or too easy handwriting samples. Below, we assess how the size of a teacher’s training set, the level of regularization and the training set bias affect the the learning outcomes.

4.1 The importance of big ideas

I first consider a student whose learning rate in the gradient descent rule (2) is equal to $\alpha=0.5$. I use 500 random handwritten samples for training the student. (For the teacher, I consider three values of $\lambda$, namely, 0, 5 and 10; and I use 500 handwritten samples for its training). The results are shown in Figure 2.

In the figure, we see that the highest predictive accuracy and lowest cost is obtained when the student learns from the teacher with $\lambda=5$ and $\lambda=10$. That is, the student learns faster from the appropriately regularized teachers than both from the data and from overfit teachers.

Why is learning from an appropriately regularized teacher more effective than learning from the data? This is because the teacher transmits to the student its life-long experience rather than just one data point. The teacher has analyzed many samples; it has distinguished the trends, and it now supplies to the student the probabilities that a given image is every single possible digit rather than just one correct digit value.

Why is the effectiveness of learning from an unregularized teacher lower and comparable to learning from the data? When $\lambda=0$, the teacher does not try to generalize the tendencies but tries to simply find parameters that explain its own sample. Such an overfit teacher understands how to deal with its own samples but gives a bad advice on other, unseen samples. Because there is no overlap between the teacher’s training set and the students’ training set, the effectiveness of learning from such a teacher is similar to the one of the raw data.

As we argued in Section 2, modern psychological studies highlight the importance of providing students with learning experiences that are designed to enhance the student’s abilities to recognize meaningful patterns of information. The knowledge of expert teachers is not simply a list of facts and formulas that are relevant to their domain; instead, their knowledge is organized around core concepts or “big ideas” that guide their thinking about their domains. This is exactly what the regularization parameter represents in our analysis.

4.2 Novice teacher versus experienced teacher

In the previous section, we analyzed learning from the teacher that was trained with 500 samples. But what happens if we increase the teacher’s training set? Would the student perform better when learning from a more trained teacher? In Appendix A.1, we show the results for a very experienced teacher trained with 3,500 samples, and we obtain results that are very similar to those in our baseline Figure 1. Why does giving the teacher more samples not lead to visibly better learning outcomes? In our experiments, the student is trained on 500 samples and reaches predictive accuracy of 70% at the end of the training. In that range, the student’s knowledge is not yet well-refined, so its learning progress does not considerably depend on whether the teacher has the accuracy of 85% or 95%. For example, the learning of students in an 8th grade algebra class may not critically depend on whether or not their teacher took advanced Ph.D. classes in topology and multivariate calculus.

However, when the training of the teacher becomes insufficient relative to the students level, the effectiveness of teaching begins to suffer. As an illustration, in Figure 3, we consider an underqualified teacher that is trained on just 100 samples.

In that case, learning from the data is faster than learning from the teacher, which leads us to the following conclusion. The success of learning depends on the qualification of the teacher relative to the qualification of the student. Overqualification of the teacher does not improve learning outcomes but underqualification worsens such outcomes.

In fact, the student that is learning from the data will continue to progress. In turn, the students that learn from the teacher are bounded by the predictive accuracy that the teacher’s training guarantees. Recall that a higher level of regularization $\lambda$ implies that the teacher tries to generalize the data and to identify a smooth trend. In that case, the teacher has poor knowledge and insufficient experience to make generalizations – just 100 samples – and its poor judgement conditions the poor progress of the student.

4.3 The teacher with a biased perception of learning process (bad

teacher)

In some experiments, I observed that atypical handwritten samples disrupt the training of the student. To assess the effect of such disruptions, I construct two sets of atypical data: one set includes handwriting samples that are hardest to classify and the other set contains handwriting samples that are the easiest to classify. For this experiment, I train a neural network on the set of 3,500 observations until convergence and pick out the two tails data sets: 500 observations with the highest cost and 500 observations with the lowest costs.

In Figure 4, I illustrate how well the students learn if their teacher has a bias because it is trained on the set with hard handwriting.

We observe that the student that learns from the data performs better than the student taught by a teacher trained on hard data. In Appendix A.2, I show that the results are similar when the teacher is trained on 500 easiest samples, which leads us to the following conclusion: *Learning from a teacher with a biased perception of the learning process is less effective than learning from the data directly, and regularization only increases the bias. *The teacher that is trained on atypical data has an inadequate perception about typical handwriting. It becomes even worse when such a teacher tries to distinguish trends via a higher level of regularization. The distinguished trend reflects the hard data and not the typical data, so the teacher gives systematically bad advice to the students. In contrast, the student that learns from the data observes the typical samples and identifies the trend correctly.

5 The learning success depends on the students too

In the previous section, I show that the training and analytical abilities of the teacher (i.e.. its capacity to generalize the trends) are critical for the learning success of the students. But the evidence outlined in Section 2 show that the student’s characteristics play an important role in the learning outcomes as well, in particular, their abilities and socioeconomic background. For the student, we control for the level of abilities by varying the learning rate $\alpha$; and I model the student’s biased experience (due to, e.g., poverty and trauma) by using biased training data sets that do not coincide with the experience of a typical student. Below, I assess the role of the student’s characteristics for the learning outcomes.

5.1 A low ability student

In Section 2.1, we considered a student with learning rate in the gradient descent rule (2) equal to $\alpha=0.5$. We use 500 random (typical) handwritten samples for training the student; see Figure 2. We now consider the student whose learning abilities are lower $\alpha=0.1$; see Figure 5.

Here, all training schemes lead to similar learning outcomes, except of that with an overfit teacher $\lambda=10$ that produces the worst results in terms of accuracy and costs. Thus, in our experiment, low-ability students benefit less from a teacher than high-ability students. This finding accords remarkably with the evidence from Kulik and Kulik (1982)**. **

5.2 A student with a bias

Let us finally consider an experiment in which the teacher is trained on the typical data and students are trained on atypical (hard, easy) data. In Figure 6, we show the case when the students are trained with the hard samples.

We now observe that the students that learn from hard samples benefit greatly from the teacher, in particular, from the appropriately regularized teacher. In Appendix A.3, I show that we observe the same regularities when the students are learning from 500 easiest samples, which leads us to conclude the following: *Learning from a teacher with typical experience is better than learning from biased data; moderate regularization can increase the effectiveness of the teacher. *

There is simple intuition behind this result. The student that learns from biased samples has inadequate experience and makes mistakes when tested on the typical samples. But the teacher that was trained on typical samples gives correct advice to the student even when the samples are atypical, which mitigates the effect of biased samples on the learning process. Regularization reinforces the learning process as in our benchmark case. Thus, our computer simulation grossly accords with the idea that the educator can become the missing person for those students whose own experience is contaminated by negative experience (e.g., poverty and trauma) by overriding those inadequate responses that the students observed from their own biased socioeconomic experience (as shown by the finding in Section 2).

6 Conclusion

The main message of the present paper is that we can gain understanding into the determinants of human learning by simulating the interactions of artificial intelligence. The advantage of computer simulation compared to experimentation with human subjects is that we have a better control over the learning experiments, we understand how the learning process depends on the model’s parameters, and we can re-run the experiments under different parameterizations as many times as needed. This approach made it possible to illustrate the importance of big ideas in the learning process and to successfully model a variety of interesting learning situations such as expert-novice teachers, high-low ability students and atypical learning experience, among others. My results are suggestive and provide explanation to the real world learning experiences documented in the modern psychological literature.

Appendix

This appendix contains the results of supplementary experiments.

6.1 Appendix A1

In Section 3.1, we analyzed the outcome of learning when the teacher was trained with 500 samples. In Figure 7, we provide the sensitivity results in which the teacher is trained with a larger number of observations, namely, 3,500. We observe that the outcomes of learning process in Figures 2 and 7 are very similar, which indicates that excessive training of the teacher does not improve the effectiveness of learning.

6.2 Appendix A2

In Section 3.3, we show that training the teacher on the hard samples reduces the effectiveness of the teacher; see Figure 4. We now consider the opposite case when the teacher is trained on the easiest samples; see Figure 8. We observe that learning from the data is more effective than learning from such a teacher. This leads us to the conclusion in the main text: a bias (in either direction) reduces the effectiveness of teaching

6.3 Appendix A3

In Section 4.2, we consider the student that is trained on excessively hard samples. We now consider the remaining case of the student that is trained with excessively easy samples; see Figure 9.

Comparing Figures 5 and 9, we observe that in both cases (hard and easy samples), learning from the teacher dominates learning from the data. The difference is that unregularized (overfit) teacher performs poorly. This is because such a teacher learned to reproduce its specific typical sample rather than learning the trends. The advice of such a teacher is generally poor, and it is especially poor for the biased sample that the student faces.