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15

Date Added: Oct 20, 2021

Date Added: Oct 20, 2021

Abstract Characterizing consciousness in and of itself is notoriously difficult. Here, we propose an alternative approach to characterize, and eventually define, consciousness through exhaustive descriptions of consciousness’ relationships to all other consciousness. This approach is founded in category theory. Indeed, category theory can prove that two objects A and B in a category can be equivalent if and only if all the relationships that A holds with others in the category are the same as those of B; this proof is called the Yoneda lemma. To introduce the Yoneda lemma, we gradually introduce key concepts of category theory to consciousness researchers. Along the way, we propose several possible definitions of categories of consciousness, both in terms of level and contents, through the usage of simple examples. We propose to use the categorical structure of consciousness as a gold standard to formalize empirical research (e.g. color qualia structure at fovea and periphery) and, especially, the empirical testing of theories of consciousness.

2

Date Added: Jul 21, 2021

Date Added: Jul 21, 2021

We present a new technology-based paradigm to support embodied mathematics educational games, using wearable devices in the form of SmartPhones and SmartWatches for math learning, for full classes of students in formal in-school education settings. The Wearable Learning Games Engine is web based infrastructure that enables students to carry one mobile device per child, as they embark on math team-based activities that require physical engagement with the environment. These Wearable Tutors serve as guides and assistants while students manipulate, measure, estimate, discern, discard and find mathematical objects that satisfy specified constraints. Multiplayer math games that use this infrastructure have yielded both cognitive and affective benefits. Beyond math game play, the Wearable Games Engine Authoring Tool enables students to create games themselves for other students to play; in this process, students engage in computational thinking and learn about finite-state machines. We present the infrastructure, games, and results for a series of experiments on both game play and game creation.

6

Date Added: Aug 7, 2020

Date Added: Aug 7, 2020

We prove that a subgroup has the same number of left and right cosets.

112

Date Added: May 20, 2021

Date Added: May 20, 2021

We present a new paradigm for Neural ODE algorithms, called ODEtoODE, where time-dependent parameters of the main flow evolve according to a matrix flow on the orthogonal group O(d). This nested system of two flows, where the parameter-flow is constrained to lie on the compact manifold, provides stability and effectiveness of training and provably solves the gradient vanishing-explosion problem which is intrinsically related to training deep neural network architectures such as Neural ODEs. Consequently, it leads to better downstream models, as we show on the example of training reinforcement learning policies with evolution strategies, and in the supervised learning setting, by comparing with previous SOTA baselines. We provide strong convergence results for our proposed mechanism that are independent of the depth of the network, supporting our empirical studies. Our results show an intriguing connection between the theory of deep neural networks and the field of matrix flows on compact manifolds.

61

Date Added: May 20, 2021

Date Added: May 20, 2021

We present a novel view on principal component analysis (PCA) as a competitive game in which each approximate eigenvector is controlled by a player whose goal is to maximize their own utility function. We analyze the properties of this PCA game and the behavior of its gradient based updates. The resulting algorithm -- which combines elements from Oja's rule with a generalized Gram-Schmidt orthogonalization -- is naturally decentralized and hence parallelizable through message passing. We demonstrate the scalability of the algorithm with experiments on large image datasets and neural network activations. We discuss how this new view of PCA as a differentiable game can lead to further algorithmic developments and insights.

16

Date Added: Mar 20, 2021

Date Added: Mar 20, 2021

Variational quantum algorithms (VQAs) optimize the parameters θ of a parametrized quantum circuit V(θ) to minimize a cost function C. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming V(θ) is an alternating layered ansatz composed of blocks forming local 2-designs. Our first result states that defining C in terms of global observables leads to exponentially vanishing gradients (i.e., barren plateaus) even when V(θ) is shallow. Hence, several VQAs in the literature must revise their proposed costs. On the other hand, our second result states that defining C with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of V(θ) is ${\mathcal{O}}(\mathrm{log}\,n)$. Our results establish a connection between locality and trainability. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation.

6

Date Added: May 20, 2021

Date Added: May 20, 2021

We introduce Performers, Transformer architectures which can estimate regular (softmax) full-rank-attention Transformers with provable accuracy, but using only linear (as opposed to quadratic) space and time complexity, without relying on any priors such as sparsity or low-rankness. To approximate softmax attention-kernels, Performers use a novel Fast Attention Via positive Orthogonal Random features approach (FAVOR+), which may be of independent interest for scalable kernel methods. FAVOR+ can be also used to efficiently model kernelizable attention mechanisms beyond softmax. This representational power is crucial to accurately compare softmax with other kernels for the first time on large-scale tasks, beyond the reach of regular Transformers, and investigate optimal attention-kernels. Performers are linear architectures fully compatible with regular Transformers and with strong theoretical guarantees: unbiased or nearly-unbiased estimation of the attention matrix, uniform convergence and low estimation variance. We tested Performers on a rich set of tasks stretching from pixel-prediction through text models to protein sequence modeling. We demonstrate competitive results with other examined efficient sparse and dense attention methods, showcasing effectiveness of the novel attention-learning paradigm leveraged by Performers.

3

Date Added: May 19, 2021

Date Added: May 19, 2021

There are several distinct failure modes for overoptimization of systems on
the basis of metrics. This occurs when a metric which can be used to improve a
system is used to an extent that further optimization is ineffective or
harmful, and is sometimes termed Goodhart's Law. This class of failure is often
poorly understood, partly because terminology for discussing them is ambiguous,
and partly because discussion using this ambiguous terminology ignores
distinctions between different failure modes of this general type. This paper
expands on an earlier discussion by Garrabrant, which notes there are "(at
least) four different mechanisms" that relate to Goodhart's Law. This paper is
intended to explore these mechanisms further, and specify more clearly how they
occur. This discussion should be helpful in better understanding these types of
failures in economic regulation, in public policy, in machine learning, and in
Artificial Intelligence alignment. The importance of Goodhart effects depends
on the amount of power directed towards optimizing the proxy, and so the
increased optimization power offered by artificial intelligence makes it
especially critical for that field.

4

Date Added: Mar 20, 2021

Date Added: Mar 20, 2021

We show how a number of well-known uncertainty principles for the Fourier transform, such as the Heisenberg uncertainty principle, the Donoho-Stark uncertainty principle, and Meshulam's nonabelian uncertainty principle, have little to do with the structure of the Fourier transform itself. Rather, all of these results follow from very weak properties of the Fourier transform (shared by numerous linear operators), namely that it is bounded as an operator , and that it is unitary. Using a single, simple proof template, and only these (or weaker) properties, we obtain some new proofs and many generalizations of these basic uncertainty principles, to new operators and to new settings, in a completely unified way. Together with our general overview, this paper can also serve as a survey of the many facets of the phenomena known as uncertainty principles.

7

Date Added: Sep 5, 2020

Date Added: Sep 5, 2020

The ‘Polymath Project’ proved that many minds can work together to solve difficult mathematical problems. Timothy Gowers and Michael Nielsen reflect on the lessons learned for open-source science.