# Module Obandit

module Obandit: sig .. end
Ocaml Multi-Armed Bandits

Obandit is an Ocaml module for multi-armed bandits. It supports the EXP, UCB and Epsilon-greedy family of algorithms.

Version v0.3-2-g16ba7ac - homepage

# Bandit Modules

This library implements multi-armed bandits as modules. A bandit module is obtained by calling a functor with a bandit module parameter. The parameter usually gives the number $K$ of arms and the hyperparameters of the bandit.

module type Bandit = sig .. end
A bandit algorithm.



Bandit modules are instanciated using functors. Depending on the algorithm type used, the module parameter varies.

For instance, the UCB1 bandit module for 3 arms is obtained with:

module UCB1 =   let module P = struct   let k=3   end   in MakeUCB1(P)

The following algorithms are available:

## The UCB family of algorithms.

We use the viewpoint of the survey [1].

### MakeAlphaPhiUCB: $(\alpha,\psi)$-UCB

At time $t$, the $(\alpha,\psi)$-UCB algorithm[1] is taking action:

$$\argmax_{i=1,\cdots,K} \quad \left[ \widehat{\mu_i}+(\psi^{*})^{-1}\left(\frac{\alpha\ln t}{T_i} \right) \right]$$

where $\alpha > 0$ is a hyperparameter, $\widehat{\mu_i}$ is the estimate of the average reward of arm $i$, $T_i$ is the number of times arm $i$ was visited so far and $\psi^*$ denotes the Legendre-Fenchel transform of a convex function $\psi$.

The pseudo-regret $\bar{R_n}$ has the following bound at round $n$: $$\bar{R_n} \leq \sum_{i:\Delta_i > 0} \left( \frac{\alpha \Delta_i}{\psi^* (\Delta_i / 2 )} \ln n + \frac{\alpha }{\alpha-2 } \right)$$

where $\Delta_i = \mu^* - \mu_i$ is the suboptimality parameter of arm $i$.

type banditEstimates = {
    t : int; (* The index of the step *)    a : int; (* The last action taken. *)    nVisits : int list; (* The visit counts by arm. *)    u : float list; (* The cumulative arm reward observations. *)
}
The inner state of a bandit that maintains estimates of arm means.
module type AlphaPhiUCBParam = sig .. end
Use to instanciate a Bandit from MakeAlphaPhiUCB by giving $\alpha$ and $\phi$.
module MakeAlphaPhiUCB: functor (P : AlphaPhiUCBParam) -> Bandit  with type bandit = banditEstimates
The $(\alpha,\psi)$-UCB Bandit for stochastic regret minimization described in [1].

### MakeAlphaUCB: $\alpha$-UCB

The $\alpha$-UCB algorithm[5] uses $\psi(\lambda) = \lambda^2 / 8$. It chooses the action:

$$\argmax_{i=1,\cdots,K} \left[ \widehat{\mu_i} + \sqrt{ \frac{\alpha \ln t}{2 T_i} } \right]$$

This gives a pseudo-regret bound of:

$$\bar{ R_n} \leq \sum_{i:\Delta_i > 0} \left( \frac{2 \alpha} { \Delta_i } \ln n + \frac{\alpha}{\alpha - 2} \right)$$

module type AlphaUCBParam = sig .. end
Use to instanciate a Bandit from MakeAlphaUCB by giving $\alpha$.
module MakeAlphaUCB: functor (P : AlphaUCBParam) -> Bandit  with type bandit = banditEstimates
The $\alpha$-UCB Bandit for stochastic regret minimization described in [1] .

### MakeUCB1: UCB1

The UCB1 algorithm[5] uses $\alpha = 4$. It chooses the action:

$$\argmax_{i=1,\cdots,K} \left[ \widehat{\mu_i} + \sqrt{ \frac{2 \ln t}{T_i} } \right]$$

At round $n$, this gives a pseudo-regret bound of:

$$\bar{R_n} \leq \sum_{i:\Delta_i > 0} \left( \frac{8}{\Delta_i} \ln n + 2 \right)$$

module type KBanditParam = sig .. end
Use to instanciate a Bandit from MakeUCB1.
module MakeUCB1: functor (P : KBanditParam) -> Bandit  with type bandit = banditEstimates
The UCB1 Bandit for stochastic regret minimization .

## The Epsilon-Greedy family of algorithms.

### MakeParametrizableEpsilonGreedy: $\epsilon$-Greedy with a parametrizable rate.

At round $t$, the $\epsilon_t$-Greedy algorithm[5] takes action $\argmax_{i=1,\cdots,K} \widehat{\mu_i}$ with probability $1-\epsilon_t$ and an uniformly random action with probability $\epsilon_t$.

module type RateBanditParam = sig .. end
Use to instanciate algorithms that need a parametrizable rate.
module MakeParametrizableEpsilonGreedy: functor (P : RateBanditParam) -> Bandit  with type bandit = banditEstimates
The $\epsilon$-Greedy Bandit with a parametrizable exploration rate.

### MakeDecayingEpsilonGreedy: $\epsilon_n$-Greedy with the decaying rate from [5].

This uses the exploration rate decay: $$\epsilon_t = \min \left\{ 1, \frac{cK}{d^2 t} \right\}$$ where $d > 0$ should be taken as a tight lower bound on $\max_{i=1,\cdots,K} \Delta_i$ and $c > 0$ is a hyperparameter.

module type DecayingEpsilonGreedyParam = sig .. end
Use to instanciate a Bandit from MakeDecayingEpsilonGreedy .
module MakeDecayingEpsilonGreedy: functor (P : DecayingEpsilonGreedyParam) -> Bandit  with type bandit = banditEstimates
The Epsilon-Greedy Bandit with the decaying exploration rate from [5].

### MakeEpsilonGreedy: $\epsilon_n$-Greedy with a fixed exploration rate.

This uses a fixed exploration rate $\epsilon$.

module type EpsilonGreedyParam = sig .. end
Use to instanciate a Bandit from MakeEpsilonGreedy .
module MakeEpsilonGreedy: functor (P : EpsilonGreedyParam) -> Bandit  with type bandit = banditEstimates
The Epsilon-Greedy Bandit with a fixed exploration rate.

## The Exp3 family of algorithms.

### MakeExp3: EXP3 with a parametrizable rate.

At round $t$, the EXP3 algorithm[1] draws an arm from a probability distribution $p$ and updates this distribution with a softmax operator:

$p_{i,t+1} = \frac{\exp ( - \eta_t \widetilde{L_{i,t}})}{\sum{k=1}^{K}\text{exp}(-\eta_t \widetilde{L_{k,t}})}$

where $\widetilde{L_{i,t}}$ is the cumulative probability-normalized loss at time $t$ of arm $i$, $\eta_t$ is the rate at time $t$.

type banditPolicy = {
    t : int; (* The index of the step $t$. *)    a : int; (* The last action taken. *)    w : float list; (* The weights of the arm that define the policy. *)
}
The internal state of an Exp3 bandit
module MakeExp3: functor (P : RateBanditParam) -> Bandit  with type bandit = banditPolicy
The Exp3 Bandit for adversarial regret minimization with a parametrizable learning rate.

### MakeDecayingExp3: EXP3 with the decaying rate from [1].

This variant uses the learning rate decay:

$$\eta_t = \sqrt{\frac{ln K}{tK}}$$

and enjoys the pseudo-regret bound: $$\bar{R_n} \leq 2 \sqrt{nK \ln K}$$

module MakeDecayingExp3: functor (P : KBanditParam) -> Bandit  with type bandit = banditPolicy
The Exp3 Bandit for adversarial regret minimization with a decaying learning rate as per [1].

### MakeFixedExp3: EXP3 with a fixed rate.

This uses a fixed rate $\eta$.

module type FixedExp3Param = sig .. end
Use to instanciate a Bandit from MakeFixedExp3 .
module MakeFixedExp3: functor (P : FixedExp3Param) -> Bandit  with type bandit = banditPolicy
The Exp3 Bandit for adversarial regret minimization with a decaying learning rate as per [1].

### MakeHorizonExp3: EXP3 with a known horizon [1].

This variant optimizes for a known horizon $n$ and uses the learning rate:

$$\eta = \sqrt{\frac{2 ln K}{nK}}$$

It has the pseudo-regret bound:

$$\bar{R_n} \leq \sqrt{2 nK \ln K}$$

module type HorizonExp3Param = sig .. end
Use to instanciate a Bandit from MakeHorizonExp3 .
module MakeHorizonExp3: functor (P : HorizonExp3Param) -> Bandit  with type bandit = banditPolicy
The Exp3 Bandit for adversarial regret minimization with a horizon-based learning rate as per [1].

## More Functors: The doubling trick.

Reward normalization in online stochastic and/or adversarial learning is a hard problem. While this is well studied in online learning [2][3][4], there is no well studied procedure for bandits yet. The WrapRange Functors applies the heuristic solution known as the doubling trick.

The WrapRange functor wraps a bandit algorithm with the doubling trick. This heuristic allows to use a bandit algorithm without knowing the reward ranges. All rewards are linearly rescaled to a range (initially given by a RangeParam). When a value is observed above the range, the bandit algorithm is restarted and the range interval is doubled in that direction.

A convenience WrapRange01 is provided for rewards that are initially thought to lie in $\left[0,1\right]$.

module type RangeParam = sig .. end
A Reward range.
type rangedAction = 
 | Reset of int | Action of int
A ranged action: Action a in normal course of action, Reset a in case * the bandit was just restarted.
type 'b rangedBandit = {
    bandit : 'b; (* The original type of the bandit. *)    u : float; (* The upper reward bound. *)    l : float; (* The lower reward bound. *)
}
The type of a bandit with a range.
module type RangedBandit = sig .. end
The type of a bandit with reward scaling.
module WrapRange: functor (R : RangeParam) -> functor (B : Bandit) -> RangedBandit  with type bandit = B.bandit
The WrapRange functor wraps a bandit algorithm with the doubling trick.
module WrapRange01: functor (B : Bandit) -> RangedBandit  with type bandit = B.bandit
The WrapRange01 functor is a convenience aliasing of WrapRange with an initial "standard" range of $\left[ 0,1 \right]$.

# Examples

see test/test.ml for an example of bandit use.

# References

[1] Regret Analysis of Stochastic and Nonstochastic Multi-armed Bandit Problems, Sebastien Bubeck and Nicolo Cesa-Bianchi.

[2] Adaptive Subgradient methods for Online Learning and Stochastic Optimization, John Duchi , Elad Hazan and Yoram Singer.

[3] Normalized Online Learning, Stephane Ross, Paul Mineiro, John Langford

[4] Scale-Free Online Learning, Francesco Orabona, Dávid Pál

[5] Finite-time Analysis of the Multiarmed Bandit Problem, Peter Auer, Nicolo Cesa-Bianchi, Paul Fischer