Sunday, July 2, 2023


Some time ago Linkedin started sending weekly e-mail with the trends for "your kind of jobs". Every week since it's been reporting a 10% drop in job postings, then went over 10%. Well, last week it held at 0, then this week it dropped another 13%. 

At the same time the government reports tout the job growth. So, is it a drop in high-paying jobs and growth in low-paying jobs? (Which is not exactly a great thing). As it happens, I know someone who runs a job posting web site that is used predominantly for low-to-mid paying jobs. And guess what, he says that the postings on his site have also sharply dropped around April. So no, not a growth of low-paying jobs either.

 What does it all tells us about the government statistics? As they saying goes, "lies, damned lies, and statistics". Maybe they'll "revise" it a few months later as they've done with statistics on income after inflation, where the loudly touted slight growth above inflation had quietly turned into a ~5% loss.

Saturday, June 3, 2023

the first actual use of Triceps

 I've recently got Triceps used for a real application, for the first time ever, after approximately 13 years. Not really a production usage, rather a scientific demo, but it's an outside project, not part of Triceps itself. Embedded in Matlab, or all things. No, not a full Matlab API, but an application written in C++ embedded as a native function into Matlab. Which is exactly what Triceps was designed for.

But really Matlab and Triceps with a full Matlab API might be a match made in heaven. Matlab really sucks at two things: the data structures where you can append and search by value, and parallelism. And those are things where Triceps excels. Yeah, Matlab has the parfor loops but the trouble there is that the parallel threads (they're more typically processes but logically still threads) are stateless. All the input data is packed up and sent to them (at a great overhead), and then the results are packed up and sent back. You can't just preserve state in a thread between two calls, it has to be sent from scratch every time. And no, it doesn't seem to be the same constant in shared memory read in parallel by multiple threads. It actually gets copied for every thread. So parfor only works well when you send a small amount of data, process it for some long time, and then send a small result back. Not well when you want your thread to make queries to a large database. But keeping state is what a Triceps thread does. The Triceps threads are also easy to organize in pipelines. (Yeah, Matlab does have some pipelines in one of extensions, but they seem as cumbersome as parfor). And the read-only shared memory would work too if queried through Triceps and only small results of the queries get returned to Matlab. It could work really, really awesomely together. The trouble of course is that I personally don't have much of an incentive to do this work.

That's the good part. What went badly? Well, Triceps in C++ feels rather dated. It's a project that was started in 2010, before C++11, and it feels like that. I didn't even aim for C++ to be the main language, but more as an API for embedding into the other languages. But now it can be done much more smoothly straight in C++. So if I ever get to it, a modernization of the C++ API is in order. Some libraries are dated too, in particular NSPR. It also proved to be a pain in building: I haven't thought much about it before, but the definitions used in building of the applications that use Triceps have to be the same as when building Triceps itself. So if triceps is built with NSPR, and the application doesn't include the right definitions for the Triceps headers to use NSPR, it crashes in a surprising way. Fortunately, the standard C++ library now has APIs for the atomic values, so transition to that API is now in order. On the other hand, shared_ptr is a more complicated question, and keeping the Triceps Autoref might still be better for efficiency.

Saturday, April 15, 2023

Bayes 29: computation for numeric values (and Python)

As promised in the previous installment, here is the how the data science handles the Bayesian inference directly on the numeric values. Remember from that last installment, the formula for inference by weight is:

W(H[i]|E) = W(H[i]) * P(E|H[i])

So what they do is instead of using the probability P(E|H[i]), they use the probability density p(E|H[i]) (note the lowercase p) from a previously computed distribution, this time E meaning not just a specific event variable but a specific numeric value of a specific variable. Which really makes sense, since that's what the probability density means: the probability that this specific value would come out in the distribution for this variable. Since it's a conditional distribution, only the training cases for the hypothesis H[i] are used to compute the distribution.

The typical distribution used here is the normal distribution. For what I understand, it's not only because it's the typical one occurring in reality, but also because for any original distribution, once you start repeatedly pulling the examples from it, the values produced by these examples become described by the normal distribution (if I remember right, this comes out of the Central Limit Theorem). The normal distribution has two parameters: mu (the mean value) and sigma (standard deviation, equal to the square root of the variance). The probability density function then is:

p(x) = (1/sqrt(2*pi*sigma^2)) * exp( -(x-mu)^2 / (2*sigma^2) )

Or, since sigma here is always squared, we can also replace the squared sigma with the variance var:

p(x) = (1/sqrt(2*pi*var)) * exp( -(x-mu)^2 / (2*var) )

The values of mu and var would be different for each hypothesis H[i]. So we'd split all the training cases into subsets by mutually exclusive hypotheses, and then compute:

mu[E][i] = sum(cases[i][E]) / N[i]
var[E][i] = sum((cases[i][E] - mu[E][i])^2) / (N[i] - 1)

Here N[i] is the count of the training cases with the outcome H[i], and cases[i] are all the cases with that outcome, effectively N[i] = count(cases[i]). The reason why we use (N[i] - 1) instead of plain N[i] is that it computes not a variance but a sample variance. This comes out of a rather interesting distinction between the probability theory and statistics: the probability theory describes the random distributions where the parameters of these distributions are known in advance, while statistics describes how to deduce these parameters from looking at the samples of the data. Here we obviously don't somehow know the distribution in advance but have to deduce it from looking at the training cases, i.e. the data samples, so we have to use the statistics and its sample variance. Technically speaking, mu here is also not the true mean but the sample mean, but the formula for it is the same anyway. However the need to divide by (N[i] - 1) comes from the sample mean producing a different value that is offset from the true mean, sample variance counter-adjusting for this offset. And the reason for this offset is that we do the sampling without replacement (i.e. each training case is used at most once in the computation). If we did the sampling with replacement, for example select 1000 values at random from the training cases (each time from the whole set of the training cases, not excluding the cases that have already been selected but allowing them to be selected again), then when computing the sample variance we'd use the plain N[i], i.e. the same 1000 rather than 999. Or if we magically knew the true mean, we'd use N[i] for the sample variance too.

And here is a good spot to talk about Python. As much as I detest the language, I see why it's popular for the data science: because it has some very good libraries in it for this purpose: numpy, sklearn, matplotlib. They've even added an operator of matrix multiplication @ into the language for the convenience of these libraries. And the point of the matrix operations is not only that they allow to write down things in a form that is short and comfortable for the data scientists (although confusing as hell for the rest of us), but also that they allow to move all the tight loops into the C++ code that is way faster than the Python code, and also allows the library to internally parallelize the computations as needed. So the computations in a slow interpreted language with questionable parallelism become quite fast and quite parallel. Aside from that, the libraries provide other conveniences. Sklearn even has a bunch of popular data sets built right into it. So in Python the code to compute mu and var looks like this:

# X is a numpy matrix, where each row corresponds to a training case
# and each column to an input variable (i.e. a numeric event);
# Y is a numpy vector containing the outcomes for all the training cases

# Find the subset of the training case that have the outcome i,
# it produces a vector of the same size with True for
# each included case and False for each excluded case.
selector = (Y == i)

# Select the subset of training cases for i, keeping all the columns
subset = X[selector, :]

# Count the included cases, which is the number of rows in the subset
n = subset.shape[0]

# Compute the sample mean, by averaging across all rows (axis=0). 
# The result is a row with a column per input.
mu[i] = subset.mean(axis = 0)

# Compute the sample variance, again producing a row with a
# column per input. The tricky part is that mu[i] is a single row,
# so the subtraction operator automatically considers it as a matrix
# that has as many rows as subset, with all the rows being the same.
# Sum adds up the columns across all the rows (axis=0) into one row.
# And division by a scalar divides each column by the same scalar.
# The result is a row with a column per input.
var[i] = np.square(subset - mu[i]).sum(axis=0) / (n - 1)

But wait, there is more. Note how the probability density function has an exponent function in it. Instead of computing the weights by taking the exponents and multiplying, we could compute the logarithms of the weights, by adding up the logarithms (and the logarithm of the exponent is the argument of the exponent, saving this function computation). So the formula becomes:

logW(H[i]|E) = logW(H[i]) + log(p(E|H[i])) =
  = logW(H[i]) + log(1/sqrt(2*pi*var[i, E])) 
             - (x[E] - mu[i, E])^2 / (2*var[i, E])

The part log(1/sqrt(2*pi*var[i, E])) is a constant that can be pre-computed in advance, so the computation with a few multiplications and additions is quite efficient.

Monday, April 10, 2023

Bayes 28: computation by weight revisited

I've found out recently how the people in the data science compute the Bayesian inference for values from an arbitrary numeric range (as opposed to yes/no events). As it turns out, they do it using a computation by weights. I've been wondering for a while after discovering the computation by weight on my own, why nobody else uses it, it's so much simpler than by probabilities. So the answer is similar to what I've discovered before for the connection between the Bayes and neural networks: the people in the field do know about it and do use it, only the simplified popular explanations don't.

I wanted to write down the explanation of how they do it (at least, for myself in the future), and so I went back to read what I wrote before about the Bayesian computation by weight, and found that what I wrote before is quite complicated. I wrote it down as I discovered it when experimenting with all those extra parameters that make a Bayesian system work in reality, and so that explanation is also full of discussion of those parameters. Also, I've learned some things since then.

Before going into th enew ground, let me try a new take on the same thing: discuss the Bayesian inference by weight, only now in its most basic form.

Let's start again with an event E and hypothesis H. In the modern terms, they call this machine a Bayesian classifier, and call the mutually-exclusive hypotheses classes. The events are the inputs, and based on the values of the inputs the machine is trying to decide, which hypothesis is most likely to be true. The classic Bayesian formula for a binary event E is:

P(H|E) = P(H) * P(E|H) / P(E)

Here P(H) is the prior (i.e. previous) probability that the hypothesis is true, P(E) is the prior probability that the event E will be found true (after we learn the actual value of the event this probability collapses), P(E|H) is the conditional probability that the event would be true for the cases where hypothesis H is true, and finally P(H|E) is the new (posterior) probability of H after we learn that E is true. If we learn that E is false, we would compute P(H|~E) instead, for the complementary event ~E.

Where did that formula come from? Well, when talking about probabilities, it really helps to draw a diagram that starts with the range of all the possible cases and then splits it into parts with the appropriate weights. Let's draw this field as a square, in beautiful ASCII art. And then split it with a horizontal line so that the area above the line matches the number of the cases resulting in the hypothesis H1 being true, and below the line with H1 being false (i.e. the complementary hypothesis ~H1, which we'll also name H2, being true). This is for a very simple classification with two complementary hypotheses: H1 = ~H2 and H2 = ~H1.

|               |
|               | H1
|               |
|               |
|               | H2 = ~H1
|               |

Now let's do the same, but split the same square vertically. The left side will match the event E1, and the right side will be its complementary ~E1:

|         |     |
|         |     |
|         |     |
|         |     |
|         |     |
|         |     |
|         |     |
    E1      ~E1

Now let's take the second picture and split each vertical part horizontally, into two parts, that again match H1 on top and H2 on the bottom. I've marked these parts with the letters a, b, c, d.

| a       | b   |
|         |     |
|---------|     | H1
| c       |     |
|         |     |
|         |-----|
|         | d   | H2
    E1      ~E1

Here the parts a and b correspond to H1, and their total area is equal to the original area (within the ASCII-art limitations) of the upper part of the split in the first picture. The parts c and d correspond to H2, and again the sum of their areas is equal to the original area of the lower part. But obviously they're split differently on the left and right sides, meaning that if we know that E is true, H1 has a lower probability than H2, but if E is false, H1 has a higher probability. And if we don't differntiate based on E1, the left and right parts average out.

The areas of these parts a...d are the weights of four sub-divisions:

a: E & H1
b: ~E & H1
c: E & H2 (or equivalently E & ~H1)
d: ~E & H2 (or equivalently E & ~H1)

The reason why I prefer to use H2 instead of ~H1 is that this notation allows to generalize more obviously to more than two hypotheses: H3, H4, and so on. Each additional hypothesis would add two areas to the picture, one on the left and one on the right.

Now we can express the probabilities through relations of these areas (and areas can also be called weights):

P(H1) = (a + b) / (a + b + c + d)
P(H2) = (c + d) / (a + b + c + d)
P(E) = (a + c) / (a + b + c + d)
P(~E) = 1 - P(E) = (b + d) / (a + b + c + d)
P(E|H1) = a / (a + b)
P(E|H2) = c / (c + d)
P(H1|E) = a / (a + c)
P(H2|E) = c / (a + c)
P(H1|~E) = b / (b + d)
P(H2|~E) = d / (b + d)

The general principle for P(x|y) is that the area that satisfies both conditions x and y becomes the numerator, and the area that satisfies y becomes the denominator.

Alright, let's substitute these formulas into the Bayesian formula:

P(H1) * P(E|H1) / P(E) = P(H1) * (1/P(E)) * P(E|H1)
  = ((a + b) / (a + b + c + d))
    * ((a + b + c + d) / (a + c))
    * (a / (a + b))
  = a / (a + c)
  = P(H1|E)
P(H2) * P(E|H2) / P(E) = P(H2) * (1/P(E)) * P(E|H2)
  = ((c + d) / (a + b + c + d))
    * ((a + b + c + d) / (a + c))
    * (c / (c + d))
  = c / (a + c)
  = P(H2|E)

So that's where that formula comes from and how it gets proven. Note that the computation of both probabilities involves the final division by the same value (a + c). If we just want to compare them, this division by the same value doesn't matter and we can skip it. Instead we'll just get a and c, which are also the weights W(H1|E) and W(H2|E), and if we want to get the probabilities, we can normalize by dividing by their sum. Or I guess rather than W(H1|E) we could say W(H1 & E) but that's the beauty of workiing with the weights instead of probabilities: the probabilities require normalization at each step, dividing by the sum of all possible weights, while with weights the sum is kept implicit, and W(H1|E) = W(H1 & E) = W(E|H1). When expressed in weights, the formulas become simpler:

W(H1) = a + b
W(H1|E) = W(H) * P(E|H1) = (a + b) * (a / (a + b)) = a

That's pretty much it. But there is one more question to consider: what if we have more than one event? We usually do have multiple events. After we compute P(Hi|E1) (or W(Hi|E1)), now we have a smaller rectangle left, with the level of the horizontal divider shifted from where it was in the previous square (or rectangle). What do we do with the next event? There are multiple ways to look at it.

One way is to hope that the events are completely independent from each other. This basically means that as we look at each part produced on splitting by E1 (E1 and ~E1), and further split each of these parts by E2, the horizontal lines in each vertical quarter shift according to the current level of H in the part being split, with the result that P(E2|E1,Hi) = P(E2|Hi). It would look something like this:

| e  | f  |g |h |
|----|    |  |  |
|    |    |  |  | H1
|    |----|  |  |
|    |    |--|  |
|    |    |  |  |
|    |    |  |--| H2
 E1   E1   ~E1 ~E1
 E2   ~E2  E2  ~E2

The equality P(E2|E1,H1) = P(E2|H1) = P(E2|~E1,H1) would imply (even though it doesn't quite look like it in ASCII art):

e / (e + f) = a / (a + b) = g / (g+h)

That's a simple-minded implementation (in the scienific circles they call it naive, sometimes even spelled with two "i"s). The problem there is that when the assumption is not true and the events are strongly dependent, this can drive probabilities in weird ways.

Another way would be to build the tree of exact splits: split the slices produced by E1 into slices for E2, then for E3, and so on, and for each vertical slice after each split find the exact proportion of cases. This is obviously much more complex, and complexity grows exponentially with the number of events.

The third way (I think I've just realized it from reading the materials about data science) would be to track the splits by events pair-wise, do the vertical splits by each pair: (E1, E2), (E1, E3), ..., (E1, En), (E2, E3), (E2, E4), ..., (E2, En), and so on. I haven't quite thought through yet the adjustments that would need to be computed for each split. I guess, fundamentally it would be a representation of the covariance matrix (another word I've learned). But I guess there are two potential ways to do it: one would be to adjust the positions of the horizontal lines after the second split, another would be to adjust the positions of the vertical lines for the second split. I'm not sure which one is more correct. Maybe either way would adjust the ratio of the areas on the left and right to the same value. But either way, after you apply E1, adjust the probabilities of E2, E3, and the rest of the events according to this matrix. Then after applying E2, adjust E3, E4, and so on. It won't be a perfect match that can be produced with the exact splits but it would easily catch the events that are duplicates and near-duplicates of each other.

Monday, March 6, 2023

VNC How-to

Theoretically speaking, an X terminal can work remotely through a tunnel with ssh -X. But in practice it does that very, very slowly. I don't understand why but they do a huge number of synchronous requests, which become very slow when the RTT is high. The thing that allows to use it over a long distances is called VNC. Its documentation is surprisingly poor, and so are the recorded talks, but I've finally figured it out.

All the VNC varieties grow from one source, that used to be Open Source but produced by a commercial company (NX). At some point they've stopped opensourcing it, but the previously published Open Source version started living its own life (FreeNX). The next major branch was TightVNC, and then TigerVNC branched off (but for what I undertand, TigerVNC is still backwards-compatible with TightVNC). Nowadays either TightVNC or TigerVNC or both are included in the Linux distributions as packages.

Running is fairly easy. On the remote machine start:

xvncserver -geometry 1800x1100 -alwaysshared :1

Change the geometry and display number to taste (or just skip the display number altogether, it will pick the first free one). "-alwaysshared" means that it will allow multiple parallel connections. For some reason it doesn't allow to set dpi (the display resolution) but you can make a copy of the script and add the option -dpi to the X server command (but it also looks like almost  nothing nowadays pays any attention to the DPI set in the X server).

You can start this from an SSH session, and it will keep running after you close the session. It doesn't use SSH tunneling but opens its own socket, and does its own encryption and password on it (it asks to create the password on the first start). Caveat: apparently some 10 years ago a bug was found where VNC allowed bypassing the password on connection. So better not open it to the wide Internet, just in case, but then connect the display through an SSH tunnel that forwards to the VNC server (it's easy, just one option on the client).

To kill the server later, do

xvncserver -kill :1 

To connect to the server, run on the display side:

xvncviewer -via user@gateway host:1

Here host is the name of the host with the VNC server, and "-via" means to go through an SSH tunnel before connecting to the host. The screen and all applications stay alive between the connections, which is awesome (just like RDP).

That's it, just two commands. It's somewhat inconvenient that Alt-Tab for window switching gets consumed by the local machine, but you can redefine an alternative combination on the remote machine instead (at least in the civilized session managers like MATE or Cinnamon). 

A little on how it works: just as you could have expected, it starts two proxy X servers derived from Xnest, one on the remote server, one on the display side. So most of the synchronous requests get handled locally on one or another side. And the remote server stays alive all the time, so it preserves the session state between connections. But there is more to it, since the protocol between these two X servers is not the regular X protocol under encryption but a modified one, since the NX times. There are even multiple versions of these protocols, but fortunately the client and server are smart enough to negotiate them, and it just works.

Saturday, February 18, 2023

arrested bouncing in FloatNeuralNet

It's been a while since I wrote any updates about my experiments with optimization of neural networks. Partially, it's because I've been busy with other things, and partially because I've tried something a few weeks ago, it didn't work out well, and I wasn't sure what to do with the results. On one hand, it didn't work out, so carrying this complexity into all the future code would be annoying, and even gating this code behind an option would create unpleasant effects on the performance. On the other hand, I think these experiments were really interesting, and maybe I'd want to do another variation of them in the future, so I didn't want to just discard the code (I've already discarded some code  in the past only to re-create it later). So I've been stuck thinking what to do with it, and I couldn't do the other experiments in the meantime.

I think I have a good solution for this now: the "mini-branches". I've copied the original and modified code, and the description into cpp/nn/xperim, one subdirectory per "mini-branch", where it can be found, and diff-ed, and reapplied if needed. 

The mini-branch for those last experiments is (1802 is the root revision for this mini-branch in SVN on SourceForge).

Both of these experiments center around the momentum descent. To recap on the previous state, I've been using the momentum descent loosely based on FISTA, with some extensions. One of these extensions is that whenever the gradient changes sign in some dimension, I kill the momentum in this dimension. Another is that I use these gradient sign changes as an indicator to auto-detect the safe basic gradient descent rate: I keep growing this rate until many (for some definition of "many" that I've been trying to tune with various experiments) dimensions start flipping the gradient signs after each step, then I back it off, then continue growing again.

The premise of the first experiment started with me noticing that I've unintentionally changed some logic from FISTA. FISTA has an interesting computation for the coefficient "nu" that determines how much of the momentum is used in each step. It starts with not having any momentum to apply on the first step, then on the second step it skips the momentum too, then it bumps nu to 1, and then gradually reduces it to "apply the brakes" and avoid circling too much around the optimum. But killing the momentum on the gradient sign change serves as an alternative application of the brakes.So I've removed that gradual reduction of nu, and things worked better. 

But I've noticed that I didn't handle the second step right. It wasn't keeping the momentum at 0. My guess has been that this was done to avoid including the large first step in random direction from the random initial point into the momentum. So I've set to fix it. But it made things worse, the error reduction had slowed down. Evidently, the first step is not so random after all. Or maybe my momentum-killing code is very good at eliminating the effects from the truly random first steps, and the remaining ones are useful.

The premise of the second experiment was to detect the situation when some dimension has the sign of its gradient bouncing back and forth on each step without reducing much in absolute value. This is what happens when the model starts tearing itself apart due to the descent step being too large: it keeps overshooting the optimum by a larger and larger value. It could also happen because of the momentum, but the momentum already gets reset to zero in my code when the gradient sign changes, so when we detect a gradient dimension change signs twice in a row, and get to more than 0.75 of the original value, that's because of the large descent rate. So how about we make only half the originally intended step in this dimension? It should land us much closer to the optimum. That's what I've done in this experiment.

And it didn't work out either, the convergence became slower. The potential explanation for this effect is the following: Consider a trough with a ball rolling down it. The trough is not exactly straight, and the ball isn't started exactly straight, so as it rolls down the trough, it keeps wobbling in the trough left and right, overshooting the center one way then the other way. As long as the ball doesn't go into resonance and starts wobbling more and more (eventually falling out of the trough), this wobble is actually not such a big deal. But when I dampen the wobble, it also dampens the speed of rolling down the trough, and since the momentum also gets killed on every wobble, the whole descent slows down. 

So it's something to think about for the future. It probably means that the speed could be improved by not killing the momentum on every change of the gradient sign, but only when the absolute change of the gradient becomes too large. But I'm not sure yet, how can this "too large" be defined reliably.

It likely also means that going to the individual descent rate coefficients for each dimension (something that I contemplated from both my own experiments and from reading the papers) might not work so well, because it's likely to cause the same kind of slow-down. We'll see, I'm now unblocked to try more experiments.


 I've been reading a little more about what other people do with the MNIST recognition. Apparently, doing the transformations on the training set is fairly common, so maybe a simple linear stretching and shrinking to multiply the number of examples would be good enough.

But the most interesting discovery has been that the dataset I've been playing with is not MNIST. Surprise, surprise. :-)

 The MNIST data set is much larger, and also has a higher resolution. It might be the older NIST dataset (without the "M" that stands for "Modified"), or maybe not even that. For some reason I've thought that the MNIST set comes from the ZIP codes recognition, so that's what I was looking for, the ZIP codes dataset, and apparently I've thought wrong. An interesting thing about the original NIST dataset is that it had the training set and the test set collected from different demographies (one from schoolchildren, another from employees of a government agency), so I guess if this is what I've got, it would explain why the sets don't represent each other so well. That was apparently a known major complaint with the older dataset that got straightened in the new modified one.

Sunday, January 15, 2023

trapeze to btimap for MNIST

Continuing with the trapeze-based implementation of the handwritten images that I've described in the previous post, next I've tried to do a reverse transformation: from the set of trapezes to a bitmap that tries to preserve the significant features detected by the trapezes. And if you've read the previous post, you've probably can guess the result: it became a little worse yet!

Looking at the images that got misrecognized, I think  I understand why. Think for example of a "4" and "9". "4" is often drawn with an open top. But when we draw a "9", there is also sometimes a small opening in the top right corner. When I do the detection of trapezes, this little opening becomes a significant feature, and when I convert these trapezes back to a bitmap, this little opening becomes a larg-ish opening, so now "9" became more like "4"! I guess, with a large and representative enough training set, they could be differentiated well. But not if all the "9"s with an opening are in the test set (and yes, if I fold the test set into the training set, it gets recognized quite well).

While looking at it all, I've also noticed a few more interesting things. I've turned the classifier mode on again (and improved it a bit to make the gradients more stable), and noticed that it drives the error on the training set down a lot. The classifier mode tries to make sure that the training cases that produce the wrong result get more attention by multiplying their gradients, which is essentially equivalent to adding 99 more copies of the same training case. Turn the classifier mode on, and the training error goes down from 0.06 to 0.025 in almost no time. Which means that a lot of this error is caused by the few outliers. And maybe another way to fix it would be to take a larger power in the loss function, say the 4th power instead of square. But the error on the test set doesn't budge much, so maybe this is a misleading measure that doesn't matter much and can cause overfitting.

I've found a bug in my code that computed, how many cases got recognized with a very low confidence: if even with the right outcome the absolute value for the highest outcome was still below 0. Due to this bug, all the cases that were labeled with "0" were counted as low-confidence. So this measure wasn't 17.5% on the training set, it was more like 1%, and in the classifier mode goes to near 0. But on the MNIST test set, the total of misrecognized and low-confidence cases is still near 20%.

The auto-adjustment of the descent rate still works well, and I've bumped it up to be more aggressive: changed the step up from 1.1 to 1.2, and the step down from 0.1 to 0.2. Now I think it exhibits the behavior that I've seen with a higher manually-set rate, where once in a while the error would bump up a little, explore the surroundings, and then quickly drop down below the previous low. And the attempts to start the destructive resonance are still well-arrested.

I've added a printout of gradients by layer, and they do tend to vary in the interesting way, kind of "sloshing about". At the start of the training the high layers usually show the high gradients, and the low layers show the high gradients. But this gradually reverses, and after a few thousand training passes you get the high gradients in the low layers, low gradients in the high layers. Unless you then go and change the some training criteria, then the high layers get the higher gradients again until things settle. Which probably means that selecting separate training rates by layer might be worth a try. Or maybe even selecting them separately for each weight, like the Adam algorithm that I've mentioned before does (I haven't tried this specific algorithm yet).

Sunday, January 8, 2023

trapeze data representation for MNIST

 As I've told before, the trouble with MNIST data set is that the test set contains the cases that are substantially different from any case in the training set. So doing well on it requires some kind of generalization on the training set to be able to correctly recognize the test cases. One idea I've had is to parse the images into the logical parts that look like trapezes.

For example, consider an image of the digit 0. It can be broken somewhat like this:

_   /===\   _
   //   \\
_ //     \\ _
  \\     //
_  \\   //  _

Here going from the top we have a horizontal bar that can be seen as a horizontal band containing trapeze of whitespace, followed by a drawn trapeze, followed by another whitespace trapeze. I've separated it by the underbar marks on the sides. Then the sides go expanding: there is still whitespace on the left, then goes the bar expanding to the left, an expanding trapeze of whitespace in the middle, the bar expanding to the right, and another whitespace on the right. Then it goes symmetrically shrinking on the bottom side, and completes with another horizontal bar. There could also be a vertical part on the sides in the middle. Once we recognize the break-down like this, it doesn't matter any more, what exact size is the symbol 0, it's still the same symbol. It gets harder for the many handwritten variations but maybe we get enough samples to build a recognition of all the major ones. So I've decided to try this path wit MNIST.

Recognizing the trapezes in a strict black-and-white image is much easier than with the halftones. So the first thing I've doe was to generate a B&W image by filtering on a fixed level. I've done this, and since I've had it, I've tried to train a model on it. And it did a little worse than on the halftone image.

Well, the next simple approximation is to encode the rows of pixels as run-length. So I've done it, and I've tried to train a model, and guess what, it has done a little worse yet. Which is a bad sign. Part of the problem is that each row becomes of a different size in runs, and to keep the fixed size, the unused runs at the end have to be filled with something. Originally I've filled them with 0s, and then I've tried to fill them with -1. Which made thins a little better but still a little worse than plain B&W image.

At this point the indications were kind of bad but if I didn't try the trapezes, I wouldn't know, right? So I've tried. I've encoded each trapeze with 3 numbers: 

(1) The average slope coefficient of its left side, computed as dx/dy (since a vertical line is possible and a horizontal line is not). Which is also happens to be the slope of the right side of the previous trapeze. It's average because in reality the pixels represent the curved lines, and I just approximate them with the straight trapezes.

(2) The width of the top of the trapeze

(3) The width of the bottom of the trapeze.

To get the ordering of whitespaces and drawings right, I always start each row of trapezes and also end it with a whitespace, and whatever gets left unused gets filled with -1 for widths and 0 for slope.

There is also one exception: since the leftmost whitespace trapeze always has its left slope vertical (i.e. 0), I've reused that value to put the row height into it.

And the result? By now you've probably guessed it, it was a little worse yet (and a lot slower to compute each pass because of the increased input size) than the run-length version. So this representation has turned out to be not very good for generalization, although not terribly bad either. Maybe it can be twisted in some way to make it better. Maybe put it back into a bitmap by treating each trapeze as one pixel, but by now I'm rather pessimistic on these prospects :-)

While experimenting with it, I've tuned the autoRate2 coefficients to make them a little more aggressive, and they had proved themselves quite well. To check how my auto-scaled momentum measures against the classic stochastic descent, I've tried that one too, and the stochastic version did noticeably worse. I've also tried going to Leaky RELU activation, and that one did a little worse yet. So I think at least the descent part and my Corner activation function are working decently well.

Thursday, January 5, 2023

unmelting a neural network

I've got wondering, could a neural network that experienced a "meltdown" from a too-high training rate be restored? I've got such an example of MNIST and dumped out its contents. What happened inside is that pretty much all the weights got pushed to the values of 1 and -1, the neurons becoming very much the same. So when trying to train it again, this lack of, pardon my French, diversity, wouldn't let the optimization to progress far: it just optimizes this one neuron state per layer in many copies.

For some time I've had an idea of how this problem could be resolved: by a partial randomization of gradients. So I've done this with a simple saw-toothed "randomization" where each next weight gets its gradient reduced slightly more up to a limit, then it drops back down and the next "sawtooth" starts. The starting position of the teeth gets shifted by one on each training pass. 

I've started by "tweaking"  in this way 1% of the gradient size (I've named this option of FloatNeuralNet tweakRate_), and combined it with options momentum_ and autoRate2_. It did work some but barely made a scratch.  OK, so I've bumped it up to 30%. It did work much better but still not quite great. Over about 300K training passes it got to the mean-square error of about 0.15 (by comparison, the normal training from a random point gets to about 0.05 in 10K steps) and would not budge much any more. The verification on the test cases was much closer: mean-square error of about 0.24 instead of 0.19 and the error rate of about 7% instead of 5.5%. So it might be not that bad after all. The combination of the new auto-rate and momentum descent worked great at preventing another meltdown. Interestingly, at the start all the gradient was concentrated in the last layer of 3, and then it gradually shifted towards the first layer.

Then I've tried the same tweak rate of 0.3 for the training from a random initial state, and it didn't have any detrimental effect at all, even did slightly better than without it. So it should be safe to use in general case as a cheap preventative measure.

This also gave me an opportunity to look more into the tuning of the auto-rate algorithm, which I've made a bit better, and also look into what gradients are where. As it turns out, the highest gradient dimensions by far are at the weights that have reached the [-1, +1] boundary, and they skew the norm2 and mean-square of the gradient a lot. When I've changed the code to mark such gradients post-factum as 0, that gave me an opportunity to count them separately and to exclude them from the means. Their number grows early in the draining and then gradually reduces (but not to 0). 

How about if we raise the boundaries, this should reduce the number of the dimensions hitting them and make the training faster, right? And it also would be a good test of the new auto-rate algorithm, since as I've shown before the weights over 1 are much more susceptible to meltdowns. I've tried the boundaries of 10 and 100. The auto-rate worked great but the training got slower. For all I can tell, the higher weights trigger more often the situations where the auto-rate algorithm drops the training rate down, and the rate tends to be 10 to 100 times lower.

But the bad news for the auto-rate logic is that manually picking a just-high-enough training rate still ultimately produces a slightly better result. The auto-rate algorithm starts with a similar rate but then gradually drops it by about 3 orders of magnitude. And as I've been watching the mean-square errors pass by pass, I could see that they showed differently: the fixed-rate algorithm would periodically have the error grow and maybe even stay up a while but then drop lower than where it was before, the auto-rate algorithm tends to just chisel away at the error rate little by little, it still has the error grow a little periodically but squashes it very fast. So perhaps the conservatively low rate gets the function trapped in some local minimum, while the fixed rate breaks out of them (when it doesn't lead to a meltdown). If I let the auto-rate algorithm grow the rate more, and then drop when it gets out of control, it actually does worse. But maybe some better adaptivity could be devised. 

And/or maybe bring the stochastic descent back into the mix. I've been computing the full gradient because this way any kinds of postprocessing represent a relatively low overhead done once per training pass, doing the same after each training case would be very slow. But it's much more resistant to the too-high descent rate, and should be able to shake out of the local minimums better. So maybe they can be combined, doing a few passes stochastic then a few passes deterministic, and so on, with the rate computed at the deterministic passes fed to control the stochastic passes.

Monday, January 2, 2023

optimization 13 - using the gradient sign changes

 When I've previously experimented with FISTA momentum descent, one thing that worked quite well was detecting when some dimension of the gradient changes sign, and then resetting the momentum in this dimension to 0.

One of the typical problems of the momentum methods is that they tend to "circle the drain" around he minimum. Think of one of those coin collection bins for charity where you get a coin rolling down the trough, and then it circles the "gravity well" of the bin for quite a while before losing momentum and descending into the center hole. This happens because the speed (momentum) of the coin is initially directed at an angle from the minimum (the center hole). And the same happens with the momentum descent in optimization, the momentum usually develops at an angle, and just as the current point gets it close to the minimum, the momentum carries it by and away to the other side of the "well", where it will eventually reverse direction and come back, hopefully this time closer to the minimum. But there is a clear indication that we're getting carried past the minimum: the sign of the gradient in the direction where we're getting carried past changes. So if we kill the momentum in this direction at this point, we won't get carried past. It helps a good deal with the quicker convergence.

An overshot is not the only reason why a gradient's dimension might be changing sign. Our "virtual coin" might be rolling down a muti-dimensional trough,  oscillating a little left and right in this trough. But there killing the momentum wouldn't hurt either, it would just dampen these oscillations, which is also a good thing.

Which gave me idea that once we have this, there is no point to the gradual reduction of momentum that is embedded into FISTA through the parameter t. If the momentum reduction on overshot gets taken care off as described above, there is no point in shrinking the momentum otherwise.

And then I've thought of applying the same logic to estimating the training rate. I've been thinking about the Adam method that I've linked to in a recent post, and it doesn't really solve the issue with the training rate. It adjusts the rate between the different dimensions but it still has in it a constant that essentially represents the global training rate. In their example they had just come up with this constant empirically, but it would be different for different problems, and how do we find it? It's the same problem as finding the simple raining rate. The sign change detection to the rescue.

After all, what happens when the too-high rate starts tearing the model apart? The rate that is too high causes an algorithm step to overshoot the minimum. And not just overshoot but overshoot it so much that the gradient grows. So on the next step it overshoots back even more, and the gradient grows again, and so on, and so on, getting farther from the minimum on each step. And the momentum methods tend to exacerbate this problem.

So if we detect the dimensions that change sign, and see if the gradient in them grows, and by how much, and how it compares with the gradient in the dimensions that didn't change sign, we'd be able to detect the starting resonance and dampen it by reducing the training rate. I've tried it, and it works very well, check out, it's the FloatNeuralNet option autoRate2_. So far the tunables for it are hardcoded, and I think that I've set them a bit too conservatively, but all together it works very well, producing a little faster training than I've seen before, and without the meltdowns. 

Another thing I've changed in the current version is the logic that pushes the rarely-seen unusual cases to be boosted for a better recognition. It previously didn't work well with the momentum methods, because it was changing the direction of the gradient drastically between the passes. I've changed it to make the boosting more persistent between the passes, and instead of shrinking the gradients of the correct cases, to gradually grow the representation of the incorrect cases, expanding their gradients. It's still a work in progress but looks promising.

Oh, and BTW, one thing that didn't work out was the attempt to boost similarly the cases that give the correct answer by having the highest output point to the right digit but do this at a very low confidence, so that even the best output is below 0 (and sometimes substantially below 0, something like -0.95). All I could do was shrink the percentage of such cases slightly, from about 17.5% to about 16.5%. And I'm  not sure what can be done about it. I guess it's just another manifestation of a great variability of handwritten characters. Maybe it could be solved by vastly growing the model size and the training set, but even if it could, it would be nice to find some smarter way. Perhaps a better topological representation of the digits instead of a plain bitmap would do the trick but I don't know how to do it. One of the theories I've had was that it's caused by a natural trend towards negative numbers, because in each training case we have one output with 1 and nine outputs with -1. So it we changed the negatives to say -0.1, that would pull the numbers higher. But that's not a solution either, it just moves the average up, diluting strength of the negatives.

a quick test of theory about MNIST

 I've had this theory that the test set in MNIST just contains the digit images that are substantially different from the training set, and this is the reason for why they're not recognized well. I've come up with a quick way to test this theory: just merge the test set into the training set, and see if it make any difference.

It does. When the test set is included into training, it gets recognized very well. There still usually are a couple of images that have issues but that's down from about a hundred. I think it tells us in two ways that the test set contains different images:

(1) If they start getting recognized when trained on them, this means that the training set just doesn't train for the right thing.

(2) When the NN has a hard time even training on some images, this means that the abilities of the NN are getting stretched, that it doesn't have enough brain power to cope with all the different possibilities.

Then I've tried adding some brain-power to the NN. First I went back to the original 16x16 images instead of the shrunk 8x8 but kept the width of the NN layers the same. This grew the cost of the first layer 4-fold but the others stayed the same, so the code didn't slow down so much. This did help some, both with the original and with the mixed training set. But not spectacularly. Then I've also expanded the first layer of neurons 4-fold. This made things slower yet, and provided another improvement, but again nothing spectacular.

I think there are just too many ways to draw the digits - slant one or another stroke a little more or a little less, or use a thicker pen, and the digit suddenly has a stroke where it used to have a hole and the other way around, and the NN gets confused. Also, some digits turn out to be unexpectedly similar, such as 6 and 2. When we draw them by hand, they grow loops around the corners, and more loops where the pen doesn't quite lift between the digits. So both 6 and 2 and up looking as a vertical loop with an opening that's connected to a horizontal semi-loop. The only difference is in the direction: 6 has the vertical loop opening on the right and the horizontal loop opening on the left, while 2 has the vertical loop opening on the left and horizontal loop opening on the right. And the way the strokes shift around, it's easy for the NN to get confused.

I don't know how do they do the handwriting recognition in the reality. My guess is that some preprocessing of the images that extracts the topology of the strokes into a more explicit representation should help a lot. I guess it's also possible to generate more of the sample images for the training set by stretching the existing ones in different ways but that sounds like a dead end. A variation of it might be to use the batching again, but this time include only the images of the same digit into a batch.

Sunday, December 25, 2022

optimization 12 - adaptable methods of neural network training

 In the last post I've been wondering, why aren't the momentum descent methods widely used for the neural network training. And guess what, a quick search has showed that they are, as well as the auto-scaling methods.

In particular, here is an interesting paper:

Adam: A Method for Stochastic Optimization

What they do looks actually more like a momentum method than selecting the rate of descent. This paper has references to a few other methods:

  • AdaGrad
  • RMSProp
  • vSGD
  • AdaDelta

 And I've found this algorithm through another web  site that talks about various ML methods:

Thursday, December 22, 2022

optimization 11 - FISTA momentum descent in FloatNeuralNet

Continuing from the previous post, I've had a look at the characters in the test set that have trouble getting decoded. Well, they truly are handwritten, I've had difficulties recognizing some of them as digits at all. But a good deal of them are rather similar variations on the digit "6" that I guess are just not represented well in the training set. For many of these digits the even highest recognized class has the value deep in the negative zone, which means that they're not like anything in training. I think there should be some way to adapt the training set, such as expand it by including the digits shifted left and right by a pixel or two. Need some more thinking on that.

I've looked at the training set and counted how many training cases have their best guess negative (even if it's the correct one), and it came out at about 16.5%. I've tried to shift the training weights to prioritize  cases with the low best guesses just as I've done with the wrong guesses, and it didn't help at all, the best I could scrape out was about an extra 0.2%. Maybe it's just because for every 1 positive output there are 9 negative outputs, and on average the values tend to be pulled down.

In the meantime, I've added the momentum descent, based on the FISTA algorithm. I've learned some tricks playing with my version of FISTA that I plan to add later, but so far just a very basic version, with only one trick included: when applying the momentum, compute the saturation by the limits, and adjust the momentum for it. So that if some dimension hits the limit, the momentum won't continue pushing it past the limit again and again but will reset on that dimension. 

I've been trying to get the automatic descent rate deduction working properly before implementing the momentum but I gave up on that for now and decided to just try the momentum. It is a bit sensitive to the training rate being too high, maybe because there is a higher chance of driving into a situation where a high rate would give an overshoot. But bump the rate down, and the momentum descent works very, very, very well, doing a much faster progress. The mean square error for the training set went down from the previous-best 0.025  to 0.01. The bad news is that it didn't help with the test set at all: it again goes better up to a limit, and after that as the training set's stats get better, test set's stats get worse. So I think I really need to accept that the test set really truly contains the characters that are not well represented in the training set. And probably the convolution won't fix it either.

Given that the momentum descent works so awesomely, I don't understand why isn't everyone using it. It adds cost per pass but then it reduces more the number of passes needed to achieve the same precision, giving a large net win. Or maybe they do use it and don't tell everyone.  Or maybe it doesn't work well for all the neural networks and I've just got lucky with my example. I don't know.

The way FISTA works is basically by remembering the last step and repeating it before computing the gradient at the new position and doing a new step. Then remembering the total step (repeat + new step) for the next step. It also has a little optimization in scaling the remembered last step by a factor a little under 1, and very gradually reducing this factor. This downscaling is intended to reduce the overshoots and "circling the drain" when the position gets close to the optimum. It also limits the maximum momentum, when the downscaling starts shrinking the momentum more than the current gradient accelerates it. Another little thing it does is that it skips the repeat not only on the first step (where it just has nothing to repeat) but also on the second step. For all I can tell, the reason for this skipping of momentum on the second step is that the initial point is usually random, and if the gradient approximation is good, it would put the point after the first step into a decent vicinity of the optimum. Which means that the first step will be large, and also stepping in a random direction that is nowhere close to the direction of the following steps. Thus it's better to not include this first large random step into the momentum that will keep repeating.

 The mechanics of enabling the momentum mode in FloatNeuralNet are: set the option

options.momentum_ = true 

And now there also are methods for reading the best-result-still-below-0 stats:

getPassAbove() returns the number of cases in the last pass where the correct and best outcome is above 0

getPassNotAbove() returns the number of cases in the last pass where the best outcome is either incorrect or not above 0

Tuesday, December 20, 2022

FloatNeuralNet classifier training mode

Continuing the story from the previous post, I've been experimenting with the MNIST classifier of handwritten digits from ZIP codes some more, and I've tried to improve on it by adding a special mode for gradient computation:

The point of a classifier is that it produces a result with the highest value at the output that corresponds to the correct classification, and the exact values on the outputs don't mater that much. So when training, it makes sense to check whether a particular training case already produces the correct result, and if so then "dampen" it to give the training cases that are still producing incorrect results a greater chance to shift the weights for their benefit. I've done this "dampening" by multiplying the gradients of the correct cases by a value less than 1. I've started with this value set at 1/1000 but that tended to produce rather wild swings where a small number of incorrect cases would pull the weights towards themselves only to make a large number of formerly correct cases incorrect. I've reduced it to 1/100 and that worked out quite well. I'm not sure if the selection of this multiplier depends on the number of training cases, FWIW, the MNIST set has a little over 7000 training cases. 

And to make sure that the training doesn't get too slow, I've added the auto-rescaling of the training rate, so that the potential sum of the gradients would total up the same. This made it a bit more touchy, since there is a greater chance of the gradient getting dominated by only one training case, and a smaller base rate (I've reduced it by a factor of 3) was needed to avoid the meltdown into overshoots. Maybe I'll embed this safety factor into the auto-rescaling at some point, I'm just not sure yet how to pick it for the general case.

The end result is a bit mixed. It worked very, very well for the training set, bringing the error rate (i.e. the number of cases that produce a wrong value) to zero. And the mean square error went the same or slightly better as with the plain implementation, meaning that not only the previously neglected cases are getting attention but also they get incorporated better into the common model and don't hurt the others. The mean square error on the test set also went slightly better, the best I've seen so far at 0.193 (an improvement of almost 0.01 over the plain version). But the error rate on the test set, which I hoped to improve, still stayed at around 5.5%, not really an improvement there. I've seen it go down to 5% and then grow again on the following rounds of training, while the mean square error became better. 

So for all I can tell, the test set really does contain some characters that are substantially unlike any of the characters in the training set. And probably the only way to do better is to actually look at them and figure out what is different about them and then  manipulate the training set to introduce variability by generating multiple slightly different versions of each character. I've kind of got interested in this puzzle, so maybe I'll find time to try this. Or maybe I'll try to implement the convolution first but by now I doubt that it would help much with this specific problem on its own. Maybe a general way to approach this variability would be to do some unsupervised classification of the images of the same digit, to produce the groups  of distinct ways to write it. But that looks difficult.

Another simple thing to try would be to make the classifier preference even stronger, requiring that not only the correct output is the highest but also that it beats the next highest output by a certain margin. And maybe even vary the gradient weight in a reverse proportion to this margin. But I doubt that it would solve the problem with the stubborn 5.5% of the test set.

Yet another interesting development is that the CORNER activation seems to work quite well. I've added the third layer with it, and it produced better results. My previous impression that the deeper networks train slower was based on a limited number of training passes, the larger models are slower there, but as the pass count goes into multiple thousands, more neurons do help to squeeze out a better quality. Of course, each pass is slower with more neurons, but as far as the progress per pass is concerned, the larger networks do better on the later rounds.

Where the new feature is in the code:

Setting the FloatNeuralnet option isClassifier_ to true enables this new classifier training mode. The option correctMultiplier_ defines the penalty gradient multiplier for the cases that are already correct (the default of 0.001 there is a bit too much of a penalty, 0.01 looks better). And since the correct and incorrect cases are counted for re-scaling the training rate appropriately, these counters are now also available externally too via methods:

size_t getPassCorrect() const;
size_t getPassIncorrect() const;

This allows to compute the error rate easily from the results of a training pass. They get computed only if the classifier training mode is enabled, and they show the error rate as it was before the last training pass, it will change after applying the gradient. The classifier mode can be used with both the piecemeal steps of applying the training rate in train() method and with applyGradient(), but only applyGradient() is able to do the automatic scaling of the training rate to compensate for the shrinkage.


Sunday, December 18, 2022

FloatNeuralNet on the MNIST set, and checkpointing

 Since I've been experimenting with neural nets on an "unnatural" example, I've been wondering, how would the things I've tried work on a more natural one. So I've made a demo for the MNIST example of recognizing the handwritten digits from ZIP codes. It's implemented in the branch , in the file nn/demo/d_FnnZip.cpp.

The code looks for the MNIST files that can be downloaded from placed in any ancestor directory's subdirectory "zipnn" (so I usually put it next to Triceps's top directory). 

The original files are scans with 16x16 dots per digit but it has turned out to be rather slow in processing, so I've made the demo downscale them to 8x8 before processing. 

Another thing that was rather disappointing was that I didn't save the state of a trained NN before. It's OK for small and quick training but with a slower and longer one, the results really need to be preserved. So I've added the FloatNeuralNet methods checkpoint() to dump the state to a file and uncheckpoint() to get it back. It can also be used creatively to do some training with some settings and then load it into an instance with different settings to see if something would improve.

With 8x8, it's 64 inputs, and 10 outputs for 10 recognized digits, and as a baseline I've used 2 inner layers of 64 neurons each: 64->64->64->10. But I've experimented with more and fewer neurons, and more layers.

So, what are the observed results?

First, the RELU variations work quite decently, probably because the original data is already centered on the [-1, 1] range. 

The larger models are slower to train, nor just in execution time per pass, but also each pass makes a smaller improvement.

The web page talks that it's a hard problem and the error rate on the test set at 2.5% is very good. Well, the best I could do so far is about 5%. The problem with this data set is that the test set differs a good deal from the training set. So when the training set gets to a mean square error of 0.07 and the error rate of 0.5% (and keeps improving),  the test set gets stuck at a mean square error of 0.2 and error rate of about 5%. Worse yet, as the error rate on the training set keeps getting a little better, on the test set it gets a little worse. Which means that the NN makes a close fit for the training data, it doesn't fit the test data. So I've tried to reduce the number of neurons per layer to 32, and the result got a little worse overall, and it still does this divergence between training and test data.

The CORNER activation showed not spectacularly but decently. Since it has a "mini-layer" on top of each normal layer, I was able to remove one layer to sizes 64->64->10, and it did just a little better than LEAKY_RELY in 64->64->64->10, and each pass got computed faster. In fact, the best seen error rate of 5%  came with CORNER at about 10K passes and then deteriorated a little to about 5.3% at 20K passes, while the best I've seen with LEAKY_RELU is 5.5%. But those small differences may also be due to randomness. I've used the same starting point in the random number generation but with the different number of weights it plays out a bit different. Adding an extra layer with CORNER brings the training mean square error to below 0.05 and error rate down to below 0.3%, and even the test mean square error goes below 0.2 (which it didn't do in other combinations), but the test error rate stays at about 5.6%, so it's improving the fit but still overfitting worse than without the extra layer.

With the 32-neuron middle layers, the test error rate gets a bit worse, around 6.6%  after 20K passes of training. 

I've thought that maybe one of the sources of trouble is that with the inputs having absolute values less than 1, and with all the weights less than 1, it might be hard to produce the outputs of +-1. So I've tried to set the expected outputs to +-0.1, and it really made no difference. Increasing the maximum weight limits to +-100 didn't make a whole lot of a difference either (but made things slightly slower to train). Interestingly, taking a trained model with limits of +-100 and trying to continue training it with limits of +-1 produces a quite bad result that has a hard time digging out of the suboptimal point.

Applying the changes after each training case vs at the end of the pass didn't make a noticeable difference either. I'm thinking that applying at the end of the pass should provide an obvious way for parallelization: compute the partitioned set of training cases in multiple threads, each one summing up its own copy of the gradients, and then sum the partial per-thread results together.

Using the "floor" option that makes the sign-flipping of the weights more "structured" (on the first push it stops at the same sign but a small non-0 value, on the second push it switches the sign of that small value, and on the following pushes it would grow with the switched sign) made the training a little slower.

The automatic training rate estimation didn't work well at all and I don't know yet how to fix it.

What things to try next?

One obvious thing to combat overfitting would be to try some batching, and it should make the processing faster too.

Another idea I've got is that for a classifier that chooses only one category (the one with highest value), it's more important that the right neuron gets the highest output than to hit the target values exactly. So maybe taking a page from Adaboost's book, the training cases that produce the right highest output should have their gradients reduced (since they are "already right"), to give the training cases that are still wrong a higher relative weight for the next move.

And the next idea yet would be to try the convolution, which would be another step in adding complexity.

P.S. I've tried the batching. It made the error rate worse, and the larger the batch size the worse it gets. A kind of good news is that it worsened the training error rate more than the test error rate, so it definitely works against the overfitting, at least kind of and somewhat. Well, the training error rate is still lower than the test one, but with a lower relation between them.

Sunday, November 27, 2022

optimization 10 - Lipschitz costant and the last layer of neural networks

 After, as described in the last post, the empirical computation of Lipschitz constant for the lower layers of a neural network didn't pan out, I've tried doing it for the last layer. The last layer minimizes a honest squared function, so it should not have the same issues as with the previous layers. And I've done it in code, see

It computes 1/L empirically like the TFOCS library: keeps the weights and gradients from the last pass (including, by the way, the inputs to the last layer too, since they are variables in the formula too), finds the differences, computes the norms and eventually 1/L, and compares it with the descent rate used for the pass. If there is room, it gradually grows the rate. If 1/L had shrunk and the last rate was too high, it reduces the rate and redoes the previous step (this is somewhat tricky, as the gradient is computed only in the following step, but I've got the rollback to work). Most of the time one rollback is enough to set things straight. But if not, well, it's just one overshoot that would get straightened in the following steps, so I've limited the rollback to be done no more than once in any case.

The results are rather ambivalent. If the whole model consists of one Corner-activated neuron, it works great, finds the right rate, and converges quickly. The extra overhead is definitely worth the trouble there. Though interestingly, it first finds a solution with a slightly lower squared error, and then as it does more steps, the gradient reduces but the squared error grows a little. My guess is that it's probably caused by the adjustment to the Corner's offset weight that is not tied directly to the error, and what it thinks is the best balance doesn't produce the lowest error. Something to consider in the future.

With more neurons, things go worse. One aspect that definitely works well is preventing the runaway growth of the weights to infinity. I've tried an example that experienced such a growth before, and it didn't diverge any more, even though it produced some weights around 50. But the trouble is that the found descent rate is way too high. It works much better if I limit the rate to about 1/10 of the computed. It descends there rapidly but then gets stuck circling with about the same error. Reducing the rate to 1/100 makes it converge slower but gets to a lower error. 1/1000 produces an even lower error. Which means that it gets the rate wrong, and just deciding on the descent rate from the last layer is not enough, the lower layers also need to be taken into account. The idea with estimating the error in the previous layer from both gradient and descent rate of its successor should work here but it might involve a good deal of overhead.Needs more thinking. 

On the ReLU and Leakey ReLU it works very similarly but doesn't resolve the issue with centering the offsets, and the rate overestimation issue is particularly sensitive for the basic ReLU, since there it can easily lead to neurons becoming dead.

So I've thought alright, maybe the overshoots are caused by the weights flipping the sign too easily with the larger rate. So I've added the idea of "weight floor": defining a small number (let's call it r) as the floor and making the values (-r < weight < r) illegal for the weights. Then if the descent tries to push a weight towards 0, crossing the +-r boundary, we'd stop it at the appropriate r or -r for this pass. If on the next pass the descent keeps pushing the weight further in the same direction, we'd "teleport" it from r to -r or the other way around from -r to r. This would prevent the large jumps, and avoid getting stuck at 0, and give a chance for a new gradient to be computed before going further, so that if it was an overshoot, it would get a chance to go back without causing a change of sign.  And if different weights end up moved at a different rate, it shouldn't matter much, because they are still moved in the right direction. Overall it works, and the "teleports" do happen occasionally.  But it didn't resolve the overshooting issue. I guess the trouble is mostly with the overshoots that don't cause a sign flip.

Saturday, November 26, 2022

optimization 9 - Lipschitz constant and earlier layers of neural networks

 I've tried to follow through with the path described in, find the Lipschitz constant for the neural network to both try to make the neural network converge faster and to avoid overshooting badly when the weights at the neurons grow larger than 1. In that previous post I've looked at the last layer, so then I've figured that I should look at the previous layers: since they don't have a square, they might be easier. Or maybe not: they feed into all the following layers.

But maybe we can make them easier. We can notice that in the original formula for gradient descent every step is multiplied by the descent rate. So maybe we can just pull that descent rate factor and "put it outside the parentheses". Which means that when we propagate the gradient from the last layer backwards, we can multiply it once by the descent rate, and that would allow us to skip multiplying by it everywhere else (which is also an optimization in the software sense!). The result of this multiplication (using r for the descent rate and v for the activated output of a previous layer's neuron) 

m = r * df/dv

also has can be seen as having a "physical meaning" that says "I want to shift the output v of the previous-layer neuron by this much to achieve the next value of v" (here t is the index of the training pass): 

v[t+1] = v[t] - m. 

In reality, of course, the output v feeds multiple neurons of the previous layer, and these feedbacks get added together. But that can be folded into the same physical interpretation too: if this output feeds into N neurons, we can say that

r = R / N

where R is some "basic descent rate", and so

v[t+1] = v[t] - R * sum{i}(df[i]/dv) / N

That is, the addition turns into averaging with a different rate R. 

But wait, there also are multiple training cases (K of them), the gradients of which also get added up. That's not a problem, because the outputs from the different training cases also get added up.

An interesting corollary from this seems to be that when a neural network has a different number of neurons per layer, the descent rate should be normalized to the number of outputs, and the layers that have fewer outputs can use a higher descent rate.But I suppose there is also a counter-argument that if there are many outputs, the feedback from them is more likely to be contradictory, so the sum is more likely to be closer to 0 rather than all pointing in the same direction, so a higher rate may be useful there. That's probably why nobody bothers with adjusting the descent rate per layer.

So all right, let's build a simple example. Suppose we have a ReLU-like activation function with gradient of 1 and a neuron with 2 inputs and an offset weight

v = u[1] y[1] + u[2] y[2] + b

And since the training cases are added up linearly, we can just start with an example with one training case. Then we could write the function to minimize as

v = u[1] y[1] + u[2] y[2] +b - m

(getting ahead, this expression turns out to be wrong, but I want to get gradually to why it's wrong). The partial derivatives will be

dv/du[i] = y[i]

dv/dy[i] = u[i]

dv/db = 1

and computing at two points and finding the difference in the dimensions of gradients and arguments:

L^2 = ( (y[1, 1] - y[1, 0])^2 + (u[1, 1] - u[1, 0])^2 + (y[2, 1] - y[2, 0])^2 + (u[2, 1] - u[2, 0])^2 + (1-1)^2) /
  ( (u[1, 1] - u[1, 0])^2 + (y[1, 1] - y[1, 0])^2 + (u[2, 1] - u[2, 0])^2 + (y[2, 1] - y[2, 0])^2 + (b[1] - b[0]^2)
  <= 1

because the lower side of the fraction has all the same terms for u and y (just in different order), and an extra positive term for b. So (1/L) >= 1, and propagating the same rate of descent or more should be what we need.

To test this conclusion, I've made up a small example with concrete numbers and did what the TFOCS library does, do some steps and compute the approximation of L by dividing the difference of gradient between the steps by the difference of weights. And... it didn't work out. Not only it produced L > 1, but I'd go back and adjust the step to be smaller, and the new L would be even larger, and I'd adjust again to an even smaller step, and L will get even larger than that again.

What went wrong? I've got the formula wrong. Looking at the expression

v = u[1] y[1] + u[2] y[2] + b

it obviously has the minimum where all the variables are 0. So yeah, if we use L=1 and step straight to 0, that would minimize it. But we don't want to get to 0, we want it to get to some value (v[t-1] - m). The right formula for minimization should be

abs( u[1] y[1] + u[2] y[2] +b - (v[t-1] - m) )

The absolute value part was missing. And I don't know how to compute L with the absolute value. I guess one way to replace it would be to substitute a square for absolute value:

( u[1] y[1] + u[2] y[2] +b - (v[t-1] - m) )^2

It can probably be done just for the computation of L, even when the neural network itself stays without squares. But I'm not sure. 

Another interesting observation is what happened when my steps in the test have overshot the empirically computed L: they would flip the sign of the weights. Which means that the occasional overshoots are important, without them the signs would stay as they had been initially assigned by the random seeding. Even if we could compute and properly use the right value of L, we might not want to, because we'd lose the sign flips.

Thursday, November 3, 2022

V-shaped activation functions what-ifs

I've noticed that a simulation of the square function with a single neuron plus CORNER (V-shaped) activation function got worse after I've clamped the weights to the range [-1, 1]. It just couldn't produce a sufficient derivative to go to the full slope of the quadratic function, so it defaulted to the same shape as ReLU, going at slope 1 for the whole range of inputs [0, 1]. How about if I increase the allowed range of weights to [-10, 10]? Much better, then it could simulate the bend, producing the total maximum slope of about 1.5. So it looks like being able to go to the higher weights might be useful, just needs to be accompanied by the dynamic computation of L and the training rate from it.

Next, it looked like the right-side coefficient of the V can be fixed at 1, since the slope produced by the neuron itself would also work to produce the same effect. As it turns out, this doesn't work well. The trouble happens when the right side needs to change sign. When the right side of the V adjusts, it changes the sign easily. But without that, changing signs in the weights of the neuron itself doesn't work well, it often gets stuck. So this change doesn't work well. It's kind of interesting that when there is one variable, it changes the sign easily, if there are two, they get stuck, but if there are three, they change signs easily again. Though the real reason is probably that two variables are pushing against each other by using each other's weights as gradients, while the third variable doesn't, it just gets driven from the top. So maybe if one of two variables would be just driven from the top, that would prevent two variables from getting stuck too. Will need to think about that.

Next, what if we do a common Leaky ReLU but adjust the offset using the same formula as with the CORNER? (Leaky ReLU has to be used because with plain ReLU the gradients on the left side would be lost, and the computation for he offset won't work). This worked about as bad as normal Leaky ReLU, and maybe even worse, so it's definitely not a solution on its own.

So it looks like the CORNER function does have value. And maybe the right-side coefficient can be rid of if the main neuron weights can be made to change sign more actively, perhaps by making one of them fed the gradient of the top, or by applying the same gradient to them in both ways.

Wednesday, November 2, 2022

demo build target

 My experimental code for FloatNeuralNet has grown kind of large, and it runs kind of slow with valgrind, which interferes with the normal tests. So I've added a new entity in the C++ code: "demo", and moved all the slower experimental examples there. They are just like tests, only sit in the "demo" subdirectories, in the files starting with "d_". The target to build them is "mkdemo", to run without valgrind "demo" or "qdemo", with valgrind "vdemo" (this is different from the tests where the target "test" runs with valgrind).

BTW, the target "mktest" didn't run the library build as a dependency. It does now, in the main cpp directory (but not in the subdirectories). And the same applies to "mkdemo".