Your hypothesis seems reasonable, and I think the following proves it.
1. This is for 5e-3, giving no spikes and faster convergences:

2. Gradient descent failed to converge for multiple LRs, from 1e-2 to 1e-5. However, decreasing the LR by 1.0001 when the training error increases gave this:

It's messy, and the decrease seems to turn the jumps of the slingshot effect into causes for getting stuck in sub-optimal basins, but the trajectory was always downwards. Increasing the rate of reduction decreased spikes but convergence no longer appeared.

An increase to 2. removed the spikes entirely.

Your hypothesis seems reasonable, and I think the following proves it.
1. This is for 5e-3, giving no spikes and faster convergences:

2. Gradient descent failed to converge for multiple LRs, from 1e-2 to 1e-5. However, decreasing the LR by 1.0001 when the training error increases gave this:

It's messy, and the decrease seems to turn the jumps of the slingshot effect into causes for getting stuck in sub-optimal basins, but the trajectory was always downwards. Increasing the rate of reduction decreased spikes but convergence no longer appeared.

An increase to 2. removed the spikes entirely.

I have uploaded html files of all the animation so they can be interactive. The corresponding training graphs are in the associated notebooks.

The original learning rate was 1e-3.

For 5e-4, it failed to converge:

For 8e-4, it did converge, and the trajectory was downwards this time:

For 1 and 2 - I have. Everything is very consistent.
For 3, I have tried several optimizers, and they all failed to converge. Tweaking the original AdamW to reduce the learning rate lead to very similar results:

For 4, I have done animations for every model (besides the 2 GELU variants). I saw pretty much what I expected: a majority of relevant developments (fourier frequencies, concentration of singular values, activations and attention heads) happened quickly, in the clean-up phase. The spikes seen in SiLU and SoLU_LN were visible, though not lasting. I have uploaded the notebooks to the drive folder, and have updated the post to reflect these findings. Thank you very much, again!
 

Fair statistical point, however in reality a vast majority of serial killers did not go above 15 victims, and the crimes they commited were perpretated before their first (and last) arrest. I do not have raw numbers, but my impression is that the number of those sentenced for one murder, later paroled, and then beginning their spree of more than 3-4, is incredibly small. Serial killers are also rare in general.

Gang considerations, however, might be a larger factor here, though I still doubt it is enough to tip the scales (especially as prison gang affiliation is a factor taken into account when considering parole). 13% of homicides are gang-related, though gang members are twice as likely to re-offend (both for violent offences and not). Even if we (awkwardly) extrapolate twice as likely to re-offend to twice as many murders after parole, this still does not meaningfully change the ball-park figures.

If we have 80 murderers, of which 10 are gang members, which are released (with gang members less likely to be released, mind, so this is an over-estimation), then we would have 7 homicidal recidivists and 1 homicidal gang recidivist, who commits two crimes instead of 1. Instead of 8 murders, we have 9: 12.5% remurdering instead of 10%, at most. I have fudged the numbers, but I don't think this substantially changes what I have said.

I am sorry for being slow to understand. I hope I will internalise your advice and the linked post quickly.

I have re-done the graphs, to be for every epoch. Very large spikes for SiLU were hidden by the skipping. I have edited the post to rectify this, with additional discussion.

Again, thank you (especially your patience).

Apologies for misunderstanding. I get it now, and will be more careful from now on.

I have re-run the graphs where such misunderstandings might appear (for this and a future post), and added them here. I don't think I have made any mistakes in interpreting the data, but I am glad to have looked at the clearer graphs.

Thank you very much!

I will keep that in mind for the future. Thank you!
I have put all high-quality .pngs of the plots in the linked Drive folder.

I have major disagreements with the arguments of this post (with the understanding that it is a steelman), but I do want to say that it has made me moderately update towards the suitability of the death penalty as a punishment, from a purely utilitarian perspective (though it has not tipped the scale). It has also showcased interesting and important figures, so thank you for that.

Deterrence and recidivism

At that point killing 3 million criminals to save the lives of 2.4 million

How many of those 2.4 million were murdered by recidivists? Even if we assume that the death penalty instituted on a larger scale would save a significant chunk of these people (evidence being very uncertain and model-dependent), there is (weak) evidence that most murderers do not re-murder (less than 10%). Violent re-offenses, however, from what I have seen, range between 20 and 50%, which is definitely significant.

As for escalation, stopping violent pre-murderers might be feasible, but over 40% of offenders of all kinds had zero criminal history, so at most about half of them could have been executed before taking anyone's life. I suspect this number is much lower, because:

After all, might as well be hung for a cow as a sheep. Yes this is a likely cost of the death penalty. I do not think it comes near to tipping the scales.

My intuition is the opposite, though I have not found concrete research either way. This would likely need to be tested, as our extrapolations regarding (say) the current behaviour of serial shoplifters might not extend in a manner we would expect when under such regimes.

One minor point: homicide and violent crime in general has been decreasing quite steadily since the 90s, approaching early 60s levels, in spite of the fall of use of the death penalty (or increasing incarceration rates, for that matter). Shifting the threshold may not even be necessary in the future, if trends continue.

Costs

Imprisoning someone for one year in the USA costs in the order of 100,000 dollars.

The SSC article you link explains that this is for California, with most other states being between 30,000 and 60,000 (which can be cut further, if needed). This halving (possible in California as well) would make the calculations you have made severely less dire. 1-2% of state/federal budgets can be gathered from other sources (such as military spending), and saying that the US is "not prepared to pay" when such large death penalty reforms are even further afield of the Overton window is somewhat strange.

In general, your assertion that "we are not a sufficiently wealthy society" is not sufficiently supported by the evidence, and seems somewhat arbitrary. The issue of investment in these matters is another of the many budget-nudging issues out there, and they can be resolved from within the current political paradigm (irrespective of whether large, systemic shifts are desirable, which they can be, of course).

If the answer to either of those is no, then they are not safe to be released into society, and we are not a society wealthy enough to lock every such person up.

Same consideration. Furthermore, those with illnesses that predispose them to commit crimes are either psychopathy (which does not automatically make one a violent criminal, and as such they should be treated like any median citizen) or disorders like schizophrenia, where the person in question would have already been medicated/in an institution. The costs would still be there.

Justice and due process

I do not believe in vengeance or justice.

An utilitarian analysis cannot include solely economic or (more) easily measurable factors. A widespread belief in justice, trust towards the state, public disinclination towards violence as a means of solving societal problems are necessary for a politically stable society, and removing those might impose major costs on society as a whole (as Knight Lee pointed out). Justice cannot be simply dismissed as a factor here.

This is also related to:

Yes, if you wait 20 years and go through umpteen rounds of court cases to finally elaborately kill a small percentage of the people you originally started the process with it's not going to save you any money. We would obviously have to significantly streamline the process, such that people are executed within 6 months of being caught or so.

There seems to be a general assumption that all of the lengthy due processes, the necessary costs for lawyers, judges, impartial jurors, associated legal staff etc., and the concrete determination of culpability and the degree of severity of the acts committed, are all merely unnecessary bureaucratic fluff that can be easily dispensed with. These processes have developed for centuries in order to ensure that every citizen receives a fair hearing, someone to represent their interests, and a method to dialectically get as close as possible to a reasonable truth, not to mention moral and practical standards of guilt for every possible degree and type of action possible (which would be rendered worthless).

Exceptions cannot be made for certain citizens over others, especially not in high-stakes situations, as that would erode the presumption of innocence and rule of law, trust in institutions, amplify existing systemic problems, and veer dangerously close to authoritarian police states, which have had unpleasant social consequences for its citizens (see the general malaise and low-trust nature resulting in high crime rates of many post-communist states).

Many (though not all, of course) death row convictions drag on due to problems with previous trials, new evidence being produced, and many other considerations. There is also the fact that our society has decided that sentencing and assigning guilt should not be governed by Bayesian standards of probability: even if the evidence is above 50% for guilt, the requirement for "beyond a reasonable doubt" pushes the threshold to the 80-90% territory, due to an acknowledgement of human subjectivity, and a belief in the necessity of the presumption of innocence for a free society where its members are not scared of being wrongfully convicted by an overly-powerful state.^[1] Loosening these norms may have unintended consequences.

All of these problems would be exacerbated the lower a bar is set for the death penalty. Their impact is difficult to measure, but could be massive and potentially society-eroding (as you yourself have pointed out with the issue of scale).

But yes, some innocent people will be killed. Just like some innocent people are killed by police shootings, and numerous innocent people are killed by the US Army, murderers who were let free, and mistaken medical diagnosis. We accept that innocent people die due to our actions all the time, and making a special exception here is an isolated demand for rigour.

Most of the situations you have described are wholly technical in nature. Determining proper guilt is also a technical problem, and as such requires the resources previously mentioned. Streamlining the process would make the rate of innocents killed even higher (see Isusr's comment). Not to mention that the act of execution is very deliberate and isolated, with the moral aspect of the act being front and centre. The situations you list here are not (no one is actively trying to kill bystanders or a patient, but the person being accused is being targeted, especially if due process is eroded). Furthermore, as was mentioned, homicidal recidivism is low enough that the cost of being let free might not be offset by an execution (~100% murder rate).

I do agree that resolving the issue of the method of execution is doable, with the firing squad a (somewhat) reasonable solution. This has historically been extremely fraught, however, with many false promises of humane and efficient executions, and increased demand might make the forces that lead to those many follies resurface, even if it solvable with enough care and integrity.

1. ^{^}

   I do have to concede that the citizens of low-crime rate Japan do not seem to suffer from this, despite flagrant presumption of guilt in their legal system. Trying to implement something similar in the US, however, would rightly lead to civil rights protests.