Generalised Mixability, Constant Regret, and Bayesian Updating

Mixability of a loss is known to characterise when constant regret bounds are achievable in games of prediction with expert advice through the use of Vovk's aggregating algorithm. We provide a new interpretation of mixability via convex analysis that highlights the role of the Kullback-Leibler divergence in its definition. This naturally generalises to what we call $\Phi$-mixability where the Bregman divergence $D_\Phi$ replaces the KL divergence. We prove that losses that are $\Phi$-mixable also enjoy constant regret bounds via a generalised aggregating algorithm that is similar to mirror descent.Comment: 12 page

Generalized Mixability via Entropic Duality

Mixability is a property of a loss which characterizes when fast convergence is possible in the game of prediction with expert advice. We show that a key property of mixability generalizes, and the exp and log operations present in the usual theory are not as special as one might have thought. In doing this we introduce a more general notion of $\Phi$-mixability where $\Phi$ is a general entropy (\ie, any convex function on probabilities). We show how a property shared by the convex dual of any such entropy yields a natural algorithm (the minimizer of a regret bound) which, analogous to the classical aggregating algorithm, is guaranteed a constant regret when used with $\Phi$-mixable losses. We characterize precisely which $\Phi$ have $\Phi$-mixable losses and put forward a number of conjectures about the optimality and relationships between different choices of entropy.Comment: 20 pages, 1 figure. Supersedes the work in arXiv:1403.2433 [cs.LG

Fast rates in statistical and online learning

The speed with which a learning algorithm converges as it is presented with more data is a central problem in machine learning --- a fast rate of convergence means less data is needed for the same level of performance. The pursuit of fast rates in online and statistical learning has led to the discovery of many conditions in learning theory under which fast learning is possible. We show that most of these conditions are special cases of a single, unifying condition, that comes in two forms: the central condition for 'proper' learning algorithms that always output a hypothesis in the given model, and stochastic mixability for online algorithms that may make predictions outside of the model. We show that under surprisingly weak assumptions both conditions are, in a certain sense, equivalent. The central condition has a re-interpretation in terms of convexity of a set of pseudoprobabilities, linking it to density estimation under misspecification. For bounded losses, we show how the central condition enables a direct proof of fast rates and we prove its equivalence to the Bernstein condition, itself a generalization of the Tsybakov margin condition, both of which have played a central role in obtaining fast rates in statistical learning. Yet, while the Bernstein condition is two-sided, the central condition is one-sided, making it more suitable to deal with unbounded losses. In its stochastic mixability form, our condition generalizes both a stochastic exp-concavity condition identified by Juditsky, Rigollet and Tsybakov and Vovk's notion of mixability. Our unifying conditions thus provide a substantial step towards a characterization of fast rates in statistical learning, similar to how classical mixability characterizes constant regret in the sequential prediction with expert advice setting.Comment: 69 pages, 3 figure

Generalised Pinsker Inequalities

We generalise the classical Pinsker inequality which relates variational divergence to Kullback-Liebler divergence in two ways: we consider arbitrary f-divergences in place of KL divergence, and we assume knowledge of a sequence of values of generalised variational divergences. We then develop a best possible inequality for this doubly generalised situation. Specialising our result to the classical case provides a new and tight explicit bound relating KL to variational divergence (solving a problem posed by Vajda some 40 years ago). The solution relies on exploiting a connection between divergences and the Bayes risk of a learning problem via an integral representation.Comment: 21 pages, 3 figures, accepted to COLT 200

The Revolution Will Be Open-Source: How 3D Bioprinting Can Change 3D Cell Culture

(First paragraph) The development of three-dimensional culture scaffolds represents a revolutionary step forward for in vitro culture systems. Various synthetic and naturally occurring substrates have been developed that support 3D growth of cells. In most fields, including mammary gland biology and tumorigenesis, the two most common substrates used are the basement membrane rich extracellur matrix (ECM) isolated from EngelbrethHolm-Swarm (EHS) mouse sarcomas (e.g. Matrigel) and collagen extracted from rat-tails. The processes of 3D culture in these two substrates has remained unchanged for nearly half a century: cells are either mixed with unpolymerized matrix to disperse them randomly throughout the substrate upon polymerization or overlaid randomly on top of a preformed hydrogel. While effective in generating organoid/tumoroid structures, the random nature of these processes has many drawbacks that limit the reproducibility and tunability of the experimental design. Furthermore, random cellular distributions limit the utility of these substrates for studying interactions within the cellular microenvironment, which have been shown to be critical for the control of stem and cancer cell function [1]

Improved Models of Cable-to-Post Attachments for High-Tension Cable Barriers

Computer simulation models were developed to analyze and evaluate a new cable-to-post attachment for high-tension cable barriers. The models replicated the performance of a keyway bolt currently used in the design of a high-tension cable median barrier being developed at the Midwest Roadside Safety Facility. Component tests of the keyway bolts were simulated and compared to the component test results. Accurate friction, fracture strain, and stress-strain material properties were determined for a solid element model of the keyway bolt by applying actual load curve measured from the test to a simulated pull cable. By simulating the material properties of the solid element keyway bolt in bending, torsion, and tension of a rod, load curves were developed for a simplified beam element model of the keyway bolt as well. When material properties were finalized, the solid and beam element models of the keyway bolt were inserted in bogie test models and simulated again. By analyzing the bogie testing results, it was determined that due to the very small size of the keyway bolt and potential contact difficulties, solid element models of the keyway bolt may be impractical for full-scale simulation purposes. However, the beam element models were determined to be advantageous and had a very small computational cost in comparison

Continued Development of a Non-Proprietary, High-Tension, Cable End Terminal System

A non-proprietary, cable guardrail system is currently under development for the Midwest States Pooled Fund Program. A cable guardrail end terminal was necessary to accompany the cable guardrail system. The objective of this research project was to develop design recommendations for the cable end terminal. Bogie testing that was previously completed on a design concept indicated delayed cable release, which was an undesired performance that led to vehicle instabilities. Several design changes were recommended for better end terminal performance and to reflect the changes made to the cable median barrier. An LS-DYNA model of the modified cable end terminal was developed. Simulations of 0- and 15-degree impacts on the end of the cable anchor bracket with a bogie model indicated that the cables would release easily and not induce vehicle instabilities. This behavior still needs to be verified through bogie and full-scale crash testing. Simulations of a 25-degree reverse impact between post nos. 2 and 3 with small car models indicated that cables did not release easily and may interlock around the car resulting in excessive vehicle decelerations or instabilities. Simulations of various line post designs found that the MWP and weakened MWP have lower forces and energies during impact than the S3x5.7 posts used in previous three-cable end terminals. This finding would suggest improved performance with respect to vehicle override and instability. However, vehicle simulations with multiple line posts impacted were inconclusive. Further design modifications, evaluation, and testing are recommended

Continued Development of a Non-Proprietary, High-Tension, Cable End Terminal System

A non-proprietary, cable guardrail system is currently under development for the Midwest States Pooled Fund Program. A cable guardrail end terminal was necessary to accompany the cable guardrail system. The objective of this research project was to develop design recommendations for the cable end terminal. Bogie testing that was previously completed on a design concept indicated delayed cable release, which was an undesired performance that led to vehicle instabilities. Several design changes were recommended for better end terminal performance and to reflect the changes made to the cable median barrier. An LS-DYNA model of the modified cable end terminal was developed. Simulations of 0- and 15-degree impacts on the end of the cable anchor bracket with a bogie model indicated that the cables would release easily and not induce vehicle instabilities. This behavior still needs to be verified through bogie and full-scale crash testing. Simulations of a 25-degree reverse impact between post nos. 2 and 3 with small car models indicated that cables did not release easily and may interlock around the car resulting in excessive vehicle decelerations or instabilities. Simulations of various line post designs found that the MWP and weakened MWP have lower forces and energies during impact than the S3x5.7 posts used in previous three-cable end terminals. This finding would suggest improved performance with respect to vehicle override and instability. However, vehicle simulations with multiple line posts impacted were inconclusive. Further design modifications, evaluation, and testing are recommended