Past computerized differentiation – Google AI Weblog
Derivatives play a central function in optimization and machine studying. By regionally approximating a training loss, derivatives information an optimizer towards decrease values of the loss. Automated differentiation frameworks resembling TensorFlow, PyTorch, and JAX are an important a part of fashionable machine studying, making it possible to make use of gradient-based optimizers to coach very complicated fashions.
However are derivatives all we want? By themselves, derivatives solely inform us how a operate behaves on an infinitesimal scale. To make use of derivatives successfully, we frequently must know greater than that. For instance, to decide on a learning rate for gradient descent, we have to know one thing about how the loss operate behaves over a small however finite window. A finite-scale analogue of computerized differentiation, if it existed, might assist us make such decisions extra successfully and thereby velocity up coaching.
In our new paper “Automatically Bounding The Taylor Remainder Series: Tighter Bounds and New Applications“, we current an algorithm referred to as AutoBound that computes polynomial higher and decrease bounds on a given operate, that are legitimate over a user-specified interval. We then start to discover AutoBound’s purposes. Notably, we current a meta-optimizer referred to as SafeRate that makes use of the higher bounds computed by AutoBound to derive studying charges which are assured to monotonically cut back a given loss operate, with out the necessity for time-consuming hyperparameter tuning. We’re additionally making AutoBound obtainable as an open-source library.
The AutoBound algorithm
Given a operate f and a reference level x0, AutoBound computes polynomial higher and decrease bounds on f that maintain over a user-specified interval referred to as a belief area. Like Taylor polynomials, the bounding polynomials are equal to f at x0. The bounds turn out to be tighter because the belief area shrinks, and method the corresponding Taylor polynomial because the belief area width approaches zero.
 |
| Robotically-derived quadratic higher and decrease bounds on a one-dimensional operate f, centered at x0=0.5. The higher and decrease bounds are legitimate over a user-specified belief area, and turn out to be tighter because the belief area shrinks. |
Like computerized differentiation, AutoBound might be utilized to any operate that may be applied utilizing customary mathematical operations. Actually, AutoBound is a generalization of Taylor mode automatic differentiation, and is equal to it within the particular case the place the belief area has a width of zero.
To derive the AutoBound algorithm, there have been two fundamental challenges we needed to deal with:
- We needed to derive polynomial higher and decrease bounds for numerous elementary features, given an arbitrary reference level and arbitrary belief area.
- We needed to give you an analogue of the chain rule for combining these bounds.
Bounds for elementary features
For a wide range of commonly-used features, we derive optimum polynomial higher and decrease bounds in closed type. On this context, “optimum” means the bounds are as tight as potential, amongst all polynomials the place solely the maximum-degree coefficient differs from the Taylor sequence. Our concept applies to elementary features, resembling exp and log, and customary neural community activation features, resembling ReLU and Swish. It builds upon and generalizes earlier work that utilized solely to quadratic bounds, and just for an unbounded belief area.
 |
| Optimum quadratic higher and decrease bounds on the exponential operate, centered at x0=0.5 and legitimate over the interval [0, 2]. |
A brand new chain rule
To compute higher and decrease bounds for arbitrary features, we derived a generalization of the chain rule that operates on polynomial bounds. As an example the thought, suppose we now have a operate that may be written as
and suppose we have already got polynomial higher and decrease bounds on g and h. How can we compute bounds on f?
The important thing seems to be representing the higher and decrease bounds for a given operate as a single polynomial whose highest-degree coefficient is an interval quite than a scalar. We will then plug the sure for h into the sure for g, and convert the consequence again to a polynomial of the identical type utilizing interval arithmetic. Below appropriate assumptions in regards to the belief area over which the sure on g holds, it may be proven that this process yields the specified sure on f.
 |
| The interval polynomial chain rule utilized to the features h(x) = sqrt(x) and g(y) = exp(y), with x0=0.25 and belief area [0, 0.5]. |
Our chain rule applies to one-dimensional features, but additionally to multivariate features, resembling matrix multiplications and convolutions.
Propagating bounds
Utilizing our new chain rule, AutoBound propagates interval polynomial bounds by means of a computation graph from the inputs to the outputs, analogous to forward-mode automatic differentiation.
 |
| Ahead propagation of interval polynomial bounds for the operate f(x) = exp(sqrt(x)). We first compute (trivial) bounds on x, then use the chain rule to compute bounds on sqrt(x) and exp(sqrt(x)). |
To compute bounds on a operate f(x), AutoBound requires reminiscence proportional to the dimension of x. Because of this, sensible purposes apply AutoBound to features with a small variety of inputs. Nonetheless, as we are going to see, this doesn’t forestall us from utilizing AutoBound for neural community optimization.
Robotically deriving optimizers, and different purposes
What can we do with AutoBound that we could not do with computerized differentiation alone?
Amongst different issues, AutoBound can be utilized to robotically derive problem-specific, hyperparameter-free optimizers that converge from any place to begin. These optimizers iteratively cut back a loss by first utilizing AutoBound to compute an higher sure on the loss that’s tight on the present level, after which minimizing the higher sure to acquire the subsequent level.
 |
| Minimizing a one-dimensional logistic regression loss utilizing quadratic higher bounds derived robotically by AutoBound. |
Optimizers that use higher bounds on this approach are referred to as majorization-minimization (MM) optimizers. Utilized to one-dimensional logistic regression, AutoBound rederives an MM optimizer first published in 2009. Utilized to extra complicated issues, AutoBound derives novel MM optimizers that will be troublesome to derive by hand.
We will use an identical thought to take an present optimizer resembling Adam and convert it to a hyperparameter-free optimizer that’s assured to monotonically cut back the loss (within the full-batch setting). The ensuing optimizer makes use of the identical replace path as the unique optimizer, however modifies the training fee by minimizing a one-dimensional quadratic higher sure derived by AutoBound. We check with the ensuing meta-optimizer as SafeRate.
 |
| Efficiency of SafeRate when used to coach a single-hidden-layer neural community on a subset of the MNIST dataset, within the full-batch setting. |
Utilizing SafeRate, we will create extra strong variants of present optimizers, at the price of a single further ahead go that will increase the wall time for every step by a small issue (about 2x within the instance above).
Along with the purposes simply mentioned, AutoBound can be utilized for verified numerical integration and to robotically show sharper variations of Jensen’s inequality, a basic mathematical inequality used incessantly in statistics and different fields.
Enchancment over classical bounds
Bounding the Taylor remainder term robotically will not be a brand new thought. A classical method produces diploma okay polynomial bounds on a operate f which are legitimate over a belief area [a, b] by first computing an expression for the okayth by-product of f (utilizing computerized differentiation), then evaluating this expression over [a,b] utilizing interval arithmetic.
Whereas elegant, this method has some inherent limitations that may result in very unfastened bounds, as illustrated by the dotted blue strains within the determine beneath.
 |
| Quadratic higher and decrease bounds on the lack of a multi-layer perceptron with two hidden layers, as a operate of the preliminary studying fee. The bounds derived by AutoBound are a lot tighter than these obtained utilizing interval arithmetic analysis of the second by-product. |
Trying ahead
Taylor polynomials have been in use for over 300 years, and are omnipresent in numerical optimization and scientific computing. However, Taylor polynomials have important limitations, which might restrict the capabilities of algorithms constructed on high of them. Our work is a part of a rising literature that acknowledges these limitations and seeks to develop a brand new basis upon which extra strong algorithms might be constructed.
Our experiments thus far have solely scratched the floor of what’s potential utilizing AutoBound, and we imagine it has many purposes we now have not found. To encourage the analysis neighborhood to discover such potentialities, we now have made AutoBound obtainable as an open-source library constructed on high of JAX. To get began, go to our GitHub repo.
Acknowledgements
This publish is predicated on joint work with Josh Dillon. We thank Alex Alemi and Sergey Ioffe for helpful suggestions on an earlier draft of the publish.