Certified Machine Learning
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README.md

MLCert

MLCert is a collection of software tools and libraries for doing verified machine learning in the Coq proof assistant, where by "verified machine learning in Coq" we mean learning with certified generalization guarantees (e.g., bounds on expected accuracy of learned models). A technical report describing MLCert is forthcoming.

BUILD

Prerequisites

  • Coq 8.8

  • OUVerT

The Ohio University Verification Toolkit, containing a number of lemmas used in MLCert.

Build Instructions

  • First build OUVerT, by cloning the repository and following the instructions in [OUVerT/README.md]. Don't forget to do make install or sudo make install.

  • Then, in MLCert, do:

make
make install
  • To transfer neural networks trained in TensorFlow to Coq, follow the instructions in MLCert/NNCert.

ORGANIZATION

Following are the primary directories and files used in the development:

  • [NNCert/]: Applies MLCert to neural networks trained in TensorFlow
  • [hs/]: The directory in which Haskell programs are extracted (e.g., from linearclassifiers.v)
  • [axioms.v]: Axioms used in the development
  • [bitvectors.v]: Bitvectors implemented on top of the axiomatized arrays of axioms.v
  • [extraction_hs.v]: Haskell-specific extraction directives
  • [extraction_ocaml.v]: OCaml-specific extraction directives
  • [float32.v]: Axiomatized 32-bit floating point numbers, as used in linearclassifiers.v
  • [monads.v]: Defines the continuation monad used in learners.v
  • [learners.v]: Learners as probabilistic programs
  • [linearclassifiers.v]: Specializes learners.v to linear classifiers and Perceptron

EXAMPLES

Learners

MLCert defines supervised learners (file learners.v):

Module Learner.
  Record t (X Y Hypers Params : Type) :=
    mk { predict : Hypers -> Params -> X -> Y;
         update : Hypers -> X*Y -> Params -> Params }.
End Learner.

as pairs of

  • a prediction function that, given hyperparameters Hypers, parameters Params, and an input X, produces a label Y; and
  • and update function that maps hypers, X*Y pairs, and parameters to new parameters.

The following function:

  Definition learn_func (init:Params) (T:training_set) : Params := 
    foldrM (fun epoch p_epoch =>
      foldable_foldM (M:=Id) (fun xy p =>
        ret (Learner.update learner h xy p))
        p_epoch T)
      init (enum 'I_epochs).

defines a generic learning procedure that applies update to a training set T by folding update over T epoch times.

Linear Threshold Classifiers

Linear threshold classifiers (linearclassifiers.v) specialize the predict of generic learners to a linear prediction function of the form ax + b > 0. We implement such classifiers in MLCert as:

Section LinearThresholdClassifier.
  Variable n : nat. (*the dimensionality*)

  Definition A := float32_arr n. (*examples*)
  Definition B := bool. (*labels*)
  Definition Weights := float32_arr n.
  Definition Bias := float32.
  Definition Params := (Weights * Bias)%type.

  Section predict.
    Open Scope f32_scope.
    Definition predict (p : Params) (a : A) : B :=
      let: (w, b) := p in
      f32_dot w a + b > 0.
  End predict.
End LinearThresholdClassifier.

n is the dimensionality of the problem space. The type A:=float32_arr n defines size-n arrays of 32-bit floating-point numbers. A linear threshold classifier's parameters are pairs of Weights (also size-n floating-point numbers) and Biases.

Perceptron

We further specialize Learner by adding to predict an update rule that implements the Perceptron algorithm, defined as:

  Definition update (h:Hypers) (example_label:A*B) (p:Params) : Params :=
    let: (example, label) := example_label in
    let: predicted_label := predict p example in
    if Bool.eqb predicted_label label then p
    else let: (w, b) := p in
    (f32_map2 (fun x1 x2 => x1 + (alpha h)*label*x2) w example, b+label).

We package Perceptron update with the generic prediction rule of linear threshold classifiers by constructing the following record:

  Definition Learner : Learner.t A B Hypers Params :=
    Learner.mk
      (fun h => @predict n)
      update.

Extraction Perceptron to Haskell

To extract the Perceptron Learner to Haskell, we build an "extractible" version of a probabilistic program that first samples a training set, then uses learn_func specialized to Perceptron to learn a linear classifier.

  Definition perceptron (r:Type) := 
    @extractible_main A B Params Perceptron.Hypers 
      (Perceptron.Learner n) hypers epochs _ (@list_Foldable (A*B)%type) r
    (fun T => ret T).

The function perceptron is parametric in the sampling procedure that produces training sets (to prove generalization results of Perceptron, we further specialize to a sampler that's assumed to produced n iid examples from a distribution D).

To extract perceptron to Haskell, we do:

Extraction Language Haskell.
Extraction "hs/Perceptron.hs" perceptron.

These commands produce a new Haskell source file, Perceptron.hs in directory MLCert/hs/, that can be compiled and executed by doing:

> cd hs/
> ghc PerceptronTest
> ./PerceptronTest

PerceptronTest.hs is an unverified shim program, written directly in Haskell, that constructs a random linearly separable data set on which to train the extracted Perceptron.hs, which it imports.

From TensorFlow to Coq/MLCert

See directory [MLCert/NNCert/] for additional directions on compiling to MLCert models learned in external tools like TensorFlow.