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May. 2, 2019
Created
Mar. 13, 2017

Certigrad is a proof-of-concept for a new way to develop machine learning systems, in which the following components are developed simultaneously:

• The implementation itself.
• A library of background mathematics.
• A formal specification of what the implementation needs to do in terms of the mathematics.
• A machine-checkable proof that the implementation satisfies its specification.

Specifically, Certigrad is a system for optimizing over stochastic computation graphs, that we debugged systematically in the Lean Theorem Prover, and ultimately proved correct in terms of the underlying mathematics.

## Background: stochastic computation graphs

Stochastic computation graphs extend the computation graphs that underlie systems like TensorFlow and Theano by allowing nodes to represent random variables and by defining the loss function to be the expected value of the sum of the leaf nodes over all the random choices in the graph. Certigrad allows users to construct arbitrary stochastic computation graphs out of the primitives that we provide. The main purpose of the system is to take a program describing a stochastic computation graph and to run a randomized algorithm (stochastic backpropagation) that, in expectation, samples the gradients of the loss function with respect to the parameters.

## Correctness

#### Stochastic backpropagation

Here is the theorem stating and proving that our implementation of stochastic backpropagation is correct:

Informally, it says that for any stochastic computation graph, `backprop` computes a vector of tensors such that each element of the vector is a random variable that (in expectation) equals the gradient of the expected loss of the graph with respect to that parameter.

Even more informally, `∇ E[loss(graph)] = E[backprop(graph)]`.

#### Certified optimizations

We also implemented two stochastic-computation-graph-transformations, one to "reparameterize" a graph so that a random variable no longer depends directly on a parameter, and one to integrate out the KL-divergence of the multivariate Gaussian.

#### Verifying properties of Certigrad programs

Certigrad also includes a front-end syntax for constructing stochastic computation graphs. Here is an example program that describes a naive variational autoencoder:

We prove that the two certified optimizations mentioned above are sound to apply in sequence to the naive autoencoder:

We also prove that backpropagation will work correctly on the resulting model, i.e. that it satisfies all the necessary preconditions:

#### Formal proof

In the process of proving a theorem, Lean constructs a formal proof certificate that can be automatically verified by a small stand-alone executable, whose soundness is based on a well-established meta-theoretic argument embedding the core logic of Lean into set theory, and whose implementation has been heavily scrutinized by many developers. Thus no human needs to be able to understand why a proof is correct in order to trust that it is.

#### Impurities

We have adopted a very high standard for our proofs, but there are a few ways in which Certigrad falls short of the purist ideal.

• We axiomatize the background mathematics instead of constructing it from first principles.
• By necessity, we execute with floating-point numbers even though our correctness theorems only hold in terms of infinite-precision real numbers.
• For performance, we replace the primitive tensor operations with calls to Eigen at runtime.
• We execute in a virtual machine, which is not designed to be as trustworthy as the proof-checker for the core logic.

## Performance

Provable correctness need not come at the expense of computational efficiency: proofs need only be checked once and they introduce no ongoing costs or runtime overhead. Although the algorithms we verify in this work lack many optimizations, most of the time training machine learning systems is spent multiplying matrices, and we are able to achieve competitive performance simply by linking with an optimized library for matrix operations (Eigen). We trained an Auto-Encoding Variational Bayes (AEVB) model on MNIST using ADAM and find that our performance is competitive with TensorFlow (on CPUs).

We include a script to train an AEVB on MNIST using ADAM:

## Benefits of the methodology

Although our methodology introduces many new challenges, it offers substantial benefits as well.

#### Debugging

First, our methodology provides a way to debug machine learning systems systematically.

Implementation errors can be extremely difficult to detect in machine learning systems---let alone to localize and address---since there are many other potential causes of undesired behavior. For example, an implementation error may lead to incorrect gradients and so cause a learning algorithm to stall, but such a symptom may also be caused by noise in the training data, a poor choice of model, an unfavorable optimization landscape, an inadequate search strategy, or numerical instability. These other issues are so common that it is often assumed that any undesired behavior is caused by one of them. As a result, actual implementation errors can persist indefinitely without detection. Errors are even more difficult to detect in stochastic programs, since some errors may only distort the distributions of random variables and may require writing custom statistical tests to detect.

With our methodology, the formal specification can be used to test and debug machine learning systems exhaustively at a logical level without needing to resort to empirical testing at all. The process of proving that the specification holds will expose all implementation errors, oversights and hidden assumptions. Once it has been proved, every interested party can be certain that the implementation is correct without needing to trust any human involved or to understand how the program works.

#### Synthesis

Second, our methodology can enable some parts of the implementation to be synthesized semi-automatically.

Whereas with the status-quo methodology, the compiler has no idea what the program is supposed to do and so can only catch superficial syntactic errors, with our methodology the theorem prover knows exactly what the program is supposed to do and can provide much more useful assistance accordingly. As a simple example, suppose we want to compile a 2-layer MLP into a single primitive operator to avoid the overhead of graph-processing at runtime. Normally this would involve deriving the gradient of the compound function by hand. However, since in our methodology the theorem prover knows about the underlying mathematics, including the relevant gradient rules and the algebraic properties of tensors, it can assist in the derivation of the gradient for the new operator.

The possibilities for synthesis go way beyond simply automating algebraic derivations. When developing Certigrad, we started proving that the specification held before even implementing the difficult parts of the system, and used the resulting proof-obligations to help determine what the program needed to do. The formal specification, and ultimately the machine-checkable proof of correctness, allowed us to implement the system correctly without a coherent, global understanding of why the system was correct. Indeed, we mostly relegated that burden to the computer.

#### Aggressive optimizations

Third, our methodology may enable safely automating much more aggressive transformations than would otherwise be advisable. For example, one could write a procedure that searches for components of a stochastic computation graph that can be integrated out analytically, that makes use of large libraries of integral identities as well as procedural methods that are impossible for humans to simulate by hand. Such a procedure may be able to achieve super-human variance reduction on many models yet may be extremely difficult to implement reliably; if the procedure is able to generate a machine-checkable certificate for a given transformation, the transformation can be trusted regardless of the complexity of the procedure itself.

#### Documentation

Fourth, a formal specification (even without a formal proof) can serve as precise documentation for a system, and can make it much easier to understand what various parts of the code do, what preconditions are assumed to hold at various places, and what invariants are being maintained. Such precise documentation can be useful for any software system but can be especially useful for machine learning systems, since not all developers may have the necessary mathematical expertise to fill in the gaps of informal descriptions.

## Incrementality

Our methodology may already be economical for high-assurance systems, and yet there is still a lot of work to be done to make it practical for mainstream developments for which correctness is only "optional". However, a crucial aspect of our methodology is that it can be adopted incrementally. One can write only a little bit of the code in Lean and simply wrap and axiomatize the rest (as we did with Eigen). One can also write down shallow correctness properties, and only prove that a few of these properties hold. We hope that over time, as the tools mature, developers will find it worth the cost to pursue our methodology further and will be able to reap more of its benefits.

Lean is still under development and the foreign function interface (FFI) has not been ported to the master branch yet. We forked Lean in order to add C++ code to wrap Eigen inside Lean's virtual machine, but we will port Certigrad to the master branch of Lean once the FFI has been released.

Until then:

3. Download this repository, and in the main directory execute `leanpkg --build`.

Note: building Certigrad currently takes ~15 minutes and consumes ~7 GB of memory.

## Warning

We have formally proved that Certigrad is correct (modulo the impurities mentioned above), but this does not imply that Certigrad does what you expect. All it means is that the theorems linked to above are true, given the assumptions.

Certigrad is designed to be a proof-of-concept, not a production system. There are many features that would need to be added to make it useful as an artifact. We are more interested in addressing the barriers to adoption of our methodology than in extending and maintaining Certigrad as an end in itself.