mnoukhov

Normal RL agents in multi-agent scenarios treat their opponents as a static part of the environment, not taking into account the fact that other agents are learning as well. This paper proposes LOLA, a learning rule that should take the agency and learning of opponents into account by optimizing "return under one step look-ahead of opponent learning"

So instead of optimizing under the current parameters of agent 1 and 2 
$$V^1(\theta_i^1, \theta_i^2)$$

LOLA proposes to optimize taking into account one step of opponent (agent 2) learning
$$V^1(\theta_i^1, \theta_i^2 + \Delta \theta^2_i)$$

where we assume the opponent's naive learning update $\Delta \theta^2_i = \nabla_{\theta^2} V^2(\theta^1, \theta^2) \cdot \eta$ and we add a second-order correction term

on top of this, the authors propose
- a learning rule with policy gradients in the case that the agent does not have access to exact gradients
- a way to estimate the parameters of the opponent, $\theta^2$, from its trajectories using maximum likelihood in the case you can't access them directly
$$\hat \theta^2 = \text{argmax}_{\theta^2} \sum_t \log \pi_{\theta^2}(u_t^2|s_t)$$

LOLA is tested on iterated prisoner's dilemma and converges to a tit-for-tat strategy more frequently than the naive RL learning algorithm, and outperforms it. LOLA is tested on iterated matching pennies (similar to prisoner's dilemma) and stably converges to the Nash equilibrium whereas the naive learners do not. In testing on coin game (a higher dimensional version of prisoner's dilemma) they find that naive learners generally choose the defect option whereas LOLA agents have a mostly-cooperative strategy.

As well, the authors show that LOLA is a dominant learning rule in IPD, where both agents always do better if either is using LOLA (and even better if both are using LOLA).

Finally, the authors also propose second order LOLA, which instead of assuming the opponent is a naive learner, assumes the opponent uses a LOLA learning rule. They show that second order LOLA does not lead to improved performance so there is no need to have a $n$th order LOLA arms race.

This paper introduces a deep universal word embedding based on using a bidirectional LM (in this case, biLSTM). First words are embedded with a CNN-based, character-level, context-free, token embedding into $x_k^{LM}$ and then each sentence is parsed using a biLSTM, maximizing the log-likelihood of a word given it's forward and backward context (much like a normal language model). 

The innovation is in taking the output of each layer of the LSTM ($h_{k,j}^{LM}$ being the output at layer $j$) 
$$
\begin{align}
R_k &= \{x_k^{LM}, \overrightarrow{h}_{k,j}^{LM}, \overleftarrow{h}_{k,j}^{LM} | j = 1 \ldots L \} \\
&= \{h_{k,j}^{LM} | j = 0 \ldots L \}
\end{align}
$$
and allowing the user to learn a their own task-specific weighted sum of these hidden states as the embedding:
$$
ELMo_k^{task} = \gamma^{task} \sum_{j=0}^L s_j^{task} h_{k,j}^{LM}
$$

The authors show that this weighted sum is better than taking only the top LSTM output (as in their previous work or in CoVe) because it allows capturing syntactic information in the lower layer of the LSTM and semantic information in the higher level. Table below shows that the second layer is more useful for the semantic task of word sense disambiguation, and the first layer is more useful for the syntactic task of POS tagging.

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

On other benchmarks, they show it is also better than taking the average of the layers (which could be done by setting $\gamma = 1$)
https://i.imgur.com/f78gmKu.png

To add the embeddings to your supervised model,  ELMo is concatenated with your context-free embeddings, $\[ x_k; ELMo_k^{task} \]$. It can also be concatenated with the output of your RNN model $\[ h_k; ELMo_k^{task} \]$ which can show improvements on the same benchmarks

https://i.imgur.com/eBqLe8G.png

Finally, they show that adding ELMo to a competitive but simple baseline gets SOTA (at the time) on very many NLP benchmarks

https://i.imgur.com/PFUlgh3.png

It's all open-source and there's a tutorial [here](https://github.com/allenai/allennlp/blob/master/tutorials/how_to/elmo.md)