This summary builds extensively on my prior summary of SIRENs, so if you haven't read that summary or the underlying paper yet, I'd recommend doing that first! 

At a high level, the idea of SIRENs is to use a neural network to learn a compressed, continuous representation of an image, where the neural network encodes a mapping from (x, y) to the pixel value at that location, and the image can be reconstructed (or, potentially, expanded in size) by sampling from that function across the full range of the image. To do this effectively, they use sinusoidal activation functions, which let them match not just the output of the neural network f(x, y) to the true image, but also the first and second derivatives of the neural network to the first and second derivatives of the true image, which provides a more robust training signal. 

NERFs builds on this idea, but instead of trying to learn a continuous representation of an image (mapping from 2D position to 3D RGB), they try to learn a continuous representation of a scene, mapping from position (specified with with three coordinates) and viewing direction (specified with two angles) to the RGB color at a given point in a 3D grid (or "voxel", analogous to "pixel"), as well as the *density* or opacity of that point. 

Why is this interesting? Because if you have a NERF that has learned a good underlying function of a particular 3D scene, you can theoretically take samples of that scene from arbitrary angles, even angles not seen during training. It essentially functions as a usable 3D model of a scene, but one that, because it's stored in the weights of a neural network, and specified in a continuous function, is far smaller than actually storing all the values of all the voxels in a 3D scene (the authors give an example of 5MB vs 15GB for a NERF vs a full 3D model). To get some intuition for this, consider that if you wanted to store the curve represented by a particular third-degree polynomial function between 0 and 10,000 it would be much more space-efficient to simply store the 3 coefficients of that polynomial, and be able to sample from it at your desired granularity at will, rather than storing many empirically sampled points from along the curve. 

https://i.imgur.com/0c33YqV.png

How is a NERF model learned?

- The (x, y, z) position of each point is encoded as a combination of sine-wave, Fourier-style curves of increasingly higher frequency. This is similar to the positional encoding used by transformers. In practical turns, this means a location in space will be represented as a vector calculated as [some point on a low-frequency curve, some point on a slightly higher frequency curve..., some point on the highest-frequency curve]. This doesn't contain any more *information* than the (x, y, z) representation, but it does empirically seem to help training when you separate the frequencies like this
- You take a dataset of images for which viewing direction is known, and simulate sending a ray through the scene in that direction, hitting some line (or possibly tube?) of voxels on the way. You calculate the perceived color at that point, which is an integral of the color information and density/opacity returned by your model, for each point. Intuitively, if you have a high opacity weight early on, that part of the object blocks any voxels further in the ray, whereas if the opacity weight is lower, more of the voxels behind will contribute to the overall effective color perceived. You then compare these predicted perceived colors to the actual colors captured by the 2D image, and train on the prediction error.
- (One note on sampling: the paper proposes a hierarchical sampling scheme to help with sampling efficiently along the ray, first taking a course sample, and then adding additional samples in regions of high predicted density)
- At the end of training, you have a network that hopefully captures the information from *that particular scene*. A notable downside of this approach is that it's quite slow for any use cases that require training on many scenes, since each individual scene network takes about 1-2 days of GPU time to train

Wu et al. provide a framework (behavior regularized actor critic (BRAC)) which they use to empirically study the impact of different design choices in batch reinforcement learning (RL). Specific instantiations of the framework include BCQ, KL-Control and BEAR. 

Pure off-policy rl describes the problem of learning a policy purely from a batch $B$ of one step transitions collected with a behavior policy $\pi_b$. The setting allows for no further interactions with the environment. This learning regime is for example in high stake scenarios, like education or heath care, desirable. 

The core principle of batch RL-algorithms in to stay in some sense close to the behavior policy. The paper proposes to incorporate this firstly via a regularization term in the value function, which is denoted as **value penalty**. In this case the value function of BRAC takes the following form:

$
V_D^{\pi}(s) = \sum_{t=0}^{\infty} \gamma ^t \mathbb{E}_{s_t \sim P_t^{\pi}(s)}[R^{pi}(s_t)- \alpha D(\pi(\cdot\vert s_t) \Vert \pi_b(\cdot \vert s_t)))], 
$

where $\pi_b$ is the maximum likelihood estimate of the behavior policy based upon $B$. 
This results in a Q-function objective:
$\min_{Q} = \mathbb{E}_{\substack{(s,a,r,s') \sim D \\ a' \sim \pi_{\theta}(\cdot \vert s)}}\left[(r + \gamma \left(\bar{Q}(s',a')-\alpha D(\pi(\cdot\vert s) \Vert \pi_b(\cdot \vert s) \right) - Q(s,a) \right]  
$

and the corresponding policy update:
$
\max_{\pi_{\theta}} \mathbb{E}_{(s,a,r,s') \sim D} \left[ \mathbb{E}_{a^{''} \sim \pi_{\theta}(\cdot \vert s)}[Q(s,a^{''})] - \alpha  D(\pi(\cdot\vert s) \Vert \pi_b(\cdot \vert s)  \right] 
$
 
The second approach is **policy regularization** . Here the regularization weight $\alpha$ is set for value-objectives (V- and Q) to zero and is non-zero for the policy objective.

It is possible to instantiate for example the following batch RL algorithms in this setting:
- BEAR: policy regularization with sample-based kernel MMD as D and min-max mixture of the two ensemble elements for $\bar{Q}$
- BCQ: no regularization but policy optimization over restricted space

Extensive Experiments over the four Mujoco tasks Ant, HalfCheetah,Hopper Walker show:
1. for a BEAR like instantiation there is a  modest advantage of keeping $\alpha$ fixed
2. using a mixture of a two or four Q-networks ensemble as target value yields better returns that using one Q-network
3. taking the minimum of ensemble Q-functions is slightly better than taking a mixture (for Ant, HalfCeetah & Walker, but not for Hooper
4. the use of value-penalty yields higher return than the policy-penalty
5. no choice for D (MMD, KL (primal), KL(dual) or Wasserstein (dual)) significantly outperforms the other (note that his contradicts the BEAR paper where MMD was better than KL)
6. the value penalty version consistently outperforms BEAR which in turn outperforms BCQ with improves upon a partially trained baseline. 

This large scale study of different design choices helps in developing new methods. It is however surprising to see, that most design choices in current methods are shown empirically to be non crucial. This points to the importance of agreeing upon common test scenarios within a community to prevent over-fitting new algorithms to a particular setting. 

Large-scale transformers on unsupervised text data have been wildly successful in recent years; arguably, the most successful single idea in the last ~3 years of machine learning. Given that, it's understandable that different domains within ML want to take their shot at seeing whether the same formula will work for them as well. This paper applies the principles of (1) transformers and (2) large-scale unlabeled data to the problem of learning informative embeddings of molecular graphs. 

Labeling is a problem in much of machine learning - it's costly, and narrowly defined in terms of a certain task - but that problem is even more exacerbated when it comes to labeling properties of molecules, since they typically require wetlab chemistry to empirically measure. Given that, and also given the fact that we often want to predict new properties - like effectiveness against a new targetable drug receptor - that we don't yet have data for, finding a way to learn and transfer from unsupervised data has the potential to be quite valuable in the molecular learning sphere. 

There are two main conceptual parts to this paper and its method - named GROVER, in true-to-ML-form tortured acronym style. The first is the actual architecture of their model itself, which combines both a message-passing Graph Neural Network to aggregate local information, and a Transformer to aggregate global information. The paper was a bit vague here, but the way I understand it is: 

https://i.imgur.com/JY4vRdd.png
- There are parallel GNN + Transformer stacks for both edges and nodes, each of which outputs both a node and edge embedding, for four embeddings total. I'll describe the one for nodes, and the parallel for edges operates the same way, except that hidden states live on edges rather than nodes, and attention is conducted over edges rather than nodes
- In the NodeTransformer version, a message passing NN (of I'm not sure how many layers) performs neighborhood aggregation (aggregating the hidden states of neighboring nodes and edges, then weight-transforming them, then aggregating again) until each node has a representation that has "absorbed" in information from a few hops out of its surrounding neighborhood. My understanding is that there is a separate MPNN for queries, keys, and values, and so each nodes end up with three different vectors for these three things.
- Multi-headed attention is then performed over these node representations, in the normal way, where all keys and queries are dot-product-ed together, and put into a softmax to calculate a weighted average over the values
- We now have node-level representations that combine both local and global information. These node representations are then aggregated into both node and edge representations, and each is put into a MLP layer and Layer Norm before finally outputting a node-based node and edge representation. This is then joined by an edge-based node and edge representation from the parallel stack. These are aggregated on a full-graph level to predict graph-level properties

https://i.imgur.com/NNl6v4Y.png

The other component of the GROVER model is the way this architecture is actually trained - without explicit supervised labels. The authors use two tasks - one local, and one global. The local task constructs labels based on local contextual properties of a given atom - for example, the atom here has one double-bonded Nitrogen and one single-bonded Oxygen in its local environment - and tries to predict those labels given the representations of that atom (or node). The global task uses RDKit (an analytically constructed molecular analysis kit) to identify 85 different modifs or functional groups in the molecule, and encodes those into an 85-long one-hot vector that is being predicted on a graph level. 

https://i.imgur.com/jzbYchA.png

With these two components, GROVER is pretrained on 10 million unlabeled molecules, and then evaluated in transfer settings where its representations are fine-tuned on small amounts of labeled data. The results are pretty impressive - it achieves new SOTA performance by relatively large amounts on all tasks, even relative to exist semi-supervised pretraining methods that similarly have access to more data. The authors perform ablations to show that it's important to do the graph-aggregation step before a transformer (the alternative being just doing a transformer on raw node and edge features), and also show that their architecture without pretraining (just used directly in downstream tasks) also performs worse. One thing I wish they'd directly ablated was the value-add of the local (also referred to as "contextual") and global semi-supervised tasks. Naively, I'd guess that most of the performance gain came from the global task, but it's hard to know without them having done the test directly.

[First off, full credit that this summary is essentially a distilled-for-my-own-understanding compression of Yannic Kilcher's excellent video on the topic] 

I'm interested in learning more about Neural Radiance Fields (or NERFs), a recent technique for learning a representation of a scene that lets you generate multiple views from it, and a paper referenced as a useful prerequisite for that technique was SIRENs, or Sinuisodial Representation Networks. In my view, the most complex part of understanding this technique isn't the technique itself, but the particularities of the problem being solved, and the ways it differs from a more traditional ML setup. 

Typically, the goal of machine learning is to learn a model that extracts and represents properties of a data distribution, and that can generalize to new examples drawn from that distribution. Instead, in this framing, a single network is being used to capture information about a single image, essentially creating a compressed representation of that image that brings with it some nice additional properties. 

Concretely, the neural network is representing a function that maps inputs of the form (x, y), representing coordinates within the image, to (r, g, b) values, representing the pixel values of the image at that coordinate. If you're able to train an optimal version of such a network, it would mean you have a continuous representation of the image. A good way to think about "continuous," here, is that, you could theoretically ask the model for the color value at pixel (3.5, 2.5), and, given that it's simply a numerical mapping, it could give you a prediction, even though in your discrete "sampling" of pixels, that pixel never appears.

Given this problem setting, the central technique proposed by SIRENs is to use sinusoidal non-linearities between the layers. On the face of it, this may seem like a pretty weird choice: non-linearities are generally monotonic, and a sine wave is absolutely not that. The appealing property of sinusoidal activations in this context is: if you take a derivative of a sine curve, what you get is a cosine curve (which is essentially a shifted sine curve), and the same is true in reverse. This means that you can take multiple derivatives of the learned function (where, again, "learned function" is your neural network optimized for this particular image), and have them still be networks of the same underlying format, with shifting constants. 

This allows SIRENs to use an enhanced version of what would be a typical training procedure for this setting. Simplistically, the way you'd go about training this kind of representation would be to simply give the inputs, and optimize against a loss function that reduced your prediction error in predicting the output values, or, in other words, the error on the f(x, y) function itself. When you have a model structure that makes it easy to take first and second derivatives of the function calculated by the model, you can, as this paper does, decide to train against a loss function of matching, not just the true f(x, y) function (again, the pixel values at coordinates), but also the first and second-derivatives (gradients and Laplacian) of the image at those coordinates. This supervision lets you learn a better underlying representation, since it enforces not just what comes "above the surface" at your sampled pixels, but the dynamics of the true function between those points. 

One interesting benefit of this procedure of using loss in a first or second derivative space (as pointed out in the paper), is that if you want to merge the interesting parts of multiple images, you can approximate that by training a SIREN on the sum of their gradients, since places where gradients are zero likely don't contain much contrast or interesting content (as an example: a constant color background). 

The Experiments section goes into a lot of specific applications in boundary-finding problems, which I understand at less depth, and thus won't try to explain. It also briefly mentions trying to learn a prior over the space of image functions (that is, a prior over the set of network weights that define the underlying function of an image); having such a prior is interesting in that it would theoretically let you sample both the implicit image function itself (from the prior), and then also points within that function.

This paper deals with the question what / how exactly CNNs learn, considering the fact that they usually have more trainable parameters than data points on which they are trained.

When the authors write "deep neural networks", they are talking about Inception V3, AlexNet and MLPs.

## Key contributions

* Deep neural networks easily fit random labels (achieving a training error of 0 and a test error which is just randomly guessing labels as expected). $\Rightarrow$Those architectures can simply brute-force memorize the training data.
* Deep neural networks fit random images (e.g. Gaussian noise) with 0 training error. The authors conclude that VC-dimension / Rademacher complexity, and uniform stability are bad explanations for generalization capabilities of neural networks
* The authors give a construction for a 2-layer network with $p = 2n+d$ parameters - where $n$ is the number of samples and $d$ is the dimension of each sample - which can easily fit any labeling. (Finite sample expressivity). See section 4.


## What I learned

* Any measure $m$ of the generalization capability of classifiers $H$ should take the percentage of corrupted labels ($p_c \in [0, 1]$, where $p_c =0$ is a perfect labeling and $p_c=1$ is totally random) into account: If $p_c = 1$, then $m()$ should be 0, too, as it is impossible to learn something meaningful with totally random labels.
* We seem to have built models which work well on image data in general, but not "natural" / meaningful images as we thought.


## Funny

> deep neural nets remain mysterious for many reasons

> Note that this is not exactly simple as the kernel matrix requires 30GB to store in memory. Nonetheless, this system can be solved in under 3 minutes in on a commodity workstation with 24 cores and 256 GB of RAM with a conventional LAPACK call.


## See also

* [Deep Nets Don't Learn Via Memorization](https://openreview.net/pdf?id=rJv6ZgHYg)