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We prove oscillation stability for the Banach space $\ell_\infty$: every weak-* Borel, uniformily continuous map from the unit sphere of this space to a compact metric space can be made arbitrarily close to a constant map when restricted to the unit sphere of a suitable linear isometric subcopy of $\ell_\infty$. We also give a new proof of oscillation stability for the Urysohn sphere (a result by Nguyen Van Th\'e--Sauer): every uniformily continuous map from the Urysohn sphere to a compact metric space can be made arbitrarily close to a constant map when restricted to a suitable isometric subcopy of the Urysohn sphere. Both proofs are based on Carlson-Simpson's dual Ramsey theorem., Comment: 14 pages

In the 1970s, structural Ramsey theory emerged as a new branch of combinatorics. This development came with the isolation of the concepts of the $\mathbf{A}$-Ramsey property and Ramsey class. Following the influential Ne\v{s}et\v{r}il-R\"{o}dl theorem, several Ramsey classes have been identified. In the 1980s Ne\v{s}et\v{r}il, inspired by a seminar of Lachlan, discovered a crucial connection between Ramsey classes and Fra\"{\i}ss\'{e} classes and, in his 1989 paper, connected the classification programme of homogeneous structures to structural Ramsey theory. In 2005, Kechris, Pestov, and Todor\v{c}evi\'{c} revitalized the field by connecting Ramsey classes to topological dynamics. This breakthrough motivated Ne\v{s}et\v{r}il to propose a program for classifying Ramsey classes. We review the progress made on this program in the past two decades, list open problems, and discuss recent extensions to new areas, namely the extension property for partial automorphisms (EPPA), and big Ramsey structures.

Dufaycolor, an additive color photography process produced from 1935 to the late 1950s, represents one of the most advanced iterations of this technique. This paper presents ongoing research and development of an open-source Color-Screen tool designed to reconstruct the original colors of additive color photographs. We discuss the incorporation of historical measurements of dyes used in the production of the color-screen filter (r\'eseau) to achieve accurate color recovery., Comment: 8 pages, 6 figures, 4 tables; submitted to proceedings of 3rd international conference on "Colour Photography and Film: analysis, preservation, and conservation of analogue and digital materials"

Building on our prior work on axiomatization of exact real computation by formalizing nondeterministic first-order partial computations over real and complex numbers in a constructive dependent type theory, we present a framework for certified computation on hyperspaces of subsets by formalizing various higher-order data types and operations. We first define open, closed, compact and overt subsets for generic spaces in an abstract topological way that allows short and elegant proofs with computational content coinciding with standard definitions in computable analysis and constructive mathematics. From these proofs we can extract programs for testing inclusion, overlapping of sets, et cetera. To enhance the efficiency of the extracted programs, we then focus on Polish spaces, where we give more efficient encodings based on metric properties of the space. As various computational properties depend on the continuity of the encoding functions, we introduce a nondeterministic version of a continuity principle which is natural in our formalization and valid under the standard type-2 realizability interpretation. Using this principle we further derive the computational equivalence between the generic and the metric encodings. Our theory is fully implemented in the Coq proof assistant. From proofs in this Coq formalization, we can extract certified programs for error-free operations on subsets. As an application, we provide a function that constructs fractals in Euclidean space, such as the Sierpinski triangle, from iterated function systems using the limit operation. The resulting programs can be used to draw such fractals up to any desired resolution.

We present the theoretical derivation and numerical implementation of the linear response equations for relativistic quantum electrodynamical density functional theory (QEDFT). In contrast to previous works based on the Pauli-Fierz Hamiltonian, our approach describes electrons interacting with photonic cavity modes at the four-component Dirac-Kohn-Sham level, derived from fully relativistic QED through a series of established approximations. Moreover, we show that a new type of spin-orbit-like (SO) cavity-mediated interaction appears under the relativistic description of the coupling of matter with quantized cavity modes. Benchmark calculations performed for atoms of group 12 elements (Zn, Cd, Hg) demonstrate how a relativistic treatment enables the description of exciton polaritons which arise from the hybridization of formally forbidden singlet-triplet transitions with cavity modes. For atoms in cavities tuned on resonance with a singlet-triplet transition we discover a significant interplay between SO effects and coupling to an off-resonant intense singlet-singlet transition. This dynamic relationship highlights the crucial role of ab initio approaches in understanding cavity quantum electrodynamics. Finally, using the mercury porphyrin complex as an example, we show that relativistic linear response QEDFT provides computationally feasible first-principles calculations of polaritonic states in large heavy element-containing molecules of chemical interest., Comment: 29 pages, 5 figures, supplemental material, v2: Corrected references to the Supplemental Material

We present a proof of the extension property for partial automorphisms (EPPA) for classes of finite $n$-partite tournaments for $n \in \{2,3,\ldots,\omega\}$, and for the class of finite semigeneric tournaments. We also prove that the generic $\omega$-partite tournament and the generic semigeneric tournament have ample generics., Comment: Revision mainly improves the presentation of the construction in Section 5

If $G$ is a graph, $A$ and $B$ its induced subgraphs, and $f\colon A\to B$ an isomorphism, we say that $f$ is a \emph{partial automorphism} of $G$. In 1992, Hrushovski proved that graphs have the \emph{extension property for partial automorphisms} (\emph{EPPA}, also called the \emph{Hrushovski property}), that is, for every finite graph $G$ there is a finite graph $H$, an \emph{EPPA-witness} for $G$, such that $G$ is an induced subgraph of $H$ and every partial automorphism of $G$ extends to an automorphism of $H$. The EPPA number of a graph $G$, denoted by $\mathop{\mathrm{eppa}}\nolimits(G)$, is the smallest number of vertices of an EPPA-witness for $G$, and we put $\mathop{\mathrm{eppa}}\nolimits(n) = \max\{\mathop{\mathrm{eppa}}\nolimits(G) : \lvert G\rvert = n\}$. In this note we review the state of the area, prove several lower bounds (in particular, we show that $\mathop{\mathrm{eppa}}\nolimits(n)\geq \frac{2^n}{\sqrt{n}}$, thereby identifying the correct base of the exponential) and pose many open questions. We also briefly discuss EPPA numbers of hypergraphs, directed graphs, and $K_k$-free graphs., Comment: Minor revision

We prove a general Ramsey theorem for trees with a successor operation. This theorem is a common generalization of the Carlson-Simpson Theorem and the Milliken Tree Theorem for regularly branching trees. Our theorem has a number of applications both in finite and infinite combinatorics. For example, we give a short proof of the unrestricted Ne\v{s}et\v{r}il-R\"odl theorem, and we recover the Graham-Rothschild theorem. Our original motivation came from the study of big Ramsey degrees - various trees used in the study can be viewed as trees with a successor operation. To illustrate this, we give a non-forcing proof of a theorem of Zucker on big Ramsey degrees., Comment: 37 pages, 9 figures

Recent advances in laser technology enable to follow electronic motion at its natural time-scale with ultrafast pulses, leading the way towards atto- and femtosecond spectroscopic experiments of unprecedented resolution. Understanding of these laser-driven processes, which almost inevitably involve non-linear light-matter interactions and non-equilibrium electron dynamics, is challenging and requires a common effort of theory and experiment. Real-time electronic structure methods provide the most straightforward way to simulate experiments and to gain insights into non-equilibrium electronic processes. In this Chapter, we summarize the fundamental theory underlying the relativistic particle-field interaction Hamiltonian as well as equation-of-motion for exact-state wave function in terms of the one- and two-electron reduced density matrix. Further, we discuss the relativistic real-time electron dynamics mean-field methods with an emphasis on Density-Functional Theory and Gaussian basis, starting from the four-component (Dirac) picture and continue to the two-component (Pauli) picture, where we introduce various flavours of modern exact two-component (X2C) Hamiltonians for real-time electron dynamics. We also overview several numerical techniques for real-time propagation and signal processing in quantum electron dynamics. We close this Chapter by listing selected applications of real-time electron dynamics to frequency-resolved and time-resolved spectroscopies.

We present a novel function fitting method for approximating the propagation of the time-dependent electric dipole moment from real-time electronic structure calculations. Real-time calculations of the electronic absorption spectrum require discrete Fourier transforms of the electric dipole moment. The spectral resolution is determined by the total propagation time, i.e. the trajectory length of the dipole moment, causing a high computational cost. Our developed method uses function fitting on shorter trajectories of the dipole moment, achieving arbitrary spectral resolution through extrapolation. Numerical testing shows that the fitting method can reproduce high-resolution spectra using short dipole trajectories. The method converges with as little as 100 a.u. dipole trajectories for some systems, though the difficulty converging increases with the spectral density. We also introduce an error estimate of the fit, reliably assessing its convergence and hence the quality of the approximated spectrum.

Oscillation stability is an important concept in Banach space theory which happens to be closely connected to discrete Ramsey theory. For example, Gowers proved oscillation stability for the Banach space $c_0$ using his now famous Ramsey theorem for $\mathrm{FIN}_k$ as the key ingredient. We develop the theory behind this connection and introduce the notion of compact big Ramsey degrees, extending the theory of (discrete) big Ramsey degrees. We then prove existence of compact big Ramsey degrees for the Banach space $\ell_\infty$ and the Urysohn sphere, with an explicit characterization in the case of $\ell_\infty$., Comment: 6 pages; extended abstract

We give an infinitary extension of the Ne\v{s}et\v{r}il-R\"{o}dl theorem for category of relational structures with special type-respecting embeddings., Comment: 5 pages. Extended abstract

Antiferromagnets are promising materials for future opto-spintronic applications since they show spin dynamics in the THz range and no net magnetization. Recently, layered van der Waals (vdW) antiferromagnets have been reported, which combine low-dimensional excitonic properties with complex spin-structure. While various methods for the fabrication of vdW 2D crystals exist, formation of large area and continuous thin films is challenging because of either limited scalability, synthetic complexity, or low opto-spintronic quality of the final material. Here, we fabricate centimeter-scale thin films of the van der Waals 2D antiferromagnetic material NiPS3, which we prepare using a crystal ink made from liquid phase exfoliation (LPE). We perform statistical atomic force microscopy (AFM) and scanning electron microscopy (SEM) to characterize and control the lateral size and number of layers through this ink-based fabrication. Using ultrafast optical spectroscopy at cryogenic temperatures, we resolve the dynamics of photoexcited excitons. We find antiferromagnetic spin arrangement and spin-entangled Zhang-Rice multiplet excitons with lifetimes in the nanosecond range, as well as ultranarrow emission linewidths, despite the disordered nature of our films. Thus, our findings demonstrate scalable thin-film fabrication of high-quality NiPS3, which is crucial for translating this 2D antiferromagnetic material into spintronic and nanoscale memory devices and further exploring its complex spin-light coupled states.

As a result of 33 intercontinental Zoom calls, we characterise big Ramsey degrees of the generic partial order. This is an infinitary extension of the well known fact that finite partial orders endowed with linear extensions form a Ramsey class (this result was announced by Ne\v{s}et\v{r}il and R\"odl in 1984 with first published proof by Paoli, Trotter and Walker in 1985). Towards this, we refine earlier upper bounds obtained by Hubi\v{c}ka based on a new connection of big Ramsey degrees to the Carlson-Simpson theorem and we also introduce a new technique of giving lower bounds using an iterated application of the upper-bound theorem., Comment: 30 pages, 7 figures. Minor revision incorporating suggestions of the second anonymous referee

This paper investigates big Ramsey degrees of unrestricted relational structures in (possibly) infinite languages. Despite significant progress in the study of big Ramsey degrees, the big Ramsey degrees of many classes of structures with finite small Ramsey degrees are still not well understood. We show that if there are only finitely many relations of every arity greater than one, then unrestricted relational structures have finite big Ramsey degrees, and give some evidence that this is tight. This is the first time finiteness of big Ramsey degrees has been established for a random structure in an infinite language. Our results represent an important step towards a better understanding of big Ramsey degrees for structures with relations of arity greater than two., Comment: 26 pages

First principle theoretical modeling of out-of-equilibrium processes observed in attosecond pump-probe transient absorption spectroscopy (TAS) triggering pure electron dynamics remains a challenging task, specially for heavy elements and/or core excitations containing fingerprints of scalar and spin-orbit relativistic effects. To address this, we formulate a methodology for simulating TAS within the relativistic real-time time-dependent density functional theory (RT-TDDFT) framework, for both the valence and core energy regime. Especially for TAS, full four-component (4c) RT simulations are feasible but computationally demanding. Therefore, in addition to the 4c approach, we also introduce the atomic mean-field exact two-component (amfX2C) Hamiltonian accounting for one- and two-electron picture-change corrections within RT-TDDFT. amfX2C preserves the accuracy of the parent 4c method at a fraction of its computational cost. Finally, we apply the methodology to study valence and near L$_{2,3}$-edge TAS processes of experimentally relevant systems and provide additional physical insights using relativistic non-equilibrium response theory.

X-ray absorption spectroscopy (XAS) has gained popularity in recent years as it probes matter with high spatial and elemental sensitivity. However, the theoretical modelling of XAS is a challenging task since XAS spectra feature a fine structure due to scalar (SC) and spin-orbit (SO) relativistic effects, in particular near L and M absorption edges. While full four-component (4c) calculations of XAS are nowadays feasible, there is still interest in developing approximate relativistic methods that enable XAS calculations at the two-component (2c) level while maintaining the accuracy of the parent 4c approach. In this article we present theoretical and numerical insights into two simple yet accurate 2c approaches based on an (extended) atomic mean-field exact two-component Hamiltonian framework, (e)amfX2C, for the calculation of XAS using linear eigenvalue and damped-response time-dependent density functional theory (TDDFT). In contrast to the commonly used one-electron X2C (1eX2C) Hamiltonian, both amfX2C and eamfX2C account for the SC and SO two-electron and exchange-correlation picture-change (PC) effects that arise from the X2C transformation. As we demonstrate on L- and M-edge XAS spectra of transition metal and actinide compounds, the absence of PC corrections in the 1eX2C approximation results in a substantial overestimatation of SO splittings, whereas (e)amfX2C Hamiltonians reproduce all essential spectral features such as shape, position, and SO splitting of the 4c references in excellent agreement, while offering significant computational savings. Therefore, the (e)amfX2C PC correction models presented here constitute reliable relativistic 2c quantum-chemical approaches for modelling XAS.

We give a process for verifying numerical programs against their functional specifications. Our implementation is capable of automatically verifying programs against tight error bounds featuring common elementary functions. We demonstrate and evaluate our implementation on several examples, yielding the first fully verified SPARK implementations of the sine and square root functions. The process integrates existing tools using a series of transformations and derivations, building on the proving process in SPARK where Why3 produces Verification Conditions (VCs) and tools such as SMT solvers attempt to verify them. We add steps aimed specifically at VCs that contain inequalities with both floating-point operations and exact real functions. PropaFP is our open-source implementation of these steps. The steps include symbolic simplifications, deriving bounds via interval arithmetic, and safely replacing floating-point operations with exact operations, utilizing tools such as FPTaylor or Gappa to bound the compound rounding errors of expressions. Finally, the VCs are passed to provers such as dReal, MetiTarski or LPPaver which attempt to complete the proof or suggest possible counter-examples., Comment: 26 pages, 2 figures, 24 listings, 6 tables, to be submitted to SEFM2022 (see https://sefm-conference.github.io/2022/ )

Exact real computation is an alternative to floating-point arithmetic where operations on real numbers are performed exactly, without the introduction of rounding errors. When proving the correctness of an implementation, one can focus solely on the mathematical properties of the problem without thinking about the subtleties of representing real numbers. We propose a new axiomatization of the real numbers in a dependent type theory with the goal of extracting certified exact real computation programs from constructive proofs. Our formalization differs from similar approaches, in that we formalize the reals in a conceptually similar way as some mature implementations of exact real computation. Primitive operations on reals can be extracted directly to the corresponding operations in such an implementation, producing more efficient programs. We particularly focus on the formalization of partial and nondeterministic computation, which is essential in exact real computation. We prove the soundness of our formalization with regards of the standard realizability interpretation from computable analysis and show how to relate our theory to a classical formalization of the reals. We demonstrate the feasibility of our theory by implementing it in the Coq proof assistant and present several natural examples. From the examples we have automatically extracted Haskell programs that use the exact real computation framework AERN for efficiently performing exact operations on real numbers. In experiments, the extracted programs behave similarly to native implementations in AERN in terms of running time and memory usage.

Phthalocyanines as organic semiconductors and molecular magnets provide plenty of industrial or high-tech applications from dyes and pigments up to gas sensors, molecular electronics, spintronics and quantum computing. Copper phthalocyanine (CuPc) belongs among the most used phthalocyanines, typically in the form of powder or films but self-grown nanowires are also known. Here we describe an opposite, i.e., top-down approach based on fabrication of ordered arrays of CuPc microstructures (microdisks) using electron beam lithography and other steps. Among critical points of this approach belongs a choice of a proper resist and a solvent. Fabricated CuPc microdisks have a diameter of 5 ${\mu}$m and heights from 7 up to 70 nm. Micro-Raman spectroscopy of the films and microdisks reveals a crystalline ${\beta}$ phase associated with a paramagnetic form. Additional measurements with an increasing laser power show a significant shift (${\Delta}{\omega}$ ~ 7.1 cm$^{-1}$ ) and broadening of a peak at 1532 rel$\cdot$cm$^{-1}$ corresponding to the phonon B1g mode. The observed smooth changes exclude a phase transition and confirm the thermally stable polymorph. Our versatile fabrication technique using the common lithographic resist brings new possibilities for the fabrication of various micro/nanostructures such as micromagnets, heterostructures or organic electronic devices.

Surface enhanced Raman spectroscopy (SERS) is a perspective non-destructive analytic technique enabling detection of individual nanoobjects, even single-molecules. . In the paper, we have studied the morphology of Ga islands deposited on CVD graphene by ultrahigh vacuum (UHV) evaporation and local optical response of this system by the correlative Raman Imaging and Scanning Electron Microscopy (RISE). Contrary to the previous papers, where only an integral Raman response from the whole ununiformed Ga NPs ensembles on graphene was investigated, the RISE technique has enabled us to detect graphene Raman peaks enhanced by single Ga islands and particularly to correlate the Raman signal with the shape and size of these single particles. In this way and by a support of numerical simulations, we have proved a plasmonic nature of the Raman signal enhancement related to localized surface plasmon resonances (LSPR). It has been found that this enhancement is island-size dependent and shows a maximum for medium-sized Ga islands. A reasonable agreement between the simulations of the plasmon enhancement of electric fields in the vicinity of Ga islands and the experimental intensities of corresponding Raman peaks proved the plasmonic origin of the observed effect known as the Surface Enhanced Raman Spectroscopy., Comment: 18 pages, 7 figures

As the characteristic dimensions of modern top-down devices are getting smaller, such devices reach their operational limits given by quantum mechanics. Thus, two-dimensional (2D) structures appear as one of the best solutions to meet the ultimate challenges of modern optoelectronic and spintronic applications. The representative of III-V semiconductors, gallium nitride (GaN), is a great candidate for UV and high-power applications at a nanoscale level. We propose a new way of fabrication of 2D GaN on the Si(111) 7x7 surface using post-nitridation of Ga droplets by hyperthermal (E < 50 eV) nitrogen ions at low substrate temperatures (T < 220{\deg}C). Both deposition of Ga droplets and their post-nitridation is carried out using a special atom-ion beam source developed in our group. This low-temperature droplet epitaxy (LTDE) approach provides well-defined ultra-high vacuum growth conditions during the whole fabrication process resulting in high purity GaN nanostructures. A sharp interface between the GaN nanostructures and the silicon substrate together with a correct elemental composition of nanostructures was confirmed by TEM. In addition, SEM, X-ray photoelectron spectroscopy (XPS), AFM and Auger microanalysis showed unique characteristics of the fabricated GaN nanostructures., Comment: 14 pages, 5 figures

A significant bottleneck in federated learning (FL) is the network communication cost of sending model updates from client devices to the central server. We present a comprehensive empirical study of the statistics of model updates in FL, as well as the role and benefits of various compression techniques. Motivated by these observations, we propose a novel method to reduce the average communication cost, which is near-optimal in many use cases, and outperforms Top-K, DRIVE, 3LC and QSGD on Stack Overflow next-word prediction, a realistic and challenging FL benchmark. This is achieved by examining the problem using rate-distortion theory, and proposing distortion as a reliable proxy for model accuracy. Distortion can be more effectively used for optimizing the trade-off between model performance and communication cost across clients. We demonstrate empirically that in spite of the non-i.i.d. nature of federated learning, the rate-distortion frontier is consistent across datasets, optimizers, clients and training rounds.

We characterize the big Ramsey degrees of free amalgamation classes in finite binary languages defined by finitely many forbidden irreducible substructures, thus refining the recent upper bounds given by Zucker. Using this characterization, we show that the Fra\"iss\'e limit of each such class admits a strong big Ramsey structure, implying that the automorphism group of the Fra\"iss\'e limit has a metrizable universal completion flow., Comment: Revised version; formerly titled "Exact big Ramsey degrees via coding trees."

Federated learning and analytics are a distributed approach for collaboratively learning models (or statistics) from decentralized data, motivated by and designed for privacy protection. The distributed learning process can be formulated as solving federated optimization problems, which emphasize communication efficiency, data heterogeneity, compatibility with privacy and system requirements, and other constraints that are not primary considerations in other problem settings. This paper provides recommendations and guidelines on formulating, designing, evaluating and analyzing federated optimization algorithms through concrete examples and practical implementation, with a focus on conducting effective simulations to infer real-world performance. The goal of this work is not to survey the current literature, but to inspire researchers and practitioners to design federated learning algorithms that can be used in various practical applications.

We study Ramsey expansions of certain homogeneous 3-hypertournaments. We show that they exhibit an interesting behaviour and, in one case, they seem not to submit to current gold-standard methods for obtaining Ramsey expansions. This makes these examples very interesting from the point of view of structural Ramsey theory as there is a large demand for novel examples., Comment: Extended abstract accepted to EUROCOMB 2021

Using the Carlson-Simpson theorem, we give a new general condition for a structure in a finite binary relational language to have finite big Ramsey degrees, Comment: 6 pages, extended abstract accepted to EUROCOMB 2021

As a result of 33 intercontinental Zoom calls, we characterise big Ramsey degrees of the generic partial order in a similar way as Devlin characterised big Ramsey degrees of the generic linear order (the order of rationals)., Comment: 6 pages, extended abstract accepted to EUROCOMB 2021

We study a family of algorithms, which we refer to as local update methods, generalizing many federated and meta-learning algorithms. We prove that for quadratic models, local update methods are equivalent to first-order optimization on a surrogate loss we exactly characterize. Moreover, fundamental algorithmic choices (such as learning rates) explicitly govern a trade-off between the condition number of the surrogate loss and its alignment with the true loss. We derive novel convergence rates showcasing these trade-offs and highlight their importance in communication-limited settings. Using these insights, we are able to compare local update methods based on their convergence/accuracy trade-off, not just their convergence to critical points of the empirical loss. Our results shed new light on a broad range of phenomena, including the efficacy of server momentum in federated learning and the impact of proximal client updates.

We propose and evaluate a novel algorithm for computation of the Duquenne-Guigues basis which combines Close-by-One and LinClosure algorithms. This combination enables us to reuse attribute counters used in LinClosure and speed up the computation. Our experimental evaluation shows that it is the most efficient algorithm for computation of the Duquenne-Guigues basis.

LCM is an algorithm for enumeration of frequent closed itemsets in transaction databases. It is well known that when we ignore the required frequency, the closed itemsets are exactly intents of formal concepts in Formal Concept Analysis (FCA). We describe LCM in terms of FCA and show that LCM is basically the Close-by-One algorithm with multiple speed-up features for processing sparse data. We analyze the speed-up features and compare them with those of similar FCA algorithms, like FCbO and algorithms from the In-Close family., Comment: full version of a conference paper to be published in IJAR

We prove that the universal homogeneous 3-uniform hypergraph has finite big Ramsey degrees. This is the first case where big Ramsey degrees are known to be finite for structures in a non-binary language. Our proof is based on the vector (or product) form of Milliken's Tree Theorem and demonstrates a general method to carry existing results on structures in binary relational languages to higher arities., Comment: 10 pages, 2 figures. Minor corrections and presentation improvements suggested by the referees

We study a family of algorithms, which we refer to as local update methods, that generalize many federated learning and meta-learning algorithms. We prove that for quadratic objectives, local update methods perform stochastic gradient descent on a surrogate loss function which we exactly characterize. We show that the choice of client learning rate controls the condition number of that surrogate loss, as well as the distance between the minimizers of the surrogate and true loss functions. We use this theory to derive novel convergence rates for federated averaging that showcase this trade-off between the condition number of the surrogate loss and its alignment with the true loss function. We validate our results empirically, showing that in communication-limited settings, proper learning rate tuning is often sufficient to reach near-optimal behavior. We also present a practical method for automatic learning rate decay in local update methods that helps reduce the need for learning rate tuning, and highlight its empirical performance on a variety of tasks and datasets.

We investigate a variant of the fuel-based approach to modeling diverging computation in type theories and use it to abstractly capture the essence of oracle Turing machines. The resulting objects we call continuous machines. We prove that it is possible to translate back and forth between such machines and names in the standard function encoding used in computable analysis. Put differently, among the operators on Baire space, exactly the partial continuous ones are implementable by continuous machines and the data that such a machine provides is a description of the operator as a sequentially realizable functional. Continuous machines are naturally formulated in type theories and we have formalized our findings in Coq. Continuous machines, their equivalence to the standard encoding and correctness of basic operations are now part of Incone, a Coq library for computable analysis. While the correctness proofs use a classical meta-theory with countable choice, the translations and algorithms that are proven correct are all fully executable. Along the way we formally prove some known results such as existence of a self-modulating moduli of continuity for partial continuous operators on Baire space. To illustrate their versatility we use continuous machines to specify some algorithms that operate on objects that cannot be fully described by finite means, such as real numbers and functions. We present particularly simple algorithms for finding the multiplicative inverse of a real number and for composition of partial continuous operators on Baire space. Some of the simplicity is achieved by utilizing the fact that continuous machines are compatible with multivalued semantics. We also connect continuous machines to the construction of precompletions and completions of represented spaces, topics that have recently caught the attention of the computable analysis community.