The Physics Of Associative Memory
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In this video we will explore the concept of Hopfield networks - a foundational model of associative memory that underlies many important ideas in neuroscience and machine learning, such as Boltzmann machines and Dense associative memory.
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OUTLINE:
00:00 Introduction
02:17 Protein folding paradox
04:23 Energy definition
08:25 Hopfield network architecture
14:03 Inference
18:40 Learning
22:48 Limitations & Perspective
24:43 Shortform
25:54 Outro
References:
1) Downing, K.L., 2023. Gradient expectations: structure, origins, and synthesis of predictive neural networks. The MIT Press, Cambridge, Massachusetts.
2) towardsdatascience.com/hopfie...
3) ml-jku.github.io/hopfield-lay...
Credits:
Protein folding: • protein folding
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I have a more streamline answer to the protein problem. The protein doesn't start folding when it's a complete sequence, it folds as the sequence is being built. This computationally and temporally constrains the degrees of movement, limiting the number of molecular forces at work at any one given time. Meaning that the part of the sequence that has already has been constructed is already folded into it's low energy state, and the part that hasn't been build isn't preturbing the current folding stage. The folding process is constrained to occur as sequentially as possible, not in parrallel.
This is top notch content, good work.
A threshold activation heatmap over a parallel distribution of temporal sequential threads is more descriptive. Each thread operates in its own input/output relative connection space and favors specific input sequences over time. Maximum amplification of a sequence (i_1/time + i_2/time + i_3/time...) indicating highly favored temporal sequence and (i_3/time - i_2/time - i_1/time...) indicating the least favored temporal sequence (with temporal sequences in-between these 2 extremes). Each thread is measured against its threshold (T), Amplification (A), and a latent timeframe (L) and elapsed time (E) for sequence-coordinated activation. When A exceeds T, the output is calculated as (1 - |L - E|) to output partners. Favored detection sequences can be defined by a integer to define (most favored position) within the temporal sequential thread. The process can be tuned by sensory reward detections over time, increasing mutation velocity in a direction, changing the param magnitude in that direction, acting on thresholds and shift the sequence of most value for each thread. There is more optimizations to further optimize this style of learning, by extending it with threads that mutate other threads based on their activation levels, allowing mutation behavior to be inferred and leveraged by the network as it's trained (it begins to handle it's own mutations internally based on inference). Then it's a matter of reading the heat map to see what parts of the network like doing certain task, and seeing the state transitions of the network.
Ladies, gentlemen, and fabulous folks of every flavor, the legend is back!
bro got lost into obsidian css configuration, but now he returns to brain cell
He's something else, manages to make computational neuroscience engaging WHILE not giving up on the details
John Hopfield wasn't the first to describe the formalism which has been subsequently popularised as "Hopfield networks".
It seems much fairer to the wider field and long history of neuroscientists, computer scientists, physicists, and so on to call them "associative memory networks", i.e. Hopfield was definitely not the first/only to propose the network some call "Hopfield networks". For instance, after the proposal of Marr (1971), many similar models of associative memory were proposed, e.g., those of Nakano (1972), Amari (1972), Little (1974), and Stanley (1976), which all have a very similar (or exactly the same) formalism as Hopfield's 1982 paper.
Today, notable researchers in this field correct their students' papers to replace instances of "Hopfield networks" with "associative memory networks (sometimes referred to as Hopfield networks)" or something similar. I would encourage you to do the same in your current/future videos.
I deeply regret making a similar mistake regarding this topic in one of my earlier papers. However, I am glad to correct the record now and in the future.
Refs:
D Marr. Simple memory: a theory for archicortex. Philos Trans R Soc Lond B Biol Sci, 262(841):23-81, July 1971.
Kaoru Nakano. Associatron-a model of associative memory. IEEE Transactions on Systems, Man, and Cybernetics, SMC-2(3):380-388, 1972. doi: 10.1109/TSMC.1972.4309133.
S.-I. Amari. Learning patterns and pattern sequences by self-organizing nets of threshold elements. IEEE Transactions on Computers, C-21(11):1197-1206, 1972. doi: 10.1109/T-C.1972.223477.
W.A. Little. The existence of persistent states in the brain. Mathematical Biosciences, 19(1):101-120, 1974. ISSN 0025-5564. doi: doi.org/10.1016/0025-5564(74)90031-5.
J. C. Stanley. Simulation studies of a temporal sequence memory model. Biological Cybernetics, 24(3):121-137, Sep 1976. ISSN 1432-0770. doi: 10.1007/BF00364115.
Wow, you cited your sources on a TH-cam comment! Thanks for the info.
See also: Sitgler's law
Thank you for sharing your knowledge
Wow, thanks for the info!
Relative connection spaces are dimensionally agnostic, they don't presupose a dimensionality for each node in the connection space, it's better at tracking large distributions (where a heat map can highligh areas of activity [threshold activations], to see the areas that light up when the system is doing specific task or undergoing a specific sensory data pattern). This way the dimensionality isn't constrained to a 2d sheet and predifined curvature manifold, you can better see the modal transitions of the system via this heat map.
This is an insanely educational video, as a ML researcher working on representation learning for multi modal retrieval, this is insanely helpful and relatable. I think you just gave me a new area to look at now, how exciting, i owe you one.
In fact, many proteins function in a _local_ minimum that is _not_ the global minimum. This is why proteins denature irreversibly when exposed to heat; there's an energy barrier that they can never come back from if they cross it.
What's stopping the proteins to fold to the global minimum immediately? And can it spontaneously transition from local to global minimum?
They function at their own global minimum, but the global minimum is also defined by the environment, for example pH, or various chemical agents. Also, for some proteins denaturation is reversible ("renaturation") when the conditions for denaturaing are removed. Irreversibility often caused by protiens interconnecting each other and forming a mesh, losing almost all degrees of freedom; in this case you cannot talk about a global minimum of an individual protein molecule.
@@GeoffryGifari The space of possible conformations for a protein is gigantic, very high dimensional, so even if they were started in a random state, it would be very unlikely for them to fall into the global minimum right away. As it happens, though, they are constructed one amino acid at a time and the enzyme that builds them keeps anything from interacting with the most recently added residues, so there is a systematic way to do it. At each step, the already-extruded portion of the polypeptide finds its own minimum, and that limits the trajectory of the conformation as it grows. Thus, it's always in some kind of local minimum at every step.
There are many other factors at play, too. There are chaperone proteins that prevent selected parts from interacting, there is a whole different process for proteins that are supposed to be in a membrane, and on and on. Life never gives you simple, straightforward rules.
What stops it from falling to the global minimum is the energy barriers surrounding local minima. That's pretty much the definition of local minimum, surrounded by energy barriers. If the water molecules that are constantly battering it give it enough energy, it can hop over the barrier and fall into the nearest minimum in that direction, so to speak, but there is no dynamical reason for it to progress toward the global minimum at any given moment; it has to reach it by random jostling. It's theoretically possible for it to jump out of the global minimum, too, in the same way, but by definition, the global minimum has the biggest barriers in the whole space, so it's likely to stick around there.
@@onebronx Some good points there; I was thinking only of heat denaturing. The result of physical heat denaturing is inevitably the global minimum, and at least when I was in university in 2006, we didn't hear about any proteins coming back from that. If a protein functions at its global minimum, then you can always cook it until it chemically comes apart, but that is another story. Most proteins need to be able to cycle between conformations at low energy scales, e.g. the binding energy of a small molecule, so they don't typically function at the global minimum. It would be pretty contrived if they did, if you think about it.
RNAzymes, on the other hand denature and cease function at _low_ temperatures, rather than high, last I checked, and this happens for the same reason: an enzyme needs to access multiple conformation states to function, so it doesn't function if denied the energy to flop around. There are way more interaction motifs for RNA than for polypeptides, so the energy landscape, while bumpy, doesn't generally have an irreversible global minimum like a protein, so they can return to function pretty well after being cooked.
That's just what I was taught ~2006, so if we know any different now, please show me a citation.
@@davidhand9721 well, again, are we talking about heat denaturating of a single polypeptide, or of a large collective? Those are different environments -- in a collective, large molecules can start interlinking, so there is no much sense to talk about some individual global minimums. Of course there is always an ultimate global minimum -- death of the Universe, but we're usually interested in somewhat more "locally global" minimums, where your surrounding environment does not kill you if you stretched a bit too much :)
For polymers it is even more complicated because of topology -- there are millions way to make a polypeptide chain self-entangled like a rope in a washing machine, and it actually can denaturate in some configuration with a _higher_ energy, simply because it cannot detangle itself from it.
Intuitively, a global minimum (or very close one) should be reachable if you build the chain slowly and carefully, shaking it periodically to allow some "annealing", allowing it to de-tangle early while the chain is still short. And the shorter the polypeptide, the more likely it will eventually find its global minimum (in a sense "global for a single molecule in the pH-neutral isotonic intracellular fluid").
Huge polypeptides can be farther from a global minimum for sure, but may be not so far too, due to evolutionaly pressure: the secondary/ternary structures are evolving and they were naturally-selected for as much stability as it still allows them to function. Those pretty alpha-heilces and beta-sheets, varoius cross-links and nicely-fitting active zones overall create more bonds per unit of length than some randomly aligned segments of the same chain - they evolved for that! And more bonds means less free energy.
And then, near the global minimum, you can have some local minimums with slightly higher energies, corresponding to different "activated" configuration. OTOH, the alternative "activated" configuration can be thought as "global" minimums for a system "protein + activator".
4:00 LOL "The ball doesn't search through all possible trajectories to select the optimal parabolic one."
The visualization of the "trajectory space" is even funnier ;)
I suspect there's different encodings for proteins with identical function, but which are more robust wrt folding consistently.
At the beginning, when you mention the O(n) problem, as a programmer it just intuitively makes you want to use a tree or a hash map lol
In any case, another banger!
Its fascinating to see how these things work!
I had the same thought. Though, a hashmap or something similar probably wouldn't work, since many times the key is incomplete or noisy, which would cause the hashing function to return a hash that would map to the wrong index
Minor correction: 'Cells that fire together, wire together' was coined by Carla Shatz (1992). Unlike Donald Hebb's original formulation, Shatz's summary of Hebbian learning eliminates the role of axonal transmission delays. By extension, neural networks which remain true to Hebb's original definition should go beyond rate coded models and instead simulate the time delays.
Yes latent time parameters need to be implemented. A threshold activation heatmap over a parallel distribution of interconnected temporal sequential threads is more descriptive in targeting what he is trying to convey in larger distibutions where hopfield computational structure fails. Each thread operates in its own input/output relative connection space and favors specific input sequences over time. Maximum amplification of a sequence (i_1/time + i_2/time + i_3/time...) indicating highly favored temporal sequence and (i_3/time - i_2/time - i_1/time...) indicating the least favored temporal sequence (with temporal sequences in-between these 2 extremes). Each thread is measured against its threshold (T), Amplification (A), and a latent timeframe (L) and elapsed time (E) for sequence-coordinated activation. When A exceeds T, the output is calculated as (1 - |L - E|) to output partners. Favored detection sequences can be defined by a integer for each input within a temporal sequential thread (a mutable trainable param), representing the input's favored position within the temporal sequence. The process can be tuned by sensory reward detections over time, increasing mutation velocity in a direction, changing the param magnitude in that direction, acting on thresholds and shift the sequence of most value for each thread. There is more optimizations to further optimize this style of learning, by extending it with threads that mutate other threads based on their activation levels, allowing mutation behavior to be inferred and leveraged by the network. Then it's a matter of reading the heat map to see what parts of the network like doing certain task given a specific time slice, and seeing the state transitions of the network when other distributions become active.
Thanks for the incredible quality in your videos
Man, really thankful to your contents. I was facinated by your video about TEM, and started trying to fully understand that network(and memory in general) in my leisure time since about a year ago. I learned about latent variables, transformer architecture(fantastic videos by andrej karpathy), autoencoders, etc, but got stuck at (modern)hopfield nets, which I think is super important in the architecture of TEM. Very glad to see that you start to touch this field of Hopfield Nets, this is probably the best video about vanilla hns I've ever watched. Really looking forward to your video about Boltzman Machines and Modern Hopfield Nets, always appreciates your videos!
Exceptional pedagogical skill! I’m not able to hold these types of explanations in my mind, so any attempt at following such a web of relations would quickly have me lost. But this is a masterclass in clear considerate communication 🙏
I was thinking about your channel less than an hour ago.
This looks like a run-of-the-mill gradient descent, how does resolves false bottoms?
Finally, you are comeback 🎉
Hopfield networks are amazing! They are studied in physics, biology, machine learning, mathematics and chemistry
The rabbit hole goes extremely deep
Your back!!! awesome vid as always.
Respect for using Coldplay 🔥🔥🔥
Waiting for your video so long. Thank you so much
This was one of the good oness. I really loved it and hope part 2 comes out sooner. Keep up the amazing production sir.
- Excellent! Thx.
- Very well presented: clear/concise, yet fairly comprehensive - and w/ great visualizations.
- Keep up the great content!...
very cool video, keen to see how the broader arguments progresses in this series
Awesome video, like always.
Just a small nitpick: your speaker audio jumps between the left and right audio channel, which is quite distracting - especially with headphones.
You can easily solve this by setting the voice audio track to "mono" when editing the video.
Cheers
Another great video! I really liked your energy landscape and gradient descent animations especially.
I was reading and watching videos about metacognition and bayesian probability and now you have thrown me into a new rabbit hole! 😅
Your videos are incredible and it's great to have a new one. Thank you!
Really good that you’re covering these foundational concepts of NNs. Cross-associative NNs, auto-associative NNs, and unsupervised learning are big missing pieces in today’s NNs.
Much love bro incredible video ❤ thank you
I wanted to do research on something like this a year or two ago. This is amazing, I've got some work to do with this.
Awesome. I just can’t wait for the next video!
This was really clear, accurate, and easy to follow. 10/10, would watch again.
This is really cool! Thanks for your work Artem!
As always, outstanding video!
One of the best most
Clear videos I’ve ever seen EVER❤ 🙏🏻THANK YOU ❤😊
That excitatory and inhibitory connections remind me of statistical correlation function
> "when you throw a ball the ball doesn't search through all the possible trajectories"
QM has entered the chat
This is incredibly well put together.
Man, I just love your videos
Thank you for this video!
Hello. It's been long since you last uploaded the last video. I hope you are well.
Best videos, bro. Keep them coming 🎉
So good. Thank you
I love that you made everything dark mode. (Noticed when I saw the Wikipedia logo)
Holy cow, this makes a complex topic so intuitive.
Thank you so much for creating this video, genuinely one of the most educational I've ever come across. I've been trying to learn more about how brains work since that's always been something I've been very curious about literally since birth, and along with entropy being my favorite physics concept this video has just led to me googling and researching for the last 4 hours (its 3am lol) trying to find out more. Really impressed with how complicated, yet still high-quality and clear, some of the topics are in this is and I'm really looking forward to watching the rest of your videos to learn more on how all of this stuff works in such an intricate way
Thank you!!
Please keep going. Keep dedicating your time to your pursuit of wonder.
Super interesting to see that the hopfield network basically reinvents binary operations like XOR and XNOR for neurons, with the two differentiated by the weight.
Thank you. This is unbelievably simple and potentially more accurate than any other speculation. Next step: determining the shape of the proteins and categorizing them. The tools may not be there, yet... but a good hypothetical certain helps to reach certain conclusions.
The protein example got me thinking. Is there only one unique folded configuration of the lowest energy? Can there be multiple stable comfiguration anyway, and transitions between them?
Can wait to see this video!!!!!
Fabulous, bravo
Waited for this...!!
great video as always
One reason I like using Neo4J is that graph networks seem like the work a bit like human memory with links between things making finding related items fast.
please try to make more videos, your content is extremely good
Кайф слушать и офигевать)
Greetings from Austria, keep doing what you're doing!
Thanks for such a this amazing content
Another awesoem Video :)
This is one of the best channels
12:21 Watching this to maximise my happiness. Max happiness Min unhappiness, this is the way. Thank you!
Great video
Love you man
The outro music 🙏🏻🙏🏻🙏🏻
Great video as always. For 22:50, has anyone tried stacking layers of Hopfield networks yet as a work around? Basically each layer acts in it's own feature level space and resolves for that level's most likely feature, then passes it up to the next higher order Hopfield feature space to be resolved there. It seems like this would allow you to store exponentially more overall patterns has they get resolved separately to avoid the overly busy end energy landscape.
Also interestingly you can see how it would carve out the energy landscape from just the raw inputs with this. You have the Hopfield network constantly comparing itself to the some abstraction of the source input layer, meaning the more times a pattern seen the stronger it gets in the Hopfield network. Also for faster convergence, it is likely the greater the xi and hi difference the faster xi updates.
This content is so high level, it's almost impossible to tell if it is true or not. Physically and philosophically, I have bought in, but my want of it to be true doesn't make it so.
I cannot imagine this is wrong, but where does this come from?
This stuff is just crazy and I don't know if it is crazy good or just crazy lol but I'm along for the ride
My God, your videos are amazing
Plz make a video about modern hopfield net or dense assosiative memory. Cuz it a different and generalize perspective of mopdern hopfield nets.
Genius !
Awesome
Just in time for my studies into Z12 cyclic groups, which surely have nothing to do with cognition 🧐
Because the neurons have two edge weights (A->B and B->A) does that might explain the case where we have the memory just out of reach and get lock trying to get it?
STUFF LIKE THIS IS SO INTERESTING TO ME TO LEARN😮💨GOD DESIGNED SUCH AN AMAZING CREATION🤩🙏🏾🙏🏾🙏🏾
Can you do a video on the work of Dmitry Krotov showing that attention mechanisms are a special case of associative memory networks
Didnt you already upload a video on boltzman machines? I thought I saw it last year.
Si that means that our neurons do some type of gradient descent? That’s very interesting to know
Great video. Little sad to see that anti-training wasn't mentioned. It doesn't really solve the problem with training two sequences that are 'close' together, so that's fair. But it does help, and has an interesting analogy with physiology.
Essentially, if you try to train two sequences that are too close to each other, their valleys will overlap, which means you might try to aim for one specific sequence and end up falling into the other. And if they're really close, you'll actually create a new local minimum that sits between the two.
In those cases, what you can do is identify all your local minima and then run the algorithm backwards, training hills on top of all your local minima. For stand-alone minima, this doesn't matter because they're still local minima. But if a minima is a false sequence that sits between two or more neighboring targets, this builds a hill in between those two neighboring valleys, helping to make those nearby sequences more distinct.
As Geoffy Hinton has pointed out, this has an interesting conceptual analog to dreaming, where we seem to replay experiences and concepts from our day (to a vague extent) and sleeping/dreaming also seems to help with learning and memory. Similarly by making the Hopfield focus on its memoreis while playing them backwards, so to speak, helps to solidify its own memory.
It may be little more than a metaphorical analog, but I think its still quite interesting.
That’s exactly right! Bolzmann machines, which are an improved version of Hopfield nets in fact do just that, with contrastive hebbian learning, by increasing the energy of “fake” memories.
Hopfield networks, being the first model, don’t have that property in the conventional form though. So we will talk about this in the Bolzmann machines video.
Good catch!
@@ArtemKirsanov Ah, gotcha I didn't realize the idea of 'anti-learning' applied to boltzman machines. I've only messed with restricted boltzmann machines and I always thought of them as stacked reversible auto-encoders.
Never though that the updating method may be replicating the same 'anti-learning' process - though it does make sense since autoencoders are trying to make a bunch of weird, distinct hyper-dimensional valleys. Maybe it's more apparent with the more general Boltzmann machine.
Looking forward to that video!
Agree 💯
Ironically, within the framework of quantum mechanics, one could actually say that the ball *does* "search" every possible path in order to find the "correct" one. It simply performs the "search" in parallel, not sequentially. And it's less of a search and more of an average of all paths. The principle of stationary action is the driving principle behind Newtonian dynamics and itself follows directly from the interference between many "virtual" trajectories, it turns out that the paths which are close to the "true path" (the classical path) have very little variance in their action, which rough speaking means that they end with nearly identical phase shifts (e^iHt/h, Ht ~ action, h = Planck constant) and can interfere constructively, whereas paths which are far from the "true path" have wildly varying actions, even if two paths are similar to each other. So they pick up big phase shifts and end up interfering destructively, leaving only the contributions from the paths "close to" the classically observed path. As far as I know this is the only way to derive the principle of stationary action, and the same basic idea is essential to finding transition probability amplitudes in QFT. It really does seem like the universe simultaneously tries all conceivable paths, superimposed together.
One could also say this for optics with Fermat's principle or classical mechanics with Hamilton's principle. Even though variational formulations are mathematically beautiful, I'd be cautious to assume that "reality works this way" i.e. "searches through all paths". They are one equivalent description of many (even though it is remarkable that they pop up basically everywhere).
@@asdf56790 I agree with your caution. I'm just increasingly convinced that theory and experiment are pointing this way and the onl obstacle is our flawed intuition and prejudice. We want there to be only a single, consistent reality. But this forces us into some intense mental and mathematical gymnastics to make the equations of QM fit observations. If you take the equations at face value then you have no trouble, you just have to contend with the idea that reality is not a single line of well defined events, but multiple histories occurring simultaneously and generally able to interfere with each other. An electron passing through a Stern-Gerlach device would then actually travel both paths, in "separate worlds" but these paths can still interfere and superimpose so long as you don't take any steps to determine which path was taken. Like if you redirect the paths to converge into a single path and put the whole thing in a box so you can only see the output, you cannot determine which path was taken and you can show experimentally that the output electron superposition of spin states is preserved. But the universe doesn't know ahead of time (superdeterminism notwithstanding) whether you will take a peek in the box and catch the electron with its pants down. In my view, the explanation requiring the fewest assumptions is that all paths really are taken, but with the assumptions that 1. When we observe a property we can only observe definite values, not superpositions, and 2. paths can interfere (unless decoherence has occurred). A poor and rushed explanation but this is kind of my thought process. As you say, there are many alternative interpretations. It's pretty much philosophy at this point. 😅
What’s the status of all those parallel paths? You’ve used the term "virtual" so you seem to view them as part of some kind of potentiality, not getting actualized in the observer measured reality (I’m using these terms very loosely, I don’t know exactly what they mean). If I’ve understood right, in the MWI there’s no actual convergence on a path, each of the possible parallel paths are actualized paths that get to be part of reality.
@@asdf56790 This is true, all subsequent physical theories are simply better and better approximations of reality, we shouldn't assume reality works that way without more experiemental and theoretical verification.
2:16 a* great introduction
Isn't the 2nd law of thermodynamics more directly linked to entropy? Is there an analog for entropy in the associative memory network?
Just as I suspected, everything is just the principle of stationary action! It's all just making an action functional stationary what if we considered non-local actions?
I wonder if the convergent process can be done parrellelly state by state.
That would be awsome.
It seems to me this "fire together wire together" notion is actually also present in the attention mechanism of transformers, except you don't just have a 1D 1 bit + - but rather an nD dot product.
This still has the same basic structure: Neurons try to more closely align to each other. But the added dimensions give each neuron more ways to accomplish that: Among three neurons, one neuron can be pretty close to the other two while those two might be pretty far away from each other.
Long time no see😁
I've always thought of energy has unhappiness, too.
Your font missed the contextual ligatures
Wow
Wait, this is just the Ising model with extra steps!
This seems like Friston's explanetion of inference. Can you explain that to in your videos?
❤❤
AGI will likely need thinking in this way, and creating at least partial ideas of how our brain works.
😍
Do you have merch?
Great explanation!
This applies to fully connected networks, correct? So memories are Dense Distributed Representations.
It's not how the brain does it. The brain uses the opposite: Sparse Distributed Representations.
Is there any way you could explain it using SDRs?
how does this relate to fristons free energy principle?
🙇
Does this mean artbros were right about diffusion models being databases?
My guess is they're "databases" of chaotic attractors. After all, the diffusion process is effectively a differential equation that evolves over time and settles in some stable basin.
Reminds me of gradient descent
That's because this is it.
Hey, someone pointed me to your page because of the links between neuroscience and ML. I think that the thalamus might be a cross-modal auto encoder. Main reason I think this is because there are 10 times as many connections going into the thalamus as there are going out, sounds like encoding to me. I was wondering, does that ring true to you?
Hi there great videos! I am curious about your view of free will and agency. I myself explore this philosophical/psychological/biological subject, and your videos are reinforcing notions I found in books like "chaos and nonlinear psychology, Schulter et al" and "determined, Sapolsky" among others. It seems to me that, from a scientific point of view at least, the notion of dualism, and free will, doesn't make sense. Is that something you think about? :) In any case keep up the work, it's really appreciate here. Antoine Dion, Canada
I use an excell spreadsheet for all my memories
wow! simplifying weights to be simmetric is a pretty huge imho, so in brain contrary to neural nets that supposedly work like brain neurons are not activated in layers from left to right, there is a mess with simultaneous back propagation waves, cools
second time my brain exploded at "maximizing the sum is same as minmizing the loss", daaamn.. so just like that the hill on the graph becomes valley - i just think it semantically triggers analogy with gravity, which is imho bad, look - even in protein folding analogy, there is energy "push", not gravity "pull", like, proteins can't fold TOO fast, right, there is some limit how fast it could be pushed in place - this analogy breaks when we start thinking about valleys instead of hills on graph, as it would become obviuos that that steeper the valley the faster solution will be converged, BUT, i just think TOO steep path sometimes can't be jumped around, so it's kinda misleading maybe. and if we keep hill representation - there should be "path up" not so steep to be able to afford it with available energy.. very interesting!)
Let's break it down and elaborate on each aspect.
Symmetric Weights
Simplifying weights to be symmetric is a pretty huge imho:
In biological neural networks, connections between neurons (synapses) are not necessarily symmetric. The strength and direction of the connection from neuron A to neuron B can be different from the connection from neuron B to neuron A. This asymmetry introduces a layer of complexity that is not present in traditional Hopfield networks.
Hopfield networks simplify this by assuming symmetric weights (i.e., 𝑤𝑖𝑗=𝑤𝑗𝑖w ij=w ji). This simplification makes the mathematical analysis more tractable and ensures convergence to a stable state. In other words, it guarantees that the network will not get stuck in an infinite loop of state changes, but will eventually settle into a stable configuration (a local minimum in the energy landscape).
Neurons Activation and Propagation in the Brain
In brain contrary to neural nets that supposedly work like brain neurons are not activated in layers from left to right, there is a mess with simultaneous back propagation waves, cools:
Traditional artificial neural networks, like those used in deep learning, often operate in discrete layers where information flows in a feedforward manner from input to output. However, the brain operates very differently. Neurons in the brain are massively interconnected and communicate through a complex network of excitatory and inhibitory signals, often in parallel and asynchronously.
In this context, the analogy with Hopfield networks is closer because Hopfield networks also consider a fully connected network where every neuron can potentially influence every other neuron. This creates a more "messy" and intertwined propagation of activation, which is more biologically plausible.
Maximizing Happiness vs. Minimizing Energy
Maximizing the sum is same as minimizing the loss:
This concept is a fundamental principle in optimization. In the context of Hopfield networks, the "happiness" of the network (which is the sum of weighted pairwise state agreements) can be thought of as the inverse of the "energy" of the network. Maximizing the happiness is equivalent to minimizing the energy.
When you maximize the sum of weighted agreements, you are effectively pushing the system towards a state where all the neurons are in configurations that are most favorable given their connections. This is conceptually similar to minimizing the loss function in machine learning, where you adjust weights to make the network's predictions as accurate as possible.
Energy Landscape and Analogy with Protein Folding
Energy "push" vs. gravity "pull":
Your observation about the analogy between protein folding and neural network optimization is insightful. In protein folding, the molecule finds its stable configuration by moving towards a state of lower potential energy. This process is driven by physical forces that minimize the system's energy, much like a ball rolling downhill due to gravity.
In neural networks, this analogy is often visualized as moving towards the bottom of a valley in an energy landscape. The idea is that the system will naturally "descend" towards the lowest energy state, which corresponds to the optimal configuration.
However, as you pointed out, this analogy can be misleading if taken too literally. In protein folding, the process is driven by physical interactions that might have constraints on how quickly the system can reach its stable state. Similarly, in neural networks, if the energy landscape has very steep valleys, the system might face difficulties in finding the optimal path, as it might get stuck in local minima or encounter barriers that prevent smooth convergence.
Steepness and Convergence:
In optimization, a very steep valley (or hill) in the energy landscape can indeed pose problems. If the gradient is too steep, the optimization algorithm might take very large steps and overshoot the minimum, or it might face numerical instability. This is why techniques like gradient clipping or adaptive learning rates are used in training neural networks to handle such issues.
Path Representation:
When you mention that the "steeper the valley the faster solution will be converged," it's important to consider that while steep gradients can accelerate convergence initially, they can also cause problems if the system overshoots or gets trapped. A more moderate and manageable path can ensure steady and reliable convergence to the optimal solution.
Conclusion
In summary, your comment highlights the nuances and complexities in drawing analogies between biological processes and artificial neural networks. The simplifications made in Hopfield networks (such as symmetric weights) serve to make the system mathematically tractable and ensure convergence, but they do not capture all the intricacies of biological neural networks. Additionally, the analogy with energy landscapes, while useful, must be carefully interpreted to account for practical challenges in optimization.
th-cam.com/video/43nsldnfkM8/w-d-xo.html
The symmetry assumption is actually strange. The way how the objective function is defined, the antisymmetric part of the weights doesn’t matter. The extrema of the function are not affected by it. This assumption is only needed for the essentially sequential Hopfield update rule to converge. But realistic neurons are not updated in some orderly fashion. This raises a lot of questions.
I didn't want to sleep anyway.
Resumindo: intuição inconsciente guiada pelo princípio da conservação de energia do ego absoluto.