This is a mildly silly paper to summarize, since there isn't really a new mechanism to understand, but rather a number of straightforward (and interesting!) empirical results that are also quite well-explained in the paper itself. That said, for the sake of a tiny bit more brevity than the paper itself provides, I'll try to pull out some of the conclusions I found the most interesting here. 

The general goal of this paper is to better understand the contours of when self-supervised representation learning is valuable for vision (and specifically when it can compete with supervised learning), and when it doesn't. In general, the results are all using ResNet backbones, with SimCLR SSL, on image classification datasets. Some bullet-point takeaways: 

- The SSL models being tested here seem to roughly saturate at unsupervised dataset sizes of around 500K; the comparative jump from dataset sizes of 500K to 1M is fairly small.
- Once you have a supervised dataset of around 50K or more, the benefit of SSL pretraining starts to diminish, and it converges to being more similar to just supervised learning on that numbrer of labeled images. On the flip side, it's only possible to get close to "good" fully supervised performance by using 100K images or more on top of a SSL baseline.
- Even within image classification datasets, it's much better to do SSL representation on the same dataset as the one you'll use for downstream training; trying to transfer representations to different datasets leads to meaningfully worse results. Interestingly, this is even true when you add out-of-domain (i.e. other-dataset) data to an existing in-domain dataset: a dataset of 250K in-dataset images does better than a 500K dataset of images from mixed datasets, and does notably better than a 1M dataset of mixed images. In this case, adding more out-of-domain images seems to have just degraded performance
- SSL seems to perform more closely to SL on a course label set; when the label set gets more granular, the task gets harder overall, but, more specifically, the gap between SSL and SL grows
- When the authors tried different forms of dataset corruption, SSL was much more robust to adding salt-and-pepper noise than it was to removing high-frequency information in the form of reducing the images to a lower resolution.

Rakelly et al. propose a method to do off-policy meta reinforcement learning (rl). The method achieves a 20-100x improvement on sample efficiency compared to on-policy meta rl like MAML+TRPO. 

The key difficulty for offline meta rl arises from the meta-learning assumption, that meta-training and meta-test time match. However during test time the policy has to explore and sees as such on-policy data which is in contrast to the off-policy data that should be used at meta-training. The key contribution of PEARL is an algorithm that allows for online task inference in a latent variable at train and test time, which is used to train a Soft Actor Critic, a very sample efficient off-policy algorithm, with additional dependence of the latent variable.

The implementation of Rakelly et al. proposes to capture knowledge about the current task in a latent stochastic variable Z. A inference network $q_{\Phi}(z \vert c)$ is used to predict the posterior over latents given context c of the current task in from of transition tuples $(s,a,r,s')$ and trained with an information bottleneck. Note that the task inference is done on samples according to a sampling strategy sampling more recent transitions. The latent z is used as an additional input to policy $\pi(a \vert s, z)$ and Q-function $Q(a,s,z)$ of a soft actor critic algorithm which is trained with offline data of the full replay buffer. 

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


So the challenge of differing conditions at test and train times is resolved by sampling the content for the latent context variable at train time only from very recent transitions (which is almost on-policy) and at test time by construction on-policy. Sampling $z \sim q(z \vert c)$ at test time allows for posterior sampling of the latent variable, yielding efficient exploration. 

The experiments are performed across 6 Mujoco tasks with ProMP, MAML+TRPO and $RL^2$ with PPO as baselines. They show:
- PEARL is 20-100x more sample-efficient 
- the posterior sampling of the latent context variable enables deep exploration that is crucial for sparse reward settings
- the inference network could be also a RNN, however it is crucial to train it with uncorrelated transitions instead of trajectories that have high correlated transitions
- using a deterministic latent variable, i.e. reducing $q_{\Phi}(z \vert c)$ to a  point estimate, leaves the algorithm unable to solve sparse reward navigation tasks which is attributed to the lack of temporally extended exploration. 

The paper introduces an algorithm that allows to combine meta learning with an off-policy algorithm that dramatically increases the sample-efficiency compared to on-policy meta learning approaches. This increases the chance of seeing meta rl in any sort of real world applications. 

We want to find two matrices $W$ and $H$ such that $V = WH$. Often a goal is to determine underlying patterns in the relationships between the concepts represented by each row and column. $W$ is some $m$ by $n$ matrix and we want the inner dimension of the factorization to be $r$. So 

$$\underbrace{V}_{m \times n} = \underbrace{W}_{m \times r} \underbrace{H}_{r \times n}$$

Let's consider an example matrix where of three customers (as rows) are associated with three movies (the columns) by a rating value.

$$
V = \left[\begin{array}{c c c}
5 & 4 & 1  \\\\
4 & 5 & 1 \\\\
2 & 1 & 5
\end{array}\right]
$$


We can decompose this into two matrices with $r = 1$. First lets do this without any non-negative constraint using an SVD reshaping matrices based on removing eigenvalues:


$$
W = \left[\begin{array}{c c c}
-0.656 \\\
 -0.652 \\\
 -0.379
\end{array}\right],
H = \left[\begin{array}{c c c}
-6.48 & -6.26 & -3.20\\\\
\end{array}\right]
$$

We can also decompose this into two matrices with $r = 1$ subject to the constraint that $w_{ij} \ge 0$ and  $h_{ij} \ge 0$. (Note: this is only possible when $v_{ij} \ge 0$):

$$
W = \left[\begin{array}{c c c}
0.388 \\\\
0.386 \\\\
0.224
\end{array}\right],
H = \left[\begin{array}{c c c}
11.22 & 10.57 & 5.41  \\\\
\end{array}\right]
$$

Both of these $r=1$ factorizations reconstruct matrix $V$ with the same error. 

$$
V \approx WH = \left[\begin{array}{c c c}
4.36 & 4.11 & 2.10 \\\
4.33 & 4.08 & 2.09 \\\
2.52 & 2.37 & 1.21 \\\
\end{array}\right]
$$


If they both yield the same reconstruction error then why is a non-negativity constraint useful? We can see above that it is easy to observe patterns in both factorizations such as similar customers and similar movies. `TODO: motivate why NMF is better`



#### Paper Contribution 

This paper discusses two approaches for iteratively creating a non-negative $W$ and $H$ based on random initial matrices. The paper discusses a multiplicative update rule where the elements of $W$ and $H$ are iteratively transformed by scaling each value such that error is not increased. 

The multiplicative approach is discussed in contrast to an additive gradient decent based approach where small corrections are iteratively applied. The multiplicative approach can be reduced to this by setting the learning rate ($\eta$) to a ratio that represents the magnitude of the element in $H$ to the scaling factor of $W$ on $H$.



### Still a draft

[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.

Kumar et al. propose an algorithm to learn in batch reinforcement learning (RL), a setting where an agent learns purely form a fixed batch of data, $B$, without any interactions with the environments. The data in the batch is collected according to a batch policy $\pi_b$. Whereas most previous methods (like BCQ) constrain the learned policy to stay close to the behavior policy, Kumar et al. propose bootstrapping error accumulation reduction (BEAR), which constrains the newly learned policy to place some probability mass on every non negligible action. 
The difference is illustrated in the picture from the BEAR blog post:
https://i.imgur.com/zUw7XNt.png
The behavior policy is in both images the dotted red line, the left image shows the policy matching where the algorithm is constrained to the purple choices, while the right image shows the support matching. 

**Theoretical Contribution:**
The paper analysis formally how the use of out-of-distribution actions to compute the target in the Bellman equation influences the back-propagated error.

Firstly a distribution constrained backup operator is defined as
$T^{\Pi}Q(s,a) = \mathbb{E}[R(s,a) + \gamma \max_{\pi \in \Pi} \mathbb{E}_{P(s' \vert s,a)} V(s')]$ and $V(s) = \max_{\pi \in \Pi} \mathbb{E}_{\pi}[Q(s,a)]$
which considers only policies $\pi \in \Pi$. 

It is possible that the optimal policy $\pi^*$ is not contained in the policy set $\Pi$, thus there is a suboptimallity constant $\alpha (\Pi) = \max_{s,a} \vert \mathcal{T}^{\Pi}Q^{*}(s,a) - \mathcal{T}Q^{*}(s,a) ]\vert $ which captures how far $\pi^{*}$ is from $\Pi$. 

Letting $P^{\pi_i}$ be the transition-matrix when following policy $\pi_i$, $\rho_0$ the state marginal distribution of the training data in the batch and $\pi_1, \dots, \pi_k \in \Pi $. The error analysis relies upon a concentrability assumption $\rho_0 P^{\pi_1} \dots P^{\pi_k} \leq c(k)\mu(s)$, with $\mu(s)$ the state marginal. Note that $c(k)$ might be infinite if the support of $\Pi$ is not contained in the state marginal of the batch. Using the coefficients $c(k)$ a concentrability coefficient is defined as:
$C(\Pi) = (1-\gamma)^2\sum_{k=1}^{\infty}k \gamma^{k-1}c(k).$
The concentrability takes values between 1 und $\infty$, where 1 corresponds to the case that the batch data were collected by $\pi$ and $\Pi = \{\pi\}$ and $\infty$ to cases where $\Pi$ has support outside of $\pi$. 

Combining this Kumar et a. get a bound of the Bellman error for distribution constrained value iteration with the constrained Bellman operator $T^{\Pi}$:

$\lim_{k \rightarrow \infty} \mathbb{E}_{\rho_0}[\vert V^{\pi_k}(s)- V^{*}(s)]  \leq  \frac{\gamma}{(1-\gamma^2)} [C(\Pi) \mathbb{E}_{\mu}[\max_{\pi \in \Pi}\mathbb{E}_{\pi}[\delta(s,a)] + \frac{1-\gamma}{\gamma}\alpha(\Pi)  ] ]$, where $\delta(s,a)$ is the Bellman error. 

This presents the inherent batch RL trade-off between keeping policies close to the behavior policy of the batch (captured by $C(\Pi)$ and keeping $\Pi$ sufficiently large (captured by $\alpha(\Pi)$). 

It is finally proposed to use support sets to construct $\Pi$, that is $\Pi_{\epsilon} = \{\pi \vert \pi(a \vert s)=0 \text{ whenever } \beta(a \vert s) < \epsilon \}$. This amounts to the set of all policies that place probability on all non-negligible actions of the behavior policy. For this particular choice of $\Pi = \Pi_{\epsilon}$ the concentrability coefficient can be bounded. 

**Algorithm**:
The algorithm has an actor critic style, where the Q-value to update the policy is taken to be the minimum over the ensemble. The support constraint to place at least some probability mass on every non negligible action from the batch is enforced via sampled MMD. The proposed algorithm is a member of the policy regularized algorithms as the policy is updated to optimize:
$\pi_{\Phi} = \max_{\pi} \mathbb{E}_{s \sim B} \mathbb{E}_{a \sim \pi(\cdot \vert s)} [min_{j = 1 \dots, k} Q_j(s,a)] s.t. \mathbb{E}_{s \sim B}[MMD(D(s), \pi(\cdot \vert s))] \leq \epsilon$

The Bellman target to update the Q-functions is computed as the convex combination of minimum and maximum of the ensemble. 

**Experiments**
The experiments use the Mujoco environments Halfcheetah, Walker, Hopper and Ant. Three scenarios of batch collection, always consisting of 1Mio. samples,  are considered:
- completely random behavior policy
- partially trained behavior policy
- optimal policy as behavior policy

The experiments confirm that BEAR outperforms other off-policy methods like BCQ or KL-control. The ablations show further that the choice of MMD is crucial as it is sometimes on par and sometimes substantially better than choosing KL-divergence.