105,612 research outputs found
Reverse Derivative Categories
The reverse derivative is a fundamental operation in machine learning and
automatic differentiation. This paper gives a direct axiomatization of a
category with a reverse derivative operation, in a similar style to that given
by Cartesian differential categories for a forward derivative. Intriguingly, a
category with a reverse derivative also has a forward derivative, but the
converse is not true. In fact, we show explicitly what a forward derivative is
missing: a reverse derivative is equivalent to a forward derivative with a
dagger structure on its subcategory of linear maps. Furthermore, we show that
these linear maps form an additively enriched category with dagger biproducts.Comment: Extended version of paper to appear at CSL 202
Reverse Derivative Ascent: A Categorical Approach to Learning Boolean Circuits
We introduce Reverse Derivative Ascent: a categorical analogue of gradient
based methods for machine learning. Our algorithm is defined at the level of
so-called reverse differential categories. It can be used to learn the
parameters of models which are expressed as morphisms of such categories. Our
motivating example is boolean circuits: we show how our algorithm can be
applied to such circuits by using the theory of reverse differential
categories. Note our methodology allows us to learn the parameters of boolean
circuits directly, in contrast to existing binarised neural network approaches.
Moreover, we demonstrate its empirical value by giving experimental results on
benchmark machine learning datasets.Comment: In Proceedings ACT 2020, arXiv:2101.0788
An axiomatic approach to differentiation of polynomial circuits
Reverse derivative categories (RDCs) have recently been shown to be a suitable semantic framework for studying machine learning algorithms. Whereas emphasis has been put on training methodologies, less attention has been devoted to particular model classes: the concrete categories whose morphisms represent machine learning models. In this paper we study presentations by generators and equations of classes of RDCs. In particular, we propose polynomial circuits as a suitable machine learning model class. We give an axiomatisation for these circuits and prove a functional completeness result. Finally, we discuss the use of polynomial circuits over specific semirings to perform machine learning with discrete values
An axiomatic approach to differentiation of polynomial circuits
Reverse derivative categories (RDCs) have recently been shown to be a suitable semantic framework for studying machine learning algorithms. Whereas emphasis has been put on training methodologies, less attention has been devoted to particular model classes: the concrete categories whose morphisms represent machine learning models. In this paper we study presentations by generators and equations of classes of RDCs. In particular, we propose polynomial circuits as a suitable machine learning model class. We give an axiomatisation for these circuits and prove a functional completeness result. Finally, we discuss the use of polynomial circuits over specific semirings to perform machine learning with discrete values
Categorical Foundations of Gradient-Based Learning
We propose a categorical semantics of gradient-based machine learning
algorithms in terms of lenses, parametrised maps, and reverse derivative
categories. This foundation provides a powerful explanatory and unifying
framework: it encompasses a variety of gradient descent algorithms such as
ADAM, AdaGrad, and Nesterov momentum, as well as a variety of loss functions
such as as MSE and Softmax cross-entropy, shedding new light on their
similarities and differences. Our approach to gradient-based learning has
examples generalising beyond the familiar continuous domains (modelled in
categories of smooth maps) and can be realized in the discrete setting of
boolean circuits. Finally, we demonstrate the practical significance of our
framework with an implementation in Python.Comment: 14 page
CHAD for Expressive Total Languages
We show how to apply forward and reverse mode Combinatory Homomorphic
Automatic Differentiation (CHAD) to total functional programming languages with
expressive type systems featuring the combination of - tuple types; - sum
types; - inductive types; - coinductive types; - function types. We achieve
this by analysing the categorical semantics of such types in -types
(Grothendieck constructions) of suitable categories. Using a novel categorical
logical relations technique for such expressive type systems, we give a
correctness proof of CHAD in this setting by showing that it computes the usual
mathematical derivative of the function that the original program implements.
The result is a principled, purely functional and provably correct method for
performing forward and reverse mode automatic differentiation (AD) on total
functional programming languages with expressive type systems.Comment: Under review at MSC
Turnover, account value and diversification of real traders: evidence of collective portfolio optimizing behavior
Despite the availability of very detailed data on financial market,
agent-based modeling is hindered by the lack of information about real trader
behavior. This makes it impossible to validate agent-based models, which are
thus reverse-engineering attempts. This work is a contribution to the building
of a set of stylized facts about the traders themselves. Using the client
database of Swissquote Bank SA, the largest on-line Swiss broker, we find
empirical relationships between turnover, account values and the number of
assets in which a trader is invested. A theory based on simple mean-variance
portfolio optimization that crucially includes variable transaction costs is
able to reproduce faithfully the observed behaviors. We finally argue that our
results bring into light the collective ability of a population to construct a
mean-variance portfolio that takes into account the structure of transaction
costsComment: 26 pages, 9 figures, Fig. 8 fixe
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