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package obandit
Ocaml Multi-Armed Bandits
Version 0.2 — 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 ... endA 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 ... endUse to instanciate a Bandit from MakeAlphaPhiUCB by giving $\alpha$ and $\phi$.
module MakeAlphaPhiUCB
(P : AlphaPhiUCBParam) :
Bandit with type bandit = banditEstimatesThe $(\alpha,\psi)$-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 ... endUse to instanciate a Bandit from MakeAlphaUCB by giving $\alpha$.
module MakeAlphaUCB
(P : AlphaUCBParam) :
Bandit with type bandit = banditEstimatesThe $\alpha$-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 ... endUse to instanciate a Bandit from MakeUCB1.
module MakeUCB1 (P : KBanditParam) : Bandit with type bandit = banditEstimatesThe 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 ... endUse to instanciate algorithms that need a parametrizable rate.
module MakeParametrizableEpsilonGreedy
(P : RateBanditParam) :
Bandit with type bandit = banditEstimatesThe $\epsilon$-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 ... endUse to instanciate a Bandit from MakeDecayingEpsilonGreedy .
module MakeDecayingEpsilonGreedy
(P : DecayingEpsilonGreedyParam) :
Bandit with type bandit = banditEstimatesThe 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 ... endUse to instanciate a Bandit from MakeEpsilonGreedy .
module MakeEpsilonGreedy
(P : EpsilonGreedyParam) :
Bandit with type bandit = banditEstimatesThe 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 (P : RateBanditParam) : Bandit with type bandit = banditPolicyThe 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
(P : KBanditParam) :
Bandit with type bandit = banditPolicyThe 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 ... endUse to instanciate a Bandit from MakeFixedExp3 .
module MakeFixedExp3
(P : FixedExp3Param) :
Bandit with type bandit = banditPolicyThe 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 ... endUse to instanciate a Bandit from MakeHorizonExp3 .
module MakeHorizonExp3
(P : HorizonExp3Param) :
Bandit with type bandit = banditPolicyThe 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 ... endA Reward range.
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 ... endThe type of a bandit with reward scaling.
module WrapRange
(R : RangeParam)
(B : Bandit) :
RangedBandit with type bandit = B.banditThe 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.
module WrapRange01 (B : Bandit) : RangedBandit with type bandit = B.banditThe 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