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[
"Coherent Stokes Raman scattering microscopy (CSRS)",
"Coherent Stokes Raman scattering microscopy (CSRS)"
] | [
"Sandro Heuke \nInstitut Fresnel\nAix Marseille Univ\nCNRS\nCentrale Marseille\nMarseilleFrance\n",
"Hervé Rigneault \nInstitut Fresnel\nAix Marseille Univ\nCNRS\nCentrale Marseille\nMarseilleFrance\n"
] | [
"Institut Fresnel\nAix Marseille Univ\nCNRS\nCentrale Marseille\nMarseilleFrance",
"Institut Fresnel\nAix Marseille Univ\nCNRS\nCentrale Marseille\nMarseilleFrance"
] | [] | We report the first implementation of laser scanning coherent Stokes Raman scattering (CSRS) microscopy. To overcome the major challenge in CSRS imaging, we show how to suppress the fluorescence background by narrow bandpass filter and a lock-in based demodulation. Near background free CSRS imaging of polymer beads, human skin, onion cells, avocado flesh and the wing disc of a drosphila larva are presented. Finally, we explain and demonstrate numerically that CSRS solves a major obstacle of other coherent Raman techniques by sending a significant part (up to 100%) of the CSRS photons into the backward direction under tight focusing conditions. We believe that this discovery will pave the way for numerous technological advances, e.g., in epidetected coherent Raman multi-focus imaging, real-time laser scanning based spectroscopy or efficient endoscopy. | 10.1038/s41467-023-38941-4 | [
"https://export.arxiv.org/pdf/2301.03516v1.pdf"
] | 255,545,924 | 2301.03516 | b7c9301e814ff46545e641091af20b9543b957a0 |
Coherent Stokes Raman scattering microscopy (CSRS)
Sandro Heuke
Institut Fresnel
Aix Marseille Univ
CNRS
Centrale Marseille
MarseilleFrance
Hervé Rigneault
Institut Fresnel
Aix Marseille Univ
CNRS
Centrale Marseille
MarseilleFrance
Coherent Stokes Raman scattering microscopy (CSRS)
10.1038/s41467-023-38941-4Article
We report the first implementation of laser scanning coherent Stokes Raman scattering (CSRS) microscopy. To overcome the major challenge in CSRS imaging, we show how to suppress the fluorescence background by narrow bandpass filter and a lock-in based demodulation. Near background free CSRS imaging of polymer beads, human skin, onion cells, avocado flesh and the wing disc of a drosphila larva are presented. Finally, we explain and demonstrate numerically that CSRS solves a major obstacle of other coherent Raman techniques by sending a significant part (up to 100%) of the CSRS photons into the backward direction under tight focusing conditions. We believe that this discovery will pave the way for numerous technological advances, e.g., in epidetected coherent Raman multi-focus imaging, real-time laser scanning based spectroscopy or efficient endoscopy.
Conventional bright-field microscopy provides information about the refractive index and absorption properties but cannot elucidate the sample's chemical composition. Infrared absorption and linear Raman scattering can provide the sample chemical composition 1,2 , but they are incompatible with high spatial resolution or real-time imaging. Coherent Raman scattering imaging (CRS) fills this gap by combining chemical sensitivity with signal levels that permit video-rate image acquisition. Well-established CRS microscopy techniques are coherent anti-Stokes Raman scattering (CARS) 3,4 and stimulated Raman scattering (SRS) [5][6][7] . CARS owe its wide-range application to the blue-shifted anti-Stokes radiation, which greatly facilitates its separation from linear fluorescence. When working with near-infrared excitation wavelengths, the blue-shifted CARS radiation is readily detected using photo-electron multiplier tubes (PMT) of standard laser scanning microscopes. SRS's popularity arises from the heterodyne signal amplification that frees SRS images from an omnipresent non-resonant four-wave-mixing background that is present in CARS images 8 . SRS also allows for measurements under daylight conditions owing to its modulation and signal detection scheme.
Overshadowed by CARS and SRS until now, there exists a third four-wave-mixing process that can be resonant with vibrational levels termed coherent Stokes Raman scattering (CSRS, pronounced "SCiSsoRS") [8][9][10][11][12] . CSRS, as CARS and SRS, is always present in CRS experiments and also provides chemical sensitivity 13 -see Fig. 1. In analogy to the Stokes emission in linear Raman microscopy, the CSRS radiation (2ω S − ω p ) is red-shifted with respect to the excitation frequencies of the pump (ω p ) and Stokes beams (ω S ). Surprisingly, CSRS imaging has never been implemented for laser scanning microscopy (LSM). Presumably, this is due to the high degree of resemblance of CARS and CSRS spectra 13 , rendering CSRS-prima facie-to be either CARS with an added fluorescence background when working with visible light sources or CARS with a radiation wavelength offside high quantum yields of common detectors when working with nearinfrared (NIR) excitation. CSRS provides, however, some unique properties that are of high interest for imaging.
First, the CSRS spectrum differs from CARS in the presence of accessible electronic resonances. For example, pre-resonant CSRS will offer complementary information in the application of alkyne-labeled dyes 14 and standard dyes used in microbiology 15 . Second, the redshifted radiation of CSRS becomes an advantage for UV or near-UV excitation where CARS photons 16 would be too far blue-shifted to be detected efficiently while any SRS image 17 is likely to be compromised by various artifacts such as multi-photon absorption 18,19 . Thus, UVexcited CSRS holds the potential to achieve the highest possible spatial resolution (λ Stokes =½ ffiffiffi 8 p NA) in coherent Raman imaging. Third, NIRexcitation wavelength combined with CSRS may allow for deeper tissue imaging due to the reduced scattering and absorption of its radiation 20 . Last but most important: Due to a modified phasematching geometry, CSRS microscopy can be configured to radiate more light in the backward direction. This game changer would benefit the investigation of thick samples, real-time spectroscopy, multi-focus imaging, and endoscopy 21 . Within this contribution, we want to open up the field of laser scanning CSRS imaging by demonstrating CSRS microscopy within the visible excitation spectrum. To remove the major fluorescence background obstacle, we will show how linear fluorescence can be suppressed by combining a set of bandpass filters with a lock-in-based detection scheme. Furthermore, we shall investigate numerically CSRS' spatial radiation behavior under NIR excitation, paving the way towards CSRS experiments with an efficient epidetection.
Result and discussion
Experiments
The CSRS signal of biomedical samples is often overwhelmed by linear fluorescence as CSRS radiation is red-shifted as compared to the excitation lasers. Time-gating 22 , time-resolved detection using streak cameras 23 , or polarization filtering can be used to reduce or suppress any fluorescence signal. However, these methods require either a substantial modification of standard coherent Raman microscopes or do not work in the presence of strong fluorescence light backgrounds. Here, we exploit the fact that the CSRS is spectrally narrow under psexcitation. Thus, the majority of fluorescence can be readily suppressed by narrow-band filters. Filters with a spectral width below <1 nm are commercially available, but the selection of a specific center wavelength requires expensive custom solutions. Instead, we use a combination of two inexpensive bandpass filters with a spectral width of about 15 nm but with different center wavelengths. In addition, we fine-tune the filter transmissions by tilting them (<20°) with respect to the incident beam, see Fig. 2b. Thus, two tilt-adjusted bandpass filters create a sharp transmission line (FWHM < 3 nm) for the CSRS signal collection while rejecting a significant part of the autofluorescence.
As a second method for fluorescence background rejection, we take advantage of the CSRS intensity dependence on both pump and Stokes excitation colors while linear fluorescence follows either the intensity of the pump or the Stokes laser, see Fig. 2c. Consequently, modulating the pump and Stokes beams at f1 and f2 while demodulation the signal at f1-f2 (or f1 + f2) yields exclusively nonlinear signals that depend on both excitation colors. The f1-f2 demodulation, therefore, also discriminates the CSRS signal against 2-photon excited fluorescence (2PEF) under single-color excitation. It should be noted that the f1-f2 modulation is also sensitive to two-color 2-photon fluorescence (2C-2PEF). Nevertheless, we will find experimentally that the emission strength of native 2C-2PEF is negligible within our CSRS implementation using visible beams.
For the experimental implementation of CSRS LSM, we chose visible excitation wavelengths at 445 nm (pump) and 515 nm (Stokes) for the following reasons: First, CSRS under near UV excitation is a potentially important application area since CARS, and SRS encounter experimental difficulties within this spectral range: the CARS signal falls into the UV range while SRS artifacts are increased due the possible high concentration of endogenous chromophores. Second, the red-shifted CSRS radiation can be readily detected by ordinary PMTs. Third, stress test: fluorescence artifacts are enhanced as compared to a near-infrared (NIR) excitation. Thus, our approach will be viable as well for CSRS under NIR excitation if pure CSRS signals can be obtained under VIS excitation.
The experimental setup, the spectral filtering, and the double modulation are schematically shown in Fig. 2a. Our implementation resembles a standard SRS setup with the difference that we use visible excitation wavelengths, we modulate not one but both beams, and the photo-diode is replaced by a PMT which is connected to a lock-in amplifier. More information about the setup can be found in part "Methods: Experimental setup". To quantify the level of fluorescence rejection, we investigated the signal of native olive oil at 2850 cm −1 when blocking the pump or Stokes beams or when the temporal pulse overlap is removed. The output signal of the lock-in is plotted as functions of the demodulation frequencies at 0 Hz (DC), f1, f2 and f1-f2 in Fig. 2d. It can be observed that the DC channel contains significant amounts of fluorescence while this artifact is already reduced within the f1 and f2 channels. Nevertheless, only the difference frequency channel at f1-f2 approaches zero when the excitation pulses do not overlap in time (Δt ≫ 3ps). In a second experiment, we imaged with CSRS (demodulated at f1-f2) the interface between olive oil and a 20 μm sized Plexiglas (PMMA) bead to obtain an estimation of the lateral resolution with an excitation objective of NA = 1.45-see Fig. 2e. From this "knife-edge" CSRS intensity profile, we can infer a lateral resolution below 400 nm. The difference to the expected λ Stokes =½ ffiffiffi 8 p NA = 515 nm/ [ ffiffiffi 8 p 1.49] = 120 nm can be attributed to the underfilling of the excitation objective back aperture and the bent oil/bead interface. Having confirmed a high-resolved, fluorescence-free CSRS image contrast, we investigated the suitability of LSM-CSRS for vibrational imaging of various objects featuring non-negligible background fluorescence levels. Within Fig. 3, we show the CSRS images of test and biomedical samples demodulated at the DC and f1-f2 frequencies for (non-)overlapping pump and Stokes pulses. The images were organized along the ratio of the CSRS to the fluorescence signal, starting from the highest at the top. Comparing the DC and f1-f2 images in Fig. 3a, it is obvious that narrow spectral filtering is already sufficient for CSRS imaging of polymer beads in oil (see CSRS at DC). The first artifacts become visible for the DC CSRS images of the epithelium and dermis of a 20 μm thick section of human skin-see Fig. 3b, c. For the epithelium, a pronounced fluorescence artifact arises from melanin within the epidermis dermis junction. Artifacts within the dermis can be attributed to the autofluorescence of collagen and elastin 24 . The quantity of fluorescence observed within the DC channel increases stepwise further for CSRS imaging of onion cells, lipid droplets within the flesh of an avocado, and the wing disc of a Drosophila larva. From the second row of Fig. 3, we validate that almost no fluorescence is leaking into the f1-f2 CSRS channel.
In the next section, we address a non-intuitive but key feature of CSRS microscopy: the possibility to dramatically increase the CSRS backwards radiation opening the road for an effective epi-CSRS detection.
Momentum conservation and simulations
Before entering into the calculations, we want to consider CSRS from a heuristic viewpoint investigating the momentum conservation laws for CSRS and compare it to CARS. Under plane illumination, the momentum conservation laws can be written as K = k p − k S + k p − k aS for CARS 25 and K = k S − k p + k S − k cS for CSRS with K, k p , k S , k aS and k cS representing the wavevectors of the object, the pump (probe) and Stokes beam as well as the anti-Stokes and coherent Stokes radiation, respectively. Note that for homogeneous samples (K = 0) these laws are also referred to as phasematching condition and simplify to k p + k p = k S + k aS (CARS) and k S + k S = k p + k cS (CSRS). Under focusing conditions, the single wavevectors are replaced by the distribution of incident wavevectors which are distributed over a cap of a sphere . To identify those object frequencies (K) that are effectively probed, every combination of excitation and emission wavevector must be identified. This operation is equivalent to the convolution of the caps of the illumination and detection Ewald spheres 26 . Neglecting polarization effects, the result of this convolution (simplified to 3 points per arc) is shown in 2D within Fig. 4a.
Evidently, there exists no vector combination for epi-scattered CARS photons which covers the origin K(0,0,0) of the object space. Thus, a homogeneous sample, such as olive oil, does not provide any backward CARS radiation. On the contrary, structures that feature high object frequencies, such as small polymer beads or layered materials, generate Epi-CARS radiation. In the past, Epi-CARS was occasionally considered to be a size-selective contrast that would highlight exclusively small objects 27 . While this statement holds for the majority of biomedical samples, there do exist large structures, e.g., multi-layered lipids in vesicles, that also emit a strong CARS radiation in the backward direction. Hence, it is more appropriate to refer to Epi-CARS as a technique that probes high object frequencies along the z-axis instead of being considered as size-selective.
Switching the detection wavelength to the red-shifted CSRS radiation changes the covered object support significantly and includes now the origin at K(0,0,0). Due to the reduced size of the detection wavevector (|k cS | ≪ |k aS |), steep incident angle Stokes vectors, and the pump vector entering as complex conjugated, it is now possible to find vector combinations that cover the origin at K(0,0,0). Consequently, even a homogeneous object will radiate considerable amounts of Epi-CSRS. Nevertheless, since the centroid of the Epi-CSRS object support, i.e., the gray cloud within Fig. 4a, does not coincide with the K-space origin K(0,0,0), Epi-CSRS images will also highlight objects containing higher frequencies.
To address the question of how to increase the ratio of backward versus forward CSRS and which object frequencies are most efficiently probed using Epi-CSRS, we performed finite element simulations whose results are summarized in Fig. 4b-e. The equations implemented numerically, as well as parameters, are found in methods: numerical calculation. From the momentum conservation law and the vector diagrams in Fig. 4a, it is readily comprehensible that a larger wavelength difference between the pump and CSRS wavelength greatly The CSRS signal is separated from fluorescence by means of two angle-tuned narrow bandpass filters. c Additional suppression of fluorescence is achieved by intensity modulating the Stokes and pump beam at the frequencies f1 and f2, respectively. Fluorescence-free CSRS signal is obtained at f1-f2. d Measured CSRS signal at DC, f1, f2, and f1-f2 frequencies when the pump or Stokes beam is blocked (Ip = 0 or Is = 0) or when their temporal overlap is removed (Δt ≫ 3 ps). The fluorescence is strongly rejected on the f1-f2 time trace and mainly comes from the Stokes beam. e The CSRS intensity profile obtained at f1-f2 at the interface of a PMMA bead and olive oil indicates a lateral resolution of <400 nm. relaxes the necessity for extreme incident illumination angles of the Stokes beam. The wavelength difference between the pump and CSRS radiation is enhanced using NIR instead of VIS excitation wavelengths, which is why we used in our simulations the wavelength λ p = 797 nm and λ S = 1030 nm, which matches the 2850 cm −1 Raman shift. For these conditions, the coherent Stokes radiation is observed at λ cS = 1450 nm. It should be noted that our results equally apply to the visible excitation wavelength using a higher excitation angle (or thinner annular masks-see below).
To start with, we computed the radiation pattern of CSRS and CARS of a homogeneous object using an NA of 1.49 (oil immersion), corresponding to a maximum illumination angle of 80°. From Fig. 4b, it is evident that both CARS and CSRS are predominately forward directed though the CSRS' radiation distribution features a larger radiation cone. Considering the ratio of backward versus forwarddirected photons R b/f , we find numerically that less than 1 photon in 10 5 is backward-directed for CARS. Note that the momentum conservation law actually predicts R b/f = 0 for CARS. Thus, the deviation observed Since common surfaces within biomedical samples scatter more than 1%, it is still likely that in this high NA illumination scheme, epi-detected CSRS is dominated by forward-generated CSRS that is back-scattered by linear scattering at interfaces (as in the CARS case). To find an approach that increases the proportion of epi-CSRS radiation, we consider the CSRS vector diagram matching K(0,0,0) on the top left of Fig. 4a. The ratio of backward versus forward radiation is readily increased by reducing the impact of vectors combinations probing higher frequencies and favoring those that cover the origin by satisfying k S + k S = k p + k cS . This enhancement of epi-CSRS radiation can be achieved using an annular illumination of the Stokes beam. Experimentally, such an annular illumination can be generated, without power loss, using two axicons within the Stokes beam path 28,29 . Numerically, we restricted the incident angles for the Stokes between a The object spatial frequency K-support for Epi-CSRS(CARS) is found by convolving the illumination Ewald spheres of the Stokes (pump), pump (Stokes), and Stokes (probe) with the cap of detection Ewald sphere at coherent Stokes (anti-Stokes) frequency. Note that vector combinations covering the frequency of a homogeneous sample K(0,0,0) are only found for CSRS but not for CARS. A single wavevector combination that phase-matches K(0,0,0) is highlighted to the left, while a similar approach for CARS leads to a large phase-mismatch (ΔK). b CSRS and CARS radiation behavior of a homogeneous sample under standard illumination conditions, i.e., the pump and Stokes beam fill the objective aperture homogeneously (θ max = 80°). c same as in b but with an annular pupil filter applied to the Stokes beam for CSRS covering 50% of the area of the objective back-aperture. For an equitable comparison with CARS, the same pupil filter was applied to the pump beam. d same as for b (conventionally focused beams), but the homogeneous sample was replaced by a frequency object whose scatter density is described as 1 + cosð2πz=λ o Þ and λ o = 1 μm. e Plot of the ratio of backward/forward radiation (R b/f ) as a function of the object frequency λ o . Calculations were performed with λ p = 797 nm and λ S = 1030 nm. θ min = 56.5°and θ max = 80°, which corresponds to covering 50% of the area of the objective lens' back-focal plane. The pump beam remains a normally focused beam and covers the full lens' back-focal plane. With this Stokes pupil filtering, the ratio of backward to forward radiation increased for CARS to 2 in 10 4 photons while most of the CSRS radiation is backward directed (R b/f = 1.5) when the object is homogeneoussee Fig. 4c. For the CARS calculation we considered the annular illumination applied to the pump beam whereas the Stokes is a conventional focused beam.
As a second important result from the heuristic derivation of CSRS object support, we found that the presence of high spatial frequencies along K z increases the amount of backward radiation. To confirm this prediction, we investigated in Fig. 4d, e an object whose nonlinear scatterer density, i.e., the concentration of molecular groups, is modulated along the optical axis as 1 + cos(K z z) with K z = 2π/λ o being the object frequency. We now consider a conventional illumination scheme where both Stokes and the pump are tightly focused and cover the full back aperture of the objective lens. Figure 4d outlines the radiation behavior of such a z-structured object with K z = 2π/1 μm. It is found that R b/f increases to one-fourth for Epi-CSRS while Epi-CARS remains negligible weak. To identify those object frequencies which are most efficiently probed by Epi-CSRS, we computed R b/f as a function of K z . From Fig. 4e, we find that Epi-CSRS peaks at K z = 2π/1 μm whereas Epi-CARS R b/f still increases at K z = 2π/0.5 μm confirming that CARS requires larger K z , i.e., objects with higher frequency modulation of the scatterer density, to generate a strong Epi radiation.
Thus, we have found that CSRS features non-negligible backward radiation from a homogenous sample under tight-focusing conditions, while this is not the case for CARS. The amount of backward radiated CSRS can be further enhanced using a Stokes annular illumination to surpass the forward CSRS radiation.
In conclusion, we have demonstrated the first LSM CSRS experiment. As the major challenge, we were able to reduce the fluorescence background significantly using a pair of tilted bandpass filters. The remaining fluorescence contribution was removed by intensity modulating the Stokes and pump beams at the frequencies f1 and f2 and a lock-in-based demodulation of the CSRS signal. Taking advantage of CSRS' characteristic dependence on both excitation colors, near fluorescence-free CSRS images were obtained when demodulating the CSRS signal at f1-f2. Fluorescence-free LSM-CSRS imaging was demonstrated on a variety of samples showing different fluorescence levels, such as polymer beads, epithelium, and dermis of human skin, onion cells, avocado flesh, and the wing disc of Drosophila larva.
Having demonstrated the viability of CSRS imaging, we introduced and quantified numerically how CSRS can be implemented to generate a strong backward radiated signal with high NA objective lenses. CSRS' unique backward radiation ability can be understood considering the momentum conservation laws for all combinations of all contributing k-vectors and cannot be achieved with CARS or SRS. With efficient backward radiation at hand, various coherent Raman experiments become feasible, which were impossible before. For example, this is the case for Epi-detected confocal multi-focus CSRS, Epi-detected LSM-CSRS with a spectrometer at the descanned position, Epidetected CSRS image scanning microscopy, or efficient endoscopy. Thus, we believe that this discovery will open new directions for coherent Raman developments and applications.
Methods
Experimental setup
A Yb-based fiber laser (APE Emerald engine, 80 MHz, 2-3 ps) is frequency doubled, yielding 7 W of 515 nm output power. Parts of the emissions are used directly as a Stokes beam to drive the CSRS process. The major part (4 W) of the 515 nm is employed to pump an optical parametric oscillator (OPO, APE Emerald). The OPO's signal beam is tunable to 660-950 nm and coupled into an external SHG unit (APE, HarmoniXX). The latter generates up to 50 mW within the spectral range of 330-475 nm and serves as the pump beam for CSRS. Thus, the 330-475 nm pump combined with the 515 nm Stokes beam allows addressing a Raman shift range from 1630-11,000 cm −1 . The pump and Stokes beams are superimposed in space and time using a dichroic beam splitter (Semrock, FF470-Di01-25x36) and a delay stage. Both beams are coupled into a home-built laser scanning microscope and focused by a 40× water objective lens (Nikon, Plan, NA = 1.15, immersion: water) into the sample. The excitation objective lens was replaced for a 60× objective (Nikon, Plan Apo TIRF, NA 1.45, immersion:oil) to generate the bead-oil interface image within Fig. 2. The CSRS radiation is collected by a condenser lens (Nikon, Achr-Apl, NA 1.4) in the forward direction, spectrally separated from the broadband fluorescence background by means of 2 tilted bandpass filter (Semrock FF01-620/14-25 + FF01-605/15-25) and detected by a photo-electron multiplier (PMT, Thorlabs, PMT1001). We measured the CSRS and CARS (at 398 nm) radiation strength for olive oil one after another and found comparable signal levels. To avoid detector saturation for the acquisition of CSRS images, we applied the lowest possible PMT gain corresponding to an amplification of only 5 × 10 3 . For an enhanced suppression of the linear fluorescence background, two acousto-optic modulators (AOM, AA, MT200-A0.5-VIS) were applied to modulate the intensity of the Stokes and pump beams and at the frequencies f1 = 2.28 MHz and f2 = 3.75 MHz, respectively. The PMT output was demodulated simultaneously at the DC frequency, f1, f2, and at f1-f2 = 1.47 MHz using a lock-in amplifier (Zürich instruments, HF2LI). The lock-in time constant was set to 30 μs. All CSRS images shown were recorded with a pixel dwell time of 40 μs. All samples were investigated in live image acquisition mode. Thus, some areas were scanned more than 100 times. We noticed that the fluorescence background signal within the DC channel was reduced over time as a result of photo-bleaching though the f1-f2 channel remained unaffected, which indicates that our experimental conditions are below the damage threshold of ex vivo samples. Note that the demodulation at f1-f2 only removes the fluorescence background while the CSRS non-resonant four-wave-mixing background 30 that is inherent to all coherent Raman techniques is still present. Nevertheless, the removal of this non-resonant background could be achieved by a heterodyne interference of the CSRS signal with a reference beam at the same wavelength 31 or by Kramers-Kronig or Maximum entropy-based algorithms in application to CSRS spectra 12 .
Numerical calculation
In the following, we shall summarize the equations used to generate Fig. 4b-e. The meaning of the variables is summarized in Fig. 5.
The focused field at the sample is given by the angular spectrum representation 32 : Here f denotes the focal length of the objective lens, and the integrals I 0m are provided by
E x ðρ,I 0m = Z θ max θ min E inc ðθÞ sinðθÞ½cosðθÞ 1=2 g m ðθÞJ m ½kρ sinðθÞdθð2Þ
where g m equals 1 + cosðθÞ, sinðθÞ and 1 À cosðθÞ for m = 0, 1, 2, respectively. J m is the mth order Bessel function while E inc is the incoming electric field which we assumed to be x-polarized and constant within the (annular) aperture angles θ min ≤ θ ≤ θ max . The nonlinear polarization at anti-Stokes and coherent Stokes wavelength is given by:
Pð3Þ
Where a,b,c,d represent the polarization coordinates x, y, or z. Using an x-polarized excitation, it was noticed that χ ð3Þ xxxx dominates all other tensor components even under tight focusing conditions while filling the objective lens homogeneously 32 . Nevertheless, for the generation of Fig. 4c, an annular mask with θ min = 56.5°and θ max = 80°was applied, which does necessitate the inclusion of other tensor elements. For simplicity, we consider here only isotropic samples reducing the 81 susceptibility tensor elements to 21, which are nonzero 30 . Within isotropic media, these nonzero elements follow certain symmetry rules which are, χ 1111 = χ 2222 = χ 3333 , χ 1122 = χ 1133 = χ 2211 = χ 2233 = χ 3311 = χ 3322 , χ 1212 = χ 1313 = χ 2323 = χ 2121 = χ 3131 = χ 3232 , χ 1221 = χ 1331 = χ 2112 = χ 2332 = χ 3113 = χ 3223 . Further, it applies χ 1111 = χ 1122 + χ 1212 + χ 1221 30 . Within our simulations we were setting χ 1122 = χ 1212 = χ 1221 = 1 and, hence, χ 1111 = 3. The nonlinear far-field radiation distributions are obtained using a dyadic Green function approach: E q,R ðR,Θ,ΦÞ E q,Θ ðR,Θ,ΦÞ E q,Φ ðR,Θ,ΦÞ
where q is replaced by aS or cS to calculate either the anti-Stokes or coherent Stokes radiation. Within the simulations, we segmented the focal area into (121 × 121 × 121 ≈ ) 1.77 million elements of a width of 50 nm equally spaced into the x, y, and z directions. The far-field radiation sphere was discretized into (ΔΘ = 1°, ΔΦ = 2°) 32,400 elements. The coherent (anti-)Stokes radiation was qualified as either forward or backward-directed if falling into the range Θ.. 0-80°or Θ.. 100-180°, respectively.
Fig. 1 |
1Overview coherent Raman imaging techniques. In energy diagrams 33 , relative radiation wavelength and energy conservation under plane-wave illumination.
Fig. 2 |
2CSRS experimental implementation and characterization. a Scheme of the CSRS experiment. 1. Yb-fiber laser, 2. optical parametric oscillator (OPO), 3. second harmonic generation (SHG), 4. acousto-optic modulator (AOM), laser scanning microscope (LSM), 6. photo-electron multiplier (PMT), 7. lock-in amplifier. b
Fig. 3 |
3Laser scanning CSRS at 2850 cm −1 . The left and right column show the CSRS image demodulated at the frequencies f1-f2 = 1.47 MHz and 0 Hz (DC), respectively. To estimate the remaining fluorescence level, images without temporal overlap of the pump and Stokes pulses are displayed to the right (Δt ≫ 3ps). a Mixture of polystyrene (PS, 30 μm) and Poly-methyl-methacrylate (PMMA, 20 μm) beads in olive oil. b, c Epithelium and dermis of a 20 μm thick human skin section. d Cells of an onion. e Lipid droplets within the flesh of an avocado. f Wing disc of a Drosophila larva. The insets displayed in the second column are the zoomed "CSRS at f1-f2" regions of interest shown in the first column on (b, d). Pixel dwell time: 40 μs. Image acquisition time: 40 s (1000 × 1000). The white and green scale bar equals 20 and 5 μm, respectively. must be attributed to the finite number of voxels of the numerical model. For CSRS, R b/f increases dramatically to about 1 in 100 photons.
Fig. 4 |
4Object frequency support and radiation behavior of CSRS versus CARS.
Fig. 5 |
5Declaration of variables.
ϕ,zÞ E y ðρ,ϕ,zÞ E z ðρ,ϕ,zÞ2
6
4
3
7
5 =
ikf
2
expðÀikf Þ
I 00 + I 02 cosð2ϕÞ
I 02 sinð2ϕÞ
Ài2I 01 cosðϕÞ
2
6
4
3
7
5
ð1Þ
aS,a ðrÞ = 3χ ð3Þ abcd ðrÞE p,b E * S,c E p,d Pð3ÞcS,a ðrÞ = 3χ ð3Þ abcd ðrÞE S,b E * p,c E S,d
Nature Communications | (2023) 14:3337
© The Author(s) 2023
AcknowledgementsWe acknowledge financial support from the Center NationalData availabilityThe data that support the findings of this study are available from the corresponding author upon request.Author contributionsS.H. conceived the idea, and performed the experiments and numerical calculations. H.R. conceived the idea and discussed the results. S.H. and H.R. wrote the paper.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version contains supplementary material available at https://doi.org/10.1038/s41467-023-38941-4.Correspondence and requests for materials should be addressed to Sandro Heuke or Hervé. Rigneault.Peer review information Nature Communications thanks Giulio Cerullo, Marcus Cicerone and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.Reprints and permissions information is available at http://www.nature.com/reprintsPublisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/.
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| [] |
[
"APPROXIMATION BY EGYPTIAN FRACTIONS AND THE WEAK GREEDY ALGORITHM",
"APPROXIMATION BY EGYPTIAN FRACTIONS AND THE WEAK GREEDY ALGORITHM"
] | [
"Viê T Hùng ",
"Chu "
] | [] | [] | Let 0 < θ 1. A sequence of positive integers (b n ) ∞ n=1 is called a weak greedy approximation of θ if ∞ n=1 1/b n = θ. We introduce the weak greedy approximation algorithm (WGAA), which, for each θ, produces two sequences of positive integers (a n ) andc) there exists t 1 such that b n /a n t infinitely often. We then investigate when a given weak greedy approximation (b n ) can be produced by the WGAA. Furthermore, we show that for any non-decreasing (a n ) with a 1 2 and a n → ∞, there exist θ and (b n ) such that a) and b) are satisfied; whether c) is also satisfied depends on the sequence (a n ). Finally, we address the uniqueness of θ and (b n ) and apply our framework to specific sequences.2020 Mathematics Subject Classification. 11A67, 11B99. | 10.1016/j.indag.2023.05.008 | [
"https://export.arxiv.org/pdf/2302.01747v2.pdf"
] | 256,598,330 | 2302.01747 | 2bdc6fb486b2c53547403973a55bbd5dc8c454d1 |
APPROXIMATION BY EGYPTIAN FRACTIONS AND THE WEAK GREEDY ALGORITHM
30 May 2023
Viê T Hùng
Chu
APPROXIMATION BY EGYPTIAN FRACTIONS AND THE WEAK GREEDY ALGORITHM
30 May 2023
Let 0 < θ 1. A sequence of positive integers (b n ) ∞ n=1 is called a weak greedy approximation of θ if ∞ n=1 1/b n = θ. We introduce the weak greedy approximation algorithm (WGAA), which, for each θ, produces two sequences of positive integers (a n ) andc) there exists t 1 such that b n /a n t infinitely often. We then investigate when a given weak greedy approximation (b n ) can be produced by the WGAA. Furthermore, we show that for any non-decreasing (a n ) with a 1 2 and a n → ∞, there exist θ and (b n ) such that a) and b) are satisfied; whether c) is also satisfied depends on the sequence (a n ). Finally, we address the uniqueness of θ and (b n ) and apply our framework to specific sequences.2020 Mathematics Subject Classification. 11A67, 11B99.
INTRODUCTION
Throughout this paper, let θ denote a number in (0, 1] and let G : (0, 1] → N 2 be the function
G(θ) = 1 θ + 1;
that is, G(θ) gives the unique positive integer a 2 such that 1 a < θ 1 a − 1 .
An Egyptian fraction is a fraction of the form 1/n for some positive integer n. We consider the problem of representing θ as an infinite sum of Egyptian fractions. One natural method is the greedy underapproximation algorithm (GUA), which constructs a sequence of positive integers (a n ) ∞ n=1 recursively as follows: a 1 = G(θ) 2; supposing that a 1 , . . . , a n have been constructed, let
a n+1 = G θ − n i=1 1 a i .
By [Na23,(3)], the sequence (a n ) is strictly increasing and particularly, satisfies a 1 2 and a n+1 a 2 n − a n + 1.
(1.1)
Since by construction,
θ − n i=1
1 a n 1 a n+1 − 1 → 0, we have ∞ n=1 1 a n = θ.
According to [Na23,Theorem 5], if θ = p/q, where p, q are positive integers such that p divides q + 1, then the GUA produces the best approximations; i.e., the n-term approximation n i=1 1/a i outperforms any other n-term underapproximations using Egyptian fractions. This generalizes a result in [Cu22,So05,Ta21]. The proof involves an useful inequality established in [AB15] (see also [Na22].) However, such optimality does not hold for general θ (see [Na23,Section 5].)
The goal of this paper is to investigate a weak version of the GUA, which is inspired by the so-called (weak) thresholding greedy algorithm (TGA) in the area of functional analysis. We describe the (weak) TGA briefly. Let X be an infinite-dimensional, complete, normed vector space. Assume further that X has a basis B = (e n ) ∞ n=1 so that every vector x ∈ X can be represented by a formal series ∞ n=1 a n e n , where a n are scalars. (The series converges to a vector when our basis is Schauder; however, for general Markushevich bases, the series may only be formal.) In order to form an m-term approximation of x, the TGA chooses m largest coefficients a n in modulus. Formally, let A ⊂ N verify |A| = m and min n∈A |a n | max n / ∈A |a n |.
(1.2)
Then the TGA produces the m-term approximation n∈A a n e n . It is not always true that approximations produced by this method converge to the original vector x as m grows. In fact, Konyagin Inspired by the aforementioned interactions between the TGA and the WTGA, we introduce the weak greedy approximation algorithm (WGAA) as a companion of the GUA. The idea is that at the nth step of our weak algorithm, we pick a n based on the "greedy choice up to a constant". Specifically, fix t ∈ R 1 and an infinite set Λ ⊂ N. For each θ ∈ (0, 1], we define the (t, Λ)-WGAA as follows: let a 1 = G(θ). Choose b 1 a 1 . Additionally, we require b 1 ta 1 if 1 ∈ Λ. Assuming that a 1 , b 1 , . . . , a n , b n have been defined, we let
a n+1 = G θ − n i=1 1 b i . (1.4)
Choose b n+1 a n+1 . Additionally, we require b n+1 ta n+1 if n + 1 ∈ Λ. We see that the (t, Λ)-WGAA generalizes the GUA by simply setting t = 1 and Λ = N.
Definition 1.1. An infinite sequence of positive integers (b n ) ∞ n=1 is called a weak greedy approximation of θ if ∞ n=1 1/b n = θ and for all n 1,
G θ − n−1 i=1 1 b i b n .
(1.5) Inequality (1.5) indicates that a term b n is not necessarily picked by the greedy algorithm. Attentive readers may notice that (1.5) is superfluous. Indeed, suppose that for some N,
b N < G θ − N −1 i=1 1 b i =: a N . Then N i=1 1 b i = N −1 i=1 1 b i + 1 b N N −1 i=1 1 b i + 1 a N − 1 θ,
which contradicts ∞ n=1 1/b n = θ. We describe the paper's structure. In Section 2, we show that the WGAA satisfies the minimal requirement for an algorithm to be sensible; that is, for every θ, the sequence (b n ) produced by the WGAA satisfies
∞ n=1 1 b n = θ. (1.6)
This is the analog of the relation between the TGA and the WTGA. Moreover, we compute the growth rate of the sequence (b n ) produced by the (t, Λ)-WGAA when Λ = N and b n = ⌈ta n ⌉ (Proposition 2.2.) In Section 3, we carry out a deeper study of the two sequences (a n ) and (b n ) produced by the WGAA. According to Section 2, if (b n ) is produced by the WGAA applied to θ, then (b n ) is a weak greedy approximation of θ. We shall show that the converse is not true: there exist θ and (b n ) such that ∞ n=1 1/b n = θ, but (b n ) cannot be produced by the WGAA. To do so, we observe that θ, (a n ), (b n ) produced by the WGAA have three properties a) ∞ n=1 1/b n = θ (see Section 2); b) (1.4) holds for all n; c) there exists t 1 such that b n /a n t infinitely often. As we shall see, condition c) guarantees the convergence (1.6). However, even when θ, (a n ), and (b n ) verify a) and b), they do not necessarily satisfy c). As a result, in such cases, (b n ) cannot be produced by the WGAA. We then go further to characterize the situation when c) does not hold (see Proposition 3.2.) Next, we consider the following question: given a non-decreasing sequence (a n ) with a 1 2 and a n → ∞, are there θ ∈ (0, 1] and (b n ) such that a) and b) hold? According to [Na23,Corollary 3], the answer is positive if a n+1 a 2 n − a n + 1, in which case, θ = ∞ n=1 1/a n and b n = a n for all n 1. By explicit construction, we answer the aforementioned question in the affirmative for any non-decreasing sequence (a n ) with a 1 2 and a n → ∞ (see Theorem 3.5 and its Corollary 3.6.)
Section 4 gives necessary and sufficient conditions for when a sequence (a n ) gives unique θ and (b n ) (Corollary 4.2 and Proposition 4.3.) Finally, Section 5 applies the framework from previous sections to particular sequences including geometric progressions, arithmetic progressions, and the Fibonacci sequence.
CONVERGENCE OF THE WGAA
The minimal requirement we want the WGAA to satisfy is convergence, which is confirmed by the following proposition.
Proposition 2.1. If (b n ) ∞ n=1 is obtained from the (t, Λ)-WGAA applied to θ, then ∞ n=1 1 b n = θ.
Proof. Let (a n ), (b n ) be the two sequences produced by the (t, Λ)-WGAA applied to θ: for each n 1,
a n = G θ − n−1 i=1 1 b i ; equivalently, 0 < 1 a n < θ − n−1 i=1 1 b i 1 a n − 1 . (2.1)
Hence, (a n ) is non-decreasing. It suffices to prove that (a n ) is unbounded. Suppose otherwise that there is some M such that a n M for all n. Then b n Mt infinitely often, which implies that ∞ n=1 1/b n = ∞, contradicting (2.1). Next, we consider a special case of the general (t, Λ)-WGAA by requiring that Λ = N and for all n, b n = ⌈ta n ⌉. Let us denote this algorithm by G(t). Suppose that we use G(t) to obtain an n-term approximation n i=1 1/c i of θ. Then a logical choice is to have
c i = b i = ⌈ta i ⌉ for all 1 i n − 1, while c n = a n .
(It makes no sense if we do not choose the last term c n greedily.) An approximation by G(4/3) may outperform the GUA. We borrow an example from [Na23]. The GUA gives 1/3 + 1/17 as a 2-term underapproximation of 19/48, while G(4/3) gives 1/4 + 1/7. We have
1 3 + 1 17 < 1 4 + 1 7 < 19 48 .
By definition, G(1) is the greedy underapproximation algorithm. There is an interesting difference between t = 1 and t > 1.
If (b n ) is obtained by G(1) applied to θ, then [Na23, (3)] gives b n+1 b n b n − 1 + 1 b n .
Since lim n→∞ b n = ∞, we get lim n→∞ b n+1 /b n = ∞. However, the limit is finite when t > 1 as the following proposition shows.
Proposition 2.2. If (b n ) ∞ n=1 is the sequence from G(t) applied to θ, then lim n→∞ b n+1 b n = t/(t − 1) if t > 1, ∞ if t = 1.
Before proving Proposition 2.2, we record an important inequality addressing the relation between (a n ) and (b n ) produced by the WGAA. For each n 1, we have
1 a n+1 < θ − n i=1 1 b i = θ − n−1 i=1 1 b i − 1 b n 1 a n − 1 − 1 b n , and 1 a n+1 − 1 θ − n i=1 1 b i = θ − n−1 i=1 1 b i − 1 b n > 1 a n − 1 b n . Hence, 1 a n − 1 a n+1 − 1 < 1 b n < 1 a n − 1 − 1 a n+1 , ∀n ∈ N. (2.2)
Proof of Proposition 2.2. The case t = 1 is explained right before Proposition 2.2. Let t > 1. The right side of (2.2) yields
1 a n+1 < 1 a n − 1 − 1 b n = 1 a n − 1 − 1 ⌈ta n ⌉ < 1 a n − 1 − 1 ta n + 1 , ∀n ∈ N.
Therefore, 1 a n+1 < (t − 1)a n + 2 (ta n + 1)(a n − 1) =⇒ a n+1 a n > t + 1
an 1 − 1 an t − 1 + 2 an . (2.3)
The left side of (2.2) yields
1 a n+1 − 1 > 1 a n − 1 b n = 1 a n − 1 ⌈ta n ⌉ 1 a n − 1 ta n .
Hence, a n+1 a n < t t − 1 + 1 a n .
(2.4) From (2.3) and (2.4), we obtain that lim n→∞ a n+1 /a n = t/(t − 1). Since b n = ⌈ta n ⌉, we have the desired conclusion.
THE RANGE OF THE WGAA
In this section, we address the question of whether every weak greedy approximation can be obtained from the WGAA. The boundedness condition on the WGAA requires that for some t 1, b n /a n t infinitely often, which guarantees the convergence of ∞ n=1 1/b n to the desired θ (see the proof of Proposition 2.1.) However, there exist θ and (b n ) such that if (a n ) satisfies (1.4), then lim n→∞ b n /a n = ∞. By studying such a situation, we know more about the sequence (a n ) (see Corollary 3.3.) First, consider the following example.
Example 3.1. For n ∈ N, let b n = n(n + 2) and θ = 3/4. It is easy to check that ∞ n=1 1/b n = θ. We claim that if (a n ) satisfies (1.4), then a n = n + 1. Indeed, it suffices to show that
3 4 − n−1 i=1 1 i(i + 2) −1 = n, ∀n ∈ N. We have 3 4 − n−1 i=1 1 i(i + 2) −1 = ∞ i=1 1 i(i + 2) − n−1 i=1 1 i(i + 2) −1 = ∞ i=n 1 i(i + 2) −1 = 1 2 1 n + 1 n + 1 −1 by telescoping = n + n 2n + 1 = n.
Hence, a n = n + 1 and b n /a n → ∞.
The sequences (a n ) and (b n ) in Example 3.1 do not have b n /a n infinitely often bounded. In other words, a weak greedy approximation does not necessarily come from the WGAA. The next proposition provides a characterization of this situation.
Proposition 3.2. Let (b n ) ∞ n=1 be a weak greedy approximation of θ and (a n ) ∞ n=1 satisfy (1.4). The following are equivalent i) for all t 1, {n : b n /a n t} is finite. ii) lim n→∞ a n+1 /a n = 1.
Corollary 3.3. Let (b n ) ∞ n=1 be a weak greedy approximation of θ and (a n ) ∞ n=1 satisfy (1.4). Then (a n ) ∞ n=1 and (b n ) ∞ n=1 are obtained from the WGAA if and only if for some ε > 0, a n+1 > (1 + ε)a n infinitely often.
Proof of Proposition 3.2. i) =⇒ ii): Since (a n ) is non-decreasing, it suffices to show that for all ε > 0, there exists N such that a n+1 /a n < 1 + ε for all n > N. Choose M sufficiently large such that M/(M − 1) < 1 + ε/2. By i), there exists N such that for all n > N, b n > Ma n and 1/a n < ε/2. By (2.2),
1 a n+1 − 1 > 1 a n − 1 b n > 1 a n − 1 Ma n = M − 1 M 1 a n , ∀n > N,
which gives a n+1 a n < M M − 1 + 1 a n < 1 + ε, ∀n > N.
ii) =⇒ i): We prove by contrapositive. Choose t 1 and suppose that b n /a n t infinitely often. Let A be the infinite set {n : b n /a n t}. By (2.2), we have 1 a n+1 < 1 a n − 1 − 1 b n 1 a n − 1 − 1 ta n , ∀n ∈ A.
Trivial calculations give a n+1 a n > t(a n − 1) ta n − (a n − 1) = a n − 1 (a n − 1) − an−1
t + 1 = 1 1 − 1 t + 1 an−1 , ∀n ∈ A.
If t = 1, then a n+1 /a n > a n − 1 for all n ∈ A. That a n → ∞ implies that a n+1 /a n 2 infinitely often, making ii) fail. If t > 1, choose N sufficiently large such that for n > N, a n > 2t + 1. Then for all n ∈ A and n > N, a n+1 a n > 1 1 − 1 t + 1 2t = 1 1 − 1 2t , which contradicts ii).
Remark 3.4. If we replace the hypothesis " ∞ n=1 1/b n = θ" in Proposition 3.2 by " ∞ n=1 1/b n < θ", both i) and ii) in Proposition 3.2 hold. Indeed, if θ − ∞ n=1 1/b n =: c > 0, then
a n := G θ − n−1 i=1 1 b i G(c),
so (a n ) is bounded.
We state and prove the last result in this section.
Theorem 3.5. Let (a n ) ∞ n=1 ⊂ N be non-decreasing such that a 1 2 and a n → ∞. There exist θ ∈ (0, 1) and
(b n ) ∞ n=1 such that ∞ n=1 1 b n = θ,
and for every n 1,
a n = G θ − n−1 i=1 1 b i .
Proof. Since a n → ∞, we can form the infinite set A ⊂ N such that n ∈ A if and only if a n+1 − a n 1. In other words, A contains all the indices immediately before a jump in (a n ). Write A = {n 1 , n 2 , n 3 , . . .}, where n 1 < n 2 < n 3 < · · · . Note that a n j < a n j+1 for all j. We obtain the sequence (b n ) by first constructing all the b n for n ∈ A then constructing the rest.
Step 1: for each j 1, choose b n j such that a n j a n j+1 a n j+1 − a n j − 1 − 2a n j − 1 a n j+1 − a n j < b n j < a n j a n j+1 a n j+1 − a n j , (3.1) which can be done since the distance between the two ends are greater than 1. Note that (3.1) is equivalent to 1 a n j − 1 a n j+1 < 1 b n j < 1 a n j − 1 − 1 a n j+1 − 1 .
(3.2)
It follows that for each j 1, 1 a n j < ∞ i=j 1 b n i < 1 a n j − 1 .
(3.3)
Step 2: Due to (3.3), we can choose a sequence of positive numbers (θ j ) ∞ j=1 satisfying 1 a n j < ∞ i=j 1 b n i + θ j < 1 a n j − 1 .
Let n 0 = 0. For each j 1, set b n j−1 +1 = b n j−1 +2 = · · · = b n j −1 = N j , where N j is sufficiently large such that n j − n j−1 − 1 N j < min θ 1 2 j , θ 2 2 j−1 , . . . , θ j 2 .
Step 3: Set θ := ∞ n=1 1 bn . We claim that θ ∈ (0, 1). We have
∞ n=1 1 b n = ∞ j=1 1 b n j + ∞ j=1 n j −1 i=n j−1 +1 1 b i = ∞ j=1 1 b n j + ∞ j=1 n j − n j−1 − 1 N j < ∞ j=1 1 b n j + ∞ j=1 θ 1 2 j = ∞ j=1 1 b n j + θ 1 < 1 a n 1 − 1 1.
Step 4: Finally, we need to verify that 1 a n < ∞ i=n 1 b i 1 a n − 1 , ∀n 1.
Fix n 1 and choose j such that n j−1 < n n j . By (3.3), we have
∞ i=n 1 b i ∞ i=n j 1 b i ∞ i=j 1 b n i > 1 a n j = 1 a n .
On the other hand,
∞ i=n 1 b i ∞ i=j 1 b n i + ∞ i=j n i −1 n i−1 +1 1 b n = ∞ i=j 1 b n i + ∞ i=j n i − n i−1 − 1 N i < ∞ i=j 1 b n i + ∞ i=j θ j 2 i+1−j = ∞ i=j 1 b n i + θ j < 1 a n j − 1 = 1 a n − 1 .
This completes our proof.
Corollary 3.6. Let (a n ) ∞ n=1 ⊂ N be non-decreasing with a 1 2 and a n → ∞. Then lim n→∞ a n+1 /a n = 1 is equivalent to the existence of θ ∈ (0, 1) and (b n ) ∞ n=1 such that (a n ) ∞ n=1 and (b n ) ∞ n=1 are the sequences obtained from the WGAA applied to θ. Proof. Use Proposition 3.2 and Theorem 3.5.
Remark 3.7. Observe that (3.2) is stronger than (2.2). This observation is important in studying the uniqueness of θ and (b n ) in the next section.
UNIQUENESS OF θ AND (b n )
Thanks to Theorem 3.5, we know the existence of θ and (b n ) given any non-decreasing sequence (a n ) with a 1 2 and a n → ∞. We now give sufficient and necessary conditions for when (a n ) determines θ and (b n ) uniquely. By Step 2 in the proof of Theorem 3.5, a necessary condition is that (a n ) must be strictly increasing. We can then eliminate Step 2 in constructing the sequence (b n ) because A = N. We claim further that a n+1 − a n 2 for all n ∈ N. Indeed, suppose a N +1 − a N = 1 for some N. We rewrite (3.1) as
a N +1 a N − 2a N b N a N a N +1 .
(4.1) There are at least 2a N + 1 choices of b N , so θ and (b n ) are not unique. (Note that we allow equalities in (4.1) because the construction in the proof of Theorem 3.5 still works if we allow equalities in finitely many (3.1).)
Moreover, (b n ) must satisfy (2.2). The following proposition tells us precisely when (2.2) determines (b n ) unequivocally.
Proposition 4.1. Let (a n ) ∞ n=1 be non-decreasing such that a 1 2 and a n → ∞. Then (b n ) ∞ n=1 is uniquely determined by (2.2) if and only if a n+1 − 2 a n 2, ∀n 1, and for each n, one of the following holds i) a n+1 − a n − 1 divides a 2 n , and a n+1 √ 3 2 4a 2 n − 4a n + 3 + 2a n − 1 2 ;
ii) a n+1 − a n − 1 does not divide a 2 n , and a 2 n a n+1 − a n − 1 (a n − 1) 2 a n+1 − a n + 1 .
Proof of Theorem 3.5. By (2.2), (b n ) is uniquely determined if and only if each of the intervals I n := (a n − 1)a n+1 a n+1 − a n + 1 , a n (a n+1 − 1) a n+1 − a n − 1 contains exactly one positive integer. It is easy to verify that there always exists one largest integer in I n , called k n . In order that I n contains no other integers, we need k n − (a n − 1)a n+1 a n+1 − a n + 1 1.
(4.2)
We obtain a formula for k n depending on whether a n (a n+1 − 1)/(a n+1 − a n − 1) is an integer or not.
Case 1: if a n (a n+1 − 1)/(a n+1 − a n − 1) ∈ N, then k n = a n (a n+1 − 1) a n+1 − a n − 1 − 1.
Hence, (4.2) is equivalent to a n (a n+1 − 1) a n+1 − a n − 1 − (a n − 1)a n+1 a n+1 − a n + 1 2.
Equivalently, a 2 n+1 − (4a n − 1)a n+1 + (a 2 n + a n − 2) 0, giving
a n+1 2a n + √ 3 2 4a 2 n − 4a n + 3 − 1 2 .
Case 2: if a n (a n+1 − 1)/(a n+1 − a n − 1) / ∈ N, then k n = a n (a n+1 − 1) a n+1 − a n − 1 = a n + a 2 n a n+1 − a n − 1 .
Hence, (4.2) is equivalent to a n + a 2 n a n+1 − a n − 1 − (a n − 1) 2 a n+1 − a n + 1 + (a n − 1) 1,
giving a 2 n a n+1 − a n − 1 (a n − 1) 2 a n+1 − a n + 1 .
Corollary 4.2 (Sufficient condition for uniqueness). Let (a n ) ∞ n=1 be increasing with a 1 2 and a n → ∞. If i) a n+1 − 2 a n 2 for all n, and ii) for each n 1, one of the following holds a) a n+1 − a n − 1 divides a 2 n , and
a n+1 √ 3 2 4a 2 n − 4a n + 3 + 2a n − 1 2 ;
b) a n+1 − a n − 1 does not divide a 2 n , and a 2 n a n+1 − a n − 1 (a n − 1) 2 a n+1 − a n + 1 ,
then there exist unique θ ∈ (0, 1] and (b n ) ∞ n=1 such that ∞ n=1 1 b n = θ, (4.3)
and for every n 1,
a n = G θ − n−1 i=1 1 b i . (4.4)
Proof. Theorem 3.5 guarantees the existence of θ and (b n ). Suppose that there exists another pair (θ ′ , (b ′ n )) different from (θ, (b n )). Then for some N, b N = b ′ N , both of which must verify (2.2). This contradicts Proposition 4.1.
Next, we establish a necessary condition for the uniqueness of θ and (b n ) by requiring the inequalities a n a n+1 a n+1 − a n − 1 − 2a n − 1 a n+1 − a n b n a n a n+1 a n+1 − a n (4.5)
to determine exactly one solution b n . Again, (4.5) is slightly different from (3.2) as we allow equalities, because the construction in the proof of Theorem 3.5 still works if equalities appear in finitely many (3.1).
Proposition 4.3 (Necessary condition for uniqueness). Let (a n ) ∞ n=1 be non-decreasing with a 1 2 and a n → ∞. Suppose that there exist unique θ ∈ (0, 1) and (b n ) ∞ n=1 that satisfy (4.3) and (4.4), then for all n 1, we have a n+1 a n + 2, (a n+1 − a n ) does not divide a n a n+1 , and a 2 n a n+1 − a n < (a n − 1) 2 a n+1 − a n .
(4.6)
Proof. That a n+1 a n + 2 is due to the discussion at the beginning of this section. We find a sufficient and necessary condition for (4.5) to have exactly one solution b n . If a n+1 − a n divides a n a n+1 , then I n := a n a n+1 a n+1 − a n − 1 − 2a n − 1 a n+1 − a n , a n a n+1 a n+1 − a n contains at least two integers because a n a n+1 a n+1 − a n − a n a n+1 a n+1 − a n − 1 − 2a n − 1 a n+1 − a n > 1.
If a n+1 − a n does not divide a n a n+1 , then the largest integer in I n is a n a n+1 a n+1 − a n , and I n contains exactly one integer if and only if a n a n+1 a n+1 − a n − a n a n+1 a n+1 − a n − 1 − 2a n − 1 a n+1 − a n < 1.
Equivalently, a 2 n a n+1 − a n < (a n − 1) 2 a n+1 − a n .
This completes our proof.
Corollary 4.4. Let (a n ) ∞ n=1 be non-decreasing with a 1 2 and a n → ∞. Suppose that there exist unique θ ∈ (0, 1) and (b n ) ∞ n=1 that satisfy (4.3) and (4.4), then for all n 1, i) a n+1 − a n divides none of (a n − 1) 2 , a 2 n , a n a n+1 ; ii) 3a n < a n+1 .
Proof. i) By Proposition 4.3, a n+1 −a n does not divide a n a n+1 . By (4.6), a n+1 −a n does not divide a 2 n . Also by (4.6), a n+1 − a n does not divide (a n − 1) 2 . Indeed, supposing otherwise, we have a 2 n a n+1 − a n = (a n − 1) 2 + 2a n − 1 a n+1 − a n = (a n − 1) 2 a n+1 − a n + 2a n − 1 a n+1 − a n , contradicting (4.6). ii) We write (4.6) as a 2 n a n+1 − a n < a 2 n a n+1 − a n − 2a n − 1 a n+1 − a n .
Hence, 2a n − 1 a n+1 − a n < 1, which gives a n+1 3a n . However, a n+1 cannot be 3a n . Otherwise, we obtain from (4.6) that a 2 n 2a n < (a n − 1) 2 2a n .
By i), ⌊a 2 n /(2a n )⌋ = (a n − 1)/2. Hence, a n − 1 2 < (a n − 1) 2 2a n =⇒ a n < 1, a contradiction.
APPLICATIONS TO PARTICULAR SEQUENCES
In this section, we look at sequences (a n ) of special forms and find (b n ) that satisfies (3.2). We use specific sequences in [Sl23] as examples.
5.1. Geometric progressions. Let a, r ∈ N with a 2 and r 2. Let (a n ) be the sequence a, ar, ar 2 , ar 3 , . . . . By Corollary 3.6, (a n ) can be obtained from the WGAA applied to some θ.
If r − 1 divides a, we have the sequence b n = ar n /(r − 1) − 1 satisfy (3.2) and θ = ∞ n=1 1 ar n /(r − 1) − 1 .
For example, take a = 2, r = 3 to have a n = 2 · 3 n−1 (A008776), b n = 3 n − 1 (A024023), θ ≈ 0.68215 (irrational due to [Er48]).
If r − 1 does not divide a, we have the sequence b n = ⌊ar n /(r − 1)⌋ satisfy (3.2) and θ = ∞ n=1 1 ⌊ar n /(r − 1)⌋ . a n − 1
a n+1 = 1 F n+1 − 1 F n+2 = F n F n+1 F n+2 .
Using (3.2), we choose b 1 = 3 and for n > 1, choose b n = F n+1 F n+2 F n = F n F n+1 + F n−1 F n+1 + F 2 n + F n−1 F n F n = F n F n+1 + F 2 n + (−1) n + F 2 n + F n−1 F n F n by the Cassini's identity
= F n+3 − 1 if n is odd, F n+3
if n is even.
and Temlyakov[KT99] called a basis quasi-greedy if these approximations converge to the desired x. Meanwhile, Temlyakov[Te98] introduced a weaker version of the TGA, called the weak TGA (WTGA), which is more flexible in forming approximating sums. In particular, fixing a number t ∈ (0, 1], the WTGA considers sets A satisfying |A| = m andmin
n∈A
|a n |
t max
n /
∈A
|a n |.
(1.3)
Clearly, (1.3) is weaker than (1.2). In other words, the WTGA chooses the "largest
coefficients up to a constant." Surprisingly, the flexibility of the WTGA does not affect
convergence: a basis is quasi-greedy under the TGA if and only if it is quasi-greedy
under the WTGA (see [Te08, Section 1.5].)
On representations by Egyptian fractions. F Ambro, M Barcǎu, Rev. Roum. Math. Pures Appl. 60F. Ambro and M. Barcǎu, On representations by Egyptian fractions, Rev. Roum. Math. Pures Appl. 60 (2015), 331-336.
On Kellogg's diophantine problem. D R Curtis, Am. Math. Mon. 29D. R. Curtis, On Kellogg's diophantine problem, Am. Math. Mon. 29 (1922), 380-387.
On arithmetical properties of Lambert series. P Erdős, J. Indian Math. Soc. (N.S.). 12P. Erdős, On arithmetical properties of Lambert series, J. Indian Math. Soc. (N.S.) 12 (1948), 63-66.
A remark on greedy approximation in Banach spaces. S V Konyagin, V N Temlyakov, East J. Approx. 5S. V. Konyagin and V. N. Temlyakov, A remark on greedy approximation in Banach spaces, East J. Approx. 5 (1999), 365-379.
Underapproximation by Egyptian fractions. M B Nathanson, J. Number Theory. 242M. B. Nathanson, Underapproximation by Egyptian fractions, J. Number Theory 242 (2023), 208-234.
The Muirhead-Rado inequality, 2: symmetric means and inequalities. M B Nathanson, preprint (2022). Available atM. B. Nathanson, The Muirhead-Rado inequality, 2: symmetric means and inequalities, preprint (2022). Available at: https://arxiv.org/abs/2201.01270.
The On-Line Encyclopedia of Integer Sequences. N J A Sloane, N. J. A. Sloane et al., The On-Line Encyclopedia of Integer Sequences, 2023. Available at: https://oeis.org.
Approximating 1 from below using Egyptian fractions. K Soundararajan, preprintK. Soundararajan, Approximating 1 from below using Egyptian fractions, preprint (2005). Available at: https://arxiv.org/abs/math/0502247.
On an indeterminate equation. T Takenouchi, Proc. Phys. Math. Soc. Jpn. 3T. Takenouchi, On an indeterminate equation, Proc. Phys. Math. Soc. Jpn. 3 (1921), 78-92.
Greedy approximation. V N Temlyakov, Acta Numer. 17V. N. Temlyakov, Greedy approximation, Acta Numer. 17 (2008), 235-409.
The best m-term approximation and greedy algorithms. V N Temlyakov, Adv. Comput. Math. 8V. N. Temlyakov, The best m-term approximation and greedy algorithms, Adv. Comput. Math. 8, 249-265.
| [] |
[
"The Classical Aharonov-Bohm Interaction as a Relativity Paradox",
"The Classical Aharonov-Bohm Interaction as a Relativity Paradox"
] | [
"Timothy H Boyer \nDepartment of Physics\nCity College of the City University of New York\n10031New YorkNew YorkUSA\n"
] | [
"Department of Physics\nCity College of the City University of New York\n10031New YorkNew YorkUSA"
] | [] | The situation of a charged particle passing down the symmetry axis through a magnetic toroid presents a relativity paradox; different inertial frames suggest different forces on the charge and on the toroid due to the unperturbed systems. We review the charge-toroid interaction and suggest that the magnetic Aharonov-Bohm situation is misunderstood because of unfamiliarity with the acceleration fields following from the Darwin Lagrangian, which go unmentioned in recent textbooks of classical electromagnetism. | 10.1088/1361-6404/acc0e6 | [
"https://export.arxiv.org/pdf/2302.01937v1.pdf"
] | 256,615,603 | 2302.01937 | 2b4a76141141eb4030e72d1f0b686c5891f1b7e2 |
The Classical Aharonov-Bohm Interaction as a Relativity Paradox
3 Feb 2023
Timothy H Boyer
Department of Physics
City College of the City University of New York
10031New YorkNew YorkUSA
The Classical Aharonov-Bohm Interaction as a Relativity Paradox
3 Feb 2023arXiv:2302.01937v1 [physics.class-ph]
The situation of a charged particle passing down the symmetry axis through a magnetic toroid presents a relativity paradox; different inertial frames suggest different forces on the charge and on the toroid due to the unperturbed systems. We review the charge-toroid interaction and suggest that the magnetic Aharonov-Bohm situation is misunderstood because of unfamiliarity with the acceleration fields following from the Darwin Lagrangian, which go unmentioned in recent textbooks of classical electromagnetism.
I. INTRODUCTION
A. The Aharonov-Bohm Situation
The magnetic Aharonov-Bohm phase shift involving electrons passing a long solenoid has attracted great attention because it is claimed to be an effect of the vector potential involving no forces on the passing electrons and having no classical analogue. [1][2] This interpretation of the observed phenomenon is pervasive in the physics literature. [3][4] [5] In contradiction [6] to such views, it is suggested here that the classical electromagnetic interaction of a charge particle and a solenoid should be regarded as yet another example of a relativity paradox where the outcome is easily understood in one interial frame but is disguised in another.
B. Relativity Paradoxes
The appearance of relativity paradoxes is familiar to any instructor who has taught special relativity. Perhaps the most famous example is the pole-and-the-barn paradox where the barn has one open door and a sturdy back wall. [7] The description in the inertial frame of the barn is clear. The farmer claims that the fast-moving pole is Lorentz contracted and so easily fits inside the barn before he closes the door. The account in the rest-frame of the pole is misleading, because the physics in this frame requires new forces which are not mentioned in the original description of the unperturbed motions of the pole and of the barn. Similarly, the Aharonov-Bohm situation involves two unperturbed systems in relative motion, in this case a point charge and a solenoid. The description is misleading in the inertial frame where the solenoid is at rest. Conservation of energy in this inertial frame requires forces arising from particle accelerations which are not mentioned in the original description of the unperturbed motion of the moving charge and constant-current solenoid.
C. Aharonov-Bohm Situation as Relativity Paradox
The classical Aharonov-Bohm situation involves the electromagnetic interaction of a charged particle and a magnet at the relativistic 1/c 2 -level, though this relativity aspect is rarely mentioned in the literature. The interaction between the charged particle and the solenoid is calculated in the approximation that each continues its unperturbed behavior during the interaction. The interaction is much more easily understood in the interial frame where the charged particle is at rest and the solenoid is moving, because in this inertial frame, the physics requires no new forces beyond those arising from the original descriptions of the unperturbed parts of the interacting system. Indeed, in this inertial frame where the charge is at rest, it is easy to verify the energy conservation law based upon the equal-andopposite electric forces that the unperturbed charge and solenoid put on each other. On the other hand, in the inertial frame in which the solenoid is at rest, energy conservation is violated unless one introduces additional particle accelerations or external forces beyond those present in the unperturbed solenoid. If the charges of the solenoid are allowed to accelerate, they introduce back (Faraday) forces on the electron which were not included in the original description of an unperturbed solenoid. Alternatively, one may introduce external forces holding the solenoid particles at constant speed, and these external forces account for the required changes in energy, but such external forces were not part of the original description of the interaction of a charged particle and a solenoid as unperturbed systems.
D. Paradoxes Involving Particle-Magnet Interactions
The interaction of charges and magnets occurs at the relativistic 1/c 2 -level of energy and momentum. Because the interaction of charges and magnets at the relativistic level is poorly understood in classical electrodynamics, it has given rise to a whole class of "paradoxes," including the Aharonov-Bohm phase shift,[1] the Aharonov-Casher phase shift, [8] the Shockley-James paradox, [9] "hidden momentum in magnets," [10] and Mansuripur's erroneous claim. [11] All of these effects involve relativistic interactions where our familiar experience with nonrelativistic mechanics, or with electrostatics, or with magnetostatics may not be adequate. These interactions can all be treated at the level of the Darwin Lagrangian [12] which describes quasi-static classical electrodynamics which excludes radiation.
In this article, we will treat only the energy conservation aspects of the interaction between a point charge and a toroid. A more complete description of the interaction between a point charge and magnet will be published elsewhere. [13]
II. INTERACTION OF A POINT CHARGE AND A MAGNETIC MOMENT
A. Magnetic Dipole Moment
At its basic level, the problem of the classical Aharonov-Bohm situation involves the interaction of a magnetic moment m and a point charge e. We will picture the magnetic moment in its own S m rest frame as an electrically-neutral circular current loop of radius b and current I. The magnetic moment of this current loop (in Gaussian units) is
m = nπb 2 I/c,(1)
where the direction n is normal to the plane of the current loop and is connected to the direction of the current I by the right-hand rule. If the center of the current loop is at r m , we assume that the point charge e at r e is sufficiently far away that the separation is large compared to the radius b, b << |r e − r m |, and so the magnetic dipole approximation is adequate.
B. Interaction in the Inertial Frame where the Magnetic Moment is at Rest
In the S m inertial frame where the magnetic moment is at rest and the charge e is moving with constant velocity v e = v, the charge e carries (through order 1/c 2 ) both an electric field E e (r, t) = e (r − r e ) |r − r e | 3 1 +
1 2 v 2 c 2 − 3 2 (r − r e ) · v |r − r e | c 2(2)
and a magnetic field
B e (r, t) = e v c × (r − r e ) |r − r e | 3 ,(3)
so that the charge e has an interaction energy with the magnetic moment given by the magnetic field energy [14] ∆U
(B) = −m · B e (r m , t) ≈ −m· e v c × (r m −r e ) |r m −r e | 3 .(4)
In this inertial frame, the magnetic moment experiences a magnetic force
F (B) onm = −∇ m [−m · B e (r m , t)] = ∇ m m· e v c × (r m −r e ) |r m −r e | 3 = ∇ m m×e v c · (r m −r e ) |r m −r e | 3 = m×e v c · ∇ m (r m −r e ) |r m −r e | 3 = −3 [m×e (v/c)] · (r m −r e ) (r m −r e ) |r m −r e | 5 + [m×e (v/c)] |r m −r e | 3 ,(5)
while the charge e experiences a (deflecting) magnetic force due to the magnetic dipole
F (B) one = e v c × B m (r e , t) = e v c × 3m· (r e −r m ) (r e −r m ) |r e −r m | 5 − m |r e −r m | 3 .(6)
In this inertial frame, the forces between the magnetic moment and the charge are not equal in magnitude and opposite in direction; Eq. (5) involves a term in the direction (r m −r e )
whereas Eq. (6) involves a term in the dirction v× (r e −r m ).
C. Interaction in the Inertial Frame where the Charge e is at Rest
On the other hand, in the S e inertial frame where the charge particle e is at rest and the magnetic moment is moving with velocity v m = −v, the interaction between the charge and the magnetic moment involves energy in the electric fields because, in this frame where it is moving, the magnetic moment has an electric dipole moment [15]
p m ≅ −v c × m.(7)
In this S e inertial frame, the electric interaction energy is [16] ∆U
(E) = −p m · E e (r m , t) = − −v c × m · e (r m −r e ) |r m −r e | 3 ,(8)
which is the same as the magnetic energy given in Eq. (4). The electric force on the magnetic moment is accordingly
F (E) onm = −∇ m [−p m · E e (r m , t)] = (p m · ∇ m ) E e (r m , t) = −v c × m · ∇ m e (r m −r e ) |r m −r e | 3 = −3 [(−v/c) × m] · (r e −r m ) (r e −r m ) |r m −r e | 5 + [(−v/c) × m] |r m −r e | 3 .(9)
Noting the reversals of sign connected with the order in the cross products, one finds that this electric force on the magnetic dipole in Eq. (9) is the same as the magnetic force on the magnetic dipole appearing in Eq. (5). Also, the electric force on the charge e is just the negative of this expression,
F (E) one = eE m (r e , t) = e 3 [(−v/c) × m] · (r e −r m ) (r e −r m ) |r e −r m | 5 − [(−v/c) × m] |r e −r m | 3 .(10)
Through order 1/c 2 in this S e inertial frame, the electric forces that the magnetic moment and charge place on each other are equal in magnitude and opposite in direction. The change in electric field energy is accounted for by the work done by the electric force F (E) onm on the moving magnetic moment.
III. TRANSITION TO A POINT CHARGE AND TOROID
A. Forming a Toroid from Magnetic Dipoles
Although the equations which we have listed already record the basic paradox, the situation becomes far more vivid, and also simpler calculationally, if we imagine many magnetic moments arranged so as to form a toroid. And indeed, a toroid can be pictured as a solenoid (a stack of current loops) which is bent into a circular shape and so brings us to the Aharonov-Bohm situation where electrons pass a long solenoid.
Thus, we picture the magnetic moments (which are simply circular current loops of radius b) arranged in a circular pattern of (average) radius R around the z-axis so as to form a toroid located along the z-axis at z T . Each current loop lies in the plane formed by the z-axis and the displacement from the z-axis to the center of the current loop. We assume that there are N current loops, each carrying current I and that the (average) radius R of the toroid is much larger than the radius b of each current loop, b << R. The average magnetic field inside the toroid is
B T = φ 4π c NI 2πR = φ 2NI cR ,(11)
and the magnetic flux through each current loop of the toroid is
Φ = πb 2 B T = 2πb 2 NI cR .(12)
For the electrically-neutral toroidal situation, there are no toroidal electric fields, and all the magnetic fields are confined to the interior of the toroid.
We consider a charged particle e moving with velocity v e = zv along the z-axis, which is the axis of symmetry of the toroid. We want to obtain the lowest non-vanishing approximation for the interaction between the charge e and the toroid. This "lowest-nonvanishinginteraction" approximation suggests that we consider the toroid and the charge e as continuing their unperturbed motions despite their mutual interaction. Thus we consider the currents carried by the charge carriers of the toroid as constant. We also consider the velocity v e of the charge e as constant. With these assumptions, we wish to determine the forces on the charge e and on the toroid due to the toroid and the charge e respectively through order 1/c 2 .
C. Toroid at Rest
In the S T inertial frame where the toroid is at rest and the charge e is moving with velocity v e = zv, it appears that the passing charge puts a magnetic force (corresponding to Eq. (5)
onT = − z ∂ ∂z T −N πb 2 I c B e (z T , t = z ∂ ∂z T Nπb 2 I c e v c R (z T − z e ) 2 + R 2 3/2 = z Nπb 2 I c e v c R [−3 (z T − z e )] (z T − z e ) 2 + R 2 5/2 .(13)
Since the toroid is electrically neutral and all the magnetic fields of the toroid are confined to the interior of the toroid, there appears to be no force back of the unperturbed toroid on the charge e.
Since the charge particle e experiences no forces, there is no change in its kinetic energy.
Since the toroid is electrically neutral, there is no change in the electric energy as the charge e and the toroid interact. However, there is a change in the system magnetic energy associated with the overlap of the magnetic field of the charge with the magnetic field of the toroid in the volume of the toroid,
∆U (B) overlap = 1 4π d 3 rB e · B T = 1 4π evR c (z T − z e ) 2 + R 2 3/2 2NI cR 2πRπb 2 .(14)
In this inertial frame, it may appear that the relativistic conservation law of energy is violated, since there is apparently no force on the moving charge e and the currents of the toroid are assumed unperturbed.
D. Charge e at Rest
On the other hand, in the S e inertial frame in which the charge e is at rest while the unperturbed toroid is moving with velocity v T = − zv , there are electric forces between the charged particle and the toroid. In an inertial frame in which it is moving with velocity −v,
Also, the charge e will place an electric force on each electric dipole of the moving toroid, giving a net force on the toroid
F (E) onT = z {N z · [(p m · ∇ m ) E e (r m , t)]} = zN z · p m ∂ ∂r e rr + z (z T − z e ) (z T − z e ) 2 + r 2 3/2 r=R = zN p m e −3r (z T − z e ) (z T − z e ) 2 + r 2 5/2 r=R = zN v c πb 2 I c e −3R (z T − z e ) (z T − z e ) 2 + R 2 5/2 .(16)
This electric force on the toroid in Eq. (16) is exactly the same as the magnetic force as found in Eq. (13) for the previous inertial frame where the toroid was at rest and the charge e was moving. However, here in the S e inertial frame where the charge e is at rest and the toroid is moving, the electric forces on the charge e and on the toroid are equal in magnitude and and opposite in direction.
During the interaction, there is no energy change in the magnetic field energy, since the charge e is at rest and so has no magnetic field. However, during the interaction, there is a change in the electric field energy given by
∆U (E) = −Np m · E e (r m , t) = −N v c πb 2 I c E er (r m , t) = −N v c πb 2 I c e R (z T − z e ) 2 + r 2 3/2 .(17)
The electric energy change ∆U (E) in Eq. (17) is accounted for by the electric force F
(E)
onT on the moving toroid,
∆U (E) = − zm ∞ F (E) T · zdz T .(18)
Thus energy conservation involving the unperturbed parts of the system indeed holds in the S e inertial frame in which the toroid is moving with velocity −v and the charge e is at rest.
On the other hand, the change in electric energy ∆U (E) in Eq. (17)
IV. DISCUSSION OF THE RELATIVITY PARADOX
A. Contrast in Forces Between Different Inertial Frames
Thus we have our relativity "paradox." In both inertial frames, all the forces are of order 1/c 2 and so the forces cannot change in leading order in v/c when viewed from a different inertial frame. Nevertheless, different inertial frames claim that different forces appear. When described in the S T rest frame of the toroid, there is a magnetic force on the toroid, but apparently no force on the moving charge e. However, when described in the S e rest frame of the charge e, there are electric forces on the charge e and also on the toroid. Indeed, in the rest frame of the charge e, one finds exactly the same force on the toroid (now an electric force) as was found as a magnetic force in the inertial frame where the toroid is at rest, but now one also finds its partner in the electric force of the toroid on the charge e.
B. The Inertial Frame with the Unreliable Description
Just as in the relativity paradox of the pole and the barn, one must make a choice. Which description should one trust as representing accurate physics? We suggest that in each case, the accurate description involves the inertial frame in which the physics does not require the introduction of external forces and/or accelerations which were not part of the original account of the unperturbed motion. For the pole and the barn, the unreliable description involves the inertial frame in which the barn is moving, and so is Lorentz contracted; this inertial frame requires the introduction of new forces when the front of the pole encounters the back wall of the barn, before the barn door is closed. [7] These external forces alter the account given for the unperturbed motion of the pole.
In the situation of the classical Aharonov-Bohm interaction of a charged particle and a magnet, the situation involves the same basic idea. In which inertial frame does the physics require the introduction of new forces and/or accelerations which were not part of the original account of unperturbed motion? The answer is that the S T inertial frame in which the toroid is at rest is less satisfactory; specifically, the changes in magnetic energy associated with the overlap of the magnetic field of the charge e and the magnetic field of the toroid have not been accounted for satisfactorily.
C. Problems Involving Magnetic Energy Changes
Indeed, changes in magnetic energy often present problems. They are the basis of the present paradox. Electric energy changes involve work done directly by the electric forces, as is evident in the second description given for our charge-magnetic interaction where the charge e is at rest and the toroid is moving. In contrast, magnetic forces do no work. Therefore magnetic energy changes require work being done by separate electric or external forces. Magnetic energy balance in quasistatic systems requires the existence of electric forces associated with the accelerations involving changing speeds of charge particles. Such accelerations are not contained in the description of the unperturbed toroid.
D. Balancing Magnetic Energy Changes for the Toroid at Rest
The energy balance for the system of the charge e and the toroid involves three different contributions, mechanical kinetic energy, electric energy, and magnetic energy
∆U = ∆U (M ) + ∆U (E) + ∆U (B) .
The troublesome aspect, as usual, involves the magnetic energy ∆U (B) . Although the 1/c 2force on the toroid (given in Eqs. (13) and (16)) is exactly the same in either inertial frame, the 1/c 2 -energy change of the system given in Eqs. (14) and (17) is not the same, but indeed involves a relative minus sign. The difficulty here involves the same aspect which appears in any discussion of magnetic energy changes for quasistatic systems. [17] [18] There is a sharp contrast between electric and magnetic energy changes. Electric energies involve only the relative positions between charged particles. However, quasistatic magnetic energies involve moving charges. Therefore magnetic energy changes can involve changes in 1) the relative positions of the current carriers and/or in 2) the speeds of the charge carriers.
For our charge-toroid example in the S T inertial frame in which the toroid is at rest and the charge e is moving, we have both aspects of magnetic energy change,
∆U (B) = ∆U (B) overlap + ∆U (B) toroid currents .
There is a positive magnetic energy change ∆U (B) overlap associated with the overlap of the magnetic field of the charge e with the magnetic field of the toroid. However, the electric fields of the charge e act on the current carriers of the toroid. The zero-order electrostatic field of the charge e has no emf and so does not deliver net energy to the toroid currents.
It is the terms of order v 2 /c 2 in Eq. (2) which do indeed produce an emf and deliver net energy to (or remove magnetic energy from) the toroid currents . The toroid responds to the effort to change the speeds of the current carriers [19] in the fashion typical of a solenoid. The electric force on the charge e appears immediately in the unperturbed -motion discussion in the S e inertial frame in which the toroid is moving and so (according to the relativistic description of the unperturbed toroid motion) has an electric dipole moment. In the S T rest frame of the toroid, the basis for the back field on the charge e involves particle accelerations which are not part of the description of the unperturbed toroid. Thus the unperturbed description in the S T restframe of the toroid, which does not mention the fields arising from the accelerations of the current carriers, is indeed the less reliable description of the relativity paradox.
E. Absence of Quasistatic Acceleration Terms in Recent Textbooks
The back (Faraday) acceleration fields (which are unfamiliar in the interaction of a charge e and a toroid) are thoroughly familiar in the case of a solenoid with increasing currents. The back emf appearing in a solenoid when the currents are increasing is caused by these same back (Faraday) acceleration fields of the accelerating current carriers of the solenoid. [19] However, in the current textbooks of classical electromagnetism, the solenoid's back emf is calculated from a changing magnetic flux for a highly-symmetric solenoid, not from the accelerations of the current carriers.
Acceleration electric fields appear immediately from the Darwin Lagrangian. Thus, at the quasistatic 1/c 2 -level, the electric field of an accelerating charge e is not that given in Eq.
(2) for a constant-velocity charge e, but rather includes additional acceleration-dependent terms, [20][21]
E (r,t) = a q a (r − r a ) |r − r a | 3 1 + 1 2 ṙ a c 2 − 3 2 ṙ a · (r − r a ) |r − r a | c 2 − a q a 2c 2 r a |r − r a | + (r − r a ) [r a · (r − r a )] |r − r a | 3 .(19)
However, even as the Darwin Lagrangian, is barely mentioned in the recent textbooks of classical electromagnetism, the local (Faraday) acceleration fields in Eq. (19) for an accelerating charge are never mentioned. Fields due to accelerating charges appear only in the sections on radiation leading to Larmor's formula.
F. Classical Counterpart to the Aharonov-Bohm Effect
The interaction between a charged particle and a magnet is a relativistic effect of order 1/c 2 to lowest order. Therefore the interaction is adequately described by the Darwin Lagrangian which reproduces classical electrodynamics through order 1/c 2 but excludes radiation. We expect that the same basic interaction continues to hold for full classical electrodynamics, where we have the additional complications of retarded times and (very small) radiation effects.
It seems widely accepted that there is "no classical analogue to the Aharonov-Bohm effect." Statements of this sort appear in many textbooks of quantum theory [22] and in some textbooks of classical electromagnetism. [4][5] The usual argument for this no-classicalanalogue statement notes that the magnetic field vanishes outside a very long solenoid or toroid where the currents are constant, and hence concludes that there is no force on a passing charged particle. However, such unsophisticated views based upon magnetostatics do not do justice to the subtleties of classical electrodynamics. Because physicists are unfamiliar with the idea of quasistatic accelerating charges producing the electric fields associated with an emf , the claims associated with the classical Aharonov-Bohm situation have been rarely challenged. [23] V. ACKNOWLEDGEMENT
The reanalysis here of the classical interaction of a charged particle and a magnet was stimulated by a manuscript of Dr. Hanno Essén, "A classical Aharonov-Bohm effet arises when one goes beyond the test particle approximation." I wish to thank Dr. Essén for alerting me to the work included in reference 21, [2] For reviews of the Aharonov-Bohm phase shift, see for example, S. Olariu and I. Iovitzu
Popescu, "The Quantum Effects of Electromagnetic Fluxes," Rev. Mod. Phys. 57, 339-436
) on each magnetic dipole moment (circular current loop) of the toroid. By symmetry, the z-components of the forces add while the radial components cancel. The magnetic field of the charge e (assumed positive) is in the same circular pattern as that of the toroid. Taking the negative derivative of the −m · B contributions gives a total magnetic force on the toroidF (B)
an unperturbed magnetic moment m has an electric dipole moment p m = (−v/c) × m as given in Eq.(7). Thus in the S e inertial frame in which it is moving with velocity −v, the unperturbed toroid has a ring of electric dipoles which produce a net z-component of electric force on the charge e which is N times larger than the z-component of force produced by a single electric dipole in Eq. (10),F (E) one = eE T (r e , t) = ze N3R (z T − z e )c (z T − z e ) 2 + R
is exactly equal to the negative of the change in magnetic energy ∆U (B) overlap in Eq. (14) due to the overlap of the magnetic field of the charge e and the magnetic field of the toroid.
The (small) accelerations of the (many) toroid current carriers produce a back (Faraday) acceleration electric field acting on the agent causing the original emf , in this case on the charge e. The magnetic energy change due to the changing toroid currents involves B 2 T and so is twice as large and of opposite sign as the overlap magnetic energy change which involves only the first power of B T .It is the back (Faraday) acceleration electric field of these accelerating charge carriers which places a force on the charge e in the S T inertial frame where the toroid is at rest and the charge e is moving. The energy-balancing back force is or order 1/c 2 .
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Electric and magnetic forces and energies for a parallel-plate capacitor and a flattened, slip-joint solenoid. T H Boyer, Am J. Phys. 69T. H. Boyer, "Electric and magnetic forces and energies for a parallel-plate capacitor and a flattened, slip-joint solenoid," Am J. Phys. 69, 1277-1279 (2001).
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Many-Body Interactions in Atomic and Nuclear Systems. B Podolsky, K S Kunz, ; H Primakoff, T Holstein, Fundamentals of Electrodynamics. New YorkMarcel DekkerSee also, B. Podolsky and K. S. Kunz, Fundamentals of Electrodynamics (Marcel Dekker, New York 1969); and H. Primakoff and T. Holstein, "Many-Body Interactions in Atomic and Nuclear Systems," Phys. Rev. 1218-1234 (1939).
D J See For Example, Griffiths, or L. E. Balentine, Quantum Mechanics. Upper Saddle River, NJ; Englewood Cliffs, New JerseyPrentice HallIntroduction to Quantum MechanicsSee for example, D. J. Griffiths, Introduction to Quantum Mechanics 2nd ed. (Pearson Prentice Hall, Upper Saddle River, NJ 2005), pp. 384-391 or L. E. Balentine, Quantum Mechanics (Prentice Hall, Englewood Cliffs, New Jersey 07632, 1990), pp. 220-223.
A full discussion of the forces and energy changes is given in reference. 13A full discussion of the forces and energy changes is given in reference [13].
February 3 AB-asParadox4.tex. February 3 AB-asParadox4.tex
| [] |
[
"Controllability-Aware Unsupervised Skill Discovery",
"Controllability-Aware Unsupervised Skill Discovery"
] | [
"Seohong Park ",
"Kimin Lee ",
"Youngwoon Lee ",
"Pieter Abbeel "
] | [] | [] | One of the key capabilities of intelligent agents is the ability to discover useful skills without external supervision. However, the current unsupervised skill discovery methods are often limited to acquiring simple, easy-to-learn skills due to the lack of incentives to discover more complex, challenging behaviors. We introduce a novel unsupervised skill discovery method, Controllabilityaware Skill Discovery (CSD), which actively seeks complex, hard-to-control skills without supervision. The key component of CSD is a controllability-aware distance function, which assigns larger values to state transitions that are harder to achieve with the current skills. Combined with distance-maximizing skill discovery, CSD progressively learns more challenging skills over the course of training as our jointly trained distance function reduces rewards for easy-toachieve skills. Our experimental results in six robotic manipulation and locomotion environments demonstrate that CSD can discover diverse complex skills including object manipulation and locomotion skills with no supervision, significantly outperforming prior unsupervised skill discovery methods. Videos and code are available at | 10.48550/arxiv.2302.05103 | [
"https://export.arxiv.org/pdf/2302.05103v3.pdf"
] | 256,808,231 | 2302.05103 | e966cca871cef85f3bfb9a6c69cdcbec23357c1d |
Controllability-Aware Unsupervised Skill Discovery
Seohong Park
Kimin Lee
Youngwoon Lee
Pieter Abbeel
Controllability-Aware Unsupervised Skill Discovery
One of the key capabilities of intelligent agents is the ability to discover useful skills without external supervision. However, the current unsupervised skill discovery methods are often limited to acquiring simple, easy-to-learn skills due to the lack of incentives to discover more complex, challenging behaviors. We introduce a novel unsupervised skill discovery method, Controllabilityaware Skill Discovery (CSD), which actively seeks complex, hard-to-control skills without supervision. The key component of CSD is a controllability-aware distance function, which assigns larger values to state transitions that are harder to achieve with the current skills. Combined with distance-maximizing skill discovery, CSD progressively learns more challenging skills over the course of training as our jointly trained distance function reduces rewards for easy-toachieve skills. Our experimental results in six robotic manipulation and locomotion environments demonstrate that CSD can discover diverse complex skills including object manipulation and locomotion skills with no supervision, significantly outperforming prior unsupervised skill discovery methods. Videos and code are available at
Introduction
Humans are capable of autonomously learning skills, ranging from basic muscle control to complex acrobatic behaviors, which can be later combined to achieve highly complex tasks. Can machines similarly discover useful skills without any external supervision? Recently, many unsupervised skill discovery methods have been proposed to discover diverse behaviors in the absence of extrinsic rewards (Gregor et al., 2016;Eysenbach et al., 2019;Sharma et al., 2020;Achiam et al., 2018;Campos Camúñez et al., 2020;Hansen et al., 2020;Kim et al., 2021;Liu & Abbeel, 2021a; Proceedings of the 40 th International Conference on Machine Learning, Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright 2023 by the author(s).
CSD (ours)
LSD DIAYN Object (b) Skill trajectories (a) FetchPush Gripper Figure 1. Object trajectories and gripper trajectories of 2-D continuous skills discovered by three unsupervised skill discovery methods, CSD (ours), LSD , and DIAYN (Eysenbach et al., 2019), in the FetchPush environment. Trajectories with different colors represent different skills. While previous methods focus only on maneuvering the gripper, CSD discovers object manipulation skills in the absence of supervision. Laskin et al., 2022). These methods have also demonstrated efficient downstream reinforcement learning (RL) either by fine-tuning (Laskin et al., 2021; or sequentially combining (Eysenbach et al., 2019;Sharma et al., 2020; the discovered skills.
However, in complex environments, current unsupervised skill discovery methods are often limited to discovering only simple, easy-to-learn skills. For example, as illustrated in Figure 1, previous approaches (LSD and DIAYN) only learn to gain control of the agent's own 'body' (i.e., the gripper and joint angles), completely ignoring the object in the Fetch environment. This is because learning difficult skills, such as interacting with the object, has no incentive for them compared to learning easy skills. In other words, their objectives can be fully optimized with simple skills.
To mitigate this issue, prior approaches incorporate human supervision, such as limiting the agent's focus to specific dimensions of the state space of interest (Eysenbach et al., 2019;Sharma et al., 2020;Adeniji et al., 2022). However, this not only requires manual feature engineering but also significantly limits the diversity of skills. On the other hand, we humans consistently challenge ourselves to learn more complex skills after mastering simple skills in an autonomous manner.
Inspired by this, we propose a novel unsupervised skill discovery method, Controllability-aware Skill Discovery (CSD), which explicitly seeks complex, hard-to-learn behaviors that are potentially more useful for solving down-stream tasks. Our key idea is to train a controllability-aware distance function based on the current skill repertoire and combine it with distance-maximizing skill discovery. Specifically, we train the controllability-aware distance function to assign larger values to harder-to-achieve state transitions and smaller values to easier-to-achieve transitions with the current skills. Since CSD aims to maximize this controllabilityaware distance, it autonomously learns increasingly complex skills over the course of training. We highlight that, to the best of our knowledge, CSD is the first unsupervised skill discovery method that demonstrates diverse object manipulation skills in the Fetch environment without any external supervision or manual feature engineering (e.g., limiting the focus only to the object).
To summarize, the main contribution of this work is to propose CSD, a novel unsupervised skill discovery method built upon the notion of controllability. We also formulate a general distance-maximizing skill discovery approach to incorporate our controllability-aware distance function with skill discovery. We empirically demonstrate that CSD discovers various complex behaviors, such as object manipulation skills, with no supervision, outperforming previous state-of-the-art skill discovery methods in diverse robotic manipulation and locomotion environments.
Preliminaries
Unsupervised skill discovery aims at finding a potentially useful set of skills without external rewards. Formally, we consider a reward-free Markov decision process (MDP) defined as M = (S, A, µ, p), where S and A are the state and action spaces, respectively, µ : P(S) is the initial state distribution, and p : S × A → P(S) is the transition dynamics function. Each skill is defined as a skill latent vector z ∈ Z and a skill-conditioned policy π(a|s, z) that is shared across the skills. The skill space Z can be either discrete skills ({1, 2, . . . , D}) or continuous skills (R D ).
To collect a skill trajectory (behavior), we sample a skill z from a predefined skill prior distribution p(z) at the beginning of an episode. We then roll out the skill policy π(a|s, z) with the sampled z for the entire episode. For the skill prior, we use a standard normal distribution for continuous skills and a uniform distribution for discrete skills.
Throughout the paper, I(·; ·) denotes the mutual information and H(·) denotes either the Shannon entropy or differential entropy depending on the context. We use uppercase letters for random variables and lowercase letters for their values (e.g., S denotes the random variable for states s).
Related Work
In this section, we mainly discuss closely related prior unsupervised skill discovery work based on mutual information maximization or Euclidean distance maximization. A more extensive literature survey on unsupervised skill discovery and unsupervised RL can be found in Appendix A.
Mutual Information-Based Skill Discovery
Mutual information-based unsupervised skill discovery maximizes the mutual information (MI) between states S and skills Z, I(S; Z), which associates different states with different skill latent vectors so that the behaviors from different zs are diverse and distinguishable. Since computing exact MI is intractable, previous MI-based methods approximate MI in diverse ways, which can be categorized into reverse-MI and forward-MI (Campos Camúñez et al., 2020).
First, reverse-MI approaches (Gregor et al., 2016;Eysenbach et al., 2019;Achiam et al., 2018;Hansen et al., 2020) optimize MI in the form of I(S; Z) = H(Z) − H(Z|S), where H(Z) is a constant as we assume that the skill prior distribution p(z) is fixed. Thus, maximizing I(S; Z) corresponds to minimizing H(Z|S), which can be approximated with a variational distribution q θ (z|s). For instance, DIAYN (Eysenbach et al., 2019) maximizes the variational lower bound of MI as follows:
I(S; Z) = −H(Z|S) + H(Z) (1) = E z,s [log p(z|s)] − E z [log p(z)] (2) ≥ E z,s [log q θ (z|s)] + (const),(3)
where q θ (z|s) is a variational approximation of p(z|s) (Barber & Agakov, 2003). Intuitively, q θ (z|s) works as a 'skill discriminator' that tries to infer the original skill z from the state s, encouraging the skill policy to generate distinguishable skill trajectories for different zs (i.e., diverse skills).
Other reverse-MI methods optimize the MI objective similarly but computing MI on entire trajectories (Achiam et al., 2018) or only on final states (Gregor et al., 2016) rather than all intermediate states, or using von Mises-Fisher distributions (Hansen et al., 2020) for the skill prior distribution instead of Gaussian or uniform distributions.
On the other hand, forward-MI approaches (Sharma et al., 2020;Campos Camúñez et al., 2020;Liu & Abbeel, 2021a;Laskin et al., 2022) employ the other decomposition of MI:
I(S; Z) = H(S) − H(S|Z)
. This decomposition explicitly maximizes the state entropy H(S), which helps diversify skill trajectories in practice (Laskin et al., 2022). Forward-MI methods minimize the H(S|Z) term with a variational approximation (Sharma et al., 2020;Liu & Abbeel, 2021a;Campos Camúñez et al., 2020) or a contrastive estimator (Laskin et al., 2022). H(S) can be estimated using a particlebased entropy estimator (Liu & Abbeel, 2021a;Laskin et al., 2022), a state marginal matching objective (Lee et al., 2019;Campos Camúñez et al., 2020), or sampling-based approximation (Sharma et al., 2020).
One major limitation of MI-based approaches is that optimizing the MI objective does not necessarily lead to cov-(c) LSD (Euclidean distance) (d) CSD (ours) (Controllability-aware distance) (b) Skill space mappings of LSD and the MI objective (a) Two skill sets having the same MI Figure 2. Illustration of unsupervised skill discovery methods. (a) MI is invariant to traveled distances. (b) The MI objective simply seeks any mapping between Z and S, while LSD finds the largest (longest) possible mapping. (c) LSD maximizes the Euclidean traveled distance, which can lead to simple or trivial behaviors. (d) Our CSD maximizes the traveled distance with respect to our learned controllability-aware distance function that assigns larger values to harder-to-achieve state transitions. This leads to more complex skills that can be useful for downstream tasks. ering a larger region in the state space. This is because MI is invariant to traveled distances or any invertible transformation (Figure 2a), i.e., I(S; Z) = I(f (S); Z) for any invertible f (Kraskov et al., 2004). Since there is no incentive for the MI objective to further explore the state space, they often end up discovering 'static' skills with limited state coverage (Gu et al., 2021;Laskin et al., 2022).
Euclidean Distance-Maximizing Skill Discovery
To resolve the limitation of MI-based skill discovery, recently proposed Lipschitz-constrained Skill Discovery (LSD), which aims to not only establish a mapping between Z and S but also maximize the Euclidean traveled distance in the state space for each skill. Specifically, LSD maximizes the state change along the direction specified by the skill z with the following objective:
J LSD := E z,s,s ′ [(ϕ(s ′ ) − ϕ(s)) ⊤ z](4)s.t. ∀x, y ∈ S, ∥ϕ(x) − ϕ(y)∥ ≤ ∥x − y∥,(5)
where s ′ denotes the next state and ϕ : S → R D denotes a mapping function. LSD maximizes Equation (4) with respect to both the policy and ϕ. Intuitively, this objective aims to align the directions of z and (ϕ(s ′ ) − ϕ(s)) while maximizing the length ∥ϕ(s ′ ) − ϕ(s)∥, which leads to an increase in the state difference ∥s ′ − s∥ due to the Lipschitz constraint. As illustrated in Figure 2b, LSD finds the largest possible mapping in the state space by maximizing Euclidean traveled distances in the state space in diverse directions, which leads to more 'dynamic' skills. On the other hand, the MI objective finds any mapping between the skill space and the state space, being agnostic to the area of the mapped region, which often results in 'static' skills with limited state coverage.
While promising, LSD is still limited in that it maximizes Euclidean traveled distances in the state space, which often does not match the behaviors of our interests because the Euclidean distance treats all state dimensions equally. For example, in the Fetch environment in Figure 1, simply diversifying the position and joint angles of the robot arm is sufficient to achieve large Euclidean traveled distances because both the coordinates of the object and the gripper lie in the same Euclidean space ( Figure 2c). As such, LSD and any previous MI-based approaches mostly end up learning skills that only diversify the agent's own internal states, ignoring the external states (e.g., object pose).
Instead of maximizing the Euclidean distance, we propose to maximize traveled distances with respect to a learned controllability-aware distance function that 'stretches' the axes along hard-to-control states (e.g., objects) and 'contracts' the axes along easy-to-control states (e.g., joint angles), so that maximizing traveled distances results in the discovery of more complex, useful behaviors ( Figure 2d).
Unsupervised Goal-Conditioned RL
Another line of unsupervised RL focuses on discovering a wide range of goals and learning corresponding goalreaching policies, which leads to diverse learned behaviors (Warde-Farley et al., 2019;Pong et al., 2020;Pitis et al., 2020;Mendonca et al., 2021). On the other hand, unsupervised skill discovery, including our approach, (1) focuses on more general behaviors (e.g., running, flipping) not limited to goal-reaching skills, whose behaviors tend to be 'static' (Mendonca et al., 2021;Jiang et al., 2022), and (2) aims to learn a compact set of distinguishable skills embedded in a low-dimensional, possibly discrete skill space, rather than finding all possible states, making it more amenable to hierarchical RL by providing a low-dimensional high-level action space (i.e., skill space). While these two lines of approaches are not directly comparable, we provide empirical comparisons and further discussion in Appendix C.
Controllability-Aware Skill Discovery
To discover complex, useful skills without extrinsic reward and domain knowledge, we introduce the notion of controllability 1 to skill discovery -once an agent discovers 1 The term controllability in this paper describes whether an agent can manipulate hard-to-control states (e.g., external objects) or not, different from the one used in control theory (Ogata et al., 2010). easy-to-achieve skills, it continuously moves its focus to hard-to-control states and learns more diverse and complex skills. We implement this idea in our Controllability-aware Skill Discovery (CSD) by combining a distance-maximizing skill discovery approach (Section 4.1) with a jointly trained controllability-aware distance function (Section 4.2), which enables the agent to find increasingly complex skills over the course of training (Section 4.3).
General Distance-Maximizing Skill Discovery
As explained in Section 3.2, Euclidean distance-maximizing skill discovery does not necessarily maximize distances along hard-to-control states (i.e., hard-to-achieve skills). To discover more challenging skills, we propose to learn a skill policy with respect to a jointly learned controllability-aware distance function.
To this end, we first present a general Distance-maximizing Skill Discovery approach (DSD) that can be combined with any arbitrary distance function d(·, ·) : S×S → R + 0 . Specifically, we generalize the Euclidean distance-maximizing skill discovery by replacing ∥x − y∥ in Equation (5) with d(x, y) as follows:
J DSD := E z,s,s ′ [(ϕ(s ′ ) − ϕ(s)) ⊤ z](6)s.t. ∀x, y ∈ S, ∥ϕ(x) − ϕ(y)∥ ≤ d(x, y),(7)
where ϕ(·) : S → R D is a function that maps states into a D-dimensional space (which has the same dimensionality as the skill space). DSD can discover skills that maximize the traveled distance under the given distance function d in diverse directions by (1) aligning the directions of z and (ϕ(s ′ ) − ϕ(s)) and (2) maximizing its length ∥ϕ(s ′ ) − ϕ(s)∥, which also increases d(s, s ′ ) due to the constraint in Equation (7). Here, LSD can be viewed as a special case of DSD with d(x, y) = ∥x − y∥.
When dealing with a learned distance function d, it is generally not straightforward to ensure that d is a valid distance (pseudo-)metric, which must satisfy symmetry and the triangle inequality. However, DSD has the nice property that d in Equation (7) does not have to be a valid metric. This is because DSD implicitly converts the original constraint (Equation (7)) into the one with a valid pseudometricd. As a result, we can use any arbitrary non-negative function d for DSD, with the semantics being implicitly defined by its induced pseudometricd. We summarize our theoretical results as follows and the proofs are in Appendix B.1.
Theorem 4.1. Given any non-negative function d : S ×S → R + 0 , there exists a valid pseudometricd : S × S → R + 0 that satisfies the following properties:
1. Imposing Equation (7) with d is equivalent to imposing
Equation (7) withd, i.e., ∀x, y ∈ S, ∥ϕ(x) − ϕ(y)∥ ≤ d(x, y) (8) ⇐⇒ ∀x, y ∈ S, ∥ϕ(x) − ϕ(y)∥ ≤d(x, y). (9) 2.d is a valid pseudometric. 3.d is a lower bound of d, i.e., ∀x, y ∈ S, 0 ≤d(x, y) ≤ d(x, y).(10)
Training of DSD. While LSD implements the Lipshitz constraint in Equation (5) using Spectral Normalization (Miyato et al., 2018), similarly imposing DSD's constraint in Equation (7) is not straightforward because it is no longer a Euclidean Lipschitz constraint. Hence, we optimize our objective with dual gradient descent (Boyd et al., 2004): i.e., with a Lagrange multiplier λ ≥ 0, we use the following dual objectives to train DSD:
r DSD := (ϕ(s ′ ) − ϕ(s)) ⊤ z,(11)J DSD,ϕ := E[(ϕ(s ′ ) − ϕ(s)) ⊤ z + λ · min(ϵ, d(x, y) − ∥ϕ(x) − ϕ(y)∥)], (12) J DSD,λ := −λ · E[min(ϵ, d(x, y) − ∥ϕ(x) − ϕ(y)∥)],(13)
where r DSD is the intrinsic reward for the policy, and J DSD,ϕ and J DSD,λ are the objectives for ϕ and λ, respectively. x and y are sampled from some state pair distribution p cst (x, y) that imposes the constraint in Equation (7). ϵ > 0 is a slack variable to avoid the gradient of λ always being nonnegative. With these objectives, we can train DSD by optimizing the policy with Equation (11) as an intrinsic reward while updating the other components with Equations (12) and (13).
Controllability-Aware Distance Function
To guide distance-maximizing skill discovery to focus on more challenging skills, a distance function d is required to assign larger values to state transitions that are hard-toachieve with the current skills and smaller values to easyto-achieve transitions. d also needs to be adaptable to the current skill policy so that the agent continuously acquires new skills and finds increasingly difficult state transitions over the course of training.
Among many potential distance functions, we choose a negative log-likelihood of a transition from the current skill policy, − log p(s ′ |s), as a controllability-aware distance function in this paper. Accordingly, we define the degree to which a transition is "hard-to-achieve" as − log p(s ′ |s) with respect to the current skill policy's transition distribution. This suits our desiderata since (1) it assigns high values for rare transitions (i.e., low p(s ′ |s)) while assigns small values for frequently visited transitions (i.e., high p(s ′ |s));
(2) p(s ′ |s) can be approximated by training a density model q θ (s ′ |s) from policy rollouts; and (3) the density model q θ (s ′ |s) continuously adjusts to the current skill policy by jointly training it with the skill policy. Here, while it is also possible to employ multi-step transitions p(s t+k |s t ) for the distance function, we stick to the single-step version for simplicity. We note that even though we employ singlestep log-likelihoods, DSD maximizes the sum of rewards,
T −1 t=0 (ϕ(s t+1 ) − ϕ(s t )) ⊤ z = (ϕ(s T ) − ϕ(s 0 )) ⊤ z
for the trajectory (s 0 , a 0 , s 1 , . . . , s T ), which maximizes the traveled distance of the whole trajectory while maintaining the directional alignment with z.
Controllability-Aware Skill Discovery
Now, we introduce Controllability-aware Skill Discovery (CSD), a distance-maximizing skill discovery method with our controllability-aware distance function. With the distance function in Section 4.2 we can rewrite the constraint of DSD in Equation (7) as follows:
∀s, s ′ ∈ S, ∥ϕ(s) − ϕ(s ′ )∥ ≤ d CSD (s, s ′ ),(14)d CSD (s, s ′ ) ≜ (s ′ − µ θ (s)) ⊤ Σ −1 θ (s)(s ′ − µ θ (s)) (15) ∝ − log q θ (s ′ |s) + (const),(16)
where the density model is parameterized as q θ (s ′ |s) = N (µ θ (s), Σ θ (s)), which is jointly trained using (s, s ′ ) tuples collected by the skill policy. We also use the same p(s, s ′ ) distribution from the skill policy for the dual constraint distribution p cst (x, y) introduced in Section 4.1 as well. Here, we note that d CSD (·, ·) is not necessarily a valid distance metric; however, we can still use it for the constraint in Equation (7) according to Theorem 4.1, because it automatically transforms d CSD into its induced valid pseudometricd CSD . Further discussion about its implications and limitations can be found in Appendix B.2.
CSD has several main advantages. First, the agent actively seeks rare state transitions and thus acquires increasingly complex skills over the course of training, which makes the skills discovered more useful for downstream tasks. In contrast, LSD or previous MI-based approaches only maximize Euclidean distances or are even agnostic to traveled distances, which often leads to simple or static behaviors. Second, unlike LSD, the optimal behaviors of CSD are agnostic to the semantics and scales of each dimension of the state space; thus, CSD does not require domain knowledge about the state space. Instead, the objective of CSD only depends on the difficulty or sparsity of state transitions. Finally, unlike curiosity-or disagreement-based exploration methods that only seek unseen transitions (Pathak et al., 2017;Mendonca et al., 2021), CSD finds a balance between covering unseen transitions and learning maximally different skills across zs via directional alignments, which leads to diverse yet consistent skills.
Algorithm 1 Controllability-aware Skill Discovery (CSD) 1: Initialize skill policy π(a|s, z), function ϕ(s), conditional density model q θ (s ′ |s), Lagrange multiplier λ 2: for i ← 1 to (# epochs) do Only CSD learns to manipulate the object across all three tasks without supervision while other methods focus only on moving the robot arm. We refer to Appendix D for the complete qualitative results from all random seeds.
Training of CSD. We train the skill policy π(a|s, z) with Soft Actor-Critic (SAC) (Haarnoja et al., 2018b) with Equation (11) as an intrinsic reward. We train the other components with stochastic gradient descent. We summarize the training procedure of CSD in Algorithm 1 and provide the full implementation details in Appendix E.
Experiments
The goal of our experiments is to verify whether our controllability-aware skill discovery method can learn complex, useful skills without supervision in a variety of environments. We test CSD on six environments across three different domains: three Fetch manipulation environments (FetchPush, FetchSlide, and FetckPickAndPlace) (Plappert et al., 2018), Kitchen , and two MuJoCo locomotion environments (Ant and HalfCheetah) Brockman et al., 2016). We mainly compare CSD with three state-of-the-art unsupervised skill discovery methods: LSD Figure 4. Comparison of the object state coverage and downstream task performances of skill discovery methods in three Fetch manipulation environments. Only CSD learns to manipulate the object without external supervision, while the other methods mainly focus on controlling the internal states ( Figure 16) because there is little incentive for them to discover more 'challenging' skills. Figure 5. Comparison of the downstream task performances of skill discovery methods with the oracle prior, which restricts the input to the skill discriminators to the object xyz coordinates. 2019), and DADS (Sharma et al., 2020). They respectively fall into the categories of Euclidean distance-maximizing skill discovery, reverse-MI, and forward-MI (Section 3). We also compare with disagreement-based exploration used in unsupervised goal-conditioned RL, such as LEXA (Mendonca et al., 2021), in Appendix C. We evaluate state coverage and performance on downstream tasks to assess the diversity and usefulness of the skills learned by each method. For our quantitative experiments, we use 8 random seeds and present 95% confidence intervals using error bars or shaded areas. We refer to our project page for videos.
Fetch Manipulation
We first show (1) whether CSD can acquire object manipulation skills without any supervision, (2) how useful the learned skills are for the downstream tasks, and (3) which component allows CSD to learn complex skills in the Fetch manipulation environments (Plappert et al., 2018). Each Fetch environment consists of a robot arm and an object but has a unique configuration; e.g., FetchSlide has a slippery table and FetchPickAndPlace has a two-fingered gripper.
We train CSD, LSD, DIAYN, and DADS on the three Fetch environments for 80K episodes with 2-D continuous skills (FetchPush, FetchSlide) or 3-D continuous skills (Fetch-PickAndPlace). Note that we do not leverage human prior knowledge on the state space (e.g., object pose); thus, all methods are trained on the full state in this experiment. 2 2 We note that the Fetch experiments in the LSD paper are using the 'oracle' prior, which enforces an agent to only focus on the state change of the object. Figure 6. Ablation study of distance-maximizing skill discovery in three Fetch environments. This suggests that CSD's performance cannot be achieved by just applying simple tricks to the previous Euclidean distance-maximizing skill discovery method. Figure 3 illustrates the object trajectories of continuous skills learned by skill discovery methods in the absence of any supervision. CSD successfully learns to move the object in diverse directions without external supervision. On the other hand, all of the previous methods fail to learn such skills and instead focus on diversifying the joint angles of the robot arm itself. This is because there is no incentive for the previous methods to focus on challenging skills such as object manipulation, while CSD explicitly finds hard-toachieve state transitions.
Following the setup in , we evaluate two quantitative metrics: the object state coverage and goalreaching downstream task performance. Figure 4a compares the four skill discovery methods in terms of the object state coverage, which is measured by the number of 0.1 × 0.1 square bins occupied by the object at least once, in the three Fetch environments. Figure 4b shows the comparison of the goal-reaching downstream task performances, where we train a hierarchical controller π h (z|s, g) that sequentially combines skills z for the frozen skill policy π(a|s, z) to move the object to a goal position g. We additionally train the vanilla SAC baseline to verify the effectiveness of leveraging autonomously discovered skills. We refer to Appendix E.2 for further details. On both quantitative metrics, CSD outperforms the prior methods by large margins, successfully discovering diverse manipulation skills that are useful for solving downstream tasks.
Skill discovery with the oracle prior on the state space.
While our experiments show that our approach can discover useful manipulation skills without any human prior on the state space, previous unsupervised skill discovery methods (Eysenbach et al., 2019;Sharma et al., 2020; mostly do not work without the oracle state prior, which restricts the skill discriminator module's input to only the xyz coordinates of the object. To investigate how CSD and the prior methods perform in the presence of this supervision, we train them with the oracle state prior. Figure 5 demonstrates that even without the oracle state prior, our CSD is mostly comparable to the previous best method with the oracle prior. This result demonstrates the potential of our approach in scalability to more complex environments, where human prior is no longer available. Moreover, with the oracle state prior, CSD further improves its performance. We refer to Figure 17 for the full qualitative results of CSD and LSD with the oracle prior in FetchPickAndPlace.
Ablation study. To understand the importance of our controllability-aware distance function in CSD, we examine whether similar results can be achieved without some components of CSD or by just applying simple tricks to LSD, a previous Euclidean distance-maximizing skill discovery method. Specifically, we consider the following three variants: (1) LSD + preset: LSD with a normalized state space using the precomputed standard deviation of each state dimension from randomly generated trajectories, . Evolution of task-related distances and corresponding task success rates. Our learned task-related distances decrease once the agent gains control of the corresponding objects, which makes the agent focus on other new objects consistently over the course of training. Distance plots are smoothed over a window of size 10 for better visualization. gradient descent instead of spectral normalization (i.e., CSD without our learned distance function). Figure 6 compares the performances of these variants with CSD, LSD, and SAC in three downstream tasks. The results show that only CSD learns to manipulate objects, which suggests that our controllability-aware distance function is indeed necessary to discover such complex skills without supervision.
Kitchen Manipulation
To verify the scalability of unsupervised skill discovery in a complex environment with diverse objects, we evaluate our method on the Kitchen manipulation environment , which includes 13 downstream tasks in total, such as opening a microwave, turning a light switch, moving a kettle, and opening slide/hinge cabinet doors (Figure 8a). We train CSD, LSD, DIAYN, and DADS with both 2-D continuous skills and 16 discrete skills for 40K episodes without any supervision. We refer to Appendix E for further experimental details regarding the Kitchen environment. We first measure the task success rates of the skills learned by the four methods. After the unsupervised skill training, we roll out the skill policy to collect 50 trajectories with 50 randomly sampled zs and measure whether each of the 13 tasks has at least one successful trajectory. The results with 16 discrete skills in Figure 7 suggest that CSD learns on average 10 out of 13 skills, while the prior methods fail to discover such skills (2 for LSD, 4 for DIAYN, 0 for DADS) because they mainly focus on diversifying the robot state. Continuous skills in Figure 14 also show similar results.
We then evaluate the downstream task performance by training a high-level controller π h (z|s, g) with the learned 2-D continuous skills π(a|s, z) as behavioral primitives to achieve a task specified by a 13-D one-hot vector g. The high-level controller chooses a skill z every 10 steps until the episode ends. The results in Figure 8b show that CSD significantly outperforms the previous methods.
Qualitative analysis. Figure 9 illustrates how our controllability-aware distance evolves over time and how this leads to the discovery of diverse, complex skills, e.g., SlideCabinet, KettleLift, and Microwave. Over training, we measure the task-related controllability-aware distance v ⊤ Σ −1 θ (s)v for each task v using skill trajectories, where v is the one-hot task vector corresponding to each of the three tasks. At around 4K episodes (Figure 9a), our controllability-aware distance encourages the agent to control the sliding cabinet with a large distance value (i.e., high reward). Once the agent learns to manipulate the sliding cabinet door, our controllability-aware distance for that skill decreases, letting the agent move its focus to other harderto-achieve skills, e.g., lifting kettle (Figure 9b) or opening a microwave (Figure 9c). As a result, the number of successful tasks gradually increases over the course of training. Figure 11. Comparison of the state coverage and downstream task performance of skills discovery methods in Ant and HalfCheetah.
MuJoCo Locomotion
To assess whether the idea of controllability-aware skill discovery works on domains other than manipulation, we evaluate CSD mainly on two MuJoCo locomotion environments Brockman et al., 2016): Ant and HalfCheetah. We additionally employ 17-DoF Humanoid, the most complex environment in the benchmark, for a qualitative comparison between CSD and LSD. In these environments, we train skill discovery methods for 200K episodes (100K for Humanoid) with 16 discrete skills. Figure 10 shows examples of skills discovered by each method, which suggests that CSD leads to the largest state coverage thanks to our controllability-aware distance function. For quantitative evaluation, we first measure the state space coverage by counting the number of 1 × 1 bins occupied by the agent's xy coordinates (xz coordinates for 2-D HalfCheetah) at least once. Figure 11a demonstrates that CSD covers the largest area among the four methods. This is because CSD's controllability objective makes the agent mainly focus on diversifying the global position of the agent, which corresponds to the 'challenging' state transitions in these locomotion environments. We emphasize that CSD not just learns to navigate in diverse directions but also learns a variety of behaviors, such as rotating and flipping in both environments (videos). We also note that MI-based methods (DIAYN and DADS) completely fail to diversify the agent's location and only discover posing skills, because the MI objective is agnostic to the distance metric, not providing incentives to maximize traveled distances in the state space.
We also evaluate the downstream learning performance on four tasks: AntGoal, AntMultiGoals, HalfCheetahGoal, and HalfCheetahHurdle, following previous works (Eysenbach et al., 2019;Sharma et al., 2020;Kim et al., 2021;. In AntGoal and HalfCheetahGoal, the agent should reach a randomly sampled goal position, and in AntMultiGoals, the agent should follow multiple randomly sampled goals in sequence. In HalfCheetahHurdle (Qureshi et al., 2020), the agent should jump over as many hurdles as possible. With downstream task rewards, we train a high-level policy that sequentially combines the learned skills. In Figure 11b, CSD consistently demonstrates the best performance among the four methods, which suggests that the skills discovered by CSD are effective not just on locomotion tasks but also on a wide variety of tasks, such as hurdle jumping.
Conclusion
In this paper, we present Controllability-aware Skill Discovery (CSD), a novel unsupervised skill discovery method that explicitly looks for hard-to-achieve skills. Specifically, we first formulate a distance-maximizing skill discovery approach (DSD), which can be combined with any arbitrary distance function. We then propose a jointly trained controllability-aware distance function, which consistently encourages the agent to discover more complex, hard-to-achieve skills. We empirically show that the idea of controllability-awareness enables the agent to acquire diverse complex skills in the absence of supervision in a variety of robotic manipulation and locomotion environments.
Limitations and future directions. Although the general idea of controllability-aware skill discovery is still applicable to pixel domains, e.g., in combination with representation learning techniques (Hafner et al., 2020;Srinivas et al., 2020;Seo et al., 2022), where they will reveal both the object and agent representations and CSD will focus on the object representation, we did not verify the scalability of our controllability-aware distance function to pixel-based environments. We leave it as future work. Another limitation is that CSD in its current form might not discover 'slowly moving' skills because underlying DSD prefers skills with large state variations. We believe acquiring skills with diverse moving speeds is another interesting future direction.
Haarnoja, T., Zhou, A., Hartikainen, K., Tucker, G., Ha, S., Tan, J., Kumar, V., Zhu, H., Gupta, A., Abbeel, P., and Levine, S. Soft actor-critic algorithms and applications. ArXiv, abs/1812.05905, 2018b.
Hafner Rajeswar, S., Mazzaglia, P., Verbelen, T., Pich'e, A., Dhoedt, B., Courville, A. C., and Lacoste, A. Unsupervised modelbased pre-training for data-efficient control from pixels. ArXiv, abs/2209.12016, 2022.
A. Extended Related Work on Unsupervised RL
The goal of unsupervised RL is to learn useful knowledge, such as dynamics models, state representations, and behavioral primitives, without predefined tasks so that we can later utilize them to efficiently solve downstream tasks. One line of research focuses on gathering knowledge of the environment with pure exploration (Pathak et al., 2017;Burda et al., 2019;Pathak et al., 2019;Sekar et al., 2020;Liu & Abbeel, 2021b;Yarats et al., 2021;Rajeswar et al., 2022). Unsupervised skill discovery methods (Gregor et al., 2016;Co-Reyes et al., 2018;Eysenbach et al., 2019;Sharma et al., 2020;Kim et al., 2021;Kamienny et al., 2022;Strouse et al., 2022;Shafiullah & Pinto, 2022;Jiang et al., 2022;Zhao et al., 2022) aim to learn a set of temporally extended useful behaviors, and our CSD falls into this category. Another line of work focuses on discovering goals and corresponding goal-conditioned policies via pure exploration (Warde-Farley et al., 2019;Pong et al., 2020;Pitis et al., 2020;Mendonca et al., 2021) or asymmetric/curriculum self-play (Sukhbaatar et al., 2018;OpenAI et al., 2021;Du et al., 2022). Lastly, Touati & Ollivier (2021); Touati et al. (2022) aim to learn a set of policies that can be instantly adapted to task reward functions given an unsupervised exploration method or an offline dataset.
B. Theoretical Results
B.1. Proof of Theorem 4.1
We assume that we are given an arbitrary non-negative function d : S × S → R + 0 . We first introduce some additional notations. For x, y ∈ S, define d s (x, y) ≜ min(d(x, y), d(y, x)). For x, y ∈ S, let P (x, y) be the set of all finite state paths from x to y. For a state path p = (s 0 , s 1 , . . . , s t ), define D s (p) ≜ t−1 i=0 d s (s i , s i+1 ). Now, for x, y ∈ S, we define the induced pseudometricd : S × S → R + 0 as follows:
d(x, y) ≜ inf p∈P (x,y) D s (p) if x ̸ = y 0 if x = y .(17)
Then, the following theorems hold.
Lemma B.1.d is a lower bound of d, i.e., ∀x, y ∈ S, 0 ≤d(x, y) ≤ d(x, y).
Proof. If x = y, thend(x, y) = 0 by definition and thus 0 ≤d(x, y) ≤ d(x, y) always holds. Otherwise, 0 ≤d(x, y) ≤ D s ((x, y)) = d s (x, y) ≤ d(x, y) holds and this completes the proof.
Theorem B.2. For ϕ : S → R D , imposing Equation (7) with d is equivalent to imposing Equation (7) withd, i.e., ∀x, y ∈ S, ∥ϕ(x) − ϕ(y)∥ ≤ d(x, y) ⇐⇒ ∀x, y ∈ S, ∥ϕ(x) − ϕ(y)∥ ≤d(x, y).
Proof. From Lemma B.1, we know that ∥ϕ(x) − ϕ(y)∥ ≤d(x, y) implies ∥ϕ(x) − ϕ(y)∥ ≤ d(x, y). Now, we assume that ∥ϕ(x) − ϕ(y)∥ ≤ d(x, y) holds for any x, y ∈ S. First, if x = y, then ∥ϕ(x) − ϕ(y)∥ becomes 0 and thus ∥ϕ(x) − ϕ(y)∥ ≤ d(x, y) always holds. For x ̸ = y, let us consider any state path p = (s 0 = x, s 1 , s 2 , . . . , s t−1 , s t = y) ∈ P (x, y). For any i ∈ {0, 1, . . . , t − 1}, we have
∥ϕ(s i ) − ϕ(s i+1 )∥ ≤ d(s i , s i+1 ),(20)∥ϕ(s i+1 ) − ϕ(s i )∥ ≤ d(s i+1 , s i ),(21)
and thus we get ∥ϕ(s
i ) − ϕ(s i+1 )∥ = ∥ϕ(s i+1 ) − ϕ(s i )∥ ≤ min(d(s i , s i+1 ), d(s i+1 , s i )) = d s (s i , s i+1 )
. Now, we have the following inequalities:
∥ϕ(s 0 ) − ϕ(s 1 )∥ ≤ d s (s 0 , s 1 ),(22)∥ϕ(s 1 ) − ϕ(s 2 )∥ ≤ d s (s 1 , s 2 ),(23). . . ,(24)∥ϕ(s t−1 ) − ϕ(s t )∥ ≤ d s (s t−1 , s t ).(25)
From these, we obtain ∥ϕ(
x)−ϕ(y)∥ = ∥ϕ(s 0 )−ϕ(s t )∥ ≤ t−1 i=0 ∥ϕ(s i )−ϕ(s i+1 )∥ ≤ t−1 i=0 d s (s i , s i+1 ) = D s (p).
Then, by taking the infimum of the right-hand side over all possible p ∈ P (x, y), we get ∥ϕ(x) − ϕ(y)∥ ≤ inf p∈P (x,y) D s (p) = d(x, y) and this completes the proof. Theorem B.3.d is a valid pseudometric, i.e., (a) ∀x ∈ S,d(x, x) = 0.
(b) (Symmetry) ∀x, y ∈ S,d(x, y) =d(y, x).
(c) (Triangle inequality) ∀x, y, z ∈ S,d(x, y) ≤d(x, z) +d(z, y).
Proof. (a) By definition,d(x, x) = 0 always holds for all x ∈ S.
(b) If x = y, thend(x, y) =d(y, x) = 0. Otherwise, with p = (s 0 = x, s 1 , s 2 , . . . , s t−1 , s t = y) ∈ P (x, y), we can prove the symmetry ofd as follows:d
(x, y) = inf p∈P (x,y) D s (p) (26) = inf p∈P (x,y) t−1 i=0 d s (s i , s i+1 ) (27) = inf p∈P (x,y) t−1 i=0 d s (s i+1 , s i ) (28) = inf p∈P (y,x) D s (p) (29) =d(y, x).(30)
(c) If x = y, y = z, or z = x, then it can be easily seen thatd(x, y) ≤d(x, z) +d(z, y) always holds. Hence, we assume that they are mutually different from each other. Then, the following inequality holds:
d(x, y) = inf p∈P (x,y) D s (p) (31) ≤ inf p1∈P (x,z),p2∈P (z,y) D s (p 1 ) + D s (p 2 ) (32) = inf p1∈P (x,z) D s (p 1 ) + inf p2∈P (z,y) D s (p 2 )(33)
=d(x, z) +d(z, y),
which completes the proof.
B.2. Implications of Theorem 4.1
Theorem 4.1 suggests that the constraint in Equation (7) implicitly transforms an arbitrary distance function d into a tighter valid pseudometricd. Intuitively, thisd(x, y) corresponds to the minimum possible (symmetrized) path distance from x to y. Hence, if we train DSD with Equation (7), it will find long-distance transitions that cannot be equivalently achieved by taking multiple short-distance transitions. Intuitively, in the context of CSD (Section 4.2), this implies that the agent will find rare state transitions that cannot be bypassed by taking 'easy' intermediate steps, which is a desirable property.
However, there are some limitations regarding the use of our distance function d CSD (Equation (16)). First, while the DSD constraint in Equation (7) implicitly symmetrizes the distance function by taking the minimum between d(x, y) and d(y, x), this may not be ideal in highly asymmetric environments involving many irreversible transitions. In practice, this may be resolved by only imposing one-sided constraints of our interest. Second, in our implementation, we only consider a single-step transition (s, s ′ ) and a single-step density model q θ (s ′ |s) as we found this simple design choice to be sufficient for our experiments. However, in order to fully leverage the aforementioned property of the induced pseudometric, the constraint may be imposed on any state pairs with a multi-step density model, which we leave for future work.
CSD Disagreement
Ant Half Cheetah CSD Disagreement Figure 13. The agent's xy (Ant) or x (HalfCheetah) trajectories of CSD and disagreement-based exploration. While CSD seeks very consistent, directed behaviors, disagreement-based exploration only focuses on diversifying states with chaotic, random behaviors. We provide videos illustrating this difference on our project page.
C. Comparison with Unsupervised Disagreement-Based Exploration
In this section, we discuss the difference between CSD and unsupervised goal-conditioned RL and present an empirical comparison between them. Unsupervised goal-conditioned RL approaches, such as DISCERN ( (1) running an exploration method that collects diverse 'goal' states g and (2) learning a goal-conditioned policy π(a|s, g) to reach the states discovered. Hence, treating g as a |S|-dimensional skill latent vector, these approaches may be viewed as a special type of unsupervised skill discovery.
However, the main focuses of unsupervised skill discovery are different from that of unsupervised goal-conditioned RL. First, unsupervised skill discovery aims to discover more general skills not restricted to goal-reaching behaviors, which tend to be static as the agent is encouraged to stay still at the goal state (Mendonca et al., 2021;Jiang et al., 2022). For instance, our approach maximizes traveled distances, which leads to more 'dynamic' behaviors like consistently running in a specific direction ( Figure 10). Second, unsupervised skill discovery aims to build a compact set of skills, which could also be discrete, rather than finding all the possible states in the given environment. For example, if we train CSD with three discrete skills, these behaviors will be as 'distant' as possible from one another, being maximally distinguishable. As such, we can have useful behaviors with a much low-dimensional skill space, making it more amenable to hierarchical RL.
Despite the difference in goals, to better illustrate the difference between them, we make an empirical comparison between CSD and ensemble disagreement-based exploration (Pathak et al., 2019), which some previous unsupervised goal-conditioned RL methods like LEXA (Mendonca et al., 2021) use as the exploration method. Disagreement-based exploration learns an ensemble of E forward dynamics models {p i (s ′ |s, a)} i∈{1,2,...,E} , and uses its variance |S| k V[p i (· k |s, a)] as an intrinsic reward, in order to seek unexplored transitions with high epistemic uncertainty. While unsupervised goal-condition RL approaches additionally learn a goal-conditioned policy, we do not separately learn it since the state coverage metrics of the exploration policy can serve as an approximate upper bound of the corresponding optimal goal-conditioned policy's performance. Figure 12 presents the comparisons of unsupervised state coverage metrics between CSD and disagreement-based exploration in all of our six environments. The results suggest that CSD mostly outperforms disagreement-based exploration in our state coverage metrics, mainly because CSD actively diversifies hard-to-control states such as the object position or the agent location, while the pure exploration method only focuses on finding unseen transitions. This difference is especially prominent in Ant and HalfCheetah (Figure 13), in which CSD seeks very consistent, directed behaviors, such as moving in one direction, while disagreement-based exploration only focuses on diversifying states with chaotic, random behaviors. We provide videos illustrating this difference at https://seohong.me/projects/csd/.
D. Additional Results
Additional quantitative results. Figure 14 shows the task success rates of the 2-D continuous skills learned by CSD, LSD, DIAYN, and DADS. As in the discrete case, CSD outperforms the other methods by a significant margin. Figure 15 demonstrates extended learning curves in Fetch and Kitchen downstream tasks, where we train SAC for four times as long as skill discovery methods. The results suggest that, while SAC alone can solve the FetchSlideGoal task with a lot more samples, it fails at learning FetchPushGoal, FetchPickAndPlaceGoal, and Kitchen mainly because they are challenging sparse-reward tasks. In contrast, agents can quickly learn all these tasks with temporally extended skills from CSD.
Additional qualitative results. Figures 16 and 19 illustrate the skill trajectories of all runs we use for our experiments in Fetch manipulation and two MuJoCo locomotion environments (eight random seeds for each method in each environment).
In the Fetch environments, CSD is the only method that learns object manipulation skills without supervision ( Figure 16). In Ant and HalfCheetah, CSD not only learns locomotion skills but also discovers a variety of diverse skills, such as rotating and flipping in both environments (Figure 19, videos). We provide the complete qualitative results in Humanoid in Figure 18. Figure 17 shows the full results of CSD and LSD equipped with the oracle prior in FetchPickAndPlace (eight seeds each). While CSD always learns to pick up the object, LSD discovers such skills in only three out of eight runs ( Figure 17). This is because our controllability-aware distance function consistently encourages the agent to learn more challenging picking-up behaviors. As a result, CSD significantly outperforms LSD in downstream tasks ( Figure 5). . Complete qualitative results in three Fetch environments (eight runs for each method in each environment). We plot the skill trajectories of the object and the gripper with different colors. CSD is the only unsupervised skill discovery method that discovers object manipulation skills without supervision.
CSD (oracle)
LSD (oracle)
Object trajectories Figure 17. Complete qualitative results of CSD and LSD trained with the oracle prior in FetchPickAndPlace (eight runs for each method). We plot the skill trajectories of the object and the gripper with different colors. Note that while LSD mostly just throws the object away, CSD always learns to pick up the object in all eight runs. Figure 18. Complete qualitative results in Humanoid (four runs for each method in each environment). We plot the skill xy trajectories of the agent with different colors. We note that we train CSD and LSD for 100K episodes (which is a tenth of the number of episodes used in the LSD paper ). . Complete qualitative results in Ant and HalfCheetah (eight runs for each method in each environment). We plot the skill trajectories of the agent with different colors. We note that in both environments, CSD not only learns locomotion skills but also discovers a variety of diverse skills, such as rotating and flipping.
CSD
LSD
LSD
Figure 3 .
3The object trajectories in the xy plane of randomly sampled 1000 continuous skills learned by CSD, LSD, DIAYN, and DADS in three Fetch manipulation environments without any supervision. Trajectories with different colors represent different skills.
Figure 7 .Figure 8 .
78Task success rates of 16 discrete skills discovered by CSD, LSD, DIAYN, and DADS in the Kitchen environment. CSD learns to manipulate diverse objects in the kitchen without any supervision. We refer to Appendix D for the results with 2-D continuous skills. Comparison of the downstream task performances of skill discovery methods in the Kitchen environment.
Figure 9
9LSD + norm: LSD with a normalized state space using the moving average of the standard deviation of state differences (s ′ − s), and (3) LSD + dual: LSD trained with dual
Figure 10 .
10The agent's xy (Ant and Humanoid) or x (HalfCheetah) trajectories of skills discovered by CSD, LSD, DIAYN, and DADS in MuJoCo locomotion environments. Trajectories with different colors represent different skills. We refer to Appendix D for the complete qualitative results from all random seeds.
, D., Lillicrap, T. P., Ba, J., and Norouzi, M. Dream to control: Learning behaviors by latent imagination. In International Conference on Learning Representations (ICLR), 2020.Hansen, S., Dabney, W., Barreto, A., Wiele, T., Warde-Farley, D., and Mnih, V. Fast task inference with variational intrinsic successor features. In International Conference on Learning Representations (ICLR), 2020.Jiang, Z., Gao, J., and Chen, J. Unsupervised skill discovery via recurrent skill training. In Neural Information Processing Systems (NeurIPS), 2022.Kamienny, P.-A., Tarbouriech, J., Lazaric, A., and Denoyer, L. Direct then diffuse: Incremental unsupervised skill discovery for state covering and goal reaching. In International Conference on Learning Representations (ICLR), 2022. Kim, J., Park, S., and Kim, G. Unsupervised skill discovery with bottleneck option learning. In International Conference on Machine Learning (ICML), 2021. Kingma, D. P. and Ba, J. Adam: A method for stochastic optimization. In International Conference on Learning Representations (ICLR), 2015. Kraskov, A., Stögbauer, H., and Grassberger, P. Estimating mutual information. Physical review E, 69(6):066138, 2004. Laskin, M., Yarats, D., Liu, H., Lee, K., Zhan, A., Lu, K., Cang, C., Pinto, L., and Abbeel, P. Urlb: Unsupervised reinforcement learning benchmark. In Neural Information Processing Systems (NeurIPS) Datasets and Benchmarks Track, 2021. Laskin, M., Liu, H., Peng, X. B., Yarats, D., Rajeswaran, A., and Abbeel, P. Unsupervised reinforcement learning with contrastive intrinsic control. In Neural Information Processing Systems (NeurIPS), 2022. Lee, L., Eysenbach, B., Parisotto, E., Xing, E., Levine, S., and Salakhutdinov, R. Efficient exploration via state marginal matching. ArXiv, abs/1906.05274, 2019. Liu, H. and Abbeel, P. APS: Active pretraining with successor features. In International Conference on Machine Learning (ICML), 2021a. Liu, H. and Abbeel, P. Behavior from the void: Unsupervised active pre-training. In Neural Information Processing Systems (NeurIPS), 2021b. Mendonca, R., Rybkin, O., Daniilidis, K., Hafner, D., and Pathak, D. Discovering and achieving goals via world models. In Neural Information Processing Systems (NeurIPS), 2021. Miyato, T., Kataoka, T., Koyama, M., and Yoshida, Y. Spectral normalization for generative adversarial networks. In International Conference on Learning Representations (ICLR), 2018. Ogata, K. et al. Modern control engineering. Prentice hall Upper Saddle River, NJ, 2010. OpenAI, O., Plappert, M., Sampedro, R., Xu, T., Akkaya, I., Kosaraju, V., Welinder, P., D'Sa, R., Petron, A., de Oliveira Pinto, H. P., Paino, A., Noh, H., Weng, L., Yuan, Q., Chu, C., and Zaremba, W. Asymmetric selfplay for automatic goal discovery in robotic manipulation. ArXiv, abs/2101.04882, 2021.Park, S., Choi, J., Kim, J., Lee, H., and Kim, G. Lipschitzconstrained unsupervised skill discovery. In International Conference on Learning Representations (ICLR), 2022. Pathak, D., Agrawal, P., Efros, A. A., and Darrell, T. Curiosity-driven exploration by self-supervised prediction. In International Conference on Machine Learning (ICML), 2017. Pathak, D., Gandhi, D., and Gupta, A. K. Self-supervised exploration via disagreement. In International Conference on Machine Learning (ICML), 2019. Pitis, S., Chan, H., Zhao, S., Stadie, B. C., and Ba, J. Maximum entropy gain exploration for long horizon multi-goal reinforcement learning. In International Conference on Machine Learning (ICML), 2020. Plappert, M., Andrychowicz, M., Ray, A., McGrew, B., Baker, B., Powell, G., Schneider, J., Tobin, J., Chociej, M., Welinder, P., Kumar, V., and Zaremba, W. Multi-goal reinforcement learning: Challenging robotics environments and request for research. ArXiv, abs/1802.09464, 2018. Pong, V. H., Dalal, M., Lin, S., Nair, A., Bahl, S., and Levine, S. Skew-Fit: State-covering self-supervised reinforcement learning. In International Conference on Machine Learning (ICML), 2020. Qureshi, A. H., Johnson, J. J., Qin, Y., Henderson, T., Boots, B., and Yip, M. C. Composing task-agnostic policies with deep reinforcement learning. In International Conference on Learning Representations (ICLR), 2020.
Figure 12 .
12Comparison of unsupervised state coverage metrics between CSD and ensemble disagreement-based exploration(Pathak et al., 2019) in all six environments. CSD mostly outperforms disagreement-based exploration in our state coverage metrics mainly because it actively diversifies hard-to-control states such as the object position or the agent location.
Warde-Farley et al., 2019), Skew-Fit (Pong et al., 2020), MEGA (Pitis et al., 2020), and LEXA (Mendonca et al., 2021), learn diverse behaviors typically by
Figure 14 .Figure 15 .
1415Task success rates of 2-D continuous skills discovered by four methods in the Kitchen environment. Extended learning curves of the SAC baseline in Fetch and Kitchen downstream tasks.
Figure 16
16Figure 16. Complete qualitative results in three Fetch environments (eight runs for each method in each environment). We plot the skill trajectories of the object and the gripper with different colors. CSD is the only unsupervised skill discovery method that discovers object manipulation skills without supervision.
Figure 19
19Figure 19. Complete qualitative results in Ant and HalfCheetah (eight runs for each method in each environment). We plot the skill trajectories of the agent with different colors. We note that in both environments, CSD not only learns locomotion skills but also discovers a variety of diverse skills, such as rotating and flipping.
, DIAYN(Eysenbach et al., 0
20K
40K
60K
80K
# episodes
50
100
150
200
State coverage
0
20K
40K
60K
80K
# episodes
50
100
150
State coverage
0
20K
40K
60K
80K
# episodes
0
100
200
300
State coverage
FetchPush
FetchSlide
FetchPickAndPlace
0
16K
32K
48K
64K
# episodes
0.1
0.2
0.3
Return
0
16K
32K
48K
64K
# episodes
0.2
0.4
0.6
0.8
Return
0
16K
32K
48K
64K
# episodes
0.0
0.1
0.2
Return
FetchPushGoal
FetchSlideGoal
FetchPickAndPlaceGoal
6K
32K
48K
64K
# episodes
CSD
LSD
DIAYN
DADS
SAC
(a) Object state coverage
(b) Downstream task performance
Table 2 .
2Hyperparameters for locomotion environments.Hyperparameter
Table 3 .
3Hyperparameters for SAC downstream policies in manipulation environments.Hyperparameter
Value
# training epochs
4000 (Fetch), 8000 (Kitchen)
# episodes per epoch
16 (Fetch), 2 (Kitchen)
# gradient steps per epoch
4 (Fetch), 10 (Kitchen)
Replay buffer size
10 6
Skill sample frequency R
10
Skill range
[−1.5, 1.5] D
Table 4 .
4Hyperparameters for PPO downstream policies in locomotion environments.Hyperparameter
Value
Learning rate
3 × 10 −4
# training epochs
1000
# episodes per epoch
64
# gradient steps per episode
10
Minibatch size
256
Entropy coefficient
0.01
Skill sample frequency R
25
University of California, Berkeley 2 Google Research. Correspondence to: Seohong Park <seohong@berkeley.edu>.
The original LSD implementation updates the target network every epoch, not every gradient step, but we find the latter to be about 10× sample efficient in terms of the number of environment steps.
AcknowledgementWe would like to thank Amber Xie, Younggyo Seo, and Jaekyeom Kim for their insightful feedback and discussion. This work was funded in part by Darpa RACER, Komatsu, a Berkeley Graduate Fellowship, and the BAIR Industrial Consortium. Seohong Park was partly supported by Korea Foundation for Advanced Studies (KFAS).High-level controller. After unsupervised skill discovery, we train a high-level controller π h (z|s, g) that selects skills in a sequential manner for solving downstream tasks. We use SAC(Haarnoja et al., 2018a)for continuous skills and PPO(Schulman et al., 2017)for discrete skills. The high-level policy selects a new skill every R steps. We mostly follow the hyperparameters for low-level skill policies and present the specific hyperparameters used for high-level controllers inTables 3 and 4.
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Implementation Details For manipulation environments, we implement CSD on top of the publicly available codebase of MUSIC. E Zhao, E. Implementation Details For manipulation environments, we implement CSD on top of the publicly available codebase of MUSIC (Zhao et al., 2021).
We run our experiments on an internal cluster with NVIDIA Tesla V100 and NVIDIA GeForce RTX 2080 Ti GPUs. Park, For MuJoCo environments, we implement CSD based on the publicly available codebase of LSD. We mostly follow the hyperparameters used in the original implementations. Each run mostly takes a day or lessFor MuJoCo environments, we implement CSD based on the publicly available codebase of LSD (Park et al., 2022). We mostly follow the hyperparameters used in the original implementations. Our implementation can be found in the follow- ing repositories: https://github.com/seohongpark/CSD-manipulation (manipulation environments) and https://github.com/seohongpark/CSD-locomotion (locomotion environments). We run our experiments on an internal cluster with NVIDIA Tesla V100 and NVIDIA GeForce RTX 2080 Ti GPUs. Each run mostly takes a day or less.
2021) with state-based observations. We use an episode length of 200 for locomotion environments and an episode length of 50 for manipulation environments. In locomotion environments, to ensure fair comparisons, we use preset normalizers for all skill discovery methods as done in Park et. Park, In Fetch environments, unlike LSD, we do not use any supervision, such as limiting the discriminator's input only to the object. For the Kitchen environment, we use a 7-DoF end-effector controller. but we find that CSD can still discover diverse behaviors including locomotion skills without a normalizerE.1. Environments We adopt the same environment settings used in LSD (Park et al., 2022) for Fetch manipulation environments (FetchPush, FetchSlide, FetchPickAndPlace) (Plappert et al., 2018) and MuJoCo locomotion environments (Ant, HalfCheetah) (Todorov et al., 2012; Brockman et al., 2016). In Fetch environments, unlike LSD, we do not use any supervision, such as limiting the discriminator's input only to the object. For the Kitchen environment, we use a 7-DoF end-effector controller (Mendonca et al., 2021) with state-based observations. We use an episode length of 200 for locomotion environments and an episode length of 50 for manipulation environments. In locomotion environments, to ensure fair comparisons, we use preset normalizers for all skill discovery methods as done in Park et al. (2022), but we find that CSD can still discover diverse behaviors including locomotion skills without a normalizer.
We use the same downstream tasks in Park et al. (2022) for Fetch environments. In FetchPushGoal, FetchSlideGoal, and FetchPickAndPlaceGoal, a goal position is randomly sampled at the beginning of each episode. If the agent successfully places the object to the target position, a reward of 1 is given to the agent and the episode ends. We follow the original goal sampling range and reach criterion from. E , Plappert et al. Downstream Tasks Fetch environmentsE.2. Downstream Tasks Fetch environments. We use the same downstream tasks in Park et al. (2022) for Fetch environments. In FetchPushGoal, FetchSlideGoal, and FetchPickAndPlaceGoal, a goal position is randomly sampled at the beginning of each episode. If the agent successfully places the object to the target position, a reward of 1 is given to the agent and the episode ends. We follow the original goal sampling range and reach criterion from Plappert et al. (2018).
For the success criteria of the tasks, we mostly follow. Bottomleftburner, Bottomrightburner, Hingecabinet, Kettlebottomright, Kettlefall, Kettlelift, Kettletopleft, Kettletopright, Lightswitch, Microwave, Slidecabinet, Topleftburner, ; Toprightburner, Gupta, We consider the following 13 downstream tasks for the Kitchen environment. Kitchen environment. We consider the following 13 downstream tasks for the Kitchen environment: BottomLeftBurner, BottomRightBurner, HingeCabinet, KettleBottomRight, KettleFall, KettleLift, KettleTopLeft, KettleTopRight, LightSwitch, Microwave, SlideCabinet, TopLeftBurner, TopRightBurner. For the success criteria of the tasks, we mostly follow Gupta et al. (2019);
As in the Fetch tasks, the agent gets a reward of 1 when it satisfies the success criterion of each task. Mendonca, 2021) and refer to our implementation for detailed definitionsMendonca et al. (2021) and refer to our implementation for detailed definitions. As in the Fetch tasks, the agent gets a reward of 1 when it satisfies the success criterion of each task.
60]), and if the agent reaches the goal, it gets a reward of 10 and the episode ends. For these three environments, we consider the agent to have reached the goal if it enters within a radius of 3 from the goal. HalfCheetahGoal, a goal's x coordinate is randomly sampled from Unif. 20and if the agent reaches the goal, it gets a reward of 10 and the episode ends. In AntMultiGoals, the agent should follow four goals within 50 steps each, where goal positions are randomly sampled from Unif. In HalfCheetahHurdle, the agent gets a reward of 1 if it jumps over a hurdle, where we use the same hurdle positions from Qureshi et al.MuJoCo locomotion environments. In AntGoal, a goal's xy position is randomly sampled from Unif([−20, 20] 2 ), and if the agent reaches the goal, it gets a reward of 10 and the episode ends. In AntMultiGoals, the agent should follow four goals within 50 steps each, where goal positions are randomly sampled from Unif([−7.5, 7.5] 2 ) centered at the current coordinates. The agent gets a reward of 2.5 every time it reaches a goal. In HalfCheetahGoal, a goal's x coordinate is randomly sampled from Unif([−60, 60]), and if the agent reaches the goal, it gets a reward of 10 and the episode ends. For these three environments, we consider the agent to have reached the goal if it enters within a radius of 3 from the goal. In HalfCheetahHurdle, the agent gets a reward of 1 if it jumps over a hurdle, where we use the same hurdle positions from Qureshi et al. (2020).
For DADS, we follow the original implementation choices, such as the use of batch normalization and fixing the output variance of the skill dynamics model. For CSD in manipulation environments, we start training the skill policy from epoch 4000, after the initial conditional density model has stabilized. When modeling Σ θ (s) of the conditional density model, we use a diagonal covariance matrix as we found it to be practically sufficient for our experiments. Lsd Park, Gaussian distribution (for continuous skills) or a uniform distribution (for discrete skills), and fix the skill throughout the episode. For discrete skills, we use standard one-hot vectors for DIAYN and DADS, and zero-centered one-hot vectors for CSD. Also, we normalize the diagonal elements with their geometric mean at each state for further stabilitySkill policy. At the beginning of each episode, we sample a skill z from either a standard Gaussian distribution (for continuous skills) or a uniform distribution (for discrete skills), and fix the skill throughout the episode. For discrete skills, we use standard one-hot vectors for DIAYN and DADS, and zero-centered one-hot vectors for CSD and LSD, following Park et al. (2022). For DADS, we follow the original implementation choices, such as the use of batch normalization and fixing the output variance of the skill dynamics model. For CSD in manipulation environments, we start training the skill policy from epoch 4000, after the initial conditional density model has stabilized. When modeling Σ θ (s) of the conditional density model, we use a diagonal covariance matrix as we found it to be practically sufficient for our experiments. Also, we normalize the diagonal elements with their geometric mean at each state for further stability.
We present the full list of the hyperparameters used in our experiments in Tables 1 and 2, where we indicate the values considered for our hyperparameter search with curly brackets. For the intrinsic reward coefficient, we use 50 (DADS), 500 (CSD and LSD), 1500 (DIAYN). 50Disagreement Fetch. Disagreement Kitchen). For the learning rate, we useWe present the full list of the hyperparameters used in our experiments in Tables 1 and 2, where we indicate the values considered for our hyperparameter search with curly brackets. For the intrinsic reward coefficient, we use 50 (DADS), 500 (CSD and LSD), 1500 (DIAYN), 200 (Disagreement Fetch), or 50 (Disagreement Kitchen). For the learning rate, we use
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"https://export.arxiv.org/pdf/2304.11302v1.pdf"
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["Analysis of the Fed's communication by using textual entailment model of Zero- Shot classification(...TRUNCATED) | ["Yasuhiro Nakayama yasuhiro.nakayama@mizuho-rt.co.jp \nMizuho Research & Technologies, Ltd\n\n","To(...TRUNCATED) | [
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Dataset Card for "ArtifactAI/arxiv_s2orc_parsed"
Dataset Description
https://huggingface.co/datasets/AlgorithmicResearchGroup/arxiv_s2orc_parsed
Dataset Summary
AlgorithmicResearchGroup/arxiv_s2orc_parsed is a subset of the AllenAI S2ORC dataset, a general-purpose corpus for NLP and text mining research over scientific papers, The dataset is filtered strictly for ArXiv papers, including the full text for each paper. Github links have been extracted from each paper to aid in the development of AlgorithmicResearchGroup/arxiv_python_research_code
How to use it
from datasets import load_dataset
ds = load_dataset("AlgorithmicResearchGroup/arxiv_s2orc_parsed", split="train")
# dataset streaming (will only download the data as needed)
ds = load_dataset("AlgorithmicResearchGroup/arxiv_s2orc_parsed", streaming=True, split="train")
Dataset Structure
Data Instances
Each data instance corresponds to one file. The content of the file is in the text feature, and other features provide some metadata.
Data Fields
title(sequence): list of titles.author(sequence): list of authors.authoraffiliation(sequence): list of institution affiliations for each author.venue: (integer): paper publication venue.doi: (float): paper doi.pdfurls: (integer): url link to the paper.corpusid: (int): corpus ID as defined by s2orc.arxivid: (int): arxiv paper id.pdfsha: (string): unique pdf hash.text: (string): full text of the arxiv paper.- github_urls: (sequence): list of github urls referenced within the text
Data Splits
The dataset has no splits and all data is loaded as train split by default.
Additional Information
Dataset Curators
Matthew Kenney, AlgorithmicResearchGroup, matt@algorithmicresearchgroup.com
Citation Information
@misc{arxiv_s2orc_parsed,
title={arxiv_s2orc_parsed},
author={Matthew Kenney},
year={2023}
}
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