Probabilistic Guarantees for Safe Deep Reinforcement Learning

Deep reinforcement learning has been successfully applied to many control tasks, but the application of such agents in safety-critical scenarios has been limited due to safety concerns. Rigorous testing of these controllers is challenging, particularly when they operate in probabilistic environments due to, for example, hardware faults or noisy sensors. We propose MOSAIC, an algorithm for measuring the safety of deep reinforcement learning agents in stochastic settings. Our approach is based on the iterative construction of a formal abstraction of a controller's execution in an environment, and leverages probabilistic model checking of Markov decision processes to produce probabilistic guarantees on safe behaviour over a finite time horizon. It produces bounds on the probability of safe operation of the controller for different initial configurations and identifies regions where correct behaviour can be guaranteed. We implement and evaluate our approach on agents trained for several benchmark control problems

Hair cells use active zones with different voltage dependence of Ca²⁺ influx to decompose sounds into complementary neural codes.

For sounds of a given frequency, spiral ganglion neurons (SGNs) with different thresholds and dynamic ranges collectively encode the wide range of audible sound pressures. Heterogeneity of synapses between inner hair cells (IHCs) and SGNs is an attractive candidate mechanism for generating complementary neural codes covering the entire dynamic range. Here, we quantified active zone (AZ) properties as a function of AZ position within mouse IHCs by combining patch clamp and imaging of presynaptic Ca2+ influx and by immunohistochemistry. We report substantial AZ heterogeneity whereby the voltage of half-maximal activation of Ca2+ influx ranged over ∼20 mV. Ca2+ influx at AZs facing away from the ganglion activated at weaker depolarizations. Estimates of AZ size and Ca2+ channel number were correlated and larger when AZs faced the ganglion. Disruption of the deafness gene GIPC3 in mice shifted the activation of presynaptic Ca2+ influx to more hyperpolarized potentials and increased the spontaneous SGN discharge. Moreover, Gipc3 disruption enhanced Ca2+ influx and exocytosis in IHCs, reversed the spatial gradient of maximal Ca2+ influx in IHCs, and increased the maximal firing rate of SGNs at sound onset. We propose that IHCs diversify Ca2+ channel properties among AZs and thereby contribute to decomposing auditory information into complementary representations in SGNs

Irreducible decomposition for tensor prodect representations of Jordanian quantum algebras

Tensor products of irreducible representations of the Jordanian quantum algebras U_h(sl(2)) and U_h(su(1,1)) are considered. For both the highest weight finite dimensional representations of U_h(sl(2)) and lowest weight infinite dimensional ones of U_h(su(1,1)), it is shown that tensor product representations are reducible and that the decomposition rules to irreducible representations are exactly the same as those of corresponding Lie algebras.Comment: LaTeX, 14pages, no figur

Classification of the quantum deformation of the superalgebra GL(1∣1)GL(1|1)

We present a classification of the possible quantum deformations of the supergroup $GL(1|1)$ and its Lie superalgebra $gl(1|1)$. In each case, the (super)commutation relations and the Hopf structures are explicitly computed. For each $R$ matrix, one finds two inequivalent coproducts whether one chooses an unbraided or a braided framework while the corresponding structures are isomorphic as algebras. In the braided case, one recovers the classical algebra $gl(1|1)$ for suitable limits of the deformation parameters but this is no longer true in the unbraided case.Comment: 23p LaTeX2e Document - packages amsfonts,subeqn - misprints and errors corrected, one section adde

Three dimensional quantum algebras: a Cartan-like point of view

A perturbative quantization procedure for Lie bialgebras is introduced and used to classify all three dimensional complex quantum algebras compatible with a given coproduct. The role of elements of the quantum universal enveloping algebra that, analogously to generators in Lie algebras, have a distinguished type of coproduct is discussed, and the relevance of a symmetrical basis in the universal enveloping algebra stressed. New quantizations of three dimensional solvable algebras, relevant for possible physical applications for their simplicity, are obtained and all already known related results recovered. Our results give a quantization of all existing three dimensional Lie algebras and reproduce, in the classical limit, the most relevant sector of the complete classification for real three dimensional Lie bialgebra structures given by X. Gomez in J. Math. Phys. Vol. 41. (2000) 4939.Comment: LaTeX, 15 page

Stochastic electrotransport selectively enhances the transport of highly electromobile molecules

Nondestructive chemical processing of porous samples such as fixed biological tissues typically relies on molecular diffusion. Diffusion into a porous structure is a slow process that significantly delays completion of chemical processing. Here, we present a novel electrokinetic method termed stochastic electrotransport for rapid nondestructive processing of porous samples. This method uses a rotational electric field to selectively disperse highly electromobile molecules throughout a porous sample without displacing the low-electromobility molecules that constitute the sample. Using computational models, we show that stochastic electrotransport can rapidly disperse electromobile molecules in a porous medium. We apply this method to completely clear mouse organs within 1–3 days and to stain them with nuclear dyes, proteins, and antibodies within 1 day. Our results demonstrate the potential of stochastic electrotransport to process large and dense tissue samples that were previously infeasible in time when relying on diffusion.Simons Foundation. Postdoctoral FellowshipLife Sciences Research FoundationBurroughs Wellcome Fund (Career Awards at the Scientific Interface)Searle Scholars ProgramMichael J. Fox Foundation for Parkinson's ResearchUnited States. Defense Advanced Research Projects AgencyJPB FoundationNational Institutes of Health (U.S.)National Institutes of Health (U.S.) (Grant 1-U01-NS090473-01

Maps between Deformed and Ordinary Gauge Fields

In this paper, we introduce a map between the q-deformed gauge fields defined on the GL$_{q}(N)$-covariant quantum hyperplane and the ordinary gauge fields. Perturbative analysis of the q-deformed QED at the classical level is presented and gauge fixing $\grave{a}$ la BRST is discussed. An other star product defined on the hybrid $(q,h)$% -plane is explicitly constructed .Comment: 10 page

Contractions, deformations and curvature

The role of curvature in relation with Lie algebra contractions of the pseudo-ortogonal algebras so(p,q) is fully described by considering some associated symmetrical homogeneous spaces of constant curvature within a Cayley-Klein framework. We show that a given Lie algebra contraction can be interpreted geometrically as the zero-curvature limit of some underlying homogeneous space with constant curvature. In particular, we study in detail the contraction process for the three classical Riemannian spaces (spherical, Euclidean, hyperbolic), three non-relativistic (Newtonian) spacetimes and three relativistic ((anti-)de Sitter and Minkowskian) spacetimes. Next, from a different perspective, we make use of quantum deformations of Lie algebras in order to construct a family of spaces of non-constant curvature that can be interpreted as deformations of the above nine spaces. In this framework, the quantum deformation parameter is identified as the parameter that controls the curvature of such "quantum" spaces.Comment: 17 pages. Based on the talk given in the Oberwolfach workshop: Deformations and Contractions in Mathematics and Physics (Germany, january 2006) organized by M. de Montigny, A. Fialowski, S. Novikov and M. Schlichenmaie

Quantum Symmetries and Marginal Deformations

We study the symmetries of the N=1 exactly marginal deformations of N=4 Super Yang-Mills theory. For generic values of the parameters, these deformations are known to break the SU(3) part of the R-symmetry group down to a discrete subgroup. However, a closer look from the perspective of quantum groups reveals that the Lagrangian is in fact invariant under a certain Hopf algebra which is a non-standard quantum deformation of the algebra of functions on SU(3). Our discussion is motivated by the desire to better understand why these theories have significant differences from N=4 SYM regarding the planar integrability (or rather lack thereof) of the spin chains encoding their spectrum. However, our construction works at the level of the classical Lagrangian, without relying on the language of spin chains. Our approach might eventually provide a better understanding of the finiteness properties of these theories as well as help in the construction of their AdS/CFT duals.Comment: 1+40 pages. v2: minor clarifications and references added. v3: Added an appendix, fixed minor typo

Classification of bicovariant differential calculi on the Jordanian quantum groups GL_{g,h}(2) and SL_{h}(2) and quantum Lie algebras

We classify all 4-dimensional first order bicovariant calculi on the Jordanian quantum group GL_{h,g}(2) and all 3-dimensional first order bicovariant calculi on the Jordanian quantum group SL_{h}(2). In both cases we assume that the bicovariant bimodules are generated as left modules by the differentials of the quantum group generators. It is found that there are 3 1-parameter families of 4-dimensional bicovariant first order calculi on GL_{h,g}(2) and that there is a single, unique, 3-dimensional bicovariant calculus on SL_{h}(2). This 3-dimensional calculus may be obtained through a classical-like reduction from any one of the three families of 4-dimensional calculi on GL_{h,g}(2). Details of the higher order calculi and also the quantum Lie algebras are presented for all calculi. The quantum Lie algebra obtained from the bicovariant calculus on SL_{h}(2) is shown to be isomorphic to the quantum Lie algebra we obtain as an ad-submodule within the Jordanian universal enveloping algebra U_{h}(sl(2)) and also through a consideration of the decomposition of the tensor product of two copies of the deformed adjoint module. We also obtain the quantum Killing form for this quantum Lie algebra.Comment: 33 pages, AMSLaTeX, misleading remark remove