Bounding Embeddings of VC Classes into Maximum Classes

One of the earliest conjectures in computational learning theory-the Sample Compression conjecture-asserts that concept classes (equivalently set systems) admit compression schemes of size linear in their VC dimension. To-date this statement is known to be true for maximum classes---those that possess maximum cardinality for their VC dimension. The most promising approach to positively resolving the conjecture is by embedding general VC classes into maximum classes without super-linear increase to their VC dimensions, as such embeddings would extend the known compression schemes to all VC classes. We show that maximum classes can be characterised by a local-connectivity property of the graph obtained by viewing the class as a cubical complex. This geometric characterisation of maximum VC classes is applied to prove a negative embedding result which demonstrates VC-d classes that cannot be embedded in any maximum class of VC dimension lower than 2d. On the other hand, we show that every VC-d class C embeds in a VC-(d+D) maximum class where D is the deficiency of C, i.e., the difference between the cardinalities of a maximum VC-d class and of C. For VC-2 classes in binary n-cubes for 4 <= n <= 6, we give best possible results on embedding into maximum classes. For some special classes of Boolean functions, relationships with maximum classes are investigated. Finally we give a general recursive procedure for embedding VC-d classes into VC-(d+k) maximum classes for smallest k.Comment: 22 pages, 2 figure

Benign overfitting in ridge regression

Classical learning theory suggests that strong regularization is needed to learn a class with large complexity. This intuition is in contrast with the modern practice of machine learning, in particular learning neural networks, where the number of parameters often exceeds the number of data points. It has been observed empirically that such overparametrized models can show good generalization performance even if trained with vanishing or negative regularization. The aim of this work is to understand theoretically how this effect can occur, by studying the setting of ridge regression. We provide non-asymptotic generalization bounds for overparametrized ridge regression that depend on the arbitrary covariance structure of the data, and show that those bounds are tight for a range of regularization parameter values. To our knowledge this is the first work that studies overparametrized ridge regression in such a general setting. We identify when small or negative regularization is sufficient for obtaining small generalization error. On the technical side, our bounds only require the data vectors to be i.i.d. sub-gaussian, while most previous work assumes independence of the components of those vectors.Comment: 28 page

Fractionation effects in phase equilibria of polydisperse hard sphere colloids

The equilibrium phase behaviour of hard spheres with size polydispersity is studied theoretically. We solve numerically the exact phase equilibrium equations that result from accurate free energy expressions for the fluid and solid phases, while accounting fully for size fractionation between coexisting phases. Fluids up to the largest polydispersities that we can study (around 14%) can phase separate by splitting off a solid with a much narrower size distribution. This shows that experimentally observed terminal polydispersities above which phase separation no longer occurs must be due to non-equilibrium effects. We find no evidence of re-entrant melting; instead, sufficiently compressed solids phase separate into two or more solid phases. Under appropriate conditions, coexistence of multiple solids with a fluid phase is also predicted. The solids have smaller polydispersities than the parent phase as expected, while the reverse is true for the fluid phase, which contains predominantly smaller particles but also residual amounts of the larger ones. The properties of the coexisting phases are studied in detail; mean diameter, polydispersity and volume fraction of the phases all reveal marked fractionation. We also propose a method for constructing quantities that optimally distinguish between the coexisting phases, using Principal Component Analysis in the space of density distributions. We conclude by comparing our predictions to perturbative theories for near-monodisperse systems and to Monte Carlo simulations at imposed chemical potential distribution, and find excellent agreement.Comment: 21 pages, 23 figures, 2 table

Experiments with Infinite-Horizon, Policy-Gradient Estimation

In this paper, we present algorithms that perform gradient ascent of the average reward in a partially observable Markov decision process (POMDP). These algorithms are based on GPOMDP, an algorithm introduced in a companion paper (Baxter and Bartlett, this volume), which computes biased estimates of the performance gradient in POMDPs. The algorithm's chief advantages are that it uses only one free parameter beta, which has a natural interpretation in terms of bias-variance trade-off, it requires no knowledge of the underlying state, and it can be applied to infinite state, control and observation spaces. We show how the gradient estimates produced by GPOMDP can be used to perform gradient ascent, both with a traditional stochastic-gradient algorithm, and with an algorithm based on conjugate-gradients that utilizes gradient information to bracket maxima in line searches. Experimental results are presented illustrating both the theoretical results of (Baxter and Bartlett, this volume) on a toy problem, and practical aspects of the algorithms on a number of more realistic problems

Further studies of the coupled chemically reacting boundary layer and charring ablator. Part 1 - Summary Final report

Computer program development for charring ablative materials, chemically reacting laminar boundary layers, and turbulent boundary layer initiatio

Law and Corporate Governance

Pragmatic and effective research on corporate governance often turns critically on appreciating the legal institutions surrounding corporate entities – yet such nuances are often unfamiliar or poorly specified to economists and other social scientists without legal training. This chapter organizes and discusses key legal concepts of corporate governance, including statutes, regulations, and jurisprudential doctrines that “govern governance” in private and public companies, with concentration on the for-profit corporation. We review the literature concerning the nature and purpose of the corporation, the objects of fiduciary obligations, the means for decision making within the firm, as well as the overlay of state and federal law pertaining to how that decision-making authority is exercised within publicly traded companies. A core feature of this analysis is that while the basic structures pertinent to corporate law and governance are familiar and in some ways predictable, they are also in a constant state of flux, shaping and being shaped by institutional adaptations of firms, regulators and courts. This chapter is most appropriate for social science researchers and/or students who are new to the legal dimensions of firm governance

MACiE: a database of enzyme reaction mechanisms.

SUMMARY: MACiE (mechanism, annotation and classification in enzymes) is a publicly available web-based database, held in CMLReact (an XML application), that aims to help our understanding of the evolution of enzyme catalytic mechanisms and also to create a classification system which reflects the actual chemical mechanism (catalytic steps) of an enzyme reaction, not only the overall reaction. AVAILABILITY: http://www-mitchell.ch.cam.ac.uk/macie/.EPSRC (G.L.H. and J.B.O.M.), the BBSRC (G.J.B. and J.M.T.—CASE studentship in association with Roche Products Ltd; N.M.O.B. and J.B.O.M.—grant BB/C51320X/1), the Chilean Government’s Ministerio de Planificacio´n y Cooperacio´n and Cambridge Overseas Trust (D.E.A.) for funding and Unilever for supporting the Centre for Molecular Science Informatics.application note restricted to 2 printed pages web site: http://www-mitchell.ch.cam.ac.uk/macie

SXP 7.92: A Recently Rediscovered Be/X-ray Binary in the Small Magellanic Cloud, Viewed Edge On

We present a detailed optical and X-ray study of the 2013 outburst of the Small Magellanic Cloud Be/X-ray binary SXP 7.92, as well as an overview of the last 18 years of observations from OGLE (Optical Gravitational Lensing Experiment), RXTE, Chandra and XMM-Newton. We revise the position of this source to RA(J2000) = 00:57:58.4, Dec(J2000) = −72:22:29.5 with a 1σ uncertainty of 1.5 arcsec, correcting the previously reported position by Coe et al. by more than 20 arcmin. We identify and spectrally classify the correct counterpart as a B1Ve star. The optical spectrum is distinguished by an uncharacteristically deep narrow Balmer series, with the Hα line in particular having a distinctive shell profile, i.e. a deep absorption core embedded in an emission line. We interpret this as evidence that we are viewing the system edge on and are seeing self-obscuration of the circumstellar disc. We derive an optical period for the system of 40.0 ± 0.3 d, which we interpret as the orbital period, and present several mechanisms to describe the X-ray/optical behaviour in the recent outburst, in particular the ‘flares'and ‘dips’ seen in the optical light curve, including a transient accretion disc and an elongated precessing disc