From 6844796464202793d15ef3df67c08e9dbc1b9ee6 Mon Sep 17 00:00:00 2001 From: Alex Lew Date: Mon, 12 Dec 2022 21:02:43 -0500 Subject: [PATCH] Add README and more examples --- README.md | 148 +++++++++++++++++++++++++++++- adev.cabal | 21 ++++- examples/ParticleFilterExample.hs | 47 ++++++++++ figures/adev-diagram.png | Bin 0 -> 42738 bytes figures/example.png | Bin 0 -> 465509 bytes package.yaml | 13 ++- src/Numeric/ADEV/Diff.hs | 4 +- src/Numeric/ADEV/Distributions.hs | 16 ++++ src/Numeric/ADEV/Interp.hs | 2 +- 9 files changed, 242 insertions(+), 9 deletions(-) create mode 100644 examples/ParticleFilterExample.hs create mode 100644 figures/adev-diagram.png create mode 100644 figures/example.png create mode 100644 src/Numeric/ADEV/Distributions.hs diff --git a/README.md b/README.md index d24f830..ddc72f3 100644 --- a/README.md +++ b/README.md @@ -1,3 +1,147 @@ -# adev +# ADEV -Haskell prototype to accompany the paper "ADEV: Sound Automatic Differentiation of Expected Values of Probabilistic Programs" \ No newline at end of file +This repository contains the Haskell prototype that accompanies the paper "[ADEV: Sound Automatic Differentiation of Expected Values of Probabilistic Programs](https://popl23.sigplan.org/details/POPL-2023-popl-research-papers/5/ADEV-Sound-Automatic-Differentiation-of-Expected-Values-of-Probabilistic-Programs)". + +## Overview + +![Overview of ADEV](figures/adev-diagram.png) +ADEV is a method of automatically differentiating loss functions defined as *expected values* of probabilistic processes. ADEV users define a _probabilistic program_ $t$, which, given a parameter of type $\mathbb{R}$ (or a subtype), outputs a value of type $\widetilde{\mathbb{R}}$, +which represents probabilistic estimators of losses. We translate $t$ to a new probabilistic program $s$, +whose expected return value is the derivative of $s$’s expected return value. Running $s$ yields provably unbiased +estimates $x_i$ of the loss's derivative, which can be used in the inner loop of stochastic optimization algorithms like ADAM or stochastic gradient descent. + +ADEV goes beyond standard AD by explicitly supporting probabilistic primitives, like `flip`, for flipping a coin. If these probabilistic constructs are ignored, standard AD may produce incorrect results, as this figure from our paper illustrates: +![Optimizing an example loss function using ADEV](figures/example.png) +In this example, standard AD +fails to account for the parameter $\theta$'s effect on the *probability* of entering each branch. ADEV, by contrast, correctly accounts +for the probabilistic effects, generating similar code to what a practitioner might hand-derive. Correct +gradients are often crucial for downstream applications, e.g. optimization via stochastic gradient descent. + +ADEV compositionally supports various gradient estimation strategies from the literature, including: +- Reparameterization trick (Kingma & Welling 2014) +- Score function estimator (Ranganath et al. 2014) +- Baselines as control variates (Mnih and Gregor 2014) +- Multi-sample estimators that Storchastic supports (e.g. leave-one-out baselines) (van Krieken et al. 2021) +- Variance reduction via dependency tracking (Schulman et al. 2015) +- Special estimators for differentiable particle filtering (Ścibior et al. 2021) +- Implicit reparameterization (Figurnov et al. 2018) +- Measure-valued derivatives (Heidergott and Vázquez-Abad 2000) +- Reparameterized rejection sampling (Nasseth et al. 2017) + + +## Haskell Example + +ADEV extends forward-mode automatic differentiation to support *probabilistic programs*. Consider the following example: + +```haskell +import Numeric.ADEV.Class (ADEV(..)) +import Numeric.ADEV.Interp () +import Numeric.ADEV.Diff (diff) +import Control.Monad (replicateM) +import Control.Monad.Bayes.Sampler.Strict (sampleIO) + +-- Define a loss function l as the expected +-- value of a probabilistic process. +l theta = expect $ do + b <- flip_reinforce theta + if b then + return 0 + else + return (-theta / 2) + +-- Take its derivative. +l' = diff l + +-- Helper function for computing averages +mean xs = sum xs / (realToFrac $ length xs) + +-- Estimating the loss and its derivative +-- by averaging many samples +estimate_loss = fmap mean (replicateM 1000 (l 0.4)) +estimate_deriv = fmap mean (replicateM 1000 (l' 0.4)) + +main = do + loss <- sampleIO estimate_loss + deriv <- sampleIO estimate_deriv + print (loss, deriv) +``` + +**Defining a loss.** The function `l` is defined to be the *expected value* of a probabilistic process, using `expect`. The process in question involves flipping a coin, whose probability of heads is `theta`, and returning either `0` or `-theta / 2`, depending on the coin flip's result. + +**Differentiating.** ADEV's `diff` operator converts such a loss into a new function `l'` representing its derivative, with respect to the input parameter `theta`. + +**Running the estimators.** Operationally, neither `l` nor `l'` compute exact expectations (or derivatives of expecations): instead, they represent _unbiased estimators_ of the desired values, which can be run using `sampleIO`. +On one run, the above code printed `(-0.122, -0.10)`, which are very close to the correct values of $-0.12$ and $-0.1$. + +**Composing `expect` with other operators.** Note that ADEV also provides primitives for manipulating expected values, e.g. `exp_` for taking their exponents. For example, the code `fmap mean (replicateM 1000 (exp_ (l 0.4)))` yielded `0.881` on a sample run, close to the true value of $e^{-0.12} = 0.886$. This is the exponent of the expected value, not the expected value of the exponent, which is slightly different ($0.6 \times e^{-0.2} + 0.4 \times e^0 = 0.891$). + +**Optimization.** We can use ADEV's estimated derivatives to implement a stochastic optimization algorithm: + +```haskell +sgd loss eta x0 steps = + if steps == 0 then + return [x0] + else do + v <- diff loss x0 + let x1 = x0 - eta * v + xs <- sgd loss eta x1 (steps - 1) + return (x0:xs) +``` + +Running `sampleIO $ sgd l 0.2 0.2 100` finds the value of $\theta$ that minimizes $l$, namely $\theta = 0.5$. + +## Haskell Encoding of ADEV Programs + +In the ADEV paper, the program `l` above would have type $\mathbb{R} \to \widetilde{\mathbb{R}}$. +In Haskell, its type is `ADEV p m r => r -> m r`. Why? + +In general, expressions in the ADEV source language are represented by Haskell expressions with polymorphic type `ADEV p m r => ...`, where the `...` is a Haskell type that uses the three type variables `p`, `m`, and `r` as follows: + +* `r` represents real numbers, $\mathbb{R}$ in the ADEV paper. (The type of positive reals reals, $\mathbb{R}_{>0}$, is represented as `Log r`.) +* `m r` represents estimated real numbers, $\widetilde{\mathbb{R}}$ in the ADEV paper. +* `p m a` represents probabilistic programs returning `a`, $P\,a$ in the ADEV paper. + +Below, we show how `l`'s type relates to the types of its sub-expressions: +```haskell +-- `flip_reinforce` takes a real parameter and +-- probabilistically outputs a Boolean. +flip_reinforce :: ADEV p m r => r -> p m Bool + +-- Using `do`, we can build a larger computation that +-- uses the result of a flip to compute a real. +-- Its type reflects that it still takes a real parameter +-- as input, but now probabilistically outputs a real. +prog :: ADEV p m r => r -> p m r +prog theta = do + b <- flip_reinforce theta + if b then + return 0 + else + return (-theta/2) + +-- The `expect` operation turns a probabilistic computation +-- over reals (type P R) into an estimator of its expected +-- value (type R~). +expect :: ADEV p m r => p m r -> m r + +-- By composing expect and prog, we get l from above. +l :: ADEV p m r => r -> m r +l = expect . prog +``` + +## Implementation + +To understand ADEV's implementation, it is useful to first skim the ADEV paper, which explains how ADEV modularly extends standard forward-mode AD with support for probabilistic primitives. The Haskell code is a relatively direct encoding of the ideas described in the paper. Briefly: + +* All the primitives in the ADEV language, including those introduced by the extensions from Appendix B, are encoded as methods of the `ADEV` typeclass, in the [Numeric.ADEV.Class](src/Numeric/ADEV/Class.hs) module. This is like a 'specification' that a specific interpreter of the ADEV language can satisfy. It leaves open what concrete Haskell types will be used to represent the ADEV types of real numbers $\mathbb{R}$, estimated reals $\widetilde{\mathbb{R}}$, and monadic probabilistic programs $P\,\tau$ — it uses the type variables `r`, `m r`, and `p m tau` for this purpose. + +* The [Numeric.ADEV.Interp](src/Numeric/ADEV/Interp.hs) module provides one instance of the `ADEV` typeclass, implementing the standard semantics of an ADEV term. The type variables `p`, `m`, and `r` are instantiated so that the type of reals `r` is interpreted as `Double`, the type `m r` of *estimated* reals is interpreted as the type `m Double` for some `MonadDistribution` `m` (where `MonadDistribution` is the [monad-bayes](https://github.com/tweag/monad-bayes) typeclass for probabilistic programs), and the type of probabilistic programs `p m a` is interpreted as `WriterT Sum m a` (the `Sum` maintains an accumulated loss, and is described in Appendix B.2 of the ADEV paper). + +* The [Numeric.ADEV.Diff](src/Numeric/ADEV/Diff.hs) module provides built-in derivatives for each primitive. These are organized into a second instance of the `ADEV` typeclass, where now the type `r` of reals is interpreted as `ForwardDouble`, representing forward-mode dual numbers $\mathcal{D}\{\mathbb{R}\}$ from the paper; the type `m r` of estimated reals is interpreted as the type `m ForwardDouble` for some `MonadDistribution` `m`, which implements the type $\widetilde{\mathbb{R}}_\mathcal{D}$ of estimated dual numbers from the paper; and the type `p m tau` of probabilistic programs is interpreted as `ContT ForwardDouble m tau`, i.e., the type of *higher-order functions* that transform an input `loss_to_go : tau -> m ForwardDouble` into an estimated dual-number loss of type `m ForwardDouble` (this implements the type $P_\mathcal{D}\,\tau$ from the paper). + + +## Installing ADEV +1. Install `stack` (https://docs.haskellstack.org/en/stable/install_and_upgrade/). +2. Clone this repository. +3. Run the examples using `stack run ExampleName`, where `ExampleName.hs` is the name of a file from the `examples` directory. +4. Or: Run `stack ghci` to enter a REPL. diff --git a/adev.cabal b/adev.cabal index 2e9727b..4838dc3 100644 --- a/adev.cabal +++ b/adev.cabal @@ -27,6 +27,7 @@ library exposed-modules: Numeric.ADEV.Class Numeric.ADEV.Diff + Numeric.ADEV.Distributions Numeric.ADEV.Interp other-modules: Paths_adev @@ -43,10 +44,24 @@ library , vector >=0.12.3.1 && <0.12.4 default-language: Haskell2010 -executable adev-exe +executable Figure2 main-is: Figure2.hs - other-modules: - Paths_adev + hs-source-dirs: + examples + ghc-options: -Wall -Wcompat -Widentities -Wincomplete-record-updates -Wincomplete-uni-patterns -Wmissing-export-lists -Wmissing-home-modules -Wpartial-fields -Wredundant-constraints -threaded -rtsopts -with-rtsopts=-N + build-depends: + ad ==4.5.2.* + , adev + , base >=4.7 && <5 + , log-domain >=0.12 && <0.14 + , monad-bayes >=1.1.0 + , mtl ==2.2.2.* + , transformers ==0.5.6.2.* + , vector >=0.12.3.1 && <0.12.4 + default-language: Haskell2010 + +executable ParticleFilterExample + main-is: ParticleFilterExample.hs hs-source-dirs: examples ghc-options: -Wall -Wcompat -Widentities -Wincomplete-record-updates -Wincomplete-uni-patterns -Wmissing-export-lists -Wmissing-home-modules -Wpartial-fields -Wredundant-constraints -threaded -rtsopts -with-rtsopts=-N diff --git a/examples/ParticleFilterExample.hs b/examples/ParticleFilterExample.hs new file mode 100644 index 0000000..aaa7186 --- /dev/null +++ b/examples/ParticleFilterExample.hs @@ -0,0 +1,47 @@ +module Main (main) where + +import Numeric.ADEV.Class +import Numeric.ADEV.Diff (diff) +import Numeric.ADEV.Interp () +import Numeric.ADEV.Distributions (normalD) +import Numeric.AD.Mode.Forward.Double (ForwardDouble) +import Control.Monad.Bayes.Class (MonadDistribution) +import Control.Monad.Bayes.Sampler.Strict (sampleIO) +import Numeric.Log (Log(..)) + +-- smc :: ([a] -> Log r) -> D m r a -> (a -> D m r a) -> ([a] -> m r) -> Int -> Int -> m r + +dens :: (RealFrac r, Floating r) => D m r Double -> Double -> Log r +dens (D _ f) x = f x + +normalDensity :: (RealFrac r, Floating r) => r -> r -> Double -> Log r +normalDensity mu sig x = Exp $ -log(sig) - log(2*pi) / 2 - ((realToFrac x)-mu)^2/(2*sig^2) + +ys = [undefined, 1,2,3,4,5] + +l :: (MonadDistribution m, RealFloat r, Floating r, ADEV p m r) => r -> m r +l theta = smc p q0 q f 2 1000 + where + p xs = let xys = zip (map realToFrac xs) (reverse (take (length xs) ys)) in + pxys xys + pxys [] = undefined + pxys [(x, y)] = normalDensity 0 (exp theta) (realToFrac x) + pxys ((x,y):((xprev,yprev):xys)) = normalDensity (realToFrac xprev) (exp theta) (realToFrac x) * normalDensity (realToFrac x) 1 y * pxys ((xprev,yprev):xys) + q0 = normalD 0 (exp theta) + q x = normalD (realToFrac x) (exp theta) + f xs = return 1 + +sga :: MonadDistribution m => (ForwardDouble -> m ForwardDouble) -> Double -> Double -> Int -> m [Double] +sga loss eta x0 steps = + if steps == 0 then + return [x0] + else do + v <- diff loss x0 + let x1 = x0 + eta * v + xs <- sga loss eta x1 (steps - 1) + return (x0:xs) + +main :: IO () +main = do + vs <- sampleIO $ sga l 10.0 0.0 500 + print (vs) \ No newline at end of file diff --git a/figures/adev-diagram.png b/figures/adev-diagram.png new file mode 100644 index 0000000000000000000000000000000000000000..47175e66776d2145ff8e62162f5c6cea41e90b9d GIT binary patch literal 42738 zcmZ6z1ymeSvn`CfLvVM3yGwA_1PJaL+=IKj2e&|iySuwXaCg_i`J3zM7-~e^6VBi78u) zNr~D0uya(iH#9c=Vq#}xswySQ%FD|G21XliY+#@&%|JV3h+$wbG)_;4;OM3r78a#y z;5*O-8AOhxgQOKc+z%ZC4laZw0#=~X-Dr&zMfz03G{hk9(dI`_RnLPb{SLl;x*L#> z0RuCDM5^7~RtVO24qhXXP}6-1P3nq1oHQzHzvV>2LESS{?|o z&)>S)nZGMZbq{q7h53VHoukd?Kz45p)4IZtg@D0T^I{+&y~*L>YQB;7$^XQ|#Xe+V zxm1z}`-8{E0(Z#Ha+;B-qxBVrGbDaE&EFr51dO&@R_nT>Rva)KkEy1NnSuh?XW%^? z7dh9C+aaFEB8OT!{bv1kx%O^1t7MV}G2y$Y~A+0}}z05&!bl9sD#4x`|NY zz7Y}CzuyKjX!E?A%0HO&v5G;_9+?Ro8#`qa-vIvn91R=M#Geu^fd6{{tXm`LsMlTg zYfhe>`EAT=B#b8sN}%Mkpg{u}oHo(S_p zkIVm!^94#=F?}c2Lh4`y^qCj^i-(-{!xcKW-GQAc9># z;3M=m>%Ydr!Qs!n?oJVf%SzsM*@K9ztMpCr0zG!ja;V1-==Ov&%Q(jK+Uj+HQ 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6150eed..59f4577 100644 --- a/src/Numeric/ADEV/Diff.hs +++ b/src/Numeric/ADEV/Diff.hs @@ -12,7 +12,7 @@ import Control.Monad.Bayes.Class ( uniformD, logCategorical, poisson, - bernoulli, + bernoulli, normal) import Control.Monad.Cont (ContT(..)) import Numeric.AD.Internal.Forward.Double ( @@ -172,7 +172,7 @@ instance MonadDistribution m => ADEV (ContT ForwardDouble) m ForwardDouble where let (D qs qd) = dq (head v) let qqd = Exp . primal . ln . qd v' <- qs - return (v':v, (pp (v':v) / pp v) / qqd v') + return (v':v, w * (pp (v':v) / pp v) / qqd v') step particles = do particles <- resample particles mapM propagate particles diff --git a/src/Numeric/ADEV/Distributions.hs b/src/Numeric/ADEV/Distributions.hs new file mode 100644 index 0000000..88f26db --- /dev/null +++ b/src/Numeric/ADEV/Distributions.hs @@ -0,0 +1,16 @@ +module Numeric.ADEV.Distributions (normalD, geometricD) where + +import Numeric.ADEV.Class + +import Control.Monad.Bayes.Class ( + MonadDistribution, + geometric, + normal) + +import Numeric.Log (Log(..)) + +normalD :: (MonadDistribution m, RealFrac r, Floating r) => r -> r -> D m r Double +normalD mu sig = D (normal (realToFrac mu) (realToFrac sig)) (\x -> Exp $ -log(sig) - log(2*pi) / 2 - ((realToFrac x)-mu)^2/(2*sig^2)) + +geometricD :: (MonadDistribution m, RealFrac r, Floating r) => r -> D m r Int +geometricD p = D (geometric (realToFrac p)) (\x -> Exp $ log(p) + (fromIntegral $ x-1) * log(1-p)) \ No newline at end of file diff --git a/src/Numeric/ADEV/Interp.hs b/src/Numeric/ADEV/Interp.hs index 6b13d0b..b97fe81 100644 --- a/src/Numeric/ADEV/Interp.hs +++ b/src/Numeric/ADEV/Interp.hs @@ -79,7 +79,7 @@ instance MonadDistribution m => ADEV (WriterT (Sum Double)) m Double where let (v, w) = particle let (D qs qd) = q (head v) v' <- qs - return (v':v, (p (v':v) / p v) / qd v') + return (v':v, w * (p (v':v) / p v) / qd v') step particles = do particles <- resample particles mapM propagate particles