# Learnability of machine learning is provably an undecidable?—part 3: closure

Update on 23 January 2019, 17:55 IST:

In this series of posts, which was just a step further from the initial, brain-storming kind of a stage, I had come to the conclusion that based on certain epistemological (and metaphysical) considerations, Ben-David et al.’s conclusion (that learnability can be an undecidable) is logically untenable.

However, now, as explained here [^], I find that this particular conclusion which I drew, was erroneous. I now stand corrected, i.e., I now consider Ben-David et al.’s result to be plausible. Obviously, it merits a further, deeper, study.

However, even as acknowledging the above-mentioned mistake, let me also hasten to clarify that I still stick to my other positions, especially the central theme in this series of posts. The central theme here was that there are certain core features of the set theory which make implications such as Godel’s incompleteness theorems possible. These features (of the set theory) demonstrably carry a glaring epistemological flaw such that applying Godel’s theorem outside of its narrow technical scope in mathematics or computer science is not permissible. In particular, Godel’s incompleteness theorem does not apply to knowledge or its validation in the more general sense of these terms. This theme, I believe, continues to hold as is.

Update over.

Gosh! I gotta get this series out of my hand—and also head! ASAP, really!! … So, I am going to scrap the bits and pieces I had written for it earlier; they would have turned this series into a 4- or 5-part one. Instead, I am going to start entirely afresh, and I am going to approach this topic from an entirely different angle—a somewhat indirect but a faster route, sort of like a short-cut. Let’s get going.

Statements:

Open any article, research paper, book or a post, and what do you find? Basically, all these consist of sentences after sentences. That is, a series of statements, in a way. That’s all. So, let’s get going at the level of statements, from a “logical” (i.e. logic-thoretical) point of view.

Statements are made to propose or to identify (or at least to assert) some (or the other) fact(s) of reality. That’s what their purpose is.

The conceptual-level consciousness as being prone to making errors:

Coming to the consciousness of man, there are broadly two levels of cognition at which it operates: the sensory-perceptual, and the conceptual.

Examples of the sensory-perceptual level consciousness would consist of reaching a mental grasp of such facts of reality as: “This object exists, here and now;” “this object has this property, to this much degree, in reality,” etc. Notice that what we have done here is to take items of perception, and put them into the form of propositions.

Propositions can be true or false. However, at the perceptual level, a consciousness has no choice in regard to the truth-status. If the item is perceived, that’s it! It’s “true” anyway. Rather, perceptions are not subject to a test of truth- or false-hoods; they are at the very base standards of deciding truth- or false-hoods.

A consciousness—better still, an organism—does have some choice, even at the perceptual level. The choice which it has exists in regard to such things as: what aspect of reality to focus on, with what degree of focus, with what end (or purpose), etc. But we are not talking about such things here. What matters to us here is just the truth-status, that’s all. Thus, keeping only the truth-status in mind, we can say that this very idea itself (of a truth-status) is inapplicable at the purely perceptual level. However, it is very much relevant at the conceptual level. The reason is that at the conceptual level, the consciousness is prone to err.

The conceptual level of consciousness may be said to involve two different abilities:

• First, the ability to conceive of (i.e. create) the mental units that are the concepts.
• Second, the ability to connect together the various existing concepts to create propositions which express different aspects of the truths pertaining to them.

It is possible for a consciousness to go wrong in either of the two respects. However, mistakes are much more easier to make when it comes to the second respect.

Homework 1: Supply an example of going wrong in the first way, i.e., right at the stage of forming concepts. (Hint: Take a concept that is at least somewhat higher-level so that mistakes are easier in forming it; consider its valid definition; then modify its definition by dropping one of its defining characteristics and substituting a non-essential in it.)

Homework 2: Supply a few examples of going wrong in the second way, i.e., in forming propositions. (Hint: I guess almost any logical fallacy can be taken as a starting point for generating examples here.)

Truth-hood operator for statements:

As seen above, statements (i.e. complete sentences that formally can be treated as propositions) made at the conceptual level can, and do, go wrong.

We therefore define a truth-hood operator which, when it operates on a statement, yields the result as to whether the given statement is true or non-true. (Aside: Without getting into further epistemological complexities, let me note here that I reject the idea of the arbitrary, and thus regard non-true as nothing but a sub-category of the false. Thus, in my view, a proposition is either true or it is false. There is no middle (as Aristotle said), or even an “outside” (like the arbitrary) to its truth-status.)

Here are a few examples of applying the truth-status (or truth-hood) operator to a statement:

• Truth-hood[ California is not a state in the USA ] = false
• Truth-hood[ Texas is a state in the USA ] = true
• Truth-hood[ All reasonable people are leftists ] = false
• Truth-hood[ All reasonable people are rightists ] = false
• Truth-hood[ Indians have significantly contributed to mankind’s culture ] = true
• etc.

For ease in writing and manipulation, we propose to give names to statements. Thus, first declaring

A: California is not a state in the USA

and then applying the Truth-hood operator to “A”, is fully equivalent to applying this operator to the entire sentence appearing after the colon (:) symbol. Thus,

Truth-hood[ A ] <==> Truth-hood[ California is not a state in the USA ] = false

Just a bit of the computer languages theory: terminals and non-terminals:

To take a short-cut through this entire theory, we would like to approach the idea of statements from a little abstract perspective. Accordingly, borrowing some terminology from the area of computer languages, we define and use two types of symbols: terminals and non-terminals. The overall idea is this. We regard any program (i.e. a “write-up”) written in any computer-language as consisting of a sequence of statements. A statement, in turn, consists of certain well-defined arrangement of words or symbols. Now, we observe that symbols (or words) can be  either terminals or non-terminals.

You can think of a non-terminal symbol in different ways: as higher-level or more abstract words, as “potent” symbols. The non-terminal symbols have a “definition”—i.e., an expansion rule. (In CS, it is customary to call an expansion rule a “production” rule.) Here is a simple example of a non-terminal and its expansion:

• P => S1 S2

where the symbol “=>” is taken to mean things like: “is the same as” or “is fully equivalent to” or “expands to.” What we have here is an example of an abstract statement. We interpret this statement as the following. Wherever you see the symbol “P,” you may substitute it using the train of the two symbols, S1 and S2, written in that order (and without anything else coming in between them).

Now consider the following non-terminals, and their expansion rules:

• P1 => P2 P S1
• P2 => S3

The question is: Given the expansion rules for P, P1, and P2, what exactly does P1 mean? what precisely does it stand for?

• P1 => (P2) P S1 => S3 (P) S1 => S3 S1 S2 S1

In the above, we first take the expansion rule for P1. Then, we expand the P2 symbol in it. Finally, we expand the P symbol. When no non-terminal symbol is left to expand, we arrive at our answer that “P1” means the same as “S3 S1 S2 S1.” We could have said the same fact using the colon symbol, because the colon (:) and the “expands to” symbol “=>” mean one and the same thing. Thus, we can say:

• P1: S3 S1 S2 S1

The left hand-side and the right hand-side are fully equivalent ways of saying the same thing. If you want, you may regard the expression on the right hand-side as a “meaning” of the symbol on the left hand-side.

It is at this point that we are able to understand the terms: terminals and non-terminals.

The symbols which do not have any further expansion for them are called, for obvious reasons, the terminal symbols. In contrast, non-terminal symbols are those which can be expanded in terms of an ordered sequence of non-terminals and/or terminals.

We can now connect our present discussion (which is in terms of computer languages) to our prior discussion of statements (which is in terms of symbolic logic), and arrive at the following correspondence:

The name of every named statement is a non-terminal; and the statement body itself is an expansion rule.

This correspondence works also in the reverse direction.

You can always think of a non-terminal (from a computer language) as the name of a named proposition or statement, and you can think of an expansion rule as the body of the statement.

Easy enough, right? … I think that we are now all set to consider the next topic, which is: liar’s paradox.

The liar paradox is a topic from the theory of logic [^]. It has been resolved by many people in different ways. We would like to treat it from the viewpoint of the elementary computer languages theory (as covered above).

The simplest example of the liar paradox is , using the terminology of the computer languages theory, the following named statement or expansion rule:

• A: A is false.

Notice, it wouldn’t be a paradox if the same non-terminal symbol, viz. “A” were not to appear on both sides of the expansion rule.

To understand why the above expansion rule (or “definition”) involves a paradox, let’s get into the game.

Our task will be to evaluate the truth-status of the named statement that is “A”. This is the “A” which comes on the left hand-side, i.e., before the colon.

In symbolic logic, a statement is nothing but its expansion; the two are exactly and fully identical, i.e., they are one and the same. Accordingly, to evaluate the truth-status of “A” (the one which comes before the colon), we consider its expansion (which comes after the colon), and get the following:

• Truth-hood[ A ] = Truth-hood[ A is false ] = false           (equation 1)

Alright. From this point onward, I will drop explicitly writing down the Truth-hood operator. It is still there; it’s just that to simplify typing out the ensuing discussion, I am not going to note it explicitly every time.

Anyway, coming back to the game, what we have got thus far is the truth-hood status of the given statement in this form:

• A: “A is false”

Now, realizing that the “A” appearing on the right hand-side itself also is a non-terminal, we can substitute for its expansion within the aforementioned expansion. We thus get to the following:

• A: “(A is false) is false”

We can apply the Truth-hood operator to this expansion, and thereby get the following: The statement which appears within the parentheses, viz., the “A is false” part, itself is false. Accordingly, the Truth-hood operator must now evaluate thus:

• Truth-hood[ A ] = Truth-hood[ A is false] = Truth-hood[ (A is false) is false ] = Truth-hood[ A is true ] = true            (equation 2)

Fun, isn’t it? Initially, via equation 1, we got the result that A is false. Now, via equation 2, we get the result that A is true. That is the paradox.

But the fun doesn’t stop there. It can continue. In fact, it can continue indefinitely. Let’s see how.

If only we were not to halt the expansions, i.e., if only we continue a bit further with the game, we could have just as well made one more expansion, and got to the following:

• A: ((A is false) is false) is false.

The Truth-hood status of the immediately preceding expansion now is: false. Convince yourself that it is so. Hint: Always expand the inner-most parentheses first.

Homework 3: Convince yourself that what we get here is an indefinitely long alternating sequence of the Truth-hood statuses that: A is false, A is true, A is false, A is true

What can we say by way of a conclusion?

Conclusion: The truth-status of “A” is not uniquely decidable.

The emphasis is on the word “uniquely.”

We have used all the seemingly simple rules of logic, and yet have stumbled on to the result that, apparently, logic does not allow us to decide something uniquely or meaningfully.

Liar’s paradox and the set theory:

The importance of the liar paradox to our present concerns is this:

Godel himself believed, correctly, that the liar paradox was a semantic analogue to his Incompleteness Theorem [^].

Go read the Wiki article (or anything else on the topic) to understand why. For our purposes here, I will simply point out what the connection of the liar paradox is to the set theory, and then (more or less) call it a day. The key observation I want to make is the following:

You can think of every named statement as an instance of an ordered set.

What the above key observation does is to tie the symbolic logic of proposition with the set theory. We thus have three equivalent ways of describing the same idea: symbolic logic (name of a statement and its body), computer languages theory (non-terminals and their expansions to terminals), and set theory (the label of an ordered set and its enumeration).

As an aside, the set in question may have further properties, or further mathematical or logical structures and attributes embedded in itself. But at its minimal, we can say that the name of a named statement can be seen as a non-terminal, and the “body” of the statement (or the expansion rule) can be seen as an ordered set of some symbols—an arbitrarily specified sequence of some (zero or more) terminals and (zero or more) non-terminals.

Two clarifications:

• Yes, in case there is no sequence in a production at all, it can be called the empty set.
• When you have the same non-terminal on both sides of an expansion rule, it is said to form a recursion relation.

An aside: It might be fun to convince yourself that the liar paradox cannot be posed or discussed in terms of Venn’s diagram. The property of the “sheet” on which Venn’ diagram is drawn is, by some simple intuitive notions we all bring to bear on Venn’s diagram, cannot have a “recursion” relation.

Yes, the set theory itself was always “powerful” enough to allow for recursions. People like Godel merely made this feature explicit, and took full “advantage” of it.

Recursion, the continuum, and epistemological (and metaphysical) validity:

In our discussion above, I had merely asserted, without giving even a hint of a proof, that the three ways (viz., the symbolic logic of statements or  propositions, the computer languages theory, and the set theory) were all equivalent ways of expressing the same basic idea (i.e. the one which we are concerned about, here).

I will now once again make a few more observations, but without explaining them in detail or supplying even an indication of their proofs. The factoids I must point out are the following:

• You can start with the natural numbers, and by using simple operations such as addition and its inverse, and multiplication and its inverse, you can reach the real number system. The generalization goes as: Natural to Whole to Integers to Rationals to Reals. Another name for the real number system is: the continuum.
• You can use the computer languages theory to generate a machine representation for the natural numbers. You can also mechanize the addition etc. operations. Thus, you can “in principle” (i.e. with infinite time and infinite memory) represent the continuum in the CS terms.
• Generating a machine representation for natural numbers requires the use of recursion.

Finally, a few words about epistemological (and metaphysical) validity.

• The concepts of numbers (whether natural or real) have a logical precedence, i.e., they come first. The entire arithmetic and the calculus must come before does the computer-representation of some of their concepts.
• A machine-representation (or, equivalently, a set-theoretic representation) is merely a representation. That is to say, it captures only some aspects or attributes of the actual concepts from maths (whether arithmetic or the continuum hypothesis). This issue is exactly like what we saw in the first and second posts in this series: a set is a concrete collection, unlike a concept which involves a consciously cast unit perspective.
• If you try to translate the idea of recursion into the usual cognitive terms, you get absurdities such as: You can be your child, literally speaking. Not in the sense that using scientific advances in biology, you can create a clone of yourself and regard that clone to be both yourself and your child. No, not that way. Actually, such a clone is always your twin, not child, but still, the idea here is even worse. The idea here is you can literally father your own self.
• Aristotle got it right. Look up the distinction between completed processes and the uncompleted ones. Metaphysically, only those objects or attributes can exist which correspond to completed mathematical processes. (Yes, as an extension, you can throw in the finite limiting values, too, provided they otherwise do mean something.)
• Recursion by very definition involves not just absence of completion but the essence of the very inability to do so.

Closure on the “learnability issue”:

Homework 4: Go through the last two posts in this series as well as this one, and figure out that the only reason that the set theory allows a “recursive” relation is because a set is, by the design of the set theory, a concrete object whose definition does not have to involve an epistemologically valid process—a unit perspective as in a properly formed concept—and so, its name does not have to stand for an abstract mentally held unit. Call this happenstance “The Glaring Epistemological Flaw of the Set Theory” (or TGEFST for short).

Homework 5: Convince yourself that any lemma or theorem that makes use of Godel’s Incompleteness Theorem is necessarily based on TGEFST, and for the same reason, its truth-status is: it is not true. (In other words, any lemma or theorem based on Godel’s theorem is an invalid or untenable idea, i.e., essentially, a falsehood.)

Homework 6: Realize that the learnability issue, as discussed in Prof. Lev Reyzin’s news article (discussed in the first part of this series [^]), must be one that makes use of Godel’s Incompleteness Theorem. Then convince yourself that for precisely the same reason, it too must be untenable.

[Yes, Betteridge’s law [^] holds.]

Other remarks:

Remark 1:

As “asymptotical” pointed out at the relevant Reddit thread [^], the authors themselves say, in another paper posted at arXiv [^] that

While this case may not arise in practical ML applications, it does serve to show that the fundamental definitions of PAC learnability (in this case, their generalization to the EMX setting) is vulnerable in the sense of not being robust to changing the underlying set theoretical model.

What I now remark here is stronger. I am saying that it can be shown, on rigorously theoretical (epistemological) grounds, that the “learnability as undecidable” thesis by itself is, logically speaking, entirely and in principle untenable.

Remark 2:

Another point. My preceding conclusion does not mean that the work reported in the paper itself is, in all its aspects, completely worthless. For instance, it might perhaps come in handy while characterizing some tricky issues related to learnability. I certainly do admit of this possibility. (To give a vague analogy, this issue is something like running into a mathematically somewhat novel way into a known type of mathematical singularity, or so.) Of course, I am not competent enough to judge how valuable the work of the paper(s) might turn out to be, in the narrow technical contexts like that.

However, what I can, and will say is this: the result does not—and cannot—bring the very learnability of ANNs itself into doubt.

Phew! First, Panpsychiasm, and immediately then, Learnability and Godel. … I’ve had to deal with two untenable claims back to back here on this blog!

… Code! I have to write some code! Or write some neat notes on ML in LaTeX. Only then will, I guess, my head stop aching so much…

Honestly, I just downloaded TensorFlow yesterday, and configured an environment for it in Anaconda. I am excited, and look forward to trying out some tutorials on it…

BTW, I also honestly hope that I don’t run into anything untenable, at least for a few weeks or so…

…BTW, I also feel like taking a break… May be I should go visit IIT Bombay or some place in konkan. … But there are money constraints… Anyway, bye, really, for now…

A song I like:

Music: Sooraj (the pen-name of “Shankar” from the Shankar-Jaikishan pair)
Lyrics: Ramesh Anavakar

[Any editing would be minimal; guess I will not even note it down separately.] Did an extensive revision by 2019.01.21 23:13 IST. Now I will leave this post in the shape in which it is. Bye for now.

# Learnability of machine learning is provably an undecidable?—part 1

Update on 23 January 2019, 17:55 IST:

In this series of posts, which was just a step further from the initial, brain-storming kind of a stage, I had come to the conclusion that based on certain epistemological (and metaphysical) considerations, Ben-David et al.’s conclusion (that learnability can be an undecidable) is logically untenable.

However, now, as explained here [^], I find that this particular conclusion which I drew, was erroneous. I now stand corrected, i.e., I now consider Ben-David et al.’s result to be plausible. Obviously, it merits a further, deeper, study.

However, even as acknowledging the above-mentioned mistake, let me also hasten to clarify that I still stick to my other positions, especially the central theme in this series of posts. The central theme here was that there are certain core features of the set theory which make implications such as Godel’s incompleteness theorems possible. These features (of the set theory) demonstrably carry a glaring epistemological flaw such that applying Godel’s theorem outside of its narrow technical scope in mathematics or computer science is not permissible. In particular, Godel’s incompleteness theorem does not apply to knowledge or its validation in the more general sense of these terms. This theme, I believe, continues to hold as is.

Update over.

This one news story has been lying around for about a week on my Desktop:

Lev Reyzin, “Unprovability comes to machine learning,” Nature, vol. 65, pp. 166–167, 10 January 2019 [^]. PDF here: [^]

(I’ve forgotten how I came to know about it though.) The story talks about the following recent research paper:

Ben-David et al., “Learnability can be undecidable,” Nature Machine Intelligence, vol. 1, pp. 44–48, January 2019 [^]. PDF here: [^]

I don’t have the requisite background in the theory of the research paper itself, and so didn’t even try to read through it. However, I did give Reyzin’s news article a try. It was not very successful; I have not yet been able to finish this story yet. However, here are a few notings which I made as I tried to progress through this news story. The quotations here all come from from Reyzin’s news story.

Before we begin, take a moment to notice that the publisher here is arguably the most reputed one in science, viz., the Nature publishing group. As to the undecidability of learnability, its apparent implications for practical machine learning, artificial intelligence, etc., are too obvious to be pointed out separately.

“During the twentieth century, discoveries in mathematical logic revolutionized our understanding of the very foundations of mathematics. In 1931, the logician Kurt Godel showed that, in any system of axioms that is expressive enough to model arithmetic, some true statements will be unprovable.”

Is it because Godel [^] assumed that any system of axioms (which is expressive enough to model arithmetic) would be based on the standard (i.e. mathematical) set theory? If so, his conclusion would not be all that paradoxical, because the standard set theory carries, from an epistemological angle, certain ill-conceived notions at its core. [BTW, throughout this (short) series of posts, I use Ayn Rand’s epistemological theory; see ITOE, 2e [^][^].]

To understand my position (that the set theory is not epistemologically sound), start with a simple concept like “table”.

According to Ayn Rand’s ITOE, the concept “table” subsumes all possible concrete instances of tables, i.e., all the tables that conceivably exist, might have ever existed, and might ever exist in future, i.e., a potentially infinite number of concrete instances of them. Ditto, for any other concept, e.g., “chair.” Concepts are mental abstractions that stand for an infinite concretes of a given kind.

Now, let’s try to run away from philosophy, and thereby come to rest in the arms of, say, a mathematical logician like Kurt Godel [^], or preferably, his predecessors, those who designed the mathematical set theory [^].

The one (utterly obvious) way to capture the fact that there exist tables, but only using the actual terms of the set theory, is to say that there is a set called “tables,” and that its elements consist of all possible tables (i.e., all the tables that might have existed, might conceivably exist, and would ever conceivably exist in future). Thus, the notion again refers to an infinity of concretes. Put into the terms of the set theory, the set of tables is an infinite set.

OK, that seems to work. How about chairs? Once again, you set up a set, now called “chairs,” and proceed to dump within its braces every possible or conceivable chair.

So far, so good. No trouble until now.

The trouble begins when you start applying operators to the sets, say by combining them via unions, or by taking their intersections, and so on—all that Venn’s diagram business [^]. But what is the trouble with the good old Venn diagrams, you ask? Well, the trouble is not so much to the Venn diagrams as it is to the basic set theory itself:

The set theory makes the notion of the set so broad that it allows you to combine any sets in any which way you like, and still be able to call the result a meaningful set—meaningful, as seen strictly from within the set theory.

Here is an example. You can not only combine (take union of) “tables” and “chairs” into a broader set called “furniture,” you are also equally well allowed, by the formalism of the set theory, to absorb into the same set all unemployed but competent programmers, Indian HR managers, and venture capitalists from the San Francisco Bay Area. The set theory does not by itself have anything in its theoretical structure, formalism or even mathematical application repertoire, using which it could possibly so much as raise a finger in such matters. This is a fact. If in doubt, refer to the actual set theory ([^] and links therein), take it strictly on its own terms, in particular, desist mixing into it any extra interpretations brought in by you.

Epistemology, on the other hand, does have theoretical considerations, including explicitly formulated rules at its core, which together allow us to distinguish between proper and improper formulations of concepts. For example, there is a rule that the concrete instances being subsumed under a concept must themselves be conceptually commensurate, i.e., they must possess the same defining characteristics, even if possibly to differing degrees. Epistemology prevents the venture capitalists from the San Francisco Bay Area from being called pieces of furniture because it clarifies that they are people, whereas pieces of furniture are inanimate objects, and for this crucial reason, the two are conceptually incommensurate—they cannot be integrated together into a common concept.

To come back to the set theory, it, however, easily allows for every abstractly conceivable “combination” for every possible operand set(s). Whether the operation has any cognitive merit to it or not, whether it results into any meaningful at all or not, is not at all a consideration—not by the design of the set theory itself (which, many people suppose, is more fundamental to every other theory).

So—and get this right—calling the collection of QC scientists as either politicians or scoundrels is not at all an abuse of the mathematical structure, content, and meaning of the set theory. The ability to take an intersection of the set of all mathematicians who publish papers and the set of all morons is not a bug, it is very much a basic and core feature of the set theory. There is absolutely nothing in the theory itself which says that the intersection operator cannot be applied here, or that the resulting set has to be an empty set. None.

Set theory very much neglects the considerations of the kind of a label there is to a set, and the kind of elements which can be added to it.

More on this, later. (This post has already gone past 1000 words.)

The songs section will come at the conclusion of this (short) series of posts, to be completed soon enough; stay tuned…

# Data science code snippets—2: Using world’s simplest ANN to visualize that deep learning is hard

Good morning! [LOL!]

In the context of machine learning, “learning” only means: changes that get effected to the weights and biases of a network. Nothing more, nothing less.

I wanted to understand exactly how the learning propagates through the layers of a network. So, I wrote the following piece of code. Actually, the development of the code went through two distinct stages.

First stage: A two-layer network (without any hidden layer); only one neuron per layer:

In the first stage, I wrote what demonstrably is world’s simplest possible artificial neural network. This network comes with only two layers: the input layer, and the output layer. Thus, there is no hidden layer at all.

The advantage with such a network (“linear” or “1D” and without hidden layer) is that the total error at the output layer is a function of just one weight and one bias. Thus the total error (or the cost function) is a function of only two independent variables. Therefore, the cost function-surface can (at all) be directly visualized or plotted. For ease of coding, I plotted this function as a contour plot, but a 3D plot also should be possible.

Here are couple of pictures it produces. The input randomly varies in the interval [0.0, 1.0) (i.e., excluding 1.0), and the target is kept as 0.15.

The following plot shows an intermittent stage during training, a stage when the gradient descent algorithm is still very much in progress.

SImplest network with just input and output layers, with only one neuron per layer: Gradient descent in progress.

The following plot shows when the gradient descent has taken the configuration very close to the local (and global) minimum:

SImplest network with just input and output layers, with only one neuron per layer: Gradient descent near the local (and global) minimum.

Second stage: An n-layer network (with many hidden layers); only one neuron per layer

In the second stage, I wanted to show that even if you add one or more hidden layers, the gradient descent algorithm works in such a way that most of the learning occurs only near the output layer.

So, I generalized the code a bit to have an arbitrary number of hidden layers. However, the network continues to have only one neuron per layer (i.e., it maintains the topology of the bogies of a train). I then added a visualization showing the percent changes in the biases and weights at each layer, as learning progresses.

Here is a representative picture it produces when the total number of layers is 5 (i.e. when there are 3 hidden layers). It was made with both the biases and the weights all being set to the value of 2.0:

It is clearly seen that almost all of learning is limited only to the output layer; the hidden layers have learnt almost nothing!

Now, by way of a contrast, here is what happens when you have all initial biases of 0.0, and all initial weights of 2.0:

Here, the last hidden layer has begun learning enough that it shows some visible change during training, even though the output layer learns much more than does the last hidden layer. Almost all learning is via changes to weights, not biases.

Next, here is the reverse situation: when you have all initial biases of 2.0, but all initial weights of 0.0:

The bias of the hidden layer does undergo a slight change, but in an opposite (positive) direction. Compared to the case just above, the learning now is relatively more concentrated in the output layer.

Finally, here is what happens when you initialize both biases and weights to 0.0.

The network does learn (the difference in the predicted vs. target does go on diminishing as the training progresses). However, the percentage change is too small to be visually registered (when plotted to the same scale as what was used earlier).

The code:

Here is the code which produced all the above plots (but you have to suitably change the hard-coded parameters to get to each of the above cases):

'''
SimplestANNInTheWorld.py
-- Implements the case-study of the simplest possible 1D ANN.
-- Each layer has only neuron. Naturally, there is only one target!
-- It even may not have hidden layers.
-- However, it can have an arbitrary number of hidden layers. This
feature makes it a good test-bed to see why and how the neurons
in the hidden layers don't learn much during deep learning, during
a straight-forward'' application of the gradient descent algorithm.
-- Please do drop me a comment or an email
if you find this code useful in any way,
say, in a corporate training setup or
-- History:
* 30 December 2018 09:27:57  IST:
Project begun
* 30 December 2018 11:57:44  IST:
First version that works.
* 01 January 2019 12:11:11  IST:
last layer, for no. of layers = 2 (i.e., no hidden layers).
* 01 January 2019 18:54:36  IST:
Added visualizations for percent changes in biases and
weights, for no. of layers &amp;amp;amp;amp;gt;=3 (i.e. at least one hidden layer).
* 02 January 2019 08:40:17  IST:
The version as initially posted on my blog.
'''
import numpy as np
import matplotlib.pyplot as plt

################################################################################
# Functions to generate the input and test data

def GenerateDataRandom( nTrainingCases ):
# Note: randn() returns samples from the normal distribution,
# but rand() returns samples from the uniform distribution: [0,1).

def GenerateDataSequential( nTrainingCases ):
adInput = np.linspace( 0.0, 1.0, nTrainingCases )

def GenerateDataConstant( nTrainingCases, dVal ):
adInput = np.full( nTrainingCases, dVal )

################################################################################
# Functions to generate biases and weights

def GenerateBiasesWeightsRandom( nLayers ):

def GenerateBiasesWeightsConstant( nLayers, dB, dW ):

################################################################################
# Other utility functions

def Sigmoid( dZ ):
return 1.0 / ( 1.0 + np.exp( - dZ ) )

def SigmoidDerivative( dZ ):
dA = Sigmoid( dZ )
dADer = dA * ( 1.0 - dA )

# Server function. Called with activation at the output layer.
# In this script, the target value is always one and the
# same, i.e., 1.0).
# Assumes that the form of the cost function is:
#       C_x = 0.5 * ( dT - dA )^2
# where, note carefully, that target comes first.
# Hence the partial derivative is:
#       \partial C_x / \partial dA = - ( dT - dA ) = ( dA - dT )
# where note carefully that the activation comes first.
def CostDerivative( dA, dTarget ):
return ( dA - dTarget )

def Transpose( dA ):
np.transpose( dA )
return dA

################################################################################
# Feed-Forward

def FeedForward( dA ):
## print( "\tFeed-forward" )
l_dAllZs = []
# Note, this makes l_dAllAs have one extra data member
# as compared to l_dAllZs, with the first member being the
# supplied activation of the input layer
l_dAllAs = [ dA ]
nL = 1
dZ = w * dA + b
l_dAllZs.append( dZ )
# Notice, dA has changed because it now refers
# to the activation of the current layer (nL)
dA = Sigmoid( dZ )
l_dAllAs.append( dA )
## print( "\tLayer: %d, Z: %lf, A: %lf" % (nL, dZ, dA) )
nL = nL + 1
return ( l_dAllZs, l_dAllAs )

################################################################################
# Back-Propagation

def BackPropagation( l_dAllZs, l_dAllAs ):
## print( "\tBack-Propagation" )
# Step 1: For the Output Layer
dZOP = l_dAllZs[ -1 ]
dAOP = l_dAllAs[ -1 ]
dZDash = SigmoidDerivative( dZOP )
dDelta = CostDerivative( dAOP, dTarget ) * dZDash

# Since the last hidden layer has only one neuron, no need to take transpose.
dAPrevTranspose = Transpose( l_dAllAs[ -2 ] )
dGradW = np.dot( dDelta, dAPrevTranspose )

# Step 2: For all the hidden layers
for nL in range( 2, nLayers ):
dZCur = l_dAllZs[ -nL ]
dZCurDash = SigmoidDerivative( dZCur )
dWNextTranspose = Transpose( dWNext )
dDot = np.dot( dWNextTranspose, dDelta )
dDelta = dDot * dZCurDash

dAPrev = l_dAllAs[ -nL-1 ]
dAPrevTrans = Transpose( dAPrev )
dGradWCur = np.dot( dDelta, dAPrevTrans )

def PlotLayerwiseActivations( c, l_dAllAs, dTarget ):
plt.subplot( 1, 2, 1 ).clear()
dPredicted = l_dAllAs[ -1 ]
sDesc = "Activations at Layers. Case: %3d\nPredicted: %lf, Target: %lf" % (c, dPredicted, dTarget)
plt.xlabel( "Layers" )
plt.ylabel( "Activations (Input and Output)" )
plt.title( sDesc )

nLayers = len( l_dAllAs )
dES = 0.2	# Extra space, in inches
plt.axis( [-dES, float(nLayers) -1.0 + dES, -dES, 1.0+dES] )

# Plot a vertical line at the input layer, just to show variations
plt.plot( (0,0), (0,1), "grey" )

# Plot the dots for the input and hidden layers
for i in range( nLayers-1 ):
plt.plot( i, l_dAllAs[ i ], 'go' )
# Plot the dots for the output layer
plt.plot( nLayers-1, dPredicted, 'bo' )
plt.plot( nLayers-1, dTarget, 'ro' )

def PlotGradDescent( c, dOrigB, dOrigW, dB, dW ):
plt.subplot( 1, 2, 2 ).clear()

d = 5.0
ContourSurface( d )
plt.axis( [-d, d, -d, d] )
plt.plot( dOrigB, dOrigW, 'bo' )
plt.plot( dB, dW, 'ro' )
plt.grid()
plt.xlabel( "Biases" )
plt.ylabel( "Weights" )
sDesc = "Gradient Descent for the Output Layer.\n" \
"Case: %3d\nWeight: %lf, Bias: %lf" % (c, dW, dB)
plt.title( sDesc )

def ContourSurface( d ):
nDivs = 10
dDelta = d / nDivs
w = np.arange( -d, d, dDelta )
b = np.arange( -d, d, dDelta )
W, B = np.meshgrid( w, b )
A = Sigmoid( W + B )
plt.imshow( A, interpolation='bilinear', origin='lower',
cmap=plt.cm.Greys, # cmap=plt.cm.RdYlBu_r,
extent=(-d, d, -d, d), alpha=0.8 )
CS = plt.contour( B, W, A )
plt.clabel( CS, inline=1, fontsize=7 )

plt.clf()

plt.axis( [-0.2, nComputeLayers+0.7, -320.0, 320.0] )

print( "Case: %3d" \
"\nPercent Changes in Biases:\n%s" \
"\nPercent Changes in Weights:\n%s\n" \
adx = np.linspace( 0.0, nComputeLayers-1, nComputeLayers )
plt.grid()
plt.xlabel( "Layer Number" )
plt.ylabel( "Percent Change in Weight (Red) and Bias (Blue)" )
sTitle = "How most learning occurs only at an extreme layer\n" \
"Percent Changes to Biases and Weights at Each Layer.\n" \
"Training case: %3d, Target: %lf, Predicted: %lf" % (c, dTarget, dPredicted)
plt.title( sTitle )

dSmall = 1.0e-10
if all( abs( adDiff ) ) &amp;amp;amp;amp;gt; dSmall and all( abs(adOrig) ) &amp;amp;amp;amp;gt; dSmall:

################################################################################
# The Main Script
################################################################################

dEta = 1.0 # The learning rate
nTrainingCases = 100
nTestCases = nTrainingCases // 5
adInput = GenerateDataRandom( nTrainingCases ) #, 0.0 )

## print( "Data:\n %s" % (adInput) )

# Must be at least 2. Tested up to 10 layers.
nLayers = 2
# Just a single target! Keep it in the interval (0.0, 1.0),
# i.e., excluding both the end-points of 0.0 and 1.0.

dTarget = 0.15

# The input layer has no biases or weights. Even the output layer
# here has only one target, and hence, only one neuron.
# Hence, the weights matrix for all layers now becomes just a
# vector.
# For visualization with a 2 layer-network, keep biases and weights
# between [-4.0, 4.0]

## print( "Initial Biases\n", adAllBs )
## print( "Initial Weights\n", adAllWs )

plt.figure( figsize=(10,5) )

# Do the training...
# For each input-target pair,
for c in range( nTrainingCases ):
## print( "Case: %d. Input: %lf" % (c, dInput) )

# Do the feed-forward, initialized to dA = dInput
l_dAllZs, l_dAllAs = FeedForward( dInput )

# Do the back-propagation

## print( "Updating the network biases and weights" )
adAllBs = [ dB - dEta * dDeltaB
adAllWs = [ dW - dEta * dDeltaW

## print( "The updated network biases:\n", adAllBs )
## print( "The updated network weights:\n", adAllWs )

if 2 == nLayers:
PlotLayerwiseActivations( c, l_dAllAs, dTarget )
PlotGradDescent( c, dOrigB, dOrigW, dB, dW )
else:
# Plot in case of many layers: Original and Current Weights, Biases for all layers
# and Activations for all layers
dPredicted = l_dAllAs[ -1 ]
plt.pause( 0.1 )

plt.show()

# Do the testing
print( "\nTesting..." )
for c in range( nTestCases ):

print( "\tTest Case: %d, Value: %lf" % (c, dInput) )

l_dAllZs, l_dAllAs = FeedForward( dInput )
dPredicted = l_dAllAs[ -1 ]
dDiff = dTarget - dPredicted
dCost = 0.5 * dDiff * dDiff
print( "\tInput: %lf, Predicted: %lf, Target: %lf, Difference: %lf, Cost: %lf\n" % (dInput, dPredicted, dTarget, dDiff, dCost) )

print( "Done!" )



Things you can try:

• Change one or more of the following parameters, and see what happens:
• Target value
• Values of initial weights and biases
• Number of layers
• The learning rate, dEta
• Change the cost function; e.g., try the linear function instead of the Sigmoid. Change the code accordingly.
• Also, try to conceptually see what would happen when the number of neurons per layer is 2 or more…

Have fun!

A song I like:

(Marathi) “pahaaTe pahaaTe malaa jaag aalee”
Music and Singer: C. Ramchandra
Lyrics: Suresh Bhat

# Data science code snippets—1: My first ANN (a copy of Nielsen’s)

I took the liberty to re-type Nielsen’s[^] code [^], in the process introducing the Hungarian naming convention for variables, and also many other non-Pythonic and non-Nielsenic whimsies. Here is the end-result, for whatever worth it is. (Its sole intended purpose was to run the feed-forward and back-propagation algorithms in IDE.) Tested just a slight bit on Ubuntu Release 18.04.1 LTS (Bionic Beaver) 64-bit + Python 3.7.0 64-bit in conda 4.5.11 + MS VSCode 1.30.1.

It works. (Has to. Except for the cosmetic changes, it’s basically Nielsen’s code.) It produces this graph as an output:

Total Errors vs. Epochs

“Enjoy!”

'''
ANN.Nielsen.SmallerAndEasiestDataVectors.py
-- Simple Artificial Neural Network.
-- This code very closely follows "network.py given in Nielsen's book
(chapter 1). It was modified a bit by Ajit R. Jadhav.
-- Meant for developing understanding of ANNs through debugging in IDE
(with ability to inspect values live at run-time that also can be
verified manually, by taking *smaller* and *simpler* vectors for the input
and target data-sets).
-- Includes functions for generation of small and simple 1D input vectors
and their corresponding target vectors. The input data generated here is,
effectively, just a randomized version of Nielsen's "vectorized_result"
function.
-- Other features of this code:
* Writes the total error for each epoch, to a file.
* Plots the total error vs. epoch number.
* Follows the Hungarian naming convention. However, the variable names
may not reflect the best possible choices; this code was written in a hurry.
-- Do whatever you like with it, but 100% on your own
responsibility.
-- But if you use this code for academic or corporate
training purposes, then please do let me know by an
email or a (confidential) comment here.
-- History
* Begun: 23 Dec. 2018
* Completed first version that works: 26 Dec. 2018
* This version: Sunday 30 December 2018 11:54:52  IST
'''

import random
import numpy as np
import matplotlib.pyplot as plt

################################################################################
# Global helper functions
################################################################################

################################################################################
# Helper function. Uses numpy.random.randn() to create inputs and targets,
# whether for training or for testing. Useful for very simple 1D studies only.
# (The input data as generated here very easily predict the targets! Even just
# a single call to the softmax function would have been sufficient!!)
def GenerateData( nNeursInInput, nLenOPLayer, nCasesPerTarget ):
# Python lists for inputs and targets, returned at the end by this function.
l_mdInput = []
l_mdTarget = []
for t in range( nLenOPLayer ):
for i in range( nCasesPerTarget ):
mdInput = np.random.rand( nNeursInInput, 1 )*0.1
mdInput[t][0] = mdInput[t][0] + np.random.random_sample()*0.1 + 0.8
l_mdInput.append( mdInput )

mdTarget = np.zeros( (nLenOPLayer,1) )
mdTarget[t][0] = 1.0
l_mdTarget.append( mdTarget )
# pickle returns data as arrays; let's simulate the same thing here
amdInput = np.array( l_mdInput )
amdOutput = np.array( l_mdTarget )
# Python 3(+?) thing. Convert zip to list either here or later.
# Better to do it here.
Data = list( zip( amdInput, amdOutput ) )
return Data

################################################################################
# Helper function. Computes the sigmoid activation function
def Sigmoid( mdZ ):
return 1.0 / ( 1.0 + np.exp( - mdZ ) )

################################################################################
# Helper function. Computes the derivative of the sigmoid activation function
def SigmoidDerivative( mdZ ):
mdA = Sigmoid( mdZ )
mdADer = mdA * ( 1.0 - mdA )

################################################################################
# Helper function. Called with a single activation vector with its
# target vector. Assumes that the form of the cost function for each
# neuron is (note carefully that target comes first):
#       C_x = 0.5 * ( dT - dA )^2
# so that
#       \partial C_x / \partial dA = - ( dT - dA ) = ( dA - dT )
# (where note that the activation comes first).
def CostDerivative( mdA_OP, mdT ):
return ( mdA_OP - mdT )

################################################################################
# Class Network begins here
################################################################################

class Network(object):

############################################################################
# Constructor.
# Supply an array containing number of neurons for each layer
# e.g., [5,4,3]
def __init__(self, anNIL):
self.m_nLayers = len( anNIL )
self.m_anNIL = np.array( anNIL )
# anNIL[1:] means: An array with all entries except the first
# anNIL[:-1] means: An array with all entries except for the last
# We allocate (N,1)-shaped matrices rather than (N)-shaped vectors
# because later on in back-propagation, to have a vectorized code,
# we perform np.dot() in its tensor-product-like avatar, and there,
# we need both the operands to be matrices.
self.m_amdAllBs = [ np.random.randn( nNeursCur, 1 )
for nNeursCur in anNIL[1:] ]
self.m_amdAllWs = [ np.random.randn( nNeursCur, nNeursPrev )
for nNeursPrev, nNeursCur in
zip( anNIL[:-1], anNIL[1:] ) ]
pass

############################################################################
# For each (input, target) tuple representing a row in the
# mini-batch, perform the training. We compute the changes (delta's)
# to be effected to biases and weights through feedback and
# backpropagation, add them up until the mini-batch is over, and then
# apply them to update the weights and biases only once at the end of
# this function. Unlike in Nielsen's code, here we have a separate
# function each for feed-forward and back-propagation
def UpdateMiniBatch( self, dEta, nMiniBatchSize, MiniBatch ):

# Allocate a local matrix each for holding the GradB and GradW matrices
# for all the layers. Both are initialized to zero's. Can't directly
# use np.zeros() because these matrices are not of uniform sizes;
# the no. of neurons in each layer is arbitrarily different
l_mdErrorsMB = []

# For each (input, target) tuple representing a row in the
# mini-batch, perform the training.

for x, y in MiniBatch:

# Feed-Forward Pass: Feed the next input vector to the network
# using the existing weights and biases, and find the net-input
# and activations it produces at all the network layers.

l_mdAllZs, l_mdAllAs = self.FeedForward( x )

# Compute the error vector
mdDiff = y - l_mdAllAs[ -1 ]
mdErrors = 0.5 * mdDiff * mdDiff
l_mdErrorsMB.append( mdErrors )

# Back-Propagation Pass: Back-propagate the change in the total cost
# function (total error) to the local gradients to be applied to
# each weight (of a synaptic connection between current and previous
# layers), and to each bias (of the current layer).

# Add the changes to the local copies of the biases and weights.
# The zip function takes iterable elements as input, and returns an
# iterator for the tuple. Thus, zipping here makes it iterate through
# each layer of the network.

# Processing for all the rows in the current mini-batch is now over.
# Now, update the network data-members from the local copies.

# Note, we take an average of the changes over all rows.
dEtaAvgOverMB = dEta / nMiniBatchSize

# w means: the old weights matrix for a *single* layer only
# nw means: \sum_{i}^{all rows of mini-batch} GradW_{i-th row}
self.m_amdAllBs = [ mdB - dEtaAvgOverMB * mdGradB

self.m_amdAllWs = [ mdW - dEtaAvgOverMB * mdGradW

# Return the average error vector for this mini-batch
return l_mdErrorsMB

############################################################################
# Called for a single input vector (i.e., "x" in Nielsen's code)
def FeedForward( self, mdA ):

# Python list of all activations, layer-by-layer
l_mdAllAs = [ mdA ]
# Python list for z vectors, layer-by-layer
l_mdAllZs = []

# For the weight and bias matrices of each layer...
for mdW, mdB in zip( self.m_amdAllWs, self.m_amdAllBs ):
# Compute the net input vector to activate
# the neurons of this layer
mdZ = np.dot( mdW, mdA ) + mdB
l_mdAllZs.append( mdZ )

# Compute the activations for all the neurons
# of this layer
mdA = Sigmoid( mdZ )
l_mdAllAs.append( mdA )

return ( l_mdAllZs, l_mdAllAs )

############################################################################
# Called with inputs and activations for all the layers,
# i.e., for the entire network in one go.
def BackProp( self, l_mdAllAs, l_mdAllZs, mdTarget ):

# Allocate a local matrix each for holding the GradB and GradW matrices
# for all the layers. Both are initialized to zero's. Can't use
# np.zeros() because these matrices are not of uniform sizes;
# the no. of neurons in each layer is arbitrarily different
amdAllGradBs = [ np.zeros( mdB.shape ) for mdB in self.m_amdAllBs ]
amdAllGradWs = [ np.zeros( mdW.shape ) for mdW in self.m_amdAllWs ]

# Back-propagation occurs in two distinct stages:
# (i) In the first stage, we begin from the end (the output layer), and
# use the layer just before that, in order to update the biases and
# weights of the connections lying between the two (which are, in this
# class, are stored in the OP layer).
# (ii) In the second stage, we iterate back successively through the
# network, starting from the last hidden layer, up to the first hidden
# layer.
# The split-up into two stages is necessary because the activation of a
# neuron from the input- or hidden-layers flows through many (all)
# neurons in the immediately next layer, and the total cost function
# (i.e. the total error) is affected by *all* these paths. In contrast,
# the output layer has no next layer, and so, the activation of a
# neuron in the output layer affects the total cost function only
# through its own connection to the total cost, not that of other
# layers. So, the Delta at the OP layer can be directly
# computed from a knowledge of the OP layer activations (predictions)
# and the prescribed targets. But to compute the Delta at an
# intermediate layer, we do need the Delta for the immediately
# next layer.

##########

# The OP layer is the last, accessed through the -1 index. Get the
# activations (A's) on its output side, and the net inputs (Z's) on its
# input side.
mdA_OP = l_mdAllAs[ -1 ]
mdZ_OP = l_mdAllZs[ -1 ]

# Compute the partial derivatives, and use them to find the Grad for
# all B's in the OP layer.
mdZ_OP_Der = SigmoidDerivative( mdZ_OP )
# As an intermediate product, find the delta for the OP layer. It is
# subsequently used in the second stage as an input, thereby
# back-propagating the changes to the total cost function back
# throughout the network.
mdDelta = CostDerivative( mdA_OP, mdTarget ) * mdZ_OP_Der
# Compute the Grad for all B's in the OP layer

# Compute the partial derivatives, and find the Grad for all W's in
# the layer just before the OP layer. Index -2 means: just before OP.
# We use "LHL" to mean the last hidden layer.
mdA_LHL_Transpose = l_mdAllAs[ -2 ].transpose()
# Here, we build the GradW *matrix* from what essentially are two
# vectors (or (N,1) matrices). Even if the numpy function says
# "dot", it actually works like the tensor product---it outputs an
# (N X N) matrix, not a single number as a scalar. It is for this step
# that we made even B's an (N,1) shaped matrix in the constructor.
amdAllGradWs[ -1 ] = np.dot( mdDelta, mdA_LHL_Transpose )

###########
# Stage II: Compute mdGradB and mdGradB for the last hidden layer
#           and go back, all the way up to the first hidden layer

# Note that only the negative of the index ever gets used in the
# following loop.
# a[-2] means the second-last layer (i.e. the last hidden layer).
# a[-1] means the last layer (i.e., the OP layer).
# a[-2-1] means the layer just before the second-last layer.
# The current layer starts from the output layer. The previous
# layer is the one before the current.
# If the indexing gets confusing, notice that Nielsen uses the range
# (2, nLayers), rather than that of (1, nLayers-1) which would have
# been a bit more natural, at least to me. But of course, his choice
# is slightly more consistent from a mathematical viewpoint---and
# from the fact that there are no weights and biases for the IP
# layer.
for nL in range( 2, self.m_nLayers ):
# Find the mdDelta for the previous layer, using that for the
# current layer...
mdZCur = l_mdAllZs[ -nL ]
mdZCurDer = SigmoidDerivative( mdZCur )
mdWNextTrans = self.m_amdAllWs[ -nL+1 ].transpose()
# Note, for the very first pass through the loop, the mdDelta being
# used on the RHS is what was earlier computed in the Stage I given
# above (viz. for the output layer). It now gets updated here, so
# that it can act as the "previous" mdDelta in the next pass
# through this loop.
mdDelta = np.dot( mdWNextTrans, mdDelta ) * mdZCurDer
# mdDelta now refers to the current layer, not to the next

mdAPrevTrans = l_mdAllAs[ -nL-1 ].transpose()
amdAllGradWs[ -nL ] = np.dot( mdDelta, mdAPrevTrans )

# Explaining the above four lines is left as an exercise
# for the reader. (Hints: (i) Write out the matrix equation
# "straight," i.e., as used in the feed-forward pass. (ii)
# Use the rules for the matrix multiplications and the rules
# for taking transposes.)

############################################################################
# Implements the stochastic gradient descent algorithm
def SGD( self, sErrorsFileName, nEpochs,
TrainingData, nMiniBatchSize, TestData ):

# This variable is used only for the x-axis during plotting the
# total activation errors at the OP layer.

# To have a small file-size, we persist the total error for the
# output layer only as averaged over the entire epoch (i.e., the
# entire training data, randomly shuffled).
foErrors = open( sErrorsFileName, 'w' )

nTrainingCases = len( TrainingData )

# An epoch is defined for the entire training data. However, each epoch
# begins by randomly shuffling the training data. Effectively, we
# change the initial position from which to begin descending down the
# higher-dimensional total-cost-function surface.
for e in range( nEpochs ):

# Create Mini-Batches covering the entire training data
np.random.shuffle( TrainingData )
MiniBatches = [ TrainingData[ k : k + nMiniBatchSize ]
for k in range( 0, nTrainingCases, nMiniBatchSize ) ]

dTotalErrorEpoch = 0
# For each mini-batch
for MiniBatch in MiniBatches:
# Conduct the training over the entire mini-batch,
# and collect the errors accumulated over it. Add them
# to the accumulated error for the current epoch.
l_mdErrorsMB = self.UpdateMiniBatch( dEta, nMiniBatchSize, MiniBatch )
dAvgErrorMB = self.ComputeAvgErrorForMiniBatch( l_mdErrorsMB )
dTotalErrorEpoch = dTotalErrorEpoch + dAvgErrorMB

dAvgErrorEpoch = dTotalErrorEpoch / nMiniBatchSize
# Write to file
sLine = "%E\n" % (dAvgErrorEpoch)
foErrors.write( sLine )

# self.Evaluate( TestData )
print( "Epoch %d: Avg. Error: %lf" % (e, dAvgErrorEpoch) )

foErrors.close()

plt.xlabel( "Epochs" )
plt.ylabel( "Total Error at the OP Layer\n(Epoch Avg. of Mini-Batch Avgs.)" )
plt.title( "Training Errors" )
plt.show()

############################################################################
# Helper function
def ComputeAvgErrorForMiniBatch( self, l_mdErrorsMB ):
nSize = len( l_mdErrorsMB )
dTotalErrorMB = 0
# For each training case in this mini-batch
for i in range( nSize ):
# Get the error vector for the i-th training case
mdE = l_mdErrorsMB[ i ]
dTotalErrorCase = mdE.sum()
dTotalErrorMB = dTotalErrorMB + dTotalErrorCase
# The average of the total errors for all cases in the current mini-batch
dAvgErrorMB = dTotalErrorMB / nSize
return dAvgErrorMB

############################################################################
# This function is mostly useless in this simple a scenario; it predicts
# 100 % accuracy right from 1st epoch! Obvious. With the kind of input
# data we generate, the prediction would have been 100 % accurate even
# with a single function call to softmax!
#
# def Evaluate( self, TestData ):
#     nTestCases = len( TestData )
#     anPredictedRes = []
#     for x, y in TestData:
#         l_mdAllZs, l_mdAllAs = self.FeedForward( x )

#         mdActivOP = l_mdAllAs[ -1 ]
#         nPredIdx = np.argmax( mdActivOP )
#         nTargetIdx = np.argmax( y )
#         anPredictedRes.append( int( nPredIdx == nTargetIdx ) )
#     print( anPredictedRes )
#     pass
################################################################################
# Class Network ends here
################################################################################

################################################################################
# main script
################################################################################

# Prepare Input Data

# The learning rate
dEta = 0.5

# Number of Neurons contained In each successive Layer
anNIL = [10,10,10]

# Size of the input layer
nNeursInInput = anNIL[ 0 ]
# Size of the target layer
nTargets = anNIL[ -1 ]
# Total number of cases to have for training
nTotalNumCasesForTraining = 10000
# Total number of cases to have for testing
nTotalNumCasesForTesting = nTotalNumCasesForTraining // 5

# Number of cases to generate for each target for training
nCasesPerTargetForTraining = nTotalNumCasesForTraining // nTargets
# Number of cases to generate for each target for testing
nCasesPerTargetForTesting = nTotalNumCasesForTesting // nTargets

# For repeatability, seed the RNG in NumPy. For "true" random-ness
# at each program invocation, comment out the next line. Notice that
# seeding, if any, must occur before generating data.
np.random.seed( 0 )

TrainingData = GenerateData( anNIL[ 0 ], anNIL[ -1 ], nCasesPerTargetForTraining )
TestingData = GenerateData( anNIL[ 0 ], anNIL[ -1 ], nCasesPerTargetForTesting )

# Number of Epochs
nEpochs = 10
# Mini-Batch Size
nMiniBatchSize = 10

# Instantiate the network
# Optionally, we can have a re-seeding also here
# np.random.seed( 10000 )
nn = Network( anNIL )
# Let the network find the most optimum combination of values
# for all the weights and biases it contains.
nn.SGD( "EpochErrors.txt", nEpochs, TrainingData, nMiniBatchSize, TestingData )

print( "Done!" )



Things for you to add and/or to try:

• Add the code to persist the network (weights and biases) to a plain-text CSV file, and to re-init it from that file.
• Add the code to show a simple (real-time) diagrammatic visualization of the training process for a tiny (say [4,3,2]) network, using a colormap-based shades being used for the lines of the synaptic weights and similarly colormap-based circles for neuron background for biases.
• Write a code for a more ambitious code for generation of input data for n-dimensional data. It would have `m’ number of islands of volumes (of simplest topology) of randomly varying dark shades which are embedded into a randomly varying background of light shades. That is, it would be an n-dimensional generalization of taking the Windows Paint (or similar) program, having a nonuniform but light-gray fill-color, with dark-shaded blobs of arbitrary shapes randomly placed in it, sort of like a two-phase materials micro-structure. You have to take a configuration of such blobs and introduce random but small variations into its shape and position too. Then, take this small group of variations and assign a target value to it. Repeat for N number of targets.

No songs section this time. Will be back on 31st/1st, as promised (see the post just below).

A song I like:

[Realizing that you wouldn’t be in a mood to listen to (and be able to hum) a neat, melodious, golden oldie, may as well run it right away. … Yes, I will come back with my new year’s resolutions on the 31st/1st (see the post just below), but it’s likely that I won’t run any songs section at that time.]

(Hindi) “dhaDakne lage dil ke taron ki duniya…”
Singers: Mahendra Kapoor, Asha Bhosale
Music: N. Dutta
Lyrics: Sahir Ludhianvi

….Done. Take care and bye for now. See you on the 31st, with the new post…

History:
First published: 2018.12.27 23:29 IST
Updated (also the code): 2018.12.28 09.08 IST

# The quantum mechanical features of my laptop…

My laptop has developed certain quantum mechanical features after its recent repairs [^]. In particular, if I press the “power on” button, it does not always get “measured” into the “power-on” state.

That’s right. In starting the machine, it is not possible to predict when the power-on button may work, when it may lead to an actual boot-up. Sometimes it does, sometimes it doesn’t.

For instance, the last time I shut it down was on the last night, just before dinner. Then, after dinner, when I tried to restart it, the quantum mechanical features kicked in and the associated randomness was such that it simply refused the request. Ditto, this morning. Ditto, early afternoon today. But now (at around 18:00 hrs on 09 October), it somehow got up and going!

Fortunately, I have taken backup of crucial data (though not all). So, I can afford to look at it with a sense of humour.

But still, if I don’t come back for a somewhat longer period of time than is usual (about 8–10 days), then know that, in all probability, I was just waiting helplessly in getting this thing repaired, once again. (I plan to take it to the repairsman tomorrow morning.) …

…The real bad part isn’t this forced break in browsing or blogging. The real bad part is: my inability to continue with my ANN studies. It’s not possible to maintain any tempo in studies in this now-on-now-off sort of a manner—i.e., when the latter is not chosen by you.

Yes, I do like browsing, but once I get into the mood of studying a new topic (and, BTW, just reading through pop-sci articles does not count as studies) and especially if the studies also involve programming, then having these forced breaks is really bad. …

Anyway, bye for now, and take care.

PS: I added that note on browsing and then it struck me. Check out a few resources while I am gone and following up with the laptop repairs (and no links because right while writing this postscript, the machine crashed, and so I am somehow completing it using smartphone—I hate this stuff, I mean typing using at most two fingers, modtly just one):

1. As to Frauchiger and Renner’s controversial much-discussed result, Chris Lee’s account at ArsTechnica is the simplest to follow. Go through it before any other sources/commentaries, whether to the version published recently in Nature Comm. or the earlier ones, since 2016.
2. Carver Mead’s interview in the American Spectator makes for an interesting read even after almost two decades.
3. Vinod Khosla’s prediction in 2017 that AI will make radiologists obsolete in 5 years’ time. One year is down already. And that way, the first time he made remarks to that sort of an effect were some 6+ years ago, in 2012!
4. As to AI’s actual status today, see the Quanta Magazine article: “Machine learning confronts the elephant in the room” by Kevin Hartnett. Both funny and illuminating (esp. if you have some idea about how ML works).
5. And, finally, a pretty interesting coverage of something about which I didn’t have any idea beforehand whatsoever: “New AI strategy mimics how brains learn to smell” by Jordana Cepelwicz in Quanta Mag.

Ok. Bye, really, for now. See you after the laptop begins working.

A Song I Like:
Indian, instrumental: Theme song of “Malgudi Days”
Music: L. Vaidyanathan

# Some running thoughts on ANNs and AI—1

Go, see if you want to have fun with the attached write-up on ANNs [^] (but please also note the version time carefully—the write-up could change without any separate announcement).

The write-up is more in the nature of a very informal blabber of the kind that goes when people work out something on a research blackboard (or while mentioning something about their research to friends, or during brain-storming session, or while jotting things on the back of the envelop, or something similar).

A “song” I don’t like:

“Credits”: Go, figure [^]. E.g., here [^]. Yes, the video too is (very strongly) recommended.

Update on 05 October 2018 10:31 IST:

Psychic attack on 05 October 2018 at around 00:40 IST (i.e. the night between 4th and 5th October, IST).

# Caste Brahmins, classification, and ANN

1. Caste Brahmins:

First, a clarification: No, I was not born in any one of the Brahmin castes, particularly, not at all in the Konkanastha Brahmins’ caste.

Second, a suggestion: Check out how many caste-Brahmins have made it to the top in the Indian and American IT industry, and what sort of money they have made—already.

No, really.

If you at all bother visiting this blog, then I do want you to take a very serious note of both these matters.

No. You don’t have to visit this blog. But, yes, if you are going to visit this blog, to repeat, I do want you to take  matters like these seriously.

Some time ago, perhaps a year ago or so, a certain caste-Brahmin in Pune from some place (but he didn’t reveal his shakha, sub-caste, gotra, pravar, etc.) had insulted me, while maintaining a perfectly cool demeanor for himself, saying how he had made so much more money than me. Point taken.

But my other caste-Brahmin “friends” kept quiet at that time; not a single soul from them interjected.

In my off-the-cuff replies, I didn’t raise this point (viz., why these other caste-Brahmins were keeping quiet), but I am sure that if I were to do that, then, their typical refrain would have been (Marathi) “tu kaa chiDatos evhaDa, to tar majene bolat hotaa.” … English translation: Why do you get so angry? He was just joking.

Note the usual caste-Brahmin trick: they skillfully insert an unjustified premise; here, that you are angry!

To be blind to the actual emotional states or reactions of the next person, if he comes from some other caste, is a caste-habit with the caste-Brahmins. The whole IT industry is full of them—whether here in India, or there in USA/UK/elsewhere.

And then, today, another Brahmin—a Konkanastha—insulted me. Knowing that I am single, he asked me if I for today had taken the charge of the kitchen, and then, proceeded to invite my father to a Ganesh Pooja—with all the outward signs of respect being duly shown to my father.

Well, coming back to the point which was really taken:

Why have caste-Brahmins made so much money—to the point that they in one generation have begun very casually insulting the “other” people, including people of my achievements?

Or has it been the case that the people of the Brahmin castes always were this third-class, in terms of their culturally induced convictions, but that we did not come to know of it from our childhood, because the elderly people around us kept such matters, such motivations, hidden from us? May be in the naive hope that we would thereby not get influenced in a bad manner? Possible.

And, of course, how come these caste-Brahmins have managed to attract as much money as they did (salaries in excess of Rs. 50 lakhs being averagely normal in Pune) even as I was consigned only to receive “attract” psychic attacks (mainly from abroad) and insults (mainly from those from this land) during the same time period?

Despite all my achievements?

Do take matters like these seriously, but, of course, as you must have gathered by now, that is not the only thing I would have, to talk about. And, the title of this post anyway makes this part amply clear.

2. The classification problem and the ANNs:

I have begun my studies of the artificial neural networks (ANNs for short). I have rapidly browsed through a lot of introductory articles (as also the beginning chapters of books) on the topic. (Yes, including those written by Indians who were born in the Brahmin castes.) I might have gone through 10+ such introductions. Many of these, I had browsed through a few years ago (I mean only the introductory parts). But this time round, of course, I picked them up for a more careful consideration.

And soon enough (i.e. over just the last 2–3 days), I realized that no one in the field (AI/ML) was talking about a good explanation of this question:

Why is it that the ANN really succeeds as well as it does, when it comes to the classification tasks, but not others?

If you are not familiar with Data Science, then let me note that it is known that ANN does not do well on all the AI tasks. It does well only on one kind of them, viz., the classification tasks. … Any time you mention the more general term Artificial Intelligence, the layman is likely to think of the ANN diagram. However, ANNs are just one type of a tool that the Data Scientist may use.

But the question here is this: why does the ANN do so well on these tasks?

I formulated this question, and then found an answer too, and I would sure like to share it with you (whether the answer I found is correct or not). However, before sharing my answer, I want you to give it a try.

It would be OK by me if you answer this question in reference to just one or two concrete classification tasks—whichever you find convenient. For instance, if you pick up OCR (optical character recognition, e.g., as explained in Michael Nielson’s free online book [^]), then you have to explain why an ANN-based OCR algorithm works in classifying those MNIST digits / alphabets.

Hint: Studies of Vedic literature won’t help. [I should know!] OTOH, studies of good books on epistemology, or even just good accounts covering methods of science, should certainly come in handy.

I will give you all some time before I come back on that question.

In the meanwhile, have fun—if you wish to, and of course, if you are able to. With questions of this kind. (Translating the emphasis in the italics into chaste Marathi: “laayaki asali tar.” Got it?)

A song I like:
(Marathi) “ooncha nicha kaahi neNe bhagawant”
Lyrics: Sant Tukaram
Music and Singer: Snehal Bhatkar

# Machine “Learning”—An Entertainment [Industry] Edition

Yes, “Machine ‘Learning’,” too, has been one of my “research” interests for some time by now. … Machine learning, esp. ANN (Artificial Neural Networks), esp. Deep Learning. …

Yesterday, I wrote a comment about it at iMechanica. Though it was made in a certain technical context, today I thought that the comment could, perhaps, make sense to many of my general readers, too, if I supply a bit of context to it. So, let me report it here (after a bit of editing). But before coming to my comment, let me first give you the context in which it was made:

Context for my iMechanica comment:

It all began with a fellow iMechanician, one Mingchuan Wang, writing a post of the title “Is machine learning a research priority now in mechanics?” at iMechanica [^]. Biswajit Banerjee responded by pointing out that

“Machine learning includes a large set of techniques that can be summarized as curve fitting in high dimensional spaces. [snip] The usefulness of the new techniques [in machine learning] should not be underestimated.” [Emphasis mine.]

Then Biswajit had pointed out an arXiv paper [^] in which machine learning was reported as having produced some good DFT-like simulations for quantum mechanical simulations, too.

A word about DFT for those who (still) don’t know about it:

DFT, i.e. Density Functional Theory, is “formally exact description of a many-body quantum system through the density alone. In practice, approximations are necessary” [^]. DFT thus is a computational technique; it is used for simulating the electronic structure in quantum mechanical systems involving several hundreds of electrons (i.e. hundreds of atoms). Here is the obligatory link to the Wiki [^], though a better introduction perhaps appears here [(.PDF) ^]. Here is a StackExchange on its limitations [^].

Trivia: Kohn and Sham received a Physics Nobel for inventing DFT. It was a very, very rare instance of a Physics Nobel being awarded for an invention—not a discovery. But the Nobel committee, once again, turned out to have put old Nobel’s money in the right place. Even if the work itself was only an invention, it did directly led to a lot of discoveries in condensed matter physics! That was because DFT was fast—it was fast enough that it could bring the physics of the larger quantum systems within the scope of (any) study at all!

And now, it seems, Machine Learning has advanced enough to be able to produce results that are similar to DFT, but without using any QM theory at all! The computer does have to “learn” its “art” (i.e. “skill”), but it does so from the results of previous DFT-based simulations, not from the theory at the base of DFT. But once the computer does that—“learning”—and the paper shows that it is possible for computer to do that—it is able to compute very similar-looking simulations much, much faster than even the rather fast technique of DFT itself.

OK. Context over. Now here in the next section is my yesterday’s comment at iMechanica. (Also note that the previous exchange on this thread at iMechanica had occurred almost a year ago.) Since it has been edited quite a bit, I will not format it using a quotation block.

[An edited version of my comment begins]

A very late comment, but still, just because something struck me only this late… May as well share it….

I think that, as Biswajit points out, it’s a question of matching a technique to an application area where it is likely to be of “good enough” a fit.

I mean to say, consider fluid dynamics, and contrast it to QM.

In (C)FD, the nonlinearity present in the advective term is a major headache. As far as I can gather, this nonlinearity has all but been “proved” as the basic cause behind the phenomenon of turbulence. If so, using machine learning in CFD would be, by the simple-minded “analysis”, a basically hopeless endeavour. The very idea of using a potential presupposes differential linearity. Therefore, machine learning may be thought as viable in computational Quantum Mechanics (viz. DFT), but not in the more mundane, classical mechanical, CFD.

But then, consider the role of the BCs and the ICs in any simulation. It is true that if you don’t handle nonlinearities right, then as the simulation time progresses, errors are soon enough going to multiply (sort of), and lead to a blowup—or at least a dramatic departure from a realistic simulation.

But then, also notice that there still is some small but nonzero interval of time which has to pass before a really bad amplification of the errors actually begins to occur. Now what if a new “BC-IC” gets imposed right within that time-interval—the one which does show “good enough” an accuracy? In this case, you can expect the simulation to remain “sufficiently” realistic-looking for a long, very long time!

Something like that seems to have been the line of thought implicit in the results reported by this paper: [(.PDF) ^].

Machine learning seems to work even in CFD, because in an interactive session, a new “modified BC-IC” is every now and then is manually being introduced by none other than the end-user himself! And, the location of the modification is precisely the region from where the flow in the rest of the domain would get most dominantly affected during the subsequent, small, time evolution.

It’s somewhat like an electron rushing through a cloud chamber. By the uncertainty principle, the electron “path” sure begins to get hazy immediately after it is “measured” (i.e. absorbed and re-emitted) by a vapor molecule at a definite point in space. The uncertainty in the position grows quite rapidly. However, what actually happens in a cloud chamber is that, before this cone of haziness becomes too big, comes along another vapor molecule, and “zaps” i.e. “measures” the electron back on to a classical position. … After a rapid succession of such going-hazy-getting-zapped process, the end result turns out to be a very, very classical-looking (line-like) path—as if the electron always were only a particle, never a wave.

Conclusion? Be realistic about how smart the “dumb” “curve-fitting” involved in machine learning can at all get. Yet, at the same time, also remain open to all the application areas where it can be made it work—even including those areas where, “intuitively”, you wouldn’t expect it to have any chance to work!

[An edited version of my comment is over. Original here at iMechanica [^]]

“Boy, we seem to have covered a lot of STEM territory here… Mechanics, DFT, QM, CFD, nonlinearity. … But where is either the entertainment or the industry you had promised us in the title?”

You might be saying that….

Well, the CFD paper I cited above was about the entertainment industry. It was, in particular, about the computer games industry. Go check out SoHyeon Jeong’s Web site for more cool videos and graphics [^], all using machine learning.

And, here is another instance connected with entertainment, even though now I am going to make it (mostly) explanation-free.

Check out the following piece of art—a watercolor landscape of a monsoon-time but placid sea-side, in fact. Let me just say that a certain famous artist produced it; in any case, the style is plain unmistakable. … Can you name the artist simply by looking at it? See the picture below:

A sea beach in the monsoons. Watercolor.

If you are unable to name the artist, then check out this story here [^], and a previous story here [^].

A Song I Like:

And finally, to those who have always loved Beatles’ songs…

Here is one song which, I am sure, most of you had never heard before. In any case, it came to be distributed only recently. When and where was it recorded? For both the song and its recording details, check out this site: [^]. Here is another story about it: [^]. And, if you liked what you read (and heard), here is some more stuff of the same kind [^].

Endgame:

I am of the Opinion that 99% of the “modern” “artists” and “music composers” ought to be replaced by computers/robots/machines. Whaddya think?

[Credits: “Endgame” used to be the way Mukul Sharma would end his weekly Mindsport column in the yesteryears’ Sunday Times of India. (The column perhaps also used to appear in The Illustrated Weekly of India before ToI began running it; at least I have a vague recollection of something of that sort, though can’t be quite sure. … I would be a school-boy back then, when the Weekly perhaps ran it.)]