Tangential comment, but one where I’d be interested in how people in this community feel: When you wrote about the part with the meetup and sign saying “I might be lying”, I immediately thought how little fun it must have been, and even how badly it might have felt for others attending. In my mind, people attending a meetup don’t want to be lied to, even if you semi-communicated it (I say “semi” because the statement on the sign was trivially true for every person. You did not clearly state you were definitely going to lie about certain things) and in the context of a “social experiment”. To me it seems quite similar to people wearing signs saying “I might be rude” and then actually being rude.
Steveot
Karma: 44
Another intuition I often found useful: KL-divergence behaves more like the square of a metric than a metric.
The clearest indicator of this is that KL-divergence satisfies a kind of Pythagorean theorem established in a paper by Csiszár (1975), see https://www.jstor.org/stable/2959270#metadata_info_tab_contents . The intuition is exactly the same as for the euclidean case: If we project a point A onto a convex set S (say the projection is B), and if C is another point in the set S, then the standard Pythagorean theorem would tell us that the angle of the triangle ABC at B is larger than 90 degree, or in other words . And the same holds if we project with respect to KL divergence, and we end up having .
This has implications if you think about things like sample efficiency (instead of a square root rate as usual, convergence rates with KL divergence usually behave like 1/n).
This is also reflected in the relation between KL divergence and other distances for probability measures, like total variation or Wasserstein distance. The most prominent example would be Pinsker’s inequality in this regard, stating that the total variation norm between two measures is bounded by a constant times the square root of the KL-divergence between the measures.
It’s not a mathematical argument, but here I first came across such an analogy drawn between training of neural networks and evolution, and a potential interpretation of what it means in terms of sample-(in)efficiency.
I thought about Agency Q4 (counterargument to Pearl) recently, but couldn’t come up with anything convincing. Does anyone have a strong view/argument here?
I like the idea a lot.
However, I really need simple systems in my work routine. Things like “hitting a stopwatch, dividing by three, and carrying over previous rest time” already feels like it’s a lot. Even though it’s just a few seconds, I prefer if these systems take as little energy as possible to maintain.
What I thought was using a simple shell script: Just start it at the beginning of work, and hit a random key whenever I switch from work to rest or vice versa. It automatically keeps track of my break times.
I don’t have Linux at home, but what I tried online ( https://www.onlinegdb.com/online_bash_shell ) is the following: (I am terrible at shell script so this is definitely not optimal, but I want to try something like this in the coming weeks. Perhaps one may want an additional warning or alarm sound if the break time gets below 0, but for me just “keeping track” is enough I think)
convertsecs() {
((h=${1}/3600))
((m=(${1}%3600)/60))
((s=${1}%60))
printf “%02d:%02d:%02d\n” $h $m $s
}function flex_pomo() {
current=0
resttime=0
total=0while true; do
until read -s -n 1 -t 0.01; do
sleep 3
current=$(( $current + 3 ))
resttime=$(( $resttime + 1 ))
total=$(( $total + 3 ))
printf “\rCurrently working: Current interval: $(convertsecs $current), accumulated rest: $(convertsecs $resttime), total worktime: $(convertsecs $total) ”
done
printf “\nSwitching to break\n”
current=0
until read -s -n 1 -t 0.01; do
sleep 3
current=$(( $current + 3 ))
resttime=$(( $resttime − 3 ))
printf “\rCurrently resting: Current interval: $(convertsecs $current), accumulated rest: $(convertsecs $resttime), total worktime: $(convertsecs $total) ”
done
printf “\nSwitching to work\n”
current=0
done
}flex_pomo
Thanks, I finally got it. What I just now fully understood is that the final inequality holds with high probability (i.e., as you say, is the data), while the learning bound or loss reduction is given for .
Thanks, I was wondering what people referred to when mentioning PAC-Bayes bounds. I am still a bit confused. Could you explain how and depend on (if they do) and how to interpret the final inequality in this light? Particularly I am wondering because the bound seems to be best when . Minor comment: I think ?
The main thing that caught my attention was that random variables are often assumed to be independent. I am not sure if it is already included, but if one wants to allow for adding, multiplying, taking mixtures etc of random variables that are not independent, one way to do it is via copulas. For sampling based methods, working with copulas is a way of incorporating a moderate variety of possible dependence structures with little additional computational cost.
The basic idea is to take a given dependence structure of some tractable multivariate random variable (e.g., one where we can produce samples quickly, like a multivariate Gaussian) and transfer its dependence structure to the individual one-dimensional distributions one likes to add, multiply, etc.
Thanks, I really like these concepts, especially Grice’s maxims were new to me and they seem very useful. Your list also got me thinking and I feel like I also have some (obvious) concepts in mind which I often usefully apply but which may not be so well known:
Data-processing inequality
Public good games
Evolutionarily stable equilibrium
Don’t be results oriented (when acting in stochastic environments)
The data-processing inequality is often useful, especially when thinking about automated tools, like LLMs. It states that for any fixed channel K, the mutual information between X and Z is always larger than that between Y and Z, if Y is the output of X processed by the channel K. E.g., if you tell an LLM simple “refine the following paragraph”, and the goal of the paragraph is to transmit information from your brain towards a reader, then using the LLM with this prompt can only destroy information (because the information is processed by a fixed channel which does not know the contents of your mind). Important are also the cases where the data processing inequality does not directly apply, e.g. with a prompt like “refine the following paragraph, note that what I want to communicate is [more detailed description than the paragraph alone]”.
2. and 3. are just particularly useful concepts from game theory. I see public good games everywhere (e.g., climate, taking on duties in a community, etc.) and actually think many situations are sufficiently well explained by a very simple modeling as a public good game. Evolutionarily stable equilibrium is a stronger concept than Nash equilibrium and useful to think about which equilibria actually occur in society. E.g., especially for large games with many players and mixed strategies, it’s a useful concept to think about cultural or group norms, globalization, etc.
The last one is probably well known to everyone who seriously played online poker or did sports betting, but it applies more generally. Roughly speaking, if you get money into the pot while having an 80% chance at winning, don’t focus on whether you lose or win the hand eventually. The feedback for your actions should be the assessed correctness of your actions, without including factors completely independent of your actions. So basically: De-noise as much as possible (without destroying information).
Edit: Just to clarify the details of the data-processing inequality because I noticed Wikipedia uses different notation: The (Markov) model is Z-X-Y in my description, and in the example Z is the reader, X the brain and Y the output of the LLM.