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. 

In the last three years, Transformers, or models based entirely on attention for aggregating information from across multiple places in a sequence, have taken over the world of NLP. In this paper, the authors propose using a Transformer to learn a molecular representation, and then building a model to predict drug/target interaction on top of that learned representation. A drug/target interaction model takes in two inputs - a protein involved in a disease pathway, and a (typically small) molecule being considered as a candidate drug - and predicts the binding affinity they'll have for one another. If binding takes place between the two, that protein will be prevented from playing its role in the chain of disease progression, and the drug will be effective.

The mechanics of the proposed Molecular Transformer DTI (or MT-DTI) model work as follows: 
https://i.imgur.com/ehfjMK3.png

- Molecules are represented as SMILES strings, a character-based representation of atoms and their structural connections. Proteins are represented as sequences of amino acids.
- A Transformer network is constructed over the characters in the SMILES string, where, at each level, the representation of each character comes from taking an attention-weighted average of the representations at other positions in the character string at that layer. At the final layer, the aggregated molecular representation is taken from a special "REP" token.
- The molecular transformer is pre-trained in BERT-like way: by taking a large corpus (97M molecules) of unsupervised molecular representations, masking or randomizing tokens within the strings, and training the model to predict the true correct token at each point. The hope is that this task will force the representations to encode information relevant to the global structures and chemical constraints of the molecule in question
- This pre-trained Transformer is then plugged into the DTI model, alongside a protein representation model in the form of multiple layers convolutions acting on embedded representations of amino acids. The authors noted that they considered a similar pretrained transformer architecture for the protein representation side of the model, but that they chose not to because (1) the calculations involved in attention are N^2 in the length of the sequence, and proteins are meaningfully longer than the small molecules being studied, and (2) there's no comparably large dataset of protein sequences that could be effectively used for pretraining
- The protein and molecule representations are combined with multiple dense layers, and then produce a final affinity score. Although the molecular representation model starts with a set of pretrained weights, it also fine tunes on top of them.

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

When evaluated empirically on two DTI datasets, this attention based model outperforms the prior SOTA, using a convolutional architecture, by a small but consistent amount across all metrics. Interestingly, their model is reasonably competitive even if they don't fine-tune the molecular representation for the DTI task, but only pretraining and fine-tuning together get the MT-DTI model over the threshold of prior work.

Format-preserving encryption is a deterministic encryption scheme that encrypts plaintext of some specified format into ciphertext of the same format. This has a lot of practical use cases such as storing SSN or credit card information, without having to change the underlying schematics of the database or application that stores the data. The protected data is in-differentiable from unprotected data, and still enables some analytics over it, such as with masking (ie, displaying last four digits of a format). For other analytics, and depending on the use cases, differential privacy or FHE should be considered. 

This paper primarily describes construction of a short-space FPE scheme. Starting off with earlier constructions based off of work done by Terence Spies and M. Bellare. The authors have developed an FPE scheme that is tweakable, to enhance security and this value is imperative for ciphertext alphabets of small range. A tweak value is like an initialization vector, chosen by the user. This ensures that with two different tweak values, the same plaintext encrypted twice will encipher to different ciphertexts. 

Of particular interest for practical applications is 'cycle-walking' and 'rank-then-encipher'. These are "meta-techniques" for building an FPE scheme that essentially ensures you get a ciphertext in the desired ciphertext space. 

misc (pg 9):

cycle-walking: 

If you wanted to encipher on some message space $X$, you would make an FPE scheme:
$$E: K \times \chi \rightarrow \chi$$

If you know how to encipher onto a superset of $X'$, with an FPE scheme:
$$E' : K \times \chi' \rightarrow \chi'\\$$
All you would need to do is keep enciphering the message $X \in \chi'$ until you get $X \in \chi$

Rank-then-Encipher:

Encipher a point $X \in \chi$
map it to a corresponding point $X' \in \chi'$, encipher that point in $\chi'$ to get a ciphertext $Y' \in \chi'$, then map $Y'$ to its corresponding point $Y$ in $\chi$.

In this tutorial paper, Carl E. Rasmussen gives an introduction to Gaussian Process Regression focusing on the definition, the hyperparameter learning and future research directions.

A Gaussian Process is completely defined by its mean function $m(\pmb{x})$ and its covariance function (kernel) $k(\pmb{x},\pmb{x}')$. The mean function $m(\pmb{x})$ corresponds to the mean vector $\pmb{\mu}$ of a Gaussian distribution whereas the covariance function $k(\pmb{x}, \pmb{x}')$ corresponds to the covariance matrix $\pmb{\Sigma}$. Thus, a Gaussian Process $f \sim \mathcal{GP}\left(m(\pmb{x}), k(\pmb{x}, \pmb{x}')\right)$ is a generalization of a Gaussian distribution over vectors to a distribution over functions. A random function vector $\pmb{\mathrm{f}}$ can be generated by a Gaussian Process through the following procedure:

1. Compute the components $\mu_i$ of the mean vector $\pmb{\mu}$ for each input $\pmb{x}_i$ using the mean function $m(\pmb{x})$
2. Compute the components $\Sigma_{ij}$ of the covariance matrix $\pmb{\Sigma}$ using the covariance function $k(\pmb{x}, \pmb{x}')$
3. A function vector $\pmb{\mathrm{f}} = [f(\pmb{x}_1), \dots, f(\pmb{x}_n)]^T$ can be drawn from the Gaussian distribution $\pmb{\mathrm{f}} \sim \mathcal{N}\left(\pmb{\mu}, \pmb{\Sigma} \right)$

Applying this procedure to regression, means that the resulting function vector $\pmb{\mathrm{f}}$ shall be drawn in a way that a function vector $\pmb{\mathrm{f}}$ is rejected if it does not comply with the training data $\mathcal{D}$. This is achieved by conditioning the distribution on the training data $\mathcal{D}$ yielding the posterior Gaussian Process $f \rvert \mathcal{D} \sim \mathcal{GP}(m_D(\pmb{x}), k_D(\pmb{x},\pmb{x}'))$ for noise-free observations with the posterior mean function $m_D(\pmb{x}) = m(\pmb{x}) + \pmb{\Sigma}(\pmb{X},\pmb{x})^T \pmb{\Sigma}^{-1}(\pmb{\mathrm{f}} - \pmb{\mathrm{m}})$  and the posterior covariance function $k_D(\pmb{x},\pmb{x}')=k(\pmb{x},\pmb{x}') - \pmb{\Sigma}(\pmb{X}, \pmb{x}')$ with $\pmb{\Sigma}(\pmb{X},\pmb{x})$ being a vector of covariances between every training case of $\pmb{X}$ and $\pmb{x}$.

Noisy observations $y(\pmb{x}) = f(\pmb{x}) + \epsilon$ with $\epsilon \sim \mathcal{N}(0,\sigma_n^2)$ can be taken into account with a second Gaussian Process with mean $m$ and covariance function $k$ resulting in $f \sim \mathcal{GP}(m,k)$ and $y \sim \mathcal{GP}(m, k + \sigma_n^2\delta_{ii'})$. The figure illustrates the cases of noisy observations (variance at training points) and of noise-free observationshttps://i.imgur.com/BWvsB7T.png (no variance at training points).

In the Machine Learning perspective, the mean and the covariance function are parametrised by hyperparameters and provide thus a way to include prior knowledge e.g. knowing that the mean function is a second order polynomial. To find the optimal hyperparameters $\pmb{\theta}$, 

1. determine the log marginal likelihood $L= \mathrm{log}(p(\pmb{y} \rvert \pmb{x}, \pmb{\theta}))$,
2. take the first partial derivatives of $L$ w.r.t. the hyperparameters, and
3. apply an optimization algorithm.

It should be noted that a regularization term is not necessary for the log marginal likelihood $L$ because it already contains a complexity penalty term. Also, the tradeoff between data-fit and penalty is performed automatically.

Gaussian Processes provide a very flexible way for finding a suitable regression model. However, they require the high computational complexity $\mathcal{O}(n^3)$ due to the inversion of the covariance matrix. In addition, the generalization of Gaussian Processes to non-Gaussian likelihoods remains complicated. 

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.