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Learn more about our use of cookies and your information. By interacting with this site, you agree to our use of cookies. We use cookies why? You can change cookie preferences ; by clicking accept, you accept all cookies. Contact us. Netflix Sign In. Netflix Netflix. Watch all you want. More Details. Watch offline. Available to download. This movie is Intimate, Inspiring. Arthur B. This is sufficiently fast for all values tested that it is not worth memoizing using the relatively slow memoise.

What does the range of possible bets in each state look like? Does it look bitonic and like an inverted V one could do binary search on? We can sample some random states, compute all the possible bets, and plot them by index to see if the payoffs form curves:.

For the most part they do generally follow a quadratic or inverted-V like shape, with the exception of several which wiggle up and down while following the overall curve numeric or approximation error? We can generate many iterates, interpolate, and graph:.

A value function or the policy function computed by decision trees is bulky and hard to carry around, weighing perhaps gigabytes in size. A random forest works fine, getting 0. It can also be approximated with NN. Some experimenting with fitting polynomial regressions did not give any decent fits for less than quintic, but probably some combination of logs and polynomials could compactly encode the value function.

The implementations here could be wrong, or decision trees not actually optimal like claimed. Simulating 56 games, we can see how while the optimal strategy looks fairly risky, it usually wins in the end:. We can also examine how the strategy does for all the other possible horizons and compare with the Kelly. How do you make decisions as coin flips happen and provide information on p?

Then our expected values in f simply change from 0. Going back to the R implementation, we add in the two additional parameters, the beta estimation, and estimate the hardwired probabilities:. Ideally we would use the prior of the participants in the experiment, to compare how well they do compared to the Bayesian decision tree, and estimate how much mistaken beliefs cost them. This prior leads to somewhat more aggressive betting in short games as the EV is higher, but overall performs much like the decision tree.

So the price of ignorance regret in this scenario is surprisingly small. Why is the Bayesian decision tree able to perform so close to the known-probability decision tree? What is sp as a distribution? Discounting in general could be seen as reflecting a probability of stopping at each stage.

The problem is more that this expands the states out enough to again make it unevaluable, so it helps to put an upper bound. So the problem becomes, playing the game knowing that the game might stop with a certain sp eg.

This can be extended to scenarios where we do learn something about the stopping time, similarly to the coin-flip probability, by doing Bayesian updates on sp. Or if sp changes over rounds. In this case, the maximum observed wealth is the sufficient statistic. It will probably perform reasonably well since it performs optimally at least one possible setting, and serve as a baseline:.

Since the Bayesian decision tree is too hard to compute in full, we need a different approach. To create a deep RL agent for the generalized Kelly coin-flip game, I will use the keras-rl library of agents, which is based on the Keras deep learning framework backed by TensorFlow ; deep learning is often sensitive to choice of hyperparameters, so for hyperparameter optimization, I use the hyperas wrapper around hyperopt paper.

I tried prototyping DQN for the simple Kelly coin-flip game using the keras-rl DQN example Cartpole , but after hours, it still had not received any rewards—it repeatedly bet more than its current wealth, equivalent to its current wealth, and busting out, never managing to go rounds to receive any reward and begin learning.

This sort of issue with sparse rewards is a common problem with DQN and deep RL in general, as the most common forms of exploration will flail around at random rather than make deep targeted exploration of particular strategies or try to seek out new regions of state-space which might contain rewards. For exploration, some random noise is added to the final action; not epsilon-random noise like in DQN but a sort of random walk, to make the continuous action consistently lower or higher and avoid canceling out.

This trick might also be usable in DQN if we discretize 0—1 into, say, percentiles. This resulted in meaningful progress but was sensitive to hyperparameters—for example, the DDPG appeared to train worse with a very large experience replay buffer than with the Lillicrap et al setting of 10, Hyperparameter-wise, Lillicrap et al used, for the low-dimensional non-ALE screen RL problems similar to the general Kelly coin-flip, the settings:.

So one ought to be able to do a binary search over actions or something even better. I also tried its CEM implementation, but after trying a variety of settings, CEM never performed better than an average reward of 0. This follows the original Mnih et al implementation. The NN is constructed with the screen observation state as the input and a final top layer outputting n reals corresponding to the q-value of the n actions, in that case the fixed number of buttons on the Atari controller.

In the Atari setting, the DQN agent interacts exclusively through the controller which has only a few buttons and a directional pad; and the environments are complex enough that it would be hard to define in advance what are the valid actions for each possible screen, which is something that must be learned, so this fixed-action trick is both necessary and efficient.

Thus, it is also used in most implementations or extensions of DQN. In the Kelly coinflip, on the other hand, the discretizing is problematic since the majority of actions will be invalid, using an invalid action is damaging since it leads to overbetting, and we can easily define the range of valid actions in each state 0- w , so the trick is much less compelling.

Submit Anonymous Feedback. Environment getArgs import Data. Function iterate import Data. Function fix import Data. Memoize memoFix2 import Data. Vector Vector import qualified Data. Vector as V generate,! Direct recursion. Memoised recursion. Memoised recursion using 'memoize' library; slower.

Mean of squared residuals: 1. Environment import Data.